# FRB/US Equation Documentation

November 2018

### Table of Contents

I. Introduction

II. Equations ordered by sector:

- Household expenditures
- Business investment
- Foreign trade and the current account
- Aggregate output and potential output identities
- Labor market
- Aggregate income
- Wages and prices
- Government
- Financial sector
- Foreign activity
- Expectations

III. List of variables

# Introduction

This document lists the roughly 375 equations in the FRB/US macroeconomic model. Many of these equations are identities. The number of "core" stochastic equations is around 50, of which about a third are based on formal models of optimization and contain explicit expectations of future economic conditions. Equations are grouped into 11 sectors, each of which is introduced by a brief discussion of its major features. Important equations are accompanied by a small amount of explanatory text. Equations in the expectations sector have two forms, one for backward-looking VAR-based expectations and one for forward-looking model-consistent expectations. The rest of this introduction provides a brief overview of the equations in FRB/US.

## Equations based on optimization

New Keynesian Phillips curve (NKPC): In the price-wage sector, the equations for core consumer price inflation and the ECI measure of hourly labor compensation are based on the NKPC framework.

PAC equations: The polynomial adjustment cost (PAC) approach underlies the other nonfinancial equations that are based on optimization. These include the equations for most components of consumption and investment, labor hours, and dividends. PAC is a specification that is used to characterize dynamic behavior based on an explicit cost minimization framework. In this framework, agents seek to minimize the expected present value of squared deviations of current and future values of the decision variable (y) from a target path (y*) plus adjustment costs associated with squared changes in the level, the growth rate, and higher-order time derivatives of the decision variable. Expressed as a decision rule, the PAC first-order condition has the following error-correction form.

    (1)       Δyt = a0(y*t-1 - yt-1) + Σk=1,m-1akΔyt-k + Et-1Σj=0,∞ djΔy*t+j + εt

Δ is the first difference operator, Et-1 represents expectations based on information available at t-1, and ε is an error term that is assumed to be serially uncorrelated. The equation decomposes the determinants of Δy into three elements:

- the lagged deviation of the decision variable from its target;
- lagged changes in the decision variable; and
- a weighted sum of expected future changes in the target variable.

The dj coefficients on leads of Δy* are transformations of the ak (k = 0,…,m-1) coefficients on lags of y and a fixed discount factor. When m=1, PAC is equivalent to the case of quadratic adjustment costs. In estimating the FRB/US PAC equations, m is chosen on the basis of the number of lags of Δy that have statistically significant coefficients, which usually leads to residuals that are free of serial correlation. Based on this criterion, m is found to be either 2 or 3. (In some cases the target variable y* is decomposed into separate stationary and non-stationary parts.)

For more information on PAC see Peter A. Tinsley, "Rational Error Correction," Computational Economics (2002). Information is also available here and here.

Permanent income: In the PAC equations for components of consumption and for residential investment, the target levels of spending depend on expectations of future income. An important characteristic of these expectations is that they are computed using a 25 percent annual rate of discount, a value that is based on micro evidence about individuals' income uncertainty.

PV equations: Financial markets are assumed to be free of adjustment costs. This applies to the equations for five-, ten-, and thirty-year government bonds, BBB corporate bonds, and the price of equity. Long-term bond rates equal the present discounted value of expected future short-term rates, adjusted for risk/term premiums. Similarly, equity prices equal the present discounted value of future dividend payments, after adjustment for a risk premium.

## Other equations

Error-correction and other behavioral equations: For the remaining core equations and other more peripheral relationships as well, conventional error-correction specifications without explicit expectations terms are used in the majority of instances. Equations of this type include export and import volumes, inventories, labor force participation, and the price of imports.

Identities: Two comments are in order concerning the numerous identities contained in FRB/US. First, the use of chain-aggregated price and quantity indexes in the National Income and Product Accounts makes the relationship between lower- and higher-level aggregates quite complex. In FRB/US, these relationships are closely approximated by identities that aggregate using Divisia indexes; the chain-aggregation formula is used for real GDP, because the value of its inventory investment component may be negative. Second, many other types of accounting identities are simplified in order to reduce the number of variables included in the system; for these quasi-identities, exogenous multiplicative scaling variables are used to link the variable in question with its major components.

## Estimation

The large size of FRB/US makes it infeasible to estimate all of its equations simultaneously. Estimation of the optimization-based equations involves the estimation of a set of independent sub-models, each of which typically combines one of the structural equations with a condensed model of the overall economy that features a VAR. Projections of the VAR provide proxies for the explicit expectations terms in the structural equation. Each VAR model shares a core set of macro variables: the federal funds rate and the value expected to prevail in the long run, consumer price inflation and the value expected to prevail in the long run, and the output gap. The structure of the core VAR model is such that interest rate and inflation expectations converge to long-run expectations as the forecast horizon lengthens. Auxiliary variables are added to individual VARs as needed to form proxies for expectations of variables not in the core set.

An iterated OLS approach is used to estimate the sub-models associated with most of the optimization-based equations. The only exceptions are the price and wage NKPC equations, which are estimated simultaneously using a mixture of maximum likelihood and Bayesian methods. The estimates other equations are obtained by standard OLS or, in a few cases, with the Kalman filter.

## Expectations

FRB/US contains roughly 30 expectations variables, most of which appear in equations whose basis includes the assumption that some or all economic agents make decisions optimally. The design of FRB/US permits each expectations variable to have either a backward-looking VAR solution or a forward-looking model-consistent solution.

The documentation of each expectations variable contains a pair of formulas. The first formula is the identity used when expectations are VAR-based. The form of each identity of this type is determined by the structure and parameters of the sub-model that is used to estimate the structural equation in which the expectation appears. Two types of parameters determine the coefficient values in the typical VAR-expectations equation: discounting weights that specify the horizon of each expectation and coefficients of the VAR system used to generate the forecasts that proxy for expectations in the estimation sub-model. The second formula is the identity used for model-consistent expectations (MCE). The coefficient values in the typical MCE equation depend only on discounting weights.

The design of FRB/US permits simulations in which some expectations have VAR-based solutions and others have model-consistent solutions.

## Equation statistics

For most equations, the equation text shows coefficients and t-statistics (in brackets) immediately before each explanatory variable. For explanatory variables that enter as a set of lags, however, the equation text gives the coefficient sum, and the individual lag coefficients and t-statistics are reported beneath the equation text. Coefficients that are constrained in estimation do not have t-statistics. Below the equation text and any distributed lag coefficients, standard estimation statistics are shown.

# Household Expenditures

In FRB/US, optimizing households seek to maximize expected utility subject to lifetime resources. Because of risk aversion and uninsurable income uncertainty, consumption depends on the present value of expected future income discounted not at the real rate of interest, but at a higher rate that is fixed at 25 percent per annum. Heterogeneity across age groups in wealth holdings and the propensity to consume makes aggregate desired spending a function of both the level and composition of expected lifetime resources. Adjustment costs cause actual spending to deviate from desired spending.

The household optimization problem has two parts. The first is the calculation of desired spending in the absence of adjustment costs. The second applies the polynomial adjustment cost (PAC) framework to find the optimal path of actual spending given the desired path. Expectations play a role in both optimization steps.

This general paradigm applies to consumer spending on nondurable goods and non-housing services (actual = ECO, desired = QECO), investment in durable goods (ECD, QECD), and investment in residential structures (EH, QEH). The equation for ECO also allows for the presence of some rule-of-thumb or liquidity constrained households whose consumption moves in proportion to their income. The desired level of spending on each of the two categories of investment depends on a real rental rate in addition to lifetime resources. A final and relatively minor consumption category, consumption of housing services (ECH), is modeled as a simple reduced form.

The modeling of desired consumption centers on variable QEC, which is the desired level of spending associated with consumption of nondurable goods and services plus the service flow from the stock of durables (EC). QEC is measured historically as the fitted values of a regression of EC on constructed measures of discounted future income and Flow of Funds estimates of the net stock of tangible assets held by the household sector. The FRB/US equation for QEC is this regression equation with QEC substituted for EC as its dependent variable.

### EC: Consumption, cw 2012$ (FRB/US definition)

The FRB/US concept of total consumer spending is approximated by a fixed-weighted aggregation of its components.

log(ec) = |
| log(ect-1) + 0.731*log(eco/ecot-1) + 0.157*log(ech/echt-1)
| +0.112*log((yhpcd+jkcd)/(yhpcdt-1+jkcdt-1))

### ECD: Consumer expenditures on durable goods, cw 2012$

Household investment in consumer durables is modeled using the polynomial adjustment cost (PAC) framework, whose general structure is shown below in the PAC Overview section. To translate the general PAC specification into the form in which it appears in the ECD equation, associate log(ECD) with y, log(QECD) with y1*, ZECO with the expected weighted sum of future Δy1*, and 8.02*ZGAPC2/400 with the expected weighed sum of future y0*. The order of adjustment costs (m) is 2.

PAC Overview: Let y denote the decision variable, and y1* and y0* be the nonstationary and stationary components of its desired level in the absence of adjustment frictions. The decision rule derived from the PAC Euler equation has the following form.

(1) Δyt = a0(y1*t-1 - yt-1) + ∑i=1,m-1 aiΔyt-i + Et-1∑i=0,∞diΔy1*t+i + Et-1∑i=0,∞hiy0*t+i

The order of adjustment costs is given by parameter m. The forward weights, di and hi, {i=0,∞}, are functions of the estimated values of ai {i=0,m-1} and a fixed discount factor (0.98). Estimation imposes the growth neutrality restriction ∑i=1,m-1ai + ∑i=0,∞di = 1. For more information, see Pac Basics.

dlog(ecd) = |
| 0.156[3.82] * log(qecdt-1/ecdt-1)
| + -0.0600[-0.88] * dlog(ecdt-1)
| + 1 * zecd
| + 11.872[3.11] * zgapc2 / 400

Regression Statistics
Adjusted R2: | 0.14
S.E. of regression: | 0.027
Sample period: | 1969Q2 2017Q4

### ECH: Consumer expenditures on housing services, cw 2012$

d(ech/kht-1) = |
| 0.000528[1.98]
| + -0.00504[-1.87]*echt-1/kht-2
| + 0.718[14.42]*d(echt-1/kht-2)

Regression Statistics
Adjusted R2: | 0.53
S.E. of regression: | 0.00019
Sample period: | 1970Q2 2017Q4

### ECNIA: Personal consumption expenditures, cw 2012$ (NIPA definition)

NIPA total consumer spending is approximated by a fixed-weighted aggregation of its components.

log(ecnia) = |
| log(ecniat-1)
| + 0.735*log(eco/ecot-1)
| + 0.106*log(ecd/ecdt-1)
| + 0.160*log(ech/echt-1)

Regression Statistics
Adjusted R2: | 0.100
S.E. of regression: | 1.32E-05
Sample period: | 2016Q4 2017Q4

### ECNIAN: Personal consumption expenditures, current $ (NIPA definition)

ecnian | = .01*pcnia*ecnia

### ECO: Consumer expenditures on non-durable goods and non-housing services, cw 2012$

Consumer spending on nondurables goods and non-housing services is a weighted average of the spending of optimizing households and rule-of-thumb (ROT) households. The consumption decisions of the former are modeled using the polynomial adjustment cost (PAC) framework. ROT consumption moves in proportion with the sum of labor and transfer income (YHL+YHT). The estimated fraction of ROT consumption is 0.18. The structure of the PAC component of the equation is based on equation (1) shown below in the PAC Overview section. To translate the general PAC specification into the form in which it appears in the ECO equation, associate log(ECO) with y, log(QECO) with y*, and ZECO with the expected weighted sum of future Δy*. The order of adjustment costs (m) is 2.

PAC Overview: Let y denote the decision variable and y* its desired level in the absence of adjustment frictions. The decision rule derived from the PAC Euler equation has the following form.

(1) Δytpac = a0(y*t-1 - yt-1) + ∑i=1,m-1 aiΔyt-i + Et-1∑i=0,∞diΔy*t+i

The order of adjustment costs is given by parameter m. The forward weights, di {i=0,∞}, are functions of the estimated values of ai {i=0,m-1} and a fixed discount factor (0.98). Estimation imposes the growth neutrality restriction ∑i=1,m-1ai + ∑i=0,∞di = 1. For more information, see PAC Basics.

dlog(eco) = |
| ( 0.158[4.069] * log(qecot-1/ecot-1)
| + 0.412[4.33] * dlog(ecot-1)
| + 1 * zeco) * (1
| -0.308[2.86])
| + 0.308[2.86] * dlog(yhl+yht)

Regression Statistics
Adjusted R2: | 0.36
S.E. of regression: | 0.0040
Sample period: | 1968Q3 2017Q4

### EH: Residential investment expenditures, cw 2012$

Household investment in residential structures is modeled using the polynomial adjustment cost (PAC) framework, whose general structure is shown below in the PAC Overview section. To translate the general PAC specification into the form in which it appears in the EH equation, associate log(EH) with y, log(QEH) with y*, and ZEH with the expected weighted sum of future Δy*. The order of adjustment costs (m) is 3.

Besides the standard PAC terms, the equation also includes the first lag of the change in the nominal mortgage rate. This term is included to capture (approximately) the temporary effects of downpayment requirements and other borrowing constraints on housing investment when nominal interest rates change. A shift in the coefficient on this term is allowed between 1982:Q4 and 1983:Q1, because the final repeal of Reg. Q in the early 1980s changed the sensitivity of residential construction to interest rates.

PAC Overview: Let y denote the decision variable and y* its desired level in the absence of adjustment frictions. The decision rule derived from the PAC Euler equation has the following form.

(1) Δytpac = a0(y*t-1 - yt-1) + ∑i=1,m-1 aiΔyt-i + Et-1∑i=0,∞diΔy*t+i

The order of adjustment costs is given by parameter m. The forward weights, di {i=0,∞}, are functions of the estimated values of ai {i=0,m-1} and a fixed discount factor (0.98). Estimation imposes the growth neutrality restriction ∑i=1,m-1ai + ∑i=0,∞di = 1. For more information, see Pac Basics.

dlog(eh) = |
| 0.0131[1.022] * log(qeht-1/eht-1)
| + 0.371[5.61] * dlog(eht-1)
| + 0.206[3.19] * dlog(eht-2)
| + 1 * zeh
| + -0.0562[-6.56] * d(rmet-1)
| + 0.0279[2.65] * d83 * d(rmet-1)

Regression Statistics
Adjusted R2: | 0.49
S.E. of regression: | 0.033
Sample period: | 1970Q2 2017Q4

### EHN: Residential investment expenditures

ehn | = .01 * phr * pxp * eh

### JKCD: Consumption of fixed capital, consumer durables

jkcd | = jrcd * kcdt-1

### KCD: Stock of consumer durables, cw 2012$

kcd | = .25*ecd + (1-jrcd/4)*kcdt-1

### KH: Stock of residential structures, cw 2012$

kh | = .25*eh + (1-jrh/4)*kht-1

### QEC: Desired level of consumption (FRBUS definition)

QEC is the desired level of spending associated with consumption of nondurable goods and services plus the imputed service flow from the stock of durables (EC). The estimated coefficient values and equation statistics come from a cointegrating regression of EC on the set of explanatory variables. QEC is measured historically as the fitted values of this regression.

log(qec) = |
| 0.365[9.77]
| + 0.494[31.92]*log(zyh-zyht-zyhp)
| + 0.186[27.66]*log(zyht) + (1
| -0.494[31.92]
| -0.186[27.66]
| -0.187[16.46])*log(zyhp)
| + 0.187[16.46]*log(wpo+wps)

Regression Statistics
Adjusted R2: | 0.100
S.E. of regression: | 0.013
Sample period: | 1967Q2 2017Q4

### QECD: Target level of consumption of durable goods, trending component

The ratio of the target level of spending on consumer durable goods to the target level of overall consumption depends on two factors. The first factor is the relative rental rate, which is the product of the relative purchase price of consumer durables, PCDR, and the real financial cost of capital (plus depreciation) for such goods, RCCD. The relative rental rate is associated with the steady-state condition for the stock of consumer durables. The second factor converts the steady-state stock condition to one for gross investment by multiplying the stock condition by the sum of two factors -- the depreciation rate for durable goods, and the steady-state growth rate of the target capital stock. The latter factor equals the sum of trend output growth (HGGDPT) and the trend rate of decline in the relative price of consumer durable goods (HGPCDR, weighted by the real rental rate elasticity).

The coefficients values and equation statistics come from a cointegrating regression in which actual investment (ECD) replaces target investment (QECD) as the dependent variable.

qecd = |
| qec*(jrcd/4 + hggdpt/400
| + -0.584[-67.29]*hgpcdr/400)*exp
| (2.413[74.47]
| + -0.584*(log(pcdr) + log(rccd)))

Regression Statistics
Adjusted R2: | 0.97
S.E. of regression: | 0.054
Sample period: | 1969Q2 2017Q4

### QECO: Desired level of consumption of nondurable goods and nonhousing services

The elasticity of the desired level of consumption on nondurable goods and non-housing services is one with respect to the desired level of overall consumption and minus one with respect to the relative price (PCOR) of consumption on nondurable goods and non-housing services to overall consumption. A constant term ensures that historically the average log difference between ECO and QECO is zero.

log(qeco) = |
| -0.326[-322.50] + log(qec) - log(pcor)

Regression Statistics
Adjusted R2: | 0.100
S.E. of regression: | 0.015
Sample period: | 1965Q3 2017Q4

### QEH: Target level of residential investment

The ratio of the target level of residential investment to the target level of overall consumption depends on two factors. The first factor is the relative rental rate, which is the product of the relative purchase price of residential investment, PHR*PXP/PCNIA, and the real financial cost of capital (plus depreciation) for such goods, RCCH. The relative rental rate is associated with the steady-state condition for the stock of housing. The second factor converts the steady-state stock condition to one for gross investment by multiplying the stock condition by the sum of two terms -- the depreciation rate for residential structures, and the steady-state growth rate of the target capital stock. The latter term equals trend output growth (HGGDPT). (Note: unlike consumer durables, no adjustment is made to the steady-state growth of the target stock to control for trend movements in the relative price of housing construction, because the relative price does not show a pronounced trend.)

The coefficients values and equation statistics come from a cointegrating regression in which actual residential investment (EH) replaces target investment (QEH) as the dependent variable.

qeh = |
| qec * (jrh/4 + hggdpt/400) * exp
| (1.909 - log(phr*pxp/pcnia)
| + -0.117*log(rcch))

Regression Statistics
Adjusted R2: | 1
S.E. of regression: | 0
Sample period: | 1964Q1 2017Q4

### RCCD: Cost of capital for consumer durables

The real user cost of the stock of consumer durable goods (excluding the purchase price of new goods) equals the sum of the depreciation rate (JRCD) and the real interest rate. The latter is approximated by the new auto loan rate minus expected inflation over the next five years. A MAX function is included to prevent RCCD from taking on implausible values, improving the stability of the model in stochastic simulations. Over history, RCCD has never approached this floor.

rccd | = (max((100*jrcd + rcar - zpi5),( .01)))

### RCCH: Cost of capital for residential investment

The real user cost of housing (excluding the purchase price of new construction) equals the depreciation rate JRH, plus the real after-tax mortgage rate (1-TRFPM/100)*RME-ZPI10, plus the effective marginal property tax rate (1-TRFPM/100)*TRSPP. A MAX function is included to prevent RCCH from taking on implausible values, improving the stability of the model in stochastic simulations. Over history, RCCH has never approached this floor.

Note: TRFPM is the marginal federal income tax rate for the taxpayers with household incomes that are twice the median; this group is considered the most representative of households who itemize.

rcch | = (max((100*jrh + (1-trfpm/100)*(rme+100*trspp) - zpi10),( .1)))

### YHPCD: Imputed income of the stock of consumer durables, 2012$

log(yhpcd) | = log(0.0538) + log(kcdt-1)

# Business Expenditures

The equation for investment is based partly on the framework of optimization subject to adjustment costs and partly on an ad hoc accelerator effect of output growth on investment growth. The latter may capture the effects of cash flow on the ability of liquidity-constrained firms to finance investment projects.

In the optimization part of each equation, the level of investment that would be desired in the absence of adjustment costs is defined in two steps. In the first, the target capital-output ratio is a function of the user cost associated with that type of capital. In the second, the target investment-output ratio is formed as the product of the target capital-output ratio and the sum of the capital-specific rate of depreciation and the rate of growth of potential output. This is the rate of investment necessary to hold the capital-output ratio at its optimal level.

The target for the individual capital-output ratio is derived jointly from a nested aggregate production function. At the highest level, the production function is Cobb-Douglas with two factors of production -- quality-adjusted hours and an aggregate capital services bundle. The capital services bundle is measured as the chain-weighted aggregate of the flow of capital services from each type of capital stock. Within the capital-services bundle, the targets for the individual categories are assumed to have a unit elasticity with respect to the user cost of capital, as in a Cobb-Douglas production function. The framework deviates from the standard Cobb-Douglas specification, however, in allowing the long-run capital services shares to vary over time.

In the optimization framework, the polynomial adjustment cost (PAC) approach determines the actual path of investment. In this approach, investment responds to the lagged gap between actual and target investment, expected future growth in target investment, and lagged growth in actual investment.

Inventories

The stock of inventories is modeled with a simple error-correction equation in which the equilibrium ratio of inventories to final sales is a random walk with negative drift.

### EBFI: Business Fixed Investment, cw 2012$

Business Fixed Investment is a weighted average of the investment spending of optimizing firms and an ad hoc term in lagged output growth adjusted for the trend growth rate of the relative price of investment.

dlog(ebfi) = |
| (0.0736[2.59]*(log(qebfit-1/ebfit-1))
| + 0.522[4.29]*dlog(ebfit-1)
| + 0.302[2.17]*dlog(ebfit-2)
| + 1*zebfi)*(1
| -0.384[2.41])
| + 0.384[2.41]*(dlog(xbt-1) - hgpbfirt-1/400)

Regression Statistics
Adjusted R2: | 0.43
S.E. of regression: | 0.016
Sample period: | 1970Q1 2017Q4

### EBFIN: Investment in equipment, current $

ebfin | = 0.01*ebfi*pbfir*pxp

### EI: Change in private inventories, cw 2012$

ei | = 4*Δ(ki)

### EIN: Change in business inventories, current $

ein | = .01*pxp*pkir*ei

### HKS: Growth rate of KS, cw 2012$ (compound annual rate)

The growth rate of capital services is modeled as a weighted average of the growth rates of three capital stocks. The weights are measures of income shares earned by each type of capital. A residual component, which makes the equation an identity, accounts for the use of partially aggregated capital stocks rather than disaggregated capital stocks, omission of several types of capital (owner-occupied housing, land), and approximation error in the constructed income share weights.

hks = |
| 400 * (ykbfin * Δ(log(kbfi))
| + ykin * Δ(log(ki))) /
| (ykbfin + ykin) + hksr

### KBFI: Capital stock - BFI, 2012$

kbfi | = (pbfir/pkbfir)*ebfi/4 + (1-jrbfi/4) * kbfit-1

### KI: Stock of private inventories, cw 2012$

The KI equation has an error-correction form in which the desired inventory-sales ratio (QKIR) is a random walk with drift. The KI equation and the QKIR process are estimated using the Kalman Filter. The secular downtrend in the inventory-sales ratio is reflected in an estimated value of the drift in QKIR that is negative. In the KI equation, movements in the growth rate of final sales have temporary effects on inventories, but permanent effects on the inventory-sales ratio are precluded by restricting the coefficients on the lagged growth rates of sales and inventories to sum to unity.

dlog(ki) = |
| -0.00121[-2.71]
| + 0.0147[0.83]*(log(qkir) - log(kit-1/xfst-1))
| + 0.457[10.24]*(dlog(kit-1)
| --0.00121[-2.71])
| + 0.236[5.40]*dlog(xfst-1)
| + 0.307*dlog(xfst-2)

Regression Statistics
Sample period: | 1960Q1 2017Q4

### KPS: Capital stock - nonresidential structures, 2012$

### KS: Capital services, 2012$

log(ks) | = log(kst-1) + hks/400

### QEBFI: Desired level of investment in business investment

The target rate of investment is defined as the rate necessary to keep the capital-output ratio at its optimal value (VBFI).

qebfi | = xb*vbfi*(hxbt/100 -.01*hgpbfir+jrbfi)

### QKIR: Desired Inventory Sales Ratio

The target inventory-sales ratio is estimated along with the equation for the stock of inventories (KI) using the Kalman filter. To ensure that long-run simulations are characterized by balanced growth, exogenous variable DGLPRD can be set to 1.0 to make the target inventory-sales ratio constant.

log(qkir) | = (1-dglprd)*-0.00189 + log(qkirt-1)

### RBFI: After-tax real financial cost of capital for business investment

The firm's financing cost is measured as a weighted average of borrowing costs in debt and equity markets. The cost of debt finance is proxied by the yield on 5-year Treasury bonds plus a risk premium measured by the spread between the BBB corporate bond rate and the yield on 10-year Treasury bonds, and allows for the tax deductibility of interest payments. The expected rate of inflation over a 5-year horizon is subtracted from the after-tax nominal yield to obtain the real after-tax rate of interest. The cost of equity finance is measured as the expected real return to equity.

rbfi = |
| 0.5*(7.2 + (1-trfcim)*(rg5 + rbbb- rg10) - zpib5)
| + 0.5*req

### RTBFI: User cost of capital for business investment

The annualized rental cost, relative to the price of output, of a unit of equipment has three components: the relative purchase price of new investment (PKBFIR*PXP/PXB); the depreciation rate (JRBFI) plus the real financing rate (RBFI) less the trend growth rate of the relative price of investment (HGPBFIR); and tax adjustments for depreciation and the marginal corporate tax rate.

rtbfi = |
| (.01*rbfi + jrbfi - .01*hgpbfir)
| * ((1-.01*tritc-trfcim*(1-tapddp*.01*tritc)*tdpv)/(1-trfcim))
| * (pkbfir*pxp/pxb)

### RTINV: User cost of capital for inventories

The annualized rental cost, relative to the price of output, of a unit of inventory capital is equal to the relative purchase price of new investment (PKIR*PXP/PXB) multiplied by the after-tax financial rate (RPD) minus the trend growth rate of the relative price of inventories (HGPKIR).

rtinv = |
| (.01*rbfi - .01*hgpkir)
| * ( ( pxp*pkir + pxpt-1*pkirt-1) /2)/pxb

### VBFI: Desired investment-output ratio

The desired investment-output ratio is inversely related to the user cost of capital, implying an elasiticity of substitution of one. The multiplicative factor UVBFI has the interpretation of the equilibrium income share for nonresidential structures. UVBFI is estimated with an HP filter over history and is assumed to be exogenous in simulation.

vbfi | = uvbfi*(pkbfir/pbfir)/rtbfi

# Foreign Trade

The core of the foreign trade sector is a pair of error-correction equations for the volumes of exports and nonoil imports. An error-correction equation for domestic crude energy consumption, in conjunction with the exogenous domestic supply of crude energy, determines the volume of oil imports. In addition, the sector contains a variety of identities and quasi-identities for aggregate real and nominal exports and imports, U.S. net foreign investment income, the current account balance, and the net foreign asset position of the United States.

### EM: Imports of goods and services, cw 2012$

The volume of total imports is approximated by the Divisia aggregate of oil and non-oil imports.

log(em) = |
| log(emt-1)
| + .5 * (emon/emn + emont-1/emnt-1) * Δ(log(emo))
| + .5 * (empn/emn + empnt-1/emnt-1) * Δ(log(emp))

### EMN: Imports of goods and services, current $

emn | = emon + empn

### EMO: Imports of goods and services ex. petroleum, cw 2012$

The equation for non-petroleum imports has an error-correction format. In the long run, the volume of imports depends on the level of real domestic absorption with an elasticity of one, the relative price of non-oil imports with an elasticity of minus one, and an exogenous trend. Historical values of the trend (UEMOT) are measured by applying the HP filter to the ratio of nominal imports to nominal absorption. In the short-run, imports respond strongly to the first difference of the output gap. The effect of large dock strikes in the 1960s and 1970s is removed by including a dummy constructed by Peter Isard (IFDP No. 60, 1975).

dlog(emo) = |
| 0.0162[12.005]
| + -0.181[-6.26] * log(emot-1*(pmot-1/100)/(uemott-1*xgdpnt-1))
| + 1.358[5.040] * (xgap2-xgap2t-1)/100
| + 1.637[6.30] * (xgap2t-1-xgap2t-2)/100
| + 0.319[2.59]* log(ddockm)
| + 0.405[5.22] * dlog(ddockm)

Regression Statistics
Adjusted R2: | 0.67
S.E. of regression: | 0.020
Sample period: | 1965Q1 2017Q4

### EMON: Imports of goods and services ex. petroleum

emon | = .01 * pmo * emo

### EMP: Petroleum imports, cw 2012$

The deviation of the ratio of EMP to GDP from a long-run component (EMPTRT) is a linear function of its price relative to the business output deflator and real activity (as captured by XGAP2).

log(emp/xgdp) = |
| log(emptrt)
| + 0.0480[0.67]*d(log(pmp/pxb))
| + 0.0221[6.39]*xgap2t-1

### EMPN: Petroleum imports, current $

empn | = .01*pmp*emp

### EX: Exports of goods and services, cw 2012$

Exports is modeled in an error-correction format. In the long run, the volume of exports depends on foreign real GDP (FGDP) with an elasticity of one and the price of exports relative to foreign prices (PXR*PXP*FPX/FPC) with an elasticity of minus one. A high short-run income elasticity is captured by the positive estimated coefficients on the first difference of the foreign output gap (FXGAP). The effect of large dock strikes in the 1960s and 1970s is removed by dummying out affected quarters.

dlog(ex) = |
| 0.840[5.76]
| + -0.108[-5.67] * log(ext-1*(pxrt-1*pxpt-1*fpxt-1)/(fgdpt-1*fpct-1))
| + 1.482[5.46] * (fxgap - fxgapt-1)/100
| + 1.027[3.88] * (fxgapt-1 - fxgapt-2)/100
| + 1.015[5.39] * ddockx

Regression Statistics
Adjusted R2: | 0.51
S.E. of regression: | 0.016
Sample period: | 1973Q1 2017Q4

### EXN: Exports of goods and services, current $

exn | = .01*pxp*pxr*ex

### FCBN: US current account balance, current $

The current account balance is equal to net exports (EXN - EMN) plus net foreign investment income (FYNIN) and a discrepancy term (FCBRN).

fcbn | = exn - emn + fynin + fcbrn

### FCBRN: US current account balance residual, current $

The discrepancy in the current account balance is assumed to be proportional to nominal potential output.

fcbrn | = ufcbr*pxb*xbt/100

### FNICN: Gross stock of claims of US residents on the rest of the world, current $

Δ(fnicn)/xgdptn = |
| .54 * Δ(log(fpc))*fnicnt-1/xgdptn
| - .67 * Δ(log(fpx))*fnicnt-1/xgdptn
| + rfnict

### FNILN: Gross stock of liabilities of US residents to the rest of the world, current $

fniln | = fnicn - fnin

### FNIN: Net stock of claims of US residents on the rest of the world, current $

The change in the net foreign investment position is equal to the the sum of the current account balance expressed at a quarterly rate, estimates of capital gains and losses on the gross investment positions due to price and exchange rate changes, and a residual term.

Δ(fnin) = |
| .25*fcbn
| + .54 * (Δ(log(fpc)) * fnicnt-1)
| - .32 * (Δ(log(pgdp)) * fnilnt-1)
| - .67 * (Δ(log(fpx)) * fnicnt-1)
| + .06 * (Δ(log(fpx)) * fnilnt-1)
| + fnirn

### FNIRN: Net stock of claims of US residents on the rest of the world, residual

fnirn | = ufnir * xgdpn

### FTCIN: Corporate taxes paid to rest of world, current $

ftcin | = uftcin * ynicpn

### FYNICN: Gross investment income received from the rest of the world, current $

fynicn | = .01*rfynic*fnicnt-1

### FYNILN: Gross investment income paid to the rest of the world, current $

fyniln | = .01*rfynil*fnilnt-1

### FYNIN: Net investment income received from the rest of the world, current $

fynin | = fynicn - fyniln

### HGEMP: Petroleum imports, cw 2012$, trend growth rate

hgemp = |
| 0.9 * hgempt-1
| + 0.1 * 400*log(emp/empt-1)

### RFYNIC: Average yield earned on gross claims of US residents on the rest of the world

d(rfynic) = |
| 0.244[3.92]
| + -0.140[-4.065] * (rfynict-1 - rfynilt-1)
| + 0.144[2.38] * d(rfynict-1)
| + 0.632[9.34] * d(rfynil)

Regression Statistics
Adjusted R2: | 0.38
S.E. of regression: | 0.29
Sample period: | 1970Q1 2017Q4

### RFYNIL: Average yield earned on liabilities of US residents on the rest of the world

The average yield on U.S. liabilities to the rest of the world is modeled in an error correction format. In the long run, this yield is a function of the rates of return on a range of domestic assets. Short-run movements in these rates of return have additional effects on the average yield on U.S. liabilities.

d(rfynil) = |
| 0.177[2.37]
| + -0.248[-6.37] * rfynilt-1
| + 0.0803[3.84] * rg10t-1
| + 0.0902[4.92] * rtbt-1
| + 0.0445[3.98] * reqpt-1
| + 0.133[2.65] * d(rfynilt-1)
| + 0.0876[1.84] * d(rg10)
| + 0.261[9.81] * d(rtb)
| + 0.0179[0.75] * d(reqp)

Regression Statistics
Adjusted R2: | 0.59
S.E. of regression: | 0.21
Sample period: | 1970Q1 2017Q4

# Aggregate Output Identities

Some of the channels through which movements in real GDP are transmitted elsewhere in the model go through XGDO and XBO, which are estimates respectively of real GDP and real business output, each adjusted for measurement error.

The key production sector in FRB/US is business output. Potential output (XBT) of this sector is modeled with a two-factor Cobb-Douglas production function, whose inputs are potential quality-adjusted labor hours and capital services. Trend multifactor productivity captures a standard Solow residual. XBT is linked to potential GDP (XGDPT) through a sequence of identities that can be described most easily in terms of the relationships between various measures of the percentage gap between actual and potential output -- the output gap.

The lack of cyclical variation in non-business output also underlies the relationship between the actual levels, adjusted for measurement error, of real GDP (XGDO) and business output (XBO). Specifically, the elasticities of XBO and XGO with respect to XGDP are the inverse of their average nominal historical ratios to GDP, when potential GDP is held fixed, and 1.0 when actual and potential GDP move together.

Many of the variables that are used to calculate the two measures of potential output (XGPOT and XBT) are modeled as stochastic processes that permit both (log) level and growth rate shocks. As a result, the trend rate of growth of such a variable differs from and is generally smoother than the growth rate of its trend level. Because the same holds true by construction for the two measures of aggregate potential output, FRB/US contains separate variables for each trend level and each trend growth rate: XBT and HXBT; and XGDPT and HGGDPT.

The historical values of some of the key variables that enter the construction of potential output are estimated as latent variables in a reduced-form state-space model that is documented here and whose code and data are available in the FRB/US supply-side package. Among these latent variables are the potential levels of GDP and business sector output, as well as potential business sector employment (LEPPOT), the natural rate of unemployment (LURNAT) and the trend components of the labor-force participation rate (QLFPR) and the workweek (QLWW). The historical values of the measures of real GDP and real business output that are adjusted for measurement error (XGDO and XBO) also come from the state-space model.

### HGGDP: Growth rate of GDP, cw 2012$ (annual rate)

hggdp | = 400*Δ(log(xgdp))

### HGGDPT: Trend growth rate of XGDP, cw 2012$ (annual rate)

The trend growth rate of GDP is the sum of the trend growth rates of business output and the stochastic trend component of potential GDP. Because FRB/US permits separate log level and growth rate shocks, the trend growth rate of GDP (HGGDPT) is different (and smoother) than the growth rate of potential GDP (400*del(log(XGDPT)).

hggdpt | = hxbt + huxb

### HGPBFIR: Trend growth rate of price of business investment (relative to PXB)

hgpbfir = |
| .975*hgpbfirt-1
| + .025*400*(log(pbfir*pxp/pxb / (pbfirt-1*pxpt-1/pxbt-1)))

### HMFPT: Trend growth rate of multifactor productivity

Trend multi-factor productivity follows a random walk with drift. The drift term, HMFPT, is an AR(1) process. As with all drift terms in FRB/US supply-side stochastic trends, HMFPT is assumed to be mean-reverting at a five percent quarterly rate.

hmfpt | = 0.055 + 0.95*hmfptt-1

### HUXB: Drift term in UXBT

The stochastic trend component of potential GDP, UXBT, follows a random walk with drift. The drift term, HUXB, is an AR(1) process. As with all drift terms in FRB/US supply-side stochastic trends, HUXB is assumed to be mean-reverting at a five percent quarterly rate. In long-run simulations, the dummy variable DGLPRD can be set to 1.0 to ensure that potential GDP and potential business sector grow at the same rate.

huxb = |
| (1-dglprd)*
| (-0.0161[-440.088]
| + 0.95*huxbt-1)

Regression Statistics
Adjusted R2: | 0.100
S.E. of regression: | 0.00053
Sample period: | 1966Q1 2017Q4

### HXBT: Trend rate of growth of XB , cw 2012$ (annual rate)

Some of the components of potential output are modeled as stochastic trends wth stochastic drift terms. Because both log level shocks and growth rate shocks occur in this framework, the trend rate of growth of such a component differs from the growth rate of its trend level. The trend rate of growth of business output is calculated using the distinct trend rates of growth, rather than on the growth rate of trends, of multifactor productivity (HMFPT) and those factor inputs that have trend growth rates (HLEPT, HQLWW). The imposed coefficients are the same as those in the production function for the level of XBT. The trend growth rate of business sector output (HXBT) is smoother than the growth rate of the level of potential adjusted business sector output (400*del(log(XBT))). As for the level equation, there is an explicit residual term (HXBTR) in the equation for potential growth.

hxbt = |
| .725*(hlept + hqlww + 400*Δ(log(lqualt)))
| + .275*hks + hmfpt +hxbtr

### HXBTR: Residual Factor (Trend rate of growth of XB)

hxbtr | = 0

### MFPT: Multifactor productivity, trend level

log(mfpt) | = 0 + log(mfptt-1) + hmfpt/400

### UXBT: Stochastic component of trend ratio of XGDPT to XBT

The trend component of potential GDP is a random walk with drift. The drift component (HUXB) follows an AR(1) process.

log(uxbt) | = 0 + log(uxbtt-1) + .0025*huxb

### XB: Business output (BEA definition), cw 2012$

xb | = xbn/ (pxb/100)

### XBO: Business output, adjusted for measurement error, cw 2012$

The estimated coefficient on XGAP2 is consistent with the observation that value added outside of the business sector has little or no cyclical variation and thus that movements in the business sector output gap fully account for movements in the GDP gap.

log(xbo) = |
| log(xbt)
| + 1.325[102.75]*xgap2/100

Regression Statistics
Adjusted R2: | 0.100
S.E. of regression: | 0.0047
Sample period: | 1965Q1 2017Q4

### XBT: Potential business output, cw 2012$

The specification of potential business output (XBT) is based on a two-factor Cobb-Douglas production function. In addition to measures of factor inputs, the equation also contains trend multi-factor productivity (MFPT). Potential labor input depends on trend employment (LEPPOT), trend hours per worker (QLWW), and trend labor quality (LQUALT). Capital input is measured by capital services (KS). In addition, the equation posseses an explicit error term (XBTR).

log(xbt) = |
| 0.725 * (log(leppot) + log(qlww) + log(lqualt))
| + 0.275 * log(ks)
| + log(mfpt) + log(xbtr)

### XBTR: Residual Factor (Potential business output)

log(xbtr) | = 0.95 * log(xbtrt-1)

### XFS: Final sales of gross domestic product, cw 2012$

Real final sales of gross domestic product is approximated by a fixed-weighted aggregation of its components.

log(xfs) = |
| log(xfst-1)
| + 0.685*log(ecnia/ecniat-1)
| + 0.039*log(eh/eht-1)
| + 0.132*log(ebfi/ebfit-1)
| + 0.043*log(egfe/egfet-1)
| + 0.022*log(egfl/egflt-1)
| + 0.040*log(egse/egset-1)
| + 0.069*log(egsl/egslt-1)
| + 0.120*log(ex/ext-1)
| + -0.140*log(emo/emot-1)
| + -0.010*log(emp/empt-1)

Regression Statistics
Adjusted R2: | 0.100
S.E. of regression: | 0.00024
Sample period: | 2012Q4 2017Q4

### XGAP: Output gap for business plus oil imports (100*log(actual/potential)

xgap | = 100*log(xbo/xbt)

### XGAP2: Output gap for GDP (100*log(actual/potential)

xgap2 | = 100 * log(xgdo/xgdpt)

### XGDI: Gross domestic income, cw 2012$

Gross domestic income is modeled with an identity as the product of gross domestic product adjusted for measurement error (XGDO) and a discrepancy factor (MEI) whose logarithm follows an AR(1) process.

xgdi | = xgdo*mei

### XGDO: Gross domestic product, adjusted for measurement error, cw 2012$

XGDO is real GDP adjusted for measurement error. In simulation XGDO equals the ratio of real gross domestic product (XGDP) to a discrepancy factor (MEP) whose logarithm follows an AR(1) process. The historical values of XGDO are estimated as latent variables in the reduced-form state-space model documented here.

xgdo | = xgdp/mep

### XGDP: GDP, cw 2012$

Real gross domestic product (XGDP) is approximated by a fixed-weighted aggregation of its components.

log(xgdp) = |
| log(xgdpt-1)
| + 0.999*log(xfs/xfst-1)
| + 0.626*(log(ki) - log(kit-1))
| + -0.625*(log(kit-1)- log(kit-2))

Regression Statistics
Adjusted R2: | 0.100
S.E. of regression: | 0.00020
Sample period: | 2012Q4 2017Q4

### XGDPT: Potential GDP, cw 2012$

The difference between the logs of potential GDP and business sector output -- log(UXBT) -- is a stochastic trend that is modeled as a random walk with drift. The drift component follows as AR(1) process.

log(xgdpt) | = log(xbt) + log(uxbt)

### XGDPTN: Potential GDP, current $

xgdptn | = .01*pgdp*xgdpt

### XP: Final sales plus imports less government labor, cw 2012$

Real domestic final purchases, excluding government compensation but including exports, is approximated by a fixed-weighted aggregation of its components.

log(xp) = |
| log(xpt-1) + 0.653*log(ecnia/ecniat-1)+ 0.0361*log(eh/eht-1)
| + (0.118)*log(ebfi/ebfit-1)
| + (0.0422)*log(egfe/egfet-1)
| + (0.0366)*log(egse/egset-1)+ 0.114*log(ex/ext-1)

# Labor Market

This sector contains equations for a variety of labor market variables, including hours worked, private employment in the nonfarm business sector, household employment, labor force participation, and the unemployment rate.

A central equation in the sector is the one for total hours, which is modeled using the polynomial adjustment cost (PAC) framework. Over time, hours error-corrects to an equilibrium level consistent with aggregate output and trend labor productivity. The latter is defined in a manner consistent with the aggregate production function, and thus depends on the optimal capital/output and energy/output ratios (which in turn are functions of relative factor prices).

Another key equation is the one for the employment discrepancy (LEO), whose variation around a trend is estimated to be procyclical. An identity determines the unemployment rate.

The historical values of some of the variables in this sector are estimated as latent variables in a reduced-form state-space model that is documented here and whose code and data are available in the FRB/US supply-side package. Among these latent variables are the natural rate of unemployment (LURNAT) and the trend levels and drift terms for the labor-force participation rate (QLFPR, HQLFPR) and the workweek (QLWW, HQLWW).

### HLEPT: Trend growth rate of LEP (annual rate)

hlept | = 400*hqlfpr + 400*Δ(log(n16))

### HLPRDT: Trend growth rate of output per hour

hlprdt | = hxbt - hlept - hqlww

### HQLFPR: Drift component of change in QLFPR

The trend first difference of the workweek follows an AR(1) process. HQLFPR is assumed to revert to a mean of zero at a five percent quarterly rate.

hqlfpr | = 0 + 0.95*hqlfprt-1

### HQLWW: Trend growth rate of workweek

The trend growth rate of the workweek follows an AR(1) process. As with all drift terms in FRB/US supply-side stochastic trends, HQLWW is assumed to be mean-reverting at a five percent quarterly rate.

hqlww | = 0.95 * hqlwwt-1 + (1-0.95) * -0.313[2.43]

### LEH: Civilian employment (break adjusted)

Civilian employment from the household survey is the sum of business employment (LEP), state and local government employment (LES), federal government employment (LEF), and the employment discrepancy (LEO).

leh | = lep + leg + leo

### LEO: Difference between household and business sector payroll employment, less government employment

LEO error corrects to the discrepancy between its lagged and desired values and also varies counter-cyclically. The desired value of LEO is the product of a desired ratio (QLEOR) and the equilibrium labor force. QLEOR is measured historically as the value of LEO that is consistent with equilibrium values of the labor force, employment in various sectors, and the unemployment rate. In simulation, QLEOR is exogenous.

log(leo) = |
| log(qleor*qlf)
| + 0.757[16.94] * log(leot-1/(qleort-1*qlft-1))
| + -0.0164[-5.13] * xgap2t-1

Regression Statistics
Adjusted R2: | 0.97
S.E. of regression: | 0.060
Sample period: | 1986Q1 2017Q4

### LEP: Employment in business sector (employee and self-employed)

Employment in the business sector equals aggregate hours divided by the average workweek.

lep | = lhp / lww

### LEPPOT: Potential employment in business sector

The trend level of employment in the business sector consists of potential economy-wide employment less trend employment in other sectors. Potential economy-wide employment equals the product of the trend labor force (QLF) and the proportion of the labor force employed in equilibrium (1.0 minus the natural rate of unemployment, LURNAT). Trend employment in sectors other than business consists of trend government employment (LEST + LEFT) and trend "other" employment (QLEOR * QLF).

leppot | = (qlf*(1-.01*lurnat - qleor))/(1+adjlegrt)

### LF: Civilian labor force (break adjusted)

lf | = lfpr * n16

### LFPR: Labor force participation rate

In the long run, the participation rate moves at its trend first difference (HQLFPR). In the short run, the participation rate error corrects at a 45 percent rate to the lagged deviation of its level from its trend level (QLFPR). QLFPR is modeled as a random walk with time-varying drift. Cyclical variation in the participation rate is captured by including the gap between the unemployment rate and its natural rate (LURNAT), which is lagged to avoid coefficient bias that might arise from measurement error that is common to LUR and LFPR.

d(lfpr)-hqlfpr = |
| 0.568[10.44]*(qlfprt-1-lfprt-1)
| + -0.000875[-9.063]*(lurt-1-lurnatt-1)

Regression Statistics
Adjusted R2: | 0.34
S.E. of regression: | 0.0012
Sample period: | 1963Q3 2017Q4

### LHP: Aggregate labor hours, business sector (employee and self-employed)

Firms attempt to keep aggregate hours in line with the expected level of production adjusted for the trend level of labor productivity. This definition of the target level of hours is embedded in a version of the polynomial adjustment cost framework that is modified to allow some portion of labor hours to adjust costlessly. The portion of the equation that corresponds to the costly adjusting hours consists of the three conventional PAC terms -- the degree hours were out of equilibrium last period, lagged hours growth, and expected growth in target hours. The structure of the PAC component of the equation is outlined below in Note 1. To translate the general PAC specification into the form in which it appears in the LHP equation, associate log(LHP) with y, log(QLHP) with y*, and ZLHP with the expected weighted sum of future Δy*. The order of adjustment costs (m) is 2.

The portion of hours that adjusts costlessly is captured by the current growth in target hours (the growth rate of XBO less the trend growth rate of output per hour, HLPRDT). The coefficient on the latter indicates that more than 40 percent of hours adjust costlessly and a bit less than 60 percent of hours adjust according to the PAC specification. The aggregation of slow-adjusting and fast-adjusting hours is outlined below in Note 2.

Note 1: PAC Overview: Let y denote the decision variable and y* its desired level in the absence of adjustment frictions. The decision rule derived from the PAC Euler equation has the following form.

Δytpac = a0(y*t-1 - yt-1) + ∑i=1,m-1 aiΔyt-i + Et-1∑i=0,∞diΔy*t+i

The order of adjustment costs is given by parameter m. The forward weights, di {i=0,∞}, are functions of the estimated values of ai {i=0,m-1} and a fixed discount factor (0.98). Estimation imposes the growth neutrality restriction ∑i=1,m-1ai + ∑i=0,∞di = 1. For more information, see PAC Basics.

Note 2: To describe the aggregation of slow-adjusting PAC hours and fast-adjusting hours, let h and q be the logs of total hours and target hours, and z be the weighted sum of expected growth of target hours. Subscripts denote slow-adjusting ("1") and fast-adjusting ("2") components. Assume the shares of the two types of hours are approximately constant. Write

Δht = c*Δh1t + (1-c)*Δh2t
Δh1t = a0(q1-h1)t-1 + a1*Δh1,t-1 + zh1t
Δh2t = Δq2t = Δqt

By assumption, the total hours gap (q-h) is concentrated in the slow adjusting hours, which leads to the final equation in the following steps

(q1t-h1t) = (qt-ht)/c
Δh1t = (Δht - (1-c)*Δqt)/c
Δh1t = a0*(q-h)t-1/c + a1*(Δht-1 - (1-c)Δqt-1)/c + zh1t
Δht = a0*(q-h)t-1 + a1*Δht-1 + c*zht + (1-c)*Δqt + a1*(1-c)*Δqt-1

The final form of the equation assumes zh = zh1.

dlog(lhp) = |
| 0.202[7.26] * (log(qlhpt-1/lhpt-1))
| + 0.203[3.39] * dlog(lhpt-1)
| + 0.372[10.35] * zlhp
| + 0.628 * (dlog(xbo) - hlprdtt-1/400 )
| + -0.127 * (dlog(xbot-1) - hlprdtt-2/400)

Regression Statistics
Adjusted R2: | 0.80
S.E. of regression: | 0.0035
Sample period: | 1965Q1 2017Q4

### LPRDT: Trend labor productivity

Trend labor productivity in the adjusted business sector is the ratio of potential output in that sector to trend total hours. The latter is the product of potential employment (LEPPOT) and the trend in hours per worker (QLWW).

log(lprdt) | = log(xbt) - log(leppot) - log(qlww)

### LUR: Civilian unemployment rate (break adjusted)

lur | = 100*(1 - leh/lf)

### LURNAT: Natural rate of unemployment

The natural rate of unemployment follows a random walk with zero drift.

lurnat | = lurnatt-1

### LWW: Workweek, business sector (employee and self-employed)

In the long run, the workweek grows as its trend rate (HQLWW). In the short run, the workweek error corrects at nearly a 20 percent rate to the lagged deviation of its level from its trend level (QLWW). QLWW is modeled as a random walk with drift. About 30 percent of the contemporaneous deviation of the rate of growth of total hours from its trend shows up as a change in the workweek (and thus about 70 percent shows up as a change in employment).

dlog(lww)-hqlww/400 = |
| 0.196[10.67]*log(qlwwt-1/lwwt-1)
| + 0.318[19.42]*(dlog(lhp)-(hlept+hqlww)/400)

Regression Statistics
Adjusted R2: | 0.65
S.E. of regression: | 0.0018
Sample period: | 1963Q3 2017Q4

### QLF: Desired level of civilian labor force

The trend labor force is equal to the product of the trend participation rate and the size of the population age 16 and up.

qlf | = qlfpr * n16

### QLFPR: Trend labor force participation rate

The trend participation rate follows a random walk with time-varying drift.

qlfpr | = qlfprt-1 + hqlfpr

### QLHP: Desired level of business labor hours

The desired level of aggregate hours equals business sector output (XB) divided by trend labor productivity (LPRDT).

qlhp | = xbo/lprdt

### QLWW: Trend workweek, business sector (employee and self-employed)

The trend workweek is a random walk with drift.

log(qlww) | = log(qlwwt-1) + hqlwwt-1/400

# Nominal Income

Many of the equations in this sector are accounting identities for measures of nominal income, output, and product. Some of the identities make use of multiplicative conversion factors, so that the full set of variables that appear in the NIPA identities do not have to be included in FRB/US.

The sector also contains equations for the measures of after-tax household income and its primary components -- labor, transfer, and property -- that are are used in the consumption sector. Property income is defined more broadly than in the NIPA accounts. It includes the following additional items: imputed income from the stock of consumer durables, less consumer interest payments to business; corporate retained earnings; and inflation losses on the stock of government debt. These modifications to the definition of household income imply that households see through the corporate veil and adjust interest income to exclude that portion which compensates for inflation. (An inflation adjustment to interest earned on corporate debt is not necessary, since an offsetting adjustment would need to be made to the definition of corporate profits.)

The sector contains estimated equations for dividends, consumer interest payments to business, the net interest and rental income component of national income, and the net financial debt of nonfinancial corporations.

### JCCAN: Consumption of fixed capital, current $

jccan/xgdpn = |
| 0.0539*(1-0.821)
| + 0.821*jccant-1/xgdpnt-1
| + (1-0.821)*pkbfirt-1*kbfit-1*jrbfi*pxpt-1*.01/xgdpnt-1

### QYNIDN: Desired level of dividends

The desired level of dividends is a constant fraction of after-tax corporate profits, with a shift in the desired fraction starting in 1980. Coefficient values are taken from a regression of the log of the actual ratio of dividends to after-tax profits on the explanatory variables. A max function is used to prevent simulation problems arising from attempts to take the log of a negative number.

log(qynidn) = |
| -0.916[-46.079]
| + 0.355[15.16]*d79a
| + 1*log((max((ynicpn-tcin),(.01))))

Regression Statistics
Adjusted R2: | 0.52
S.E. of regression: | 0.15
Sample period: | 1965Q2 2017Q4

### RSPNIA: Personal saving rate

rspnia | = 100 * yhsn / ydn

### UYNICPNR: Corporate profits, residual

uynicpnr = |
| -0.0740[-192.36]*(1-0.779[14.15])
| + 0.779[14.15]*uynicpnrt-1

### XBN: Business output (BEA definition), current $

xbn | = pxb/100*xbo + xgdpn -xgdo*pgdp/100

### XFSN: Final sales of gross domestic product, current $

xfsn | = xgdpn - ein

### XGDIN: Gross domestic income, current $

xgdin | = xgdi *(pgdp/100)

### XGDPN: GDP, current $

xgdpn | = xpn + ein - emn + egfln + egsln

### XPN: Final sales plus imports less government labor, current $

xpn | = .01 * pxp * xp

### YDN: Disposable income

ydn = |
| uyd * (ypn - tpn)

### YH: Income, household, total (real after-tax)

yh | = yhl + yht + yhp

### YHGAP: Income, household, total, ratio to XGDP, cyclical component (real after-tax)

YHGAP is the percentage deviation of the actual from the trend ratio of household income to GDP (YHSRH and ZYHST, respectively).

yhgap = |
| 100*log(yhshr/zyhst)

### YHIBN: Consumer interest payments to business

yhibn = |
| uyhibn*xgdpn

### YHL: Income, household, labor compensation (real after-tax)

yhl | = (1-tryh)*yhln/(.01*pcnia)

### YHLN: Income, household, labor compensation

yhln | = uyhln * (yniln)

### YHP: Income, household, property (real after-tax)

yhp | = ((1-tryh)*yhptn+yhpntn)/(.01*pcnia)

### YHPGAP: Income, household, property, ratio to YH, cyclical component (real after-tax)

YHPGAP is the percentage deviation of the actual from the trend ratio of household property income to total household income (YHPSHR and ZYHPST, respectively).

yhpgap = |
| 100*log(yhpshr/zyhpst)

### YHPNTN: Income, household, property, non-taxable component

Household non-taxable property income in FRB/US includes several items not included in the NIPA definition of personal income: imputed income from the stock of consumer durables, less consumer interest payments to business; corporate retained earnings; and inflation losses on the stock of government debt.

yhpntn = |
| .01*pcnia*pcdr*yhpcd
| - yhibn + ynicpn - tcin - ynidn
| - .01 * zpi10 *gfdbtn

### YHPSHR: Income, household, property, ratio to YH (real after-tax)

yhpshr | = yhp/yh

### YHPTN: Income, household, property, taxable component

Household taxable property income in FRB/US includes interest and rental income, dividends, and self-employed income. The multiplicative factor UYHPTN adjusts for the difference between total dividends (YNIDN) and personal dividend income, which reflects dividends paid to state and local governments.

yhptn = |
| uyhptn*(ynirn+gfintn+ynidn+yhibn)

### YHSHR: Income, household, total, ratio to XGDP (real after-tax)

yhshr | = yh/xgdp

### YHSN: Personal saving

yhsn = |
| yhln + yhtn + yhptn - tpn - ecnian - yhibn
| + uyhsn * xgdptn

### YHT: Income, household, transfer (real after-tax), net basis

yht | = yhtn/(.01*pcnia)

### YHTGAP: Income, household, transfer, ratio to YH, cyclical component (real after-tax)

YHTGAP is the percentage deviation of the actual from the trend ratio of household transfer income to total household income (YHTSHR and ZYTHST, respectively).

yhtgap = |
| 100*log(yhtshr/zyhtst)

### YHTN: Income, household, transfer payments. net basis

The exogenous conversion factor (UYHTN) in the identity for transfer payments to persons (YHTN) reflects the omission of business transfer payments from the equation.

yhtn = |
| uyhtn*gtn

### YHTSHR: Income, household, transfer, ratio to YH (real after-tax)

yhtshr | = yht/yh

### YKBFIN: Income from stock of BFI

ykbfin | = .01*rtbfi*pxb* ( kbfi + kbfit-1) /2

### YKIN: Income from stock of inventories

ykin | = .01*rtinv*pxb* (ki + kit-1) /2

### YNICPN: Corporate profits (national income component)

Corporate profits (YNICPN) are the residual component of national income (YNIN - YNILN - YNIIN - YNISEN). To mitigate numerical problems in simulations, the max function places a positive lower bound on profits.

ynicpn = |
| max(( (ynin-yniln-ynirn+uynicpnr*xgdpn)),( (tcin+.01*xgdpn)))

### YNIDN: Dividends (national income component)

Dividends are modeled using the polynomial adjustment cost (PAC) framework. Thus growth in real dividends depends on the three standard PAC terms -- the degree to which dividends were out of equilibrium last period, lagged dividend growth, and expected growth of desired dividents. To translate the general PAC specification, whose structure is shown below in the PAC Overview section, associate log((YNIDN-YMSDN)/PXB) with y, log(QYNIDN/PXB) with y*, and ZYNID with the expected weighted sum of future Δy*. The order of adjustment costs (m) is 2. The equation adjusts dividends for the one-time Microsoft cash payout of late 2004 (YMSDN).

PAC Overview: Let y denote the decision variable and y* its desired level in the absence of adjustment frictions. The decision rule derived from the PAC Euler equation has the following form.

(1) Δyt = a0(y*t-1 - yt-1) + ∑i=1,m-1 aiΔyt-i + Et-1∑i=0,∞diΔy*t+i

The order of adjustment costs is given by parameter m. The forward weights, di {i=0,∞}, are functions of the estimated values of ai {i=0,m-1} and a fixed discount factor (0.98). Estimation imposes the growth neutrality restriction ∑i=1,m-1ai + ∑i=0,∞di = 1. For more information, see PAC Basics.

dlog((ynidn-ymsdn)/pxb) = |
| 0.107[5.91] * log(qynidnt-1/(ynidnt-1-ymsdnt-1))
| + -0.210[-3.14] * dlog((ynidnt-1-ymsdnt-1)/pxbt-1)
| + 1 * zynid

Regression Statistics
Adjusted R2: | 0.060
S.E. of regression: | 0.040
Sample period: | 1965Q2 2017Q4

### YNILN: Labor income (national income component)

The exogenous conversion factor (UYL) in the identity for labor income reflects: (1) the omission of labor income in the farm and household and institutions sectors from the equation; and (2) the use of a measure of aggregate hours that includes not only hours of employees but also hours of the self-employed.

yniln | = 0.01 * uyl * (pl*lhp + pgfl*egfl + pgsl*egsl)

### YNIN: National income

The exogenous conversion factor (UYNI) in the identity for national income (YNIN) reflects the omission of the statistical discrepancy and business transfer payments from the equation.

ynin | = uyni*(xgdin+fynin-jccan)

### YPN: Personal income

The exogenous conversion factor (UYP) used in the personal income identity reflects the ommission of a miscellaneous set of adjustments, such as the difference between total and personal dividend payments, and the omission of business transfers payments to households.

ypn | = uyp * (yhln + yhtn + yhptn)

# Wages and Prices

Main Equations

At the center of price-wage sector is a pair of equations for core PCE price inflation (PICXFE) and the rate of change of the ECI measure of hourly compensation (PIECI). The equations are estimated simultaneously. The form of each equation is based on a variant of the New Keynesian Phillips Curve (NKPC) derived by Cogley and Sbordonne (2008) for the case in which the subset of prices or wages that is not optimally reset each period is indexed to either past or trend inflation.

(1) Πt = Et-1[Πt+1 + κμt] + εt

(2) Πt ≡ πt - γπ^t - (1-γ)π*t

(3) π^t ≡ ∑(i=1,n)πt-i /n

Equation 1 is the basic NKPC expressed in terms of the composite variable Π, which is defined in equation 2 as the difference between actual price or wage inflation (π) and a weighted average of past (π^) and trend (π*) price or wage inflation. Expectations are formed on the basis of "t-1" information, β(=.98) is the discount factor, and μ is the markup gap.

Past inflation (equation 3) is measured using a single lag (n=1) in the price NKPC and a four-quarter average (n=4) in the wage NKPC. For prices, trend inflation is measured using survey data on expectations of ten-year consumer price inflation (PTR). Trend wage inflation equals the sum of trend price inflation and the trend rate of increase of labor productivity, less an estimate of the trend rate of change of the relative price of consumption.

The detrended level of the log labor share in the nonfarm business sector is a key part of both the price and wage markup gaps. The consumption price markup gap is approximately equal to a weighted average of the detrended labor share, the relative price of nonoil imports, and a few additional but minor terms, all measured in logs. The wage markup gap -- in theory, the difference between the marginal rate of substitution and the real wage -- is expressed as a function of the detrended labor share and the unemployment gap (LUR-LURNAT). This structural dependence of wage inflation on the unemployment gap leads to a dependence of price inflation on the unemployment gap through expected price inflation.

The structure of the PICXFE and PIECI equations, along with various related equations and identities, ensures that the sector has a long-run equilibrium in which the labor share of income is constant (given by the inverse of PWSTAR). Historically, PWSTAR has a trend that is estimated using a one-sided variant of the HP filter. The long-run equilibrium conditions associated with a given value of PWSTAR are embedded in the definitions of a set of desired or target price and wage levels: QPXG, QPL, QPXP, and QPCNIA. UQPCT, which is the trend in the ratio of the price of consumption (PCNIA) to the price of final sales (PXP), also enters this structure. In a long-run equilibrium, actual inflation will equal trend inflation, where the latter is usually specified to converge to the rate of inflation desired by policymakers.

More information on the estimation of the price-wage NKPC equations is available here.

Price Disaggregation

The prices of most non-consumption components of GDP are determined in two stages. The first stage consists of the equation for PIPXNC, which is the rate of price inflation for an aggregate of business fixed investment, residential investment, non-labor government purchases, and exports. The main determinant of PIPXNC is the rate of growth of consumption prices including food and energy (PICNIA), adjusted for an estimate of the trend in PIPXNC-PICNIA. In the second stage, PIPXNC is disaggregated into nine component prices using a set of equations of the form,

(4) Δlog(Pi/PXP) = Ci + Δlog(PXNC/PXP) + DPADJ

where Pi is a component price, PXNC is the price level associated with PIPXNC, and DPADJ is a correction factor that accounts for the fact that the set of equations given by (4) yields an inexact decomposition of PNXC each quarter. The equation also includes the aggregate consumption and non-consumption price, PXP, as a scaling variable and a constant. The coded equation for each Pi sets the constant to zero, but in most simulation applications the equivalent of a non-zero intercept is included via an add factor whose value is based on that equation's average historical error. The prices determined in this system are PHR, PBFIR, PEGFR, PEGSR and PXR.

Other Prices

The fundamental energy price is the average per barrel price of imported oil, POIL. The price index for nonoil imports, PMO, depends on a weighted average of the exchange-rate adjusted foreign CPI and the price index for nonfarm business output. The rest of the sector consists of a large number of identities for such variables as the GDP price index (PGDP).

### DPADJ: Price inflation aggregation adjustment

The adjustment factor for non-consumption prices equals the value in the prior quarter plus the price aggregation discrepancy in the prior quarter. Thus, component prices are adjusted to offset any aggregation discrepancy with only a one-quarter lag.

dpadj - dpadj_aerr - dpadj(-1) | = 1 * dpgapt-1

### DPGAP: Price inflation aggregation discrepancy

The price inflation aggregation discrepancy (DPGAP) equals the rate of increase of the price for adjusted final sales excluding consumption less the weighted sum of rates of price increase for the non-consumption components of adjusted final sales.

dpgap = |
| pipxnc/400 - (
| 0.104 * Δ(log( phr * pxp))
| + 0.341 * Δ(log( pbfir * pxp))
| + (0.0820+0.0393) * Δ(log( pegfr * pxp))
| + (0.0525+0.0525) * Δ(log( pegsr * pxp))
| + 0.329 * Δ(log( pxr * pxp)))

### HGPKIR: Trend growth rate of PKIR

hgpkir = |
| 0.9 * hgpkirt-1
| + 0.1 * 400*log(pkir/pkirt-1)

### HUQPCT: Drift term in stochastic component of trend ratio of PCNIA to PXP

huqpct | = 0 + 0.95*huqpctt-1

Regression Statistics
Sample period: | 1962Q1 -
Estimation date: | August 2010

### PBFIR: Price level of BFI compared to PXP

log(pbfir) | = log(pbfirt-1) + pipxnc/400 + dpadj - Δ(log(pxp))

### PCDR: Price index for consumer durables, cw (relative to to PCNIA)

The growth rate of the relative (to PCNIA) price of consumer durable goods (PCDR) is assumed to follow a first order autoregressive process.

d(log(pcdr)) = |
| -0.00333[-6.21]
| + 0.510[8.12]*d(log(pcdrt-1))

Regression Statistics
Adjusted R2: | 0.25
S.E. of regression: | 0.0045
Sample period: | 1970Q2 2017Q4

### PCER: Price index for personal consumption expenditures on energy (relative to PCXFE)

The growth rate of the consumer energy prices (relative to PXCFE) are assumed to be a fraction of that of oil imports (relative to PCXFE, as well).

Δ(log(pcer)) = |
| 0.249[19.37] * d(log(pmp/pcxfe))

Regression Statistics
Adjusted R2: | 0.64
S.E. of regression: | 0.025
Sample period: | 1965Q1 2017Q4

### PCFR: Price index for personal consumption expenditures on food (relative to PCXFE)

Growth in relative consumer food prices is modeled using a simple error-correction specification, where the level of prices in the long run is equal to an estimated trend, PCFRT, defined historically by H-P filtering the observed series.

Δ(log(pcfr)) = |
| -0.175[-6.41] * log(pcfrt-1/pcfrtt-1)
| + -5.595E-05[-0.12]
| + DL1(3){sum=0.74}Δ(log(pcfrt-1))
| + 0.318[1.31] * Δ(log(pcfrt))

Distributed Lag Coefficients

DL11 | 0.386
DL12 | 0.0203
DL13 | 0.339
DL1sum | 0.745

Regression Statistics
Adjusted R2: | 0.37
S.E. of regression: | 0.0068
Sample period: | 1965Q1 2017Q4

### PCHR: Price index for housing services, cw (relative to to PCNIA)

The growth rate of the relative (to PCNIA) price of housing services (PCHR) is assumed to follow a first order autoregressive process.

d(log(pchr)) = |
| 0.000614[1.90]
| + 0.598[10.27]*d(log(pchrt-1))

Regression Statistics
Adjusted R2: | 0.35
S.E. of regression: | 0.0043
Sample period: | 1970Q2 2017Q4

### PCNIA: Price index for personal consumption expenditures, cw (NIPA definition)

Δ(log(pcnia)) | = picnia / 400

### PCOR: Price index for non-durable goods and non-housing services, cw (relative to to PCNIA

The relative price of non-durable goods and non-housing services (PCOR) is approximated by a fixed-weighted aggregation of its components.

log(pcor) = |
| log(pcort-1)
| + -0.144*log(pcdr/pcdrt-1)
| + -0.217*log(pchr/pchrt-1)

Regression Statistics
Adjusted R2: | 0.100
S.E. of regression: | 2.27E-06
Sample period: | 2016Q4 2017Q4

### PCPI: Consumer price index,total

The overall CPI equals the product of the PCE chain-weight price index and a proportionality factor. This factor has two components, one to account for the effect of different weights on energy in the two price indices, and the other to account for all other differences.

pcpi = |
| upcpi * pcnia

### PCPIX: Consumer price index,excluding food and energy

pcpix | = upcpix * pcxfe

### PCXFE: Price index for personal consumption expendits ex. food and energy, cw (NIPA definit

Δ(log(pcxfe)) | = picxfe/400

### PGDP: Price index for GDP, cw

pgdp | = 100*xgdpn/xgdp

### PGFL: Price index for federal government employee compensation, cw

The price index for federal employee compensation (PGFL) is proportional to the economy-wide compensation rate (PL). They are linked by the exogenous conversion factor UPGFL.

log(pgfl) = |
| log(upgfl)+log(pl)-log(lprdt)

### PGSL: Price index for S&L government employee compensation, cw

The price index for federal employee compensation (PGSL) is proportional to the economy-wide compensation rate (PL). They are linked by the exogenous conversion factor UPGSL.

log(pgsl) = |
| log(upgsl)+log(pl)-log(lprdt)

### PHR: Price index for residential investment, cw (relative to PXP)

log(phr) - phr_aerr - log(phr(-1)) | = 0 + pipxnc/400 + dpadj - Δ(log(pxp))

### PIC4: Four-quarter percent change in PCE prices

pic4 | = 100*(pcnia/pcniat-4 - 1)

### PICNIA: Inflation rate, personal consumption expenditures, cw

picnia = |
| picxfe
| + 0.040*400*(log(pcer)-log(pcert-1))
| + 0.073*400*(log(pcfr)-log(pcfrt-1))

Regression Statistics
Adjusted R2: | 0.100
S.E. of regression: | 0.0087
Sample period: | 2016Q4 2017Q4

### PICX4: Four-quarter percent change core in PCE prices

picx4 | = 100*(pcxfe/pcxfet-4 - 1)

### PICXFE: Inflation rate, personal consumption expenditures, ex. food and energy, cw

This is the main price equation in FRB/US. Its specification is based on a variant of the New Keynesian Phillips Curve (NKPC) derived by Cogley and Sbordonne (2008) for the case in which the subset of prices that is not optimally reset each period is indexed to either past or trend price inflation.

(1) Πt = Et-1[βΠt+1 + κμt] + εt

(2) Πt ≡ πt - γπ^t - (1-γ)π*t

(3) π^t ≡ πt-1

Equation 1 is the basic NKPC expressed in terms of the composite variable Π, which is defined in equation 2 as the difference between actual inflation (π) and a weighted average of past (π^) and trend (π*) inflation. Expectations are formed on the basis of "t-1" information, β(=.98) is the discount factor, and μ is the price markup gap.

Past inflation is measured as the first lag (equation 3). Trend price inflation is measured using survey data on ten-year inflation expectations (PTR). The price markup gap (log(QPCNIA/PCNIA)) is measured using a set of relationships that, if linearized, would set it equal to a weighted average of the detrended labor share in the nonfarm business sector, the relative price of nonoil imports, and a few additional but minor terms. For purposes of simplicity, the markup gap enters as a t-1 observation rather than the t-1 expectation of the observation in period t.

The unemployment rate does not directly appear in the structural price equation. The sensitivity of price inflation to the unemployment rate is indirect and operates through the effect that the structural dependence of wage inflation (PIECI) on labor market conditions has on the expectation of PICXFE in period t+1 (ZPICXFE).

The PICXFE equation, which is estimated simultaneously with the equation for the rate of growth of the ECI measure of hourly compensation (PIECI), contains two estimated parameters. γ is the fraction of non-optimized prices that is indexed to past inflation, and κ is the response of inflation to the price markup gap.

More information on the estimation of the price-wage NKPC equations is available here.

picxfe = |
| (0.671*picxfet-1
| + 0.98*zpicxfe
| + (1-0.98)*(1-0.671)*ptrt-1
| + 0.00191*400*log(qpcniat-1/pcniat-1)) / (1+0.671*0.98)

Regression Statistics
Sample period: | 1988q1 -
Estimation date: | 2017

### PIECI: Annualized rate of growth of EI hourly compensation

This is the main wage equation in FRB/US. Its specification is based on a variant of the New Keynesian Phillips Curve (NKPC) derived by Cogley and Sbordonne (2008) for the case in which the subset of wages that is not optimally reset each period is indexed to either past or trend wage inflation.

(1) Π(w)t = Et-1[βΠ(w)t+1 + κμt] + εt

(2) Π(w)t ≡ π(w)t - γπ^(w)t - (1-γ)π*(w)t

(3) π(w)^t ≡ .25 ∑(i=1,4)π(w)t-i

Equation 1 is the basic NKPC expressed in terms of the composite variable Π(w), which is defined in equation 2 as the difference between actual wage inflation (π(w)) and a weighted average of past (π^(w)) and trend (π*(w)) wage inflation. Expectations are formed on the basis of "t-1" information, β(=.98) is the discount factor, and μ is the wage markup gap.

Past wage inflation is measured as a four-quarter average (equation 3). Trend wage inflation is measured as the sum of survey data on ten-year consumer price inflation expectations (PTR) and the FRB/US estimate of the trend rate of growth of labor productivity (HLPRDT), less the FRB/US estimate of trend rate of growth of the price of consumption relative to the price of final sales (HUQPCT). The wage markup gap -- in theory, the difference between the marginal rate of substitution and the real wage -- is expressed as a function of the unemployment gap (LUR-LURNAT) and the detrended labor share (log(PL/QPL)). For purposes of simplicity, each component of the markup gap enters as a t-1 observation rather than the t-1 expectation of the observation in period t. ZPIECI is the expectation of PIECI in period t+1.

The PIECI equation, which is estimated simultaneously with the equation for core PCE price inflation (PICXFE), contains three estimated parameters: the fraction of non-optimized wages that is indexed to past inflation (γ); the coefficient on the detrended labor share; and the coefficient on the unemployment gap.

More information on the estimation of the price-wage NKPC equations is available here.

pieci = |
| (.25*0.685*((1-0.98)*(piecit-1+piecit-2+piecit-3) + piecit-4)
| + 0.98*zpieci
| + (1-0.98)*(1-0.685)*(ptrt-1 + hlprdtt-1 - 400*huqpctt-1)
| + -0.0167*(lurt-1-lurnatt-1)
| + 0.000957*400*log(qplt-1/plt-1)) / (1+.25*0.685*0.98)

Regression Statistics
Sample period: | 1988q1 -
Estimation date: | 2017

### PIGDP: Inflation rate, GDP, cw

pigdp | = 400*Δ(log(pgdp))

### PIPL: Rate of growth of PL

pipl | = pieci

### PIPXNC: Inflation rate, price of adjusted final sales excluding consumption (annual rate)

The difference between nonconsumption price inflation (PIPXNC) and consumption price inflation (PICNIA), adjusted for the trend difference between the two (1.99*400*HUQPCT), depends on two lags of itself and the rate of change of the real exchange rate.

pipxnc = |
| picnia - 1.99 * 400 * huqpct+ 0.463 * (pipxnct-1 - picniat-1 + 1.99 * 400 * huqpctt-1)
| + 0.230 * (pipxnct-2 - picniat-2 + 1.99 * 400 * huqpctt-2)
| + -0.284 * 0.131 * 400 * (log(fpxr) - log(fpxrt-1))+ .025*400*log(qpxnct-1/pxnct-1)

### PKBFIR: Ratio of price of BFI stock (KBFI) to PXP

pkbfir | = upkbfir * pbfir

### PL: Compensation per hour, business

log(pl) | = log(plt-1) + pipl/400

### PMO: Price index for imports ex. petroleum, cw

The price index for non-petroleum imports (PMO) is modeled using a Kalman filter specification in which the price level gradually converges to a steady-state target. The latter has two elements: a stochastic random walk (QPMO) and a weighted average of aggregate foreign and domestic prices. Empirical work suggests that the pass-through of changes in the exchange rate (FPX) or foreign prices (FPC) into dollar-denominated import prices is incomplete. In accordance with this evidence, only 64 percent of any movement in FPC or FPX is allowed to pass through to PMO in the long run. This pass-through relationship is defined in relative terms using the price of US business output, which accounts for the presence of PXB in the equilibrium formula. The PMO equation also contains the contemporaneous rates of foreign and domestic price inflation with the two coefficients restricted to sum to one.

dlog(pmo) = |
| 0.000267[0.0057]
| + 0.378[6.057] * (log(qpmo) + .64*log(fpct-1/fpxt-1) + .36*log(pxbt-1) - log(pmot-1))
| + 0.234[15.63] * dlog(fpc/fpx) + (1
| -0.234[15.63]) * dlog(pxb)

Regression Statistics
Sample period: | 1965Q1 2017Q4

### PMP: Price index for petroleum imports

The chain-weight price index for imported petroleum products (PMP) is proportional to the per barrel price of imported crude oil (POIL).

pmp | = upmp*poil

### POIL: Price of imported oil ($ per barrel)

poil | = poilr*pxb

### POILR: Price of imported oil, relative to price index for bus. sector output

Real oil prices error-correct to their long-run trend, POILRT.

dlog(poilr) = |
| -0.219[-6.092] * log(poilrt-1/poilrtt-1)
| + -0.00720[-0.87]
| + 0.390[6.088] * Δ(log(poilrt-1))
| + 0.394[1.25] * Δ(log(poilrt))

Regression Statistics
Adjusted R2: | 0.24
S.E. of regression: | 0.12
Sample period: | 1965Q1 2017Q4

### PXB: Price index for business output

pxb | = upxb*pgdp

### PXNC: Price of adjusted final sales excluding consumption

Δ(log(pxnc)) | = pipxnc/400

### PXP: Price index for final sales plus imports less gov. labor

log(pxp) = |
| log(pxpt-1)
| +0.647*log(pcnia/pcniat-1)
| +0.353*log(pxnc/pxnct-1)

Regression Statistics
Adjusted R2: | 0.100
S.E. of regression: | 3.14E-06
Sample period: | 2016Q4 2017Q4

### PXR: Price index for exports, cw (relative to PXP)

log(pxr) - pxr_aerr - log(pxr(-1)) | = 0 + pipxnc/400 + dpadj - Δ(log(pxp))

### QPCNIA: Desired level of consumption price

log(qpcnia) | = log(qpxp) + log(uqpct)

### QPL: Desired level of compensation per hour, trending component

The target level of hourly compensation (QPL) is defined by a condition that is just a rearrangement of the relationship used to define the target price level (QPXB). As a result, the percentage wage gap, log(PL/QPL), is the negative of the percentage price gap, log(PXB/QPXB), and both measure the deviation of the labor share (inverse price markup) in the adjusted nonfarm business sector from its long-run equilibrium.

log(qpl) | = log(pl) + 1 * log(pxb/qpxb)

### QPMO: Random walk component of non-oil import prices

In the long run, the level of non-oil import prices is determined by two factors -- a weighted average of foreign consumer prices expressed in dollars and domestic output prices; and a stochastic trend component that takes account of permanent movements in the relative price of imported goods with respect to the prices of both foreign consumption and domestic output. The stochastic trend component, QPMO, is a random walk with drift and is derived from Kalman filter estimation of the non-oil import price (PMO) equation.

log(qpmo) | = log(qpmot-1) + -0.00335

### QPXB: Desired price level of private output ex. energy, housing, and farm

Given the levels of hourly compensation (PL) and trend labor productivity (LRPDT) in the nonfarm business sector, QPXB is the level of that sector's price that would set the price markup (inverse labor share) equal to its equilibrium (PWSTAR).

log(qpxb) | = log(pwstar) + 0 + 1*log(pl/lprdt)

### QPXNC: Desired level of nonconsumption price

log(qpxnc) = |
| log(pxnc)
| + 2.985 * log(qpxp/pxp)
| + -1.985 * log(qpcnia/pcnia)

### QPXP: Desired price level of adjusted final sales

qpxp | = 100*(xpn + (.01*qpxb*xb-xbn))/xp

### UQPCT: Stochastic component of trend ratio of PCNIA to PXP

log(uqpct) | = 0 + log(uqpctt-1) + huqpct

Regression Statistics
Sample period: | 1962Q1 -
Estimation date: | August 2010

# Government

Identities. Much of sector consists of identities that: (1) relate nominal purchases (consumption and investment expenditures), transfers, and grants to associated constant-dollar variables and price indexes; (2) link tax receipts to tax rates and tax bases; and (3) compute the budget surplus and stock of debt.

Estimated equations. Error-correction equations link spending on various types of purchases to trend spending in these categories. Other estimated equations account for cyclical fluctuations in transfer payments and tax rates.

Personal income tax-rate reaction functions. The sector includes three options for setting the personal income tax rate of the federal government. The choice of an option is determined by which switch variable -- DFPSRP, DFPDBT, DFPEX -- is set to one. Activation of DFPSRP causes the tax rates to adjust so that the ratio of the federal government's surplus to GDP gradually stabilizes at a specified value. Activitation of DFPDBT causes the adjustments to stabilize the ratios of government debt to GDP at specified values. No adjustments take place when DFPEX is activated, making this a suitable choice only in the short run. The switches are exogenous variables whose values can change over the course of a simulation.

### EGFE: Federal Government expenditures, CW 2012$

Δ(log(egfe)) = |
| -0.000805[-0.49]
| + -0.120[-3] * log(egfet-1/egfett-1)
| + -0.153[-2.13] * Δ(log(egfet-1))
| + -0.104[-1.49] * Δ(log(egfet-2))
| + 1.504[6.26] * Δ(log(egfet))
| + -0.000984[-0.45] * xgap2
| + 0.000726[0.33] * xgap2t-1

Regression Statistics
Adjusted R2: | 0.22
S.E. of regression: | 0.020
Sample period: | 1965Q1 2017Q4

### EGFEN: Federal Government expenditures, current $

egfen = |
| pegfr*pxp*egfe*.01

### EGFET: Federal Government expenditures, CW 2012$, Trend

Δ(log(egfet)) = |
| -0.310[0]
| + -0.1[0] * log(.01*pegfrt-1*pxpt-1*egfett-1/xgdptnt-1)
| + 1[0] * (hggdpt+hggdptt-1+hggdptt-2+hggdptt-3) / 1600

### EGFL: Federal government employee compensation, cw 2012$

Δ(log(egfl)) = |
| 0.000242[0.41]
| + -0.0796[-3.32] * log(egflt-1/egfltt-1)
| + 0.242[3.58] * Δ(log(egflt-1))
| + -0.0613[-0.89] * Δ(log(egflt-2))
| + 1.207[6.15] * Δ(log(egflt))
| + -0.00251[-2.65] * xgap2
| + 0.00235[2.48] * xgap2t-1

Regression Statistics
Adjusted R2: | 0.36
S.E. of regression: | 0.0085
Sample period: | 1965Q1 2016Q4

### EGFLN: Federal government employee compensation, current $

egfln | = .01 * pgfl * egfl

### EGFLT: Federal government employee compensation, cw 2012$, trend

Δ(log(egflt)) |
| = -0.376
| + -0.1 * log(.01*pgflt-1*egfltt-1/xgdptnt-1)
| + 1 * (hggdpt+hggdptt-1+hggdptt-2+hggdptt-3) / 1600

### EGSE: S&L Government expenditures, CW 2012$

Δ(log(egse)) = |
| -0.000321[-0.18]
| + -0.131[-3.76] * log(egset-1/egsett-1)
| + 0.132[1.91] * Δ(log(egset-1))
| + 0.00263[0.038] * Δ(log(egset-2))
| + 0.929[4.021] * Δ(log(egset))
| + 0.00158[0.91] * xgap2
| + -0.000854[-0.49] * xgap2t-1

Regression Statistics
Adjusted R2: | 0.21
S.E. of regression: | 0.016
Sample period: | 1965Q1 2017Q4

### EGSEN: S&L Government expenditures, current $

egsen = |
| pegsr*pxp*egse*.01

### EGSET: S&L Government expenditures, CW 2012$, Trend

Δ(log(egset)) = |
| -0.32[0]
| + -0.1[0] * log(.01*pegfrt-1*pxpt-1*egsett-1/xgdptnt-1)
| + 1[0] * (hggdpt+hggdptt-1+hggdptt-2+hggdptt-3) / 1600

### EGSL: S&L government employee compensation, cw 2012$

Δ(log(egsl)) = |
| 0.000368[1.082]
| + -0.111[-4.18] * log(egslt-1/egsltt-1)
| + 0.156[2.36] * Δ(log(egslt-1))
| + 0.0216[0.32] * Δ(log(egslt-2))
| + 0.745[7.035] * Δ(log(egslt))
| + -0.00143[-4.065] * xgap2
| + 0.00177[5.016] * xgap2t-1

Regression Statistics
Adjusted R2: | 0.66
S.E. of regression: | 0.0030
Sample period: | 1965Q1 2017Q4

### EGSLN: S&L government employee compensation, current $

egsln | = .01 * pgsl * egsl

### EGSLT: S&L government employee compensation, cw 2012$, trend

Δ(log(egslt)) |
| = -0.260
| + -0.1 * log(.01*pgslt-1*egsltt-1/xgdptnt-1)
| + 1 * (hggdpt+hggdptt-1+hggdptt-2+hggdptt-3) / 1600

### GFDBTNP: Federal government debt stock held by the public, current $

gfdbtnp = |
| ugfdbtp*gfdbtn

### GFDBTN: Federal government debt stock, current $

gfdbtn = |
| ugfdbt*(gfdbtnt-1
| - .25*(gfsrpn-0.36*egfen))

### UGFSRP: Federal Government budget surplus, residual

ugfsrp = |
| 0.0417*(1-0.930) + 0.930*ugfsrpt-1

### ULEG: Multiplicative factor for government civilian employment

log(uleg) = |
| log(ulegt-1) - .1*(legt-1/lept-1-adjlegrt)

### GFEXPN: Government payments

gfexpn | = egfln + 0.64*egfen + gtn + gfintn

### GFINTN: Federal government net interest payments, current $

gfintn | = rgfint*gfdbtnt-1

### GFRECN: Government receipts and residual

gfrecn | = tpn + tcin + ugfsrp*xgdpn

### GFSRPN: Federal government budget surplus, current $

gfsrpn = |
| tpn + tcin
| - egfln - 0.64*egfen - gtn - gfintn
| + ugfsrp*xgdpn

### GTN: Federal government net transfer payments, current $

gtn = |
| .01*pgdp*gtr

### GTR: Government net transfer payments, deflated by PGDP

gtr = |
| (gtrd+gtrt)*xgdpt

### GTRD: Deviation of ratio of government transfers to GDP from trend ratio

gtrd = |
| 0.559[10.56] * gtrdt-1
| + -0.000427[-5.35] * xgap2

Regression Statistics
Adjusted R2: | 0.54
S.E. of regression: | 0.0026
Sample period: | 1965Q1 2017Q4

### LEG: Government civilian employment ex. gov. enterprise

log(leg) = |
| log(uleg) + log(egfl+egsl) - log(lprdt)

### PEGFR: Price index for federal government expenditures, CW (relative to PXP)

log(pegfr) - pegfr_aerr - log(pegfr(-1)) = |
| pipxnc/400
| + dpadj - d(log(pxp))

### PEGSR: Price index for S&L government expenditures, CW (relative to PXP)

log(pegsr) = |
| log(pegsrt-1) + pipxnc/400
| + dpadj - d(log(pxp))

### TCIN: Government corporate income tax accruals, current $

tcin = |
| trci * ynicpn

### TPN: Government personal income tax and non-tax receipts, current $

tpn = |
| trp * (ypn - gtn)

### TRCI: Average government corporate income tax rate

trci = |
| trcit
| + 0.774[18.61] * (trcit-1 - trcitt-1)
| + 0.00132[3.12] * xgap2

Regression Statistics
Adjusted R2: | 0.95
S.E. of regression: | 0.014
Sample period: | 1965Q1 2017Q4

### TRP: Average government tax rate for personal income tax and non-tax receipts

trp = |
| 1 * trpt
| + 0.660[9.76] * (trpt-1-trptt-1)
| + 0.228[3.48] * (trpt-2-trptt-2)
| + 0.000391[2.85] * xgap2t-1

Regression Statistics
Adjusted R2: | 0.82
S.E. of regression: | 0.0049
Sample period: | 1965Q1 2017Q4

### TRPT: Average government tax rate for personal income tax, trend

trpt = |
| dfpex * trptx
| + dfpdbt * ( trptt-1
| + 0.05 * (gfdbtnt-1/xgdpnt-1 - gfdrtt-1)
| + 0.5 * Δ(gfdbtnt-1/xgdpnt-1 - gfdrtt-1) )
| + dfpsrp * ( trptt-1
| + -0.1 * ((gfsrpnt-1/xgdpnt-1)
| - (gfsrtt-1+0.005*xgap2t-1)))

### TRYH: Average tax rate on household income

The average tax rate on household income is constructed as the ratio of personal income taxes (TPN) to the sum of labor income (YNLN) and taxable property income (YHPTN). Transfer income is assumed not to be taxed.

tryh | = (tpn)/(yhln+yhptn)

### YNIRN: Net interest, rental and proprietors' incomes (national income components)

The share of net interest, rental income and proprietor's of nominal GDP varies with the change in the triple B corporate bond rate.

ynirn/xgdpn = |
| 0.00759
| + 0.944*(ynirnt-1/xgdpnt-1)
| + 0.0748*0.01*d(rbbb)

# Financial Sector

The financial sector of FRB/US is divided into three blocks of equations: equations determining the stance of monetary policy, defined as the value of the federal funds rate; equations for other interest rates based on arbitrage relationships; and equations for household wealth, including one that relates the value of the stock market to real bond yields and expected growth in dividends.

Monetary Policy

In the monetary policy block, there are seven basic options for setting the federal funds rate, including variants of the Taylor rule, an estimated policy rule, a pre-determined path for the funds rate that can be defined in either nominal or real terms, and a generalized policy rule that can be used, for example, to target either inflation or the price level. These options are mutually exclusive, although it is possible to switch from one option to another in multi-period simulations by varying a group of exogenous "switches."

The chosen basic policy option can be modified so that the outcome for the federal funds rate is subject to the zero lower bound (ZLB). In addition, the timing of the liftoff of the funds rate from the ZLB can be determined by a version of the threshold criteria that appeared in FOMC statements from December 2012 to January 2014. In this case, liftoff is delayed until either the unemployment rate falls below a threshold or the expected inflation rate rises above a threshold.

The choice of a basic option is determined by which of seven switch variables -- DMPEX, DMPRR, DMPTAY, DMPTLR, DMPINTAY, DMPALT, and DMPGEN -- is set to one. Activation of one of the first two switches sets the funds rate to an pre-determined nominal (RFFIX) or real path (RRFIX). Activation of any of the remaining switches sets the funds rate to the outcome of an equation whose structure corresponds to a given policy rule (eg, RFFTAY, RFFTLR, etc). The RFFRULE equation combines the seven options in a form that yields the outcome for the funds rate under the chosen option.

Setting the exogenous variable RFFMIN to zero (or a small positive value) imposes the ZLB; setting RFFMIN to a large negative number eliminates the constraint.

When the exogenous switch variable DMPTRSH = 1, liftoff from the ZLB is delayed until either the unemployment rate falls below a critical rate (LURTRSH) or expected inflation rises above a critical rate (PITRSH). Once either threshold is crossed, the policy rate is determined, with a one-quarter delay and subject to the ZLB, by the chosen basic policy option, and it continues to be set according to that policy option irrespective of the subsequent outcomes for unemployment and expected inflation. The implementation of the threshold-based policy relies on four endogenous trigger variables, DMPTLUR, DMPTPI, DMPTMAX, and DMPTR. When either of the thresholds has been crossed, the value of DMPTR becomes 1.0 and remains at that value in subsequent quarters.

The final equation in the monetary policy block determines the value of the effective federal funds rate (RFF) by modifying the value of RFFRULE for the effects of the thresholds, when they are turned on and are binding.

Other Interest Rates

In the block of equations that determines other interest rates, the most important are those for yields on BBB corporate bonds (RBBB) and 5-, 10-, and 30-year government bonds (RG5, RG10, RG30). These equations are based on the expectations theory of the term structure, whereby the yield on a long-term bond equals a weighted average of expected rates on short-term assets over the maturity of the long-term bond plus a term/risk premium. The equations employ an approximation in which the weighted average is calculated over an infinite horizon and the weights decline at a geometric rate that is based on the bond's duration. The latter is calculated as a function of the bond's maturity and the average historical level of nominal interest rates.

(1) Rt(m) = (1-w(m))Et[∑(i=0,∞)w(m)i rt+i] + φt(m)

where R(m) is the rate of interest on a bond whose maturity is m quarters, r is the federal funds rate, and φ(m) is the term/risk premium. The following formulas define the value of w(m).

(2) w(m) = (D(m)-1)/D(m) (3) D(m) = (1-gm)/(1-g) (4) g = 1/(1+(R^).25)

D(m) is the duration of a bond whose maturity is m quarters. R^ is the average nominal rate of interest.

The term/risk premiums φ(m) associated with the four bond rates consist of a constant, a serially correlated residual, and an element that varies with the output gap (x) expressed as a present value with the same form as the funds rate present value.

(5) (1-w(m))Et[∑(i=0,∞)w(m)i xt+i]

The funds rate present value in equation (1) is assigned to a variable whose name starts with the characters ZRFF (eg, ZRFF10) and the output gap present value in equation (5) is assigned to a variable whose name starts ZGAP (eg, ZGAP10).

In addition to the four main bond rates, the second block of equations contains estimated equations for yields on Treasury bills (RTB), home mortgages (RME), and new car loans (RCAR).

Wealth

The final block of the sector determines the value of household net worth, as measured by the Federal Reserve's Flow-of-Funds accounts. Household net worth is divided into two components, corporate equity (WPSN) and other (WPON). The former is determined using a linearization of the standard Gordon formula, in which stock prices depend on the level of dividends (proxied by one-half the level of after-tax corporate profits), expected future growth in dividends, the real interest rate (on 30-year government bonds), and an equity premium. The equity premium is a function of the BBB premium and a serially correlated error. The change from one quarter to the next in household net worth excluding corporate equity equals the sum of personal savings, net investment in consumer durables, and capital gains on houses and other assets.

### DELRFF: Federal funds rate, first diff

delrff | = rff - rfft-1

### DMPTLUR: Monetary policy indicator for unemployment threshold

DMPTLUR equals zero when the unemployment rate is above its threshold (LURTRSH) one when it is below. A logistic function smoothes the transition, improving solution convergence properties.

dmptlur | = 1/(1+exp(25*(lur-lurtrsh)))

### DMPTMAX: Monetary policy indicator for both thresholds

DMPTMAX equals one when either the unemployment threshold or the inflation threshold is breached.

dmptmax | = (max((dmptlur),(dmptpi)))

### DMPTPI: Monetary policy indicator for inflation threshold

DMPTPI equals zero when expected inflation is below its threshold and one when it is above. A logistic function smoothes the transition, improving solution convergence properties.

dmptpi | = 1/(1+exp(-25*(zpic58-pitrsh)))

### DMPTR: Monetary policy indicator for policy rule thresholds

DMPTR is initially zero. It remains at that value until either the unemployment threshold or the inflation threshold is breached, after which it equals one.

dmptr | = (max((dmptmax),(dmptrt-1)))

### MEI: Multiplicative discrepancy for the difference between XGDI and XGDO

log(mei) | = 1 * log(meit-1)

### MEP: Multiplicative discrepancy for the difference between XGDP and XGDO

log(mep) | = 1 * log(mept-1)

### PHOUSE: Loan Performance House Price Index

The price of owner-occupied real estate is modeled in an error-correction format. The housing price is proportional to housing rents (PCHR*PCNIA) in the long run.

dlog(phouse) = |
| 0.00516[3.94]
| + 0.902[29.95]*dlog(phouset-1)
| + -0.0116[-3.57]*log(phouset-1/(pchrt-1*pcniat-1))

Regression Statistics
Adjusted R2: | 0.85
S.E. of regression: | 0.0066
Sample period: | 1976Q1 2017Q4

### RBBB: S&P BBB corporate bond rate

rbbb | = rbbbp + rg10

### RBBBP: S&P BBB corporate bond rate, risk/term premium

The term premium on BBB corporate bonds consists of a constant, a serially correlated residual, and a counter-cyclical term that depends on a weighted average of expected future output gaps.

rbbbp = |
| 1.610[6.84]
| + -0.213[-3.53]*zgap10
| + 0.876[22.100]*( rbbbpt-1
| - 1.610
| - -0.213*zgap10t-1)

Regression Statistics
Adjusted R2: | 0.81
S.E. of regression: | 0.27
Sample period: | 1973Q1 2007Q4

### RCAR: New car loan rate at finance companies

In the long run, the rate on new car loans equals the yield on 5-year Treasury bonds, plus an exogenous risk premium. This risk premium declined over the 1960s and 1970s, but appears to have been stable since 1980; this effect is captured using the dummy variable D79A and time trend T47. The lagged value of the auto loan rate is included in the equation to capture the sluggish adjustment of bank loan rates to movements in market interest rates.

rcar = |
| 2.078[6.95]
| + -1.164[-4.56] * d79a
| + -0.00829[-3.49] * ((1-d79a)*t47)
| + 0.697[26.61] * rcart-1
| + 0.102[2.68] * rg5
| + 0.202 * rg5t-1

Regression Statistics
Adjusted R2: | 0.99
S.E. of regression: | 0.30
Sample period: | 1965Q1 2017Q4

### RCGAIN: Rate of capital gain on the non-equity portion of household wealth

RCGAIN measures the rate of capital gain on non-equity, non-housing household net worth. In addtion to an adjustment for inflation, its equation has a cyclical component and an autocorrelated residual.

rcgain = |
| picx4
| + -0.410[-1.14]
| + 0.424[3.94]*xgap2
| + 0.226[3.94]*(rcgaint-1 - picx4t-1
| - -0.410
| - 0.424*xgap2)

Regression Statistics
Adjusted R2: | 0.10
S.E. of regression: | 3.97
Sample period: | 1963Q2 2017Q4

### REQ: Real expected rate of return on equity

The rate of return on equity equals the rate on thirty-year Treasury bonds (RG30), minus the average rate of inflation expected to prevail over the coming 30 years (ZPIC30), plus an equity premium (REQP). The latter varies with the corporate bond premium (RBBBP) and also includes an AR(1) error term.

req | = rg30 - zpic30 + reqp

### REQP: Real expected rate of return on equity, premium component

The residuals from a regression of the equity term premium on an intercept and corporate bond risk premium (RBBBP) are assumed to follow an AR(1) process.

reqp = |
| 3.140[7.55]
| + 0.729[8.012] * rbbbp
| + 0.804[17.92]*( reqpt-1
| - 3.140
| - 0.729* rbbbpt-1 )

Regression Statistics
Adjusted R2: | 0.75
S.E. of regression: | 0.73
Sample period: | 1985Q1 2017Q4

### RFF: Federal funds rate

The equation for the effective federal funds rate imposes the lower bound (RFFMIN) and, when DMPTRSH = 1, imposes a version of the unemployment and inflation threshold policy that appeared in FOMC statements from December 2012 to January 2014. In a ZLB episode, DMPTR = 0 before the thresholds are reached, and the funds rate is equal to the value of RFFMIN. After one of the thresholds is breached, DMPTR = 1, and, with a one-quarter lag, the funds rate is set according to the policy rule chosen by RFFRULE.

rff = |
| (1-dmptrsh) * (max((rffrule),( rffmin)))
| + dmptrsh * (max(((dmptrt-1*rffrule + (1-dmptrt-1)*rffmin)),( rffmin)))

### RFFALT: Value of eff. federal funds rate given by estimated policy rule

rffalt = |
| 0.055
| + 1.2 * rfft-1
| + -0.39 * rfft-2
| + 0.695 * xgap2
| + -0.517 * xgap2t-1
| + 0.329 * ( ( picxfe + picxfet-1 + picxfet-2 + picxfet-3) / 4 )

### RFFGEN: Value of eff. federal funds rate given by the generalized reaction function

The equation for RFFGEN expresses a generalized description of a monetary policy reaction function. By altering the rule's coefficients, policy can target either inflation or the price level in the long-run. Similarly, parameters can be manipulated to allow the federal funds rate to respond (as desired) to transitory movements in past interest rates, inflation, prices, the output gap and the deviation of unemployment from the NAIRU. The default coefficient setting for RFFGEN makes the equation equivalent to the Taylor rule.

rffgen = |
| 0
| + ( 0 * rfft-1 + 0 * rfft-2 + 0 * rfft-3 + 0 * rfft-4)
| + ( 0.375 * picnia + 0.375 * picniat-1 + 0.375 * picniat-2 + 0.375 * picniat-3 + 0 * picniat-4)
| + ( 0.5 * xgap2 + 0 * xgap2t-1 + 0 * xgap2t-2 + 0 * xgap2t-3 + 0 * xgap2t-4)
| + ( 0 * lur + 0 * lurt-1 + 0 * lurt-2 + 0 * lurt-3 + 0 * lurt-4)
| + ( 0 * pcnia + 0 * pcniat-1 + 0 * pcniat-2 + 0 * pcniat-3 + 0 * pcniat-4)
| + ( 1 * rstar + 0 * rstart-1 + 0 * rstart-2 + 0 * rstart-3 + 0 * rstart-4)
| + ( -0.5 * pitarg + 0 * pitargt-1 + 0 * pitargt-2 + 0 * pitargt-3 + 0 * pitargt-4)
| + ( 0 * lurnat + 0 * lurnatt-1 + 0 * lurnatt-2 + 0 * lurnatt-3 + 0 * lurnatt-4)
| + ( 0 * pcstar + 0 * pcstart-1 + 0 * pcstart-2 + 0 * pcstart-3 + 0 * pcstart-4)
| + ( 0 * picxfe + 0 * picxfet-1 + 0 * picxfet-2 + 0 * picxfet-3 + 0 * picxfet-4)

### RFFINTAY: Value of eff. federal funds rate given by the inertial Taylor rule

RFFINTAY is an inertial version of the Taylor Rule. According to the equation, the nominal funds rate gradually adjusts to a value consistent with a Taylor rule in which the funds equals the sum of the equilibrium real funds rate as perceived by policymakers (RSTAR), a four-quarter moving average of actual inflation, and adjustements in response to deviations of actual inflation from the target rate of inflation (PITARG) and deviations of the level of output from potential (XGAP2).

rffintay = |
| 0.85 * rfft-1
| + (1-0.85) * (rstar
| + ( picxfe + picxfet-1 + picxfet-2 + picxfet-3) / 4
| + 0.5 * ( ( picxfe + picxfet-1 + picxfet-2 + picxfet-3) / 4 -pitarg)
| + 1 * xgap2)

### RFFRULE: Federal funds rate

The RFFRULE equation combines the seven basic options for setting the federal funds rate in a form that yields the outcome for the funds rate under the chosen option. The funds rate is exogenous in nominal terms when DMPEX is one and the other policy switches are zero. The funds rate is exogenous in real terms when DMPRR is one. Other settings select one of the policy reaction functions. The equation sets a lower limit on the nominal funds rate, based on the value of RFFMIN. Setting it to zero (or a small positive value) imposes the zero lower bound; setting it to a large negative number effectively eliminates the constraint.

rffrule = |
| max(((dmpex * rfffix
| + dmprr * (rrfix + ( picxfe + picxfet-1 + picxfet-2 + picxfet-3) / 4 )
| + dmptay * rfftay
| + dmptlr * rfftlr
| + dmpintay * rffintay
| + dmpalt * rffalt
| + dmpgen * rffgen)),(rffmin))

### RFFTAY: Value of eff. federal funds rate given by the Taylor rule with output gap

RFFTAY is a version of the Taylor Rule. According to the equation, the nominal funds rate equals the sum of the equilibrium real funds rate as perceived by policymakers (RSTAR) and a four-quarter moving average of actual inflation. This value is then adjusted in response to deviations of actual inflation from the target rate of inflation (PITARG) and deviations of the level of output from potential (XGAP2).

rfftay = |
| rstar
| + ( picxfe + picxfet-1 + picxfet-2 + picxfet-3) / 4
| + 0.5 * ( ( picxfe + picxfet-1 + picxfet-2 + picxfet-3) / 4 -pitarg)
| + 1 * xgap2

### RFFTLR: Value of eff. federal funds rate given by the Taylor rule with unemployment gap

RFFTLR is a version of the Taylor Rule. According to the equation, the nominal funds rate equals the sum of the equilibrium real funds rate as perceived by policymakers (RSTAR) and a four-quarter moving average of actual inflation. This value is then adjusted in response to deviations of actual inflation from the target rate of inflation (PITARG) and deviations of the unemployment rate from the natural rate of unemployment.

rfftlr = |
| rstar
| + -0.5 * pitarg
| + 0.375 * ( ( picxfe + picxfet-1 + picxfet-2 + picxfet-3) )
| + 1.1 * (lurnat + deuc * leuc - lur)

### RG10: 10-year Treasury bond rate

rg10 | = zrff10 + rg10p

### RG10P: 10-year Treasury bond rate, term premium

The term premium on 10-year Treasury bonds consists of a constant (which shifts upward over 1980-1995), a serially correlated residual, and a counter-cyclical term that depends on a weighted average of expected future output gaps.

rg10p = |
| 1.050[2.15]
| + -0.461[-3.40] * zgap10
| + 0.229[0.51] * d8095
| + 0.920[26.94] * (rg10pt-1
| - 1.050[2.15]
| - -0.461[-3.40]*zgap10t-1
| - 0.229[0.51]*d8095t-1)

Regression Statistics
Adjusted R2: | 0.89
S.E. of regression: | 0.42
Sample period: | 1980Q1 2017Q4

### RG30: 30-year Treasury bond rate

rg30 | = zrff30 + rg30p

### RG30P: 30-year Treasury bond rate, term premium

The term premium on 30-year Treasury bonds consists of a constant (which shifts upward over 1980-1995), a serially correlated residual, and a counter-cyclical term that depends on a weighted average of expected future output gaps.

rg30p = |
| 1.421[2.34]
| + -0.625[-2.47] * zgap30
| + 0.135[0.26] * d8095
| + 0.938[33.83] * (rg30pt-1
| - 1.421[2.34]
| - -0.625[-2.47]*zgap30t-1
| - 0.135[0.26]*d8095t-1)

Regression Statistics
Adjusted R2: | 0.90
S.E. of regression: | 0.39
Sample period: | 1980Q1 2017Q4

### RG5: 5-year Treasury note rate

rg5 | = zrff5 + rg5p

### RG5P: 5-year Treasury note rate. term premium

The term premium on 5-year Treasury bonds consists of a constant, a serially correlated residual, and a counter-cyclical term that depends on a weighted average of expected future output gaps.

rg5p = |
| 0.691[1.68]
| + -0.350[-4.50] * zgap05
| + 0.902[25.87] * (rg5pt-1
| - 0.691[1.68]
| - -0.350[-4.50]*zgap05t-1)

Regression Statistics
Adjusted R2: | 0.86
S.E. of regression: | 0.47
Sample period: | 1980Q1 2017Q4

### RGFINT: Average rate of interest on existing federal debt

The average rate of interest on federal government debt (RGFINT) is measured historically as the ratio of net interest paid to the lagged stock of debt outstanding. In this equation, RGFINT adjusts toward the weighted average rate of interest on new government debt (RGW), with the speed of adjustment depending positively on the rate of growth of debt.

rgfint = |
| (0.86*rgfintt-1+.14*rgwt-1)*gfdbtnt-2/gfdbtnt-1 + rgwt-1*(1-gfdbtnt-2/gfdbtnt-1)
| + 0.00373[22.44]

Regression Statistics
Adjusted R2: | 0.99
S.E. of regression: | 0.0024
Sample period: | 1965Q1 2017Q4

### RGW: Approximate average rate of interest on new federal debt

The weighted average of the four current treasury yields, with the weights based on the Treasury's average issue patterns over some appropriate period.

rgw = |
| 0.0050 * rtb
| + 0.0027 * rg5
| + 0.0013 * rg10
| + 0.0011 * rg30

### RME: Interest rate on conventional mortgages

Δ(rme) = |
| 0.516[6.78]
| + 0.629[12.52] * Δ(rg10)
| + 0.256[3.072] * d87 * Δ(rg10)
| + 0.289[6.94] * (rg10t-1-rmet-1)
| + 0.0507[2.095] * d87 * (rg10t-1-rmet-1)

Regression Statistics
Adjusted R2: | 0.65
S.E. of regression: | 0.27
Sample period: | 1965Q1 2017Q4

### RRFF: Real federal funds rate

The real federal funds rate (RRFF) is defined as the nominal effective funds rate (RFF) minus a 4-quarter moving average of core consumer price inflation (PICXFE).

rrff | = rff - ( picxfe + picxfet-1 + picxfet-2 + picxfet-3) / 4

### RSTAR: Equilibrium real federal funds rate (for monetary policy reaction functions)

The estimate of the equilibrium real federal funds rate used in the monetary policy rules is updated each period by 5 percent of the gap between the ex post real short rate and the prior estimate, if the switch DRSTAR is set to 1.

rstar = |
| rstart-1
| + 0.05 * ((rrff-rstart-1)*drstar)

### RTB: 3-month Treasury bill rate

rtb = |
| -0.0510[-1.95]
| + 0.800[17.84] * rtbt-1
| + 0.111[3.80] * rtbt-2
| + 0.770[31.83] * rff
| + -0.681 * rfft-1

Regression Statistics
Adjusted R2: | 0.99
S.E. of regression: | 0.30
Sample period: | 1965Q1 2017Q4

### WPO: Household property wealth ex. stock market, real

wpo | = wpon/(.01*pcnia)

### WPON: Household property wealth ex. stock market, current $

The change in the non-equity portion of household net worth has three components -- NIPA personal savings, net investment in consumer durable goods, and capital gains on housing and other assets. The capital gains on housing and other assets are weighted by their ratios to WPON. PHOUSE is scaled by a factor of 116 so that PHOUSE*KH matches housing wealth (both owner occupied and noncorporate rental real estate) from the Flow of Funds over the past decade.

wpon = |
| wpont-1*exp((1
| -0.411)*rcgain/400
| + 0.411*(log(phouse) - log(phouset-1))) + .25 * (ydn-ecnian-yhibn) + .25 * (.01*pcdr*pcnia*(ecd-jkcd))

Regression Statistics
Adjusted R2: | 1
S.E. of regression: | 1.029E-11
Sample period: | 2016Q4 2017Q4

### WPS: Household stock market wealth, real

wps | = wpsn/(.01*pcnia)

### WPSN: Household stock market wealth, current $

The equation for the market value of equities held by households (WPSN) is derived from the standard Gordon model for valuing a firm's share price. Aggregating across firms, this model implies that WPSN equals the current level of corporate cash payments, scaled up by the difference between the expected real rate of return on equity (REQ) and the expected real growth rate of dividends (ZDIVGR). Corporate cash payments are approximated by half corporate profits (YNICPN) less corporate taxes (TCIN). The equation is expressed as a semi-log approximation to improve model simulation properties. The linearization assumes a long-run average dividend-price ratio of 4 percent, which is consistent with a long-run real required return on equity of 7 percent and a real dividend growth rate of 3 percent.

log(wpsn) = |
| log((ynicpn-tcin)*.5)
| - .25 * (req-zdivgr)
| + log(25) + 1

# Foreign Activity

Key measures of foreign activity are determined in a small-scale reduced-form forecasting model. The system contains four primary equations: an estimated "IS" curve, in which the level of foreign output gap is a function of the real short-term interest rate abroad; an estimated foreign inflation equation in which inflation expectations are expressed as a weighted average of past and target inflation; an imposed monetary policy reaction function that determines the foreign short-term rate of interest; and an estimated yield curve equation in which the foreign bond rate depends on the foriegn short-term rate and the foreign output gap. Coefficient restrictions are imposed to ensure long-run stability.

The real exchange rate is determined via an open interest parity condition based on real bond rates, augmented with a term to capture country risk.

Additional identities use exogenous conversion factors to translate many of the foreign variables between different country coverages (G10 vs G39) and different aggregation weights.

### FGDP: Foreign aggregate GDP (world, bilateral export weights)

The level of foreign GDP is determined via the identity that links it to the level of potential foreign output (FGDPT) and the foreign output gap (FXGAP).

fgdp | = fgdpt*exp(fxgap/100)

### FGDPT: Foreign aggregate GDP (world, bilateral export weights), trend

In simulations, the level of trend foreign GDP moves in proportion with level of U.S potential GDP in the long run. In the short run, the rate of growth of the foreign trend varies one-for-one with the rate of growth of U.S. potential output.

Δ(log(fgdpt)) |
| = -0.458
| + -0.1 * log(fgdptt-1/xgdptt-1)
| + 1 * (hggdpt+hggdptt-1+hggdptt-2+hggdptt-3) / 1600

### FPC: Foreign aggregate consumer price (G39, import/export trade weights)

fpc | = fpct-1*exp(fpic/400)

### FPI10: Foreign consumer price inflation (G10)

Foreign CPI inflation is a function of a weighted average of past and trend foreign inflation, the foreign output gap, and current and lagged changes in the relative price of oil. Historical values of trend inflation (FPITRG) are measured as a step function whose value since 1991 is 2.1 percent.

fpi10 = |
| 0.628[10.48] * ( ( fpi10t-1 + fpi10t-2 + fpi10t-3 + fpi10t-4) / 4 )
| + 0.372 * fpitrg
| + 0.322[4.28] * fxgapt-1

Regression Statistics
Adjusted R2: | 0.85
S.E. of regression: | 1.30
Sample period: | 1967Q1 2017Q4

### FPI10T: Foreign consumer price inflation, trend (G10)

The trend component of foreign inflation adjusts at 5 percent per quarter to movements in actual foreign inflation.

fpi10t = |
| 0.95 * fpi10tt-1
| + 0.05 * fpi10

### FPIC: Foreign consumer price inflation (G39, bilateral export trade weights)

In the long run, foreign consumer price inflation as measured on a G39 basis moves one for one with foreign inflation as measured on a G10 basis.

fpic = |
| 2.027[5.89]
| + 0.679[10.63] * fpi10
| + 0.321 * fpict-1

Regression Statistics
Adjusted R2: | 0.45
S.E. of regression: | 4.17
Sample period: | 1965Q1 2017Q4

### FPX: Nominal exchange rate (G39, import/export trade weights)

fpx | = fpxr*fpc/pcpi

### FPXR: Real exchange rate (G39, import/export trade weights)

The real exchange rate is determined via an open interest parity condition based on real bond rates, augmented with a term in the ratio of the net stock of U.S. claims on the rest of the world to U.S. GDP that captures country risk. The equation is written as an identity through the inclusion of a multiplicative residual, FPXRR.

log(fpxr) - fpxr_aerr - log(fpxrr) = |
| 0.048*(rg10-zpi10f-frl10+fpi10t)
| + 0.5*(fnin/xgdpn)

### FPXRR: Real exchange rate residual

The unexplained component of the exchange rate error-corrects to its long-run exogenous trend (FPXRRT).

Δ(log(fpxrr)) = |
| 0.0274[2.12] * log(fpxrrtt-1/fpxrrt-1)
| + 0.211[3.15] * Δ(log(fpxrrt-1)) + (1
| -0.211[3.15]) * Δ(log(fpxrrt))

Regression Statistics
Adjusted R2: | 0.056
S.E. of regression: | 0.028
Sample period: | 1965Q1 2017Q4

### FRL10: Foreign long-term interest rate (G10)

The foreign long-term interest rate (FRL10) is modeled using a reduced-form error-correction specification in which the long rate converges to the foreign short-term interest rate plus a constant premium.

frl10 - frl10t-1 = |
| 0.0375[1.20]
| + -0.0770[-3.32] * (frl10t-1 - frs10t-1)
| + 0.0655[0.95] * (frl10t-1 - frl10t-2)
| + 0.369[7.044] * (frs10 - frs10t-1)
| + 0.125[2.26] * (fxgap - fxgapt-1)

Regression Statistics
Adjusted R2: | 0.37
S.E. of regression: | 0.30
Sample period: | 1980Q1 2017Q4

### FRS10: Foreign short-term interest rate (G10)

The foreign short-term interest rate (FRS10) is set either according to a version of the Taylor rule (DFMPRR = 0), or as the sum of an exogenous real interest rate (RFRS10) and a four-quarter moving average of foreign CPI inflation (DFMPRR = 1).

frs10 = |
| dfmprr * (0
| + 1 * frstart-1
| + 1 * ( ( fpi10 + fpi10t-1 + fpi10t-2 + fpi10t-3) / 4)
| + 0.5 * ( ( fpi10 + fpi10t-1 + fpi10t-2 + fpi10t-3) / 4 - fpitrg)
| + 1 * fxgap)
| + (1-dfmprr) * (rfrs10 + ( fpi10 + fpi10t-1 + fpi10t-2 + fpi10t-3) / 4)

### FRSTAR: Equilibrium real short-term interest rate used in foreign Taylor rule

The estimate of the foreign equilibrium real short-term interest rate used in the foreign Taylor rule is updated each period by 5 percent of the gap between the ex post real short rate and the prior estimate.

frstar = |
| 0.95 * frstart-1
| + 0.05 * (frs10 - ( fpi10 + fpi10t-1 + fpi10t-2 + fpi10t-3) / 4)

### FXGAP: Foreign output gap (world, bilateral export weights)

The equation for the foreign output gap is a reduced-form IS curve. The gap depends on lags of the foreign output gap, the real short-term foreign interest rate (FRS10 less a moving average of FPI10), and the U.S. output gap (XGAP2). The coefficient on the real short-term interest rate is set equal to the estimate from a similar equation estimated on U.S. data.

fxgap = |
| + 1.291[20.36] * fxgapt-1
| + -0.468[-7.76] * fxgapt-2
| + -0.05 * (@movav(frs10t-1-(fpi10t-1+fpi10t-2+fpi10t-3+fpi10t-4)/4,3)-frstar)
| + 0.0373[2.65] * xgap2t-1

Regression Statistics
Adjusted R2: | 0.85
S.E. of regression: | 0.46
Sample period: | 1965Q1 2017Q4

# Expectations

This sector contains the identities that are used for expectational variables when the assumption of "VAR" expectations is employed. The form of each identity is determined by the structure and parameters of the sub-model that is used to estimate the structural equation in which the expectation appears. In most cases, the expectation is for a weighted sum of a variable over future quarters. The exceptions are in the wage-price block (ZPICXFE, ZPIECI), where expectations are for one period ahead. In all cases, the VAR expectation is expressed as a linear function of an observable information set. Two types of parameters determine the coefficient values in the typical expectations equation: discounting weights that specify the horizon of each expectation and coefficients of the VAR system used to generate forecasts that proxy for expectations in the estimation sub-model.

In the VAR-expectations formulas, the information set includes a core set of macro variables: actual consumer price inflation (PICNIA) and the value expected to prevail in the long run (PTR); the actual federal funds rate (RFF) and the value expected to prevail in the long run (RTR); and the output gap (XGAP). (By definition, the latter is assumed to be zero in the long run.) The structure of the core VAR model is such that interest rate and inflation expectations converge to long-run expectations as the forecast horizon lengthens. The long-run expectations are modeled as random walks in the core VAR. For many expectational variables, the information set also includes one or more sector-specific variables. Expectations in financial equations assume that current-period data are in the information set, while those in nonfinancial equations assume that only lagged data are in the information set.

For expectations appearing in PAC equations, the discounting weights depend on a general discount factor (.98 per quarter) and on the estimated adjustment cost parameters as given by the PAC equation's error-correction coefficient and coefficients on lags of the dependent variable. In most cases, the effective forward horizon of these expectations is only a few years. The sum of the discounting weights in PAC expectations is not unity. For the expectational components of long-term interest rates and the required rate of return on equity, the discounting weights sum to unity and depend on the duration of the financial instrument. For the expectations of permanent income appearing in the target level of consumption, the discouting weights sum to unity and are based on a general discount factor of 25 percent per year.

For further information, see Var Expectations Basics.

### HGYNID: Growth rate of real after-tax corporate profits

hgynid | = 400*Δ(log((ynicpn-tcin)*.5/pxb))

### PTR: 10-year expected PCE price inflation (Survey of Professional Forecasters)

PTR gradually adjusts toward a weighted average of actual and target inflation.

Historical values of PTR come from several sources. Since 1991q4, the source is the Survey of Professional Forecasters (SPF), first for expected CPI inflation and then, when it becomes available in 2007, for expected PCE price inflation. PTR data from 1981q1 to 1991q3 is primarily from a survey conducted by Richard Hoey. The Hoey and SPF CPI observations are reduced by 40 basis to account for the average difference between CPI and PCE inflation. Values of PTR before 1981 are constructed in a manner similar to the one described in Kozicki and Tinsley (2001, section 3.3), "Term Structure Views of Monetary Policy under Alternative Models of Agent Expectations," Journal of Economic Dynamics and Control, 25: 149-184.

ptr | = 0.9*ptrt-1 + 0.05*picxfet-1+ 0.05*pitargt-1

### RRTR: Expected long-run real federal funds rate

The expected long-run value of the real federal funds rate (RRTR) is assumed each quarter to close 3 percent of the gap between the current ex post real funds rate and last quarter's estimate of RRTR.

rrtr = |
| 0.97 * rrtrt-1
| + 0.03 * rrff

### RTR: Expected federal funds rate in the long run (Blue Chip)

Historical values of RTR since 1984q3 are based on data from the Blue Chip Survey. Prior to that, RTR = PTR + 2.0.

rtr | = rrtr + ptr

### ZDIVGR: Expected growth rate of real dividends, for WPSN eq. (VAR exp.)

#### VAR Equation

ZDIVGR is a weighted average of growth rates of future after-tax corporate profits. The weights sum to one and decline geometrically at a rate based on the infinite maturity of corporate equity and the average historical level of the nominal rate of interest. The equation shown below is the reduced-form representation of this expectational computation based on a small-scale VAR model.

zdivgr = |
| 1.454E-14[1.66]
| + DL1(4){sum=0.017}picnia
| + DL2(4){sum=-0.12}rff
| + 0.118[44174069683861.81]*rtr
| + -0.0169[-7073915781260.91]*ptr
| + DL3(4){sum=-0.019}xgap
| + DL4(4){sum=0.019}hgynid
| + 0.981[481143671129519.6]*hxbt

Distributed Lag Coefficients

DL11 | -0.0343
DL12 | 0.0150
DL13 | 0.00814
DL14 | 0.0281
DL1sum | 0.0169

DL21 | -0.115
DL22 | 0.0238
DL23 | -0.0800
DL24 | 0.0529
DL2sum | -0.118

DL31 | -0.319
DL32 | 0.190
DL33 | 0.0587
DL34 | 0.0520
DL3sum | -0.0186

DL41 | 0.0151
DL42 | 0.00291
DL43 | 0.00182
DL44 | -0.000833
DL4sum | 0.0190

#### MCE Equation

ZDIVGR is a weighted average of growth rates of future after-tax corporate profits. The weights sum to one and decline geometrically at a rate based on the infinite maturity of corporate equity and the average historical level of the nominal rate of interest.

zdivgr = |
| 0.00976 * hgynidt+1
| + 0.990 * zdivgrt+1

### ZEBFI: Expected growth rate of business output EBFI (VAR exp.)

#### VAR Equation

The weighted average growth rate of expected future output for BFI, ZEBFI, is computed from forecasts of the small-scale VAR model used in the estimation of the equipment (EBFI) PAC equation. The forward weights are those implied by the estimated PAC coefficients. The equation shown below is the reduced-form representation of this expectational computation.

zebfi = |
| -1.836E-16[-7.69]
| + -0.000431[-320206848445540.6]*picniat-1
| + -0.000507[-353077628504199.2]*picniat-2
| + -3.882E-05[-26256646603762.47]*picniat-3
| + 0.000168[115277170083979.5]*picniat-4
| + -0.000975[-390592430755039]*rfft-1
| + 0.000417[129716164513429.8]*rfft-2
| + 9.804E-06[3144326760867.87]*rfft-3
| + 0.000403[171769141640141]*rfft-4
| + 0.000146[23432782253918.05]*rtrt-1
| + 0.000809[155152147190294.5]*ptrt-1
| + 0.000691[216118443781045.9]*xgapt-1
| + -0.00152[-284178282543195.7]*xgapt-2
| + 0.000182[33669345477756.99]*xgapt-3
| + 0.000171[51628647230980.14]*xgapt-4
| + 0.0143[190798984072385.3]*dlog(qebfit-1)
| + 0.0100[135460773436413.1]*dlog(qebfit-2)
| + 0.00501[74374706769186.19]*dlog(qebfit-3)
| + 0.00389[58285988898562.91]*dlog(qebfit-4)
| + 0.142[67782170710475.25]*hxbtt-1/400
| + -0.142[-29921689340197.61]*hgpbfirt-1/400

#### MCE Equation

ZEBFI is a weighted average of future growth rates of business output. The pattern of the weights is a function of the PAC adjustment coefficients in the estimated equipment investment (EPD) equation. The sum of the weights equals one minus the sum of the coefficients on lagged growth of actual investment in the EPD equation.

zebfi = |
| 0.0607 * Δ(log(xb))
| + -0.0301 * Δ(log(xbt+1))
| + -0.0173 * Δ(log(xbt+2))
| + 1.420 * zebfit+1
| + -0.211 * zebfit+2
| + -0.284 * zebfit+3

### ZECD: Expected growth rate of target durable consumption, for ECD eq. (VAR exp.)

#### VAR Equation

The weighted average growth rate of expected future target spending on consumer durables, ZECD, is computed from forecasts of the small-scale VAR model used in the estimation of the consumer durables (ECD) PAC equation. The forward weights are those implied by the estimated PAC coefficients. The equation shown below is the reduced-form representation of this expectational computation.

zecd = |
| DL1(4){sum=-0.0014}picniat-1
| + DL2(4){sum=0.00064}rfft-1
| + DL3(4){sum=-0.00057}xgap2t-1
| + 0.00141[60854825918550.22]*ptrt-1
| + -0.000640[-28330121768063.43]*rtrt-1
| + DL4(4){sum=0.00043}yhgapt-1
| + DL5(4){sum=0.00011}yhtgapt-1
| + DL6(4){sum=9.28E-05}yhpgapt-1
| + 1.020[132326801278756.1]*hggdptt-1/400
| + -0.752[-193627783552012.3]*hgpcdrt-1/400
| + DL7(4){sum=0.040}dlog(qecdt-1)

Distributed Lag Coefficients

DL11 | -0.000424
DL12 | -0.000566
DL13 | -0.000428
DL14 | 4.275E-06
DL1sum | -0.00141

DL21 | -0.00133
DL22 | 0.00179
DL23 | -0.000271
DL24 | 0.000460
DL2sum | 0.000640

DL31 | 0.000429
DL32 | -0.00111
DL33 | 3.611E-05
DL34 | 7.976E-05
DL3sum | -0.000568

DL41 | -0.000108
DL42 | 0.000210
DL43 | 0.000178
DL44 | 0.000147
DL4sum | 0.000427

DL51 | -0.000140
DL52 | -3.380E-05
DL53 | 0.000167
DL54 | 0.000114
DL5sum | 0.000107

DL61 | 0.000124
DL62 | -0.000204
DL63 | 5.799E-05
DL64 | 0.000114
DL6sum | 9.280E-05

DL71 | 0.0309
DL72 | 0.0288
DL73 | -0.00770
DL74 | -0.0124
DL7sum | 0.0396

#### MCE Equation

ZECD is a weighted average of future growth rates of the target level of investment in consumer durables. The pattern of the weights is a function of the PAC adjustment coefficients in the estimated consumer durables (ECD) equation. The sum of the weights equals one minus the sum of the coefficients on lagged growth of actual investment in consumer durables in the ECD equation.

zecd = |
| 0.175 * Δ(log(qecd))
| + 0.0101 * Δ(log(qecdt+1))
| + 0.768 * zecdt+1
| + 0.0576 * zecdt+2

### ZECO: Expected growth rate of target nondurables and nonhousing services, for ECO eq (VAR

#### VAR Equation

The weighted average growth rate of expected future target consumption, ZECO, is computed from forecasts of the small-scale VAR model used in the estimation of the consumption (ECO) PAC equation. The forward weights are those implied by the estimated PAC coefficients. The equation shown below is the reduced-form representation of this expectational computation.

zeco = |
| DL1(4){sum=-7.48E-05}picniat-1
| + DL2(4){sum=0.00013}rfft-1
| + DL3(4){sum=-0.00026}xgap2t-1
| + 7.481E-05[14978364966714.62]*ptrt-1
| + -0.000130[-30205301306840.91]*rtrt-1
| + DL4(4){sum=0.00016}yhgapt-1
| + DL5(4){sum=-5.72E-05}yhtgapt-1
| + DL6(4){sum=-0.00026}yhpgapt-1
| + 0.469[248194456522938.6]*hggdptt-1/400
| + DL7(4){sum=0.12}dlog(qecot-1)

Distributed Lag Coefficients

DL11 | -7.522E-05
DL12 | -7.944E-05
DL13 | -2.059E-05
DL14 | 0.000100
DL1sum | -7.481E-05

DL21 | 2.128E-05
DL22 | 1.704E-05
DL23 | 5.501E-05
DL24 | 3.681E-05
DL2sum | 0.000130

DL31 | -0.000630
DL32 | 0.000274
DL33 | 0.000133
DL34 | -3.466E-05
DL3sum | -0.000258

DL41 | -0.000575
DL42 | 0.000316
DL43 | 0.000397
DL44 | 2.606E-05
DL4sum | 0.000164

DL51 | -6.065E-05
DL52 | -5.862E-06
DL53 | 4.609E-05
DL54 | -3.675E-05
DL5sum | -5.719E-05

DL61 | 0.000206
DL62 | -0.000241
DL63 | -0.000132
DL64 | -8.998E-05
DL6sum | -0.000257

DL71 | 0.0732
DL72 | 0.0293
DL73 | 0.00715
DL74 | 0.00908
DL7sum | 0.119

#### MCE Equation

ZECO is a weighted average of future growth rates of the target level of consumption. The pattern of the weights is a function of the PAC adjustment coefficients in the estimated consumption (ECO) equation. The sum of the weights equals one minus the sum of the coefficients on lagged growth of actual consumption in the ECO equation.

zeco = |
| 0.163 * Δ(log(qec/pcor))
| + -0.0643 * Δ(log(qect+1/pcort+1))
| + 1.228 * zecot+1
| + -0.396 * zecot+2

### ZEH: Expected growth rate of target residential investment, for EH eq. (VAR exp.)

#### VAR Equation

The weighted average growth rate of expected future target housing investment, ZEH, is computed from forecasts of the small-scale VAR model used in the estimation of the housing (EH) PAC equation. The forward weights are those implied by the estimated PAC coefficients. The equation shown below is the reduced-form representation of this expectational computation.

zeh = |
| DL1(4){sum=-0.00017}picniat-1
| + DL2(4){sum=0.00034}rfft-1
| + DL3(4){sum=-0.00014}xgap2t-1
| + 0.000171[60358923307459.16]*ptrt-1
| + -0.000344[-127538174889467.4]*rtrt-1
| + DL4(4){sum=6.089E-05}yhgapt-1
| + DL5(4){sum=9.29E-05}yhtgapt-1
| + DL6(4){sum=1.42E-05}yhpgapt-1
| + 0.428[394684640489598]*hggdptt-1/400
| + DL7(4){sum=-0.0046}dlog(qeht-1)

Distributed Lag Coefficients

DL11 | -7.826E-05
DL12 | -5.222E-05
DL13 | -5.116E-05
DL14 | 1.061E-05
DL1sum | -0.000171

DL21 | 0.000152
DL22 | 0.000115
DL23 | 6.948E-05
DL24 | 6.915E-06
DL2sum | 0.000344

DL31 | 0.000502
DL32 | -0.000531
DL33 | -0.000153
DL34 | 4.422E-05
DL3sum | -0.000138

DL41 | 5.547E-05
DL42 | 4.078E-05
DL43 | -3.078E-05
DL44 | -4.585E-06
DL4sum | 6.088E-05

DL51 | 7.136E-06
DL52 | 1.471E-05
DL53 | 3.831E-05
DL54 | 3.272E-05
DL5sum | 9.287E-05

DL61 | -6.993E-05
DL62 | 4.550E-07
DL63 | 5.505E-05
DL64 | 2.864E-05
DL6sum | 1.421E-05

DL71 | 0.00427
DL72 | -0.00147
DL73 | -0.00335
DL74 | -0.00401
DL7sum | -0.00456

#### MCE Equation

ZEH is a weighted average of future growth rates of the target level of housing investment. The pattern of the weights is a function of the PAC adjustment coefficients in the estimated housing (EH) equation. The sum of the weights equals one minus the sum of the coefficients on lagged growth of actual housing investment in the EH equation.

zeh = |
| 0.0201 * Δ(log(qeh))
| + -0.00709 * Δ(log(qeht+1))
| + -0.00391 * Δ(log(qeht+2))
| + 1.330 * zeht+1
| + -0.158 * zeht+2
| + -0.194 * zeht+3

### ZGAP05: Expected output gap, for RG5 eq. (VAR exp.)

#### VAR Equation

ZGAP05 is a weighted average of expected future output gaps. The weights sum to one and decline geometrically at a rate based on the average duration of a five-year bond. The equation shown below is the reduced-form representation of this expectational computation based on the core part of the expectations VAR.

zgap05 = |
| 1.680E-14[3.13]
| + DL1(4){sum=-0.19}picnia
| + DL2(4){sum=-0.18}rff
| + 0.179[80685294984955.11]*rtr
| + 0.191[95594785391953.76]*ptr
| + DL3(4){sum=0.39}xgap

Distributed Lag Coefficients

DL11 | -0.135
DL12 | -0.0338
DL13 | -0.0391
DL14 | 0.0174
DL1sum | -0.191

DL21 | -0.375
DL22 | 0.0970
DL23 | 0.0739
DL24 | 0.0258
DL2sum | -0.179

DL31 | 0.824
DL32 | -0.286
DL33 | -0.0670
DL34 | -0.0851
DL3sum | 0.386

#### MCE Equation

ZGAP05 is a weighted average of future output gaps. The weights sum to one and decline geometrically at a rate based on the average duration of a five-year bond.

zgap05 = |
| 0.0548 * xgap
| + 0.945 * zgap05t+1

### ZGAP10: Expected output gap, for RG10 eq. (VAR exp.)

#### VAR Equation

ZGAP10 is a weighted average of expected future output gaps. The weights sum to one and decline geometrically at a rate based on the average duration of a ten-year bond. The equation shown below is the reduced-form representation of this expectational computation based on the core part of the expectations VAR.

zgap10 = |
| -1.722E-15[-0.67]
| + DL1(4){sum=-0.10}picnia
| + DL2(4){sum=-0.091}rff
| + 0.0908[86103137071094.52]*rtr
| + 0.100[105422493648269.6]*ptr
| + DL3(4){sum=0.20}xgap

Distributed Lag Coefficients

DL11 | -0.0719
DL12 | -0.0176
DL13 | -0.0201
DL14 | 0.00932
DL1sum | -0.100

DL21 | -0.195
DL22 | 0.0515
DL23 | 0.0364
DL24 | 0.0161
DL2sum | -0.0908

DL31 | 0.416
DL32 | -0.138
DL33 | -0.0351
DL34 | -0.0449
DL3sum | 0.198

#### MCE Equation

ZGAP10 is a weighted average of future output gaps. The weights sum to one and decline geometrically at a rate based on the average duration of a ten-year bond.

zgap10 = |
| 0.0301 * xgap
| + 0.970 * zgap10t+1

### ZGAP30: Expected output gap, for RG30 eq. (VAR exp.)

#### VAR Equation

ZGAP30 is a weighted average of expected future output gaps. The weights sum to one and decline geometrically at a rate based on the average duration of a thirty-year bond. The equation shown below is the reduced-form representation of this expectational computation based on the core part of the expectations VAR.

zgap30 = |
| 1.234E-14[6.62]
| + DL1(4){sum=-0.053}picnia
| + DL2(4){sum=-0.048}rff
| + 0.0484[63050766479994.02]*rtr
| + 0.0535[77265422817041]*ptr
| + DL3(4){sum=0.11}xgap

Distributed Lag Coefficients

DL11 | -0.0383
DL12 | -0.00940
DL13 | -0.0108
DL14 | 0.00495
DL1sum | -0.0535

DL21 | -0.104
DL22 | 0.0274
DL23 | 0.0196
DL24 | 0.00840
DL2sum | -0.0484

DL31 | 0.223
DL32 | -0.0747
DL33 | -0.0188
DL34 | -0.0239
DL3sum | 0.106

#### MCE Equation

ZGAP30 is a weighted average of future output gaps. The weights sum to one and decline geometrically at a rate based on the average duration of a thirty-year bond.

zgap30 = |
| 0.0141 * xgap
| + 0.986 * zgap30t+1

### ZGAPC2: Expected output gap, for ECD eq. (VAR exp.)

#### VAR Equation

The weighted average of expected future output gaps, ZGAPC2, is computed from forecasts of the small-scale VAR model used in the estimation of the consumer durables (ECD) PAC equation. The forward weights are those implied by the estimated PAC coefficients. The equation shown below is the reduced-form representation of this expectational computation.

zgapc2 = |
| DL1(4){sum=-0.023}picniat-1
| + DL2(4){sum=-0.018}rfft-1
| + DL3(4){sum=0.091}xgap2t-1
| + 0.0234[104699003743847.8]*ptrt-1
| + 0.0180[165510326897816.2]*rtrt-1

Distributed Lag Coefficients

DL11 | -0.0142
DL12 | -0.00439
DL13 | -0.00609
DL14 | 0.00127
DL1sum | -0.0234

DL21 | -0.0427
DL22 | 0.00776
DL23 | 0.0191
DL24 | -0.00221
DL2sum | -0.0180

DL31 | 0.194
DL32 | -0.0764
DL33 | -0.0113
DL34 | -0.0156
DL3sum | 0.0911

#### MCE Equation

ZGAPC2 is a weighted average of future output gaps. The pattern of the weights is a function of the PAC adjustment coefficients in the estimated consumer durables (ECD) equation.

zgapc2 = |
| 0.0272 * xgap2
| + 0.768 * zgapc2t+1
| + 0.0576 * zgapc2t+2

### ZLHP: Expected growth rate of target aggregate hours (VAR exp.)

#### VAR Equation

The weighted average growth rate of expected future target hours, ZLHP, is computed from forecasts of the small-scale VAR model used in the estimation of the hours (LHP) PAC equation. The forward weights are those implied by the estimated PAC coefficients. The equation shown below is the reduced-form representation of this expectational computation.

zlhp = |
| DL1(4){sum=-0.00041}picniat-1
| + DL2(4){sum=-0.00032}rfft-1
| + 0.000322[11401215516722.34]*rtrt-1
| + 0.000408[9184882988091.53]*ptrt-1
| + DL3(4){sum=-0.00041}xgapt-1
| + 0.686[40540039103487.41]*(dlog(xbot-1)-dlog(lprdtt-1))
| + 0.111[3130854335557.62]*(hleptt-1+hqlwwt-1)/400

Distributed Lag Coefficients

DL11 | -0.000202
DL12 | -6.547E-05
DL13 | -0.000172
DL14 | 3.139E-05
DL1sum | -0.000408

DL21 | -0.00105
DL22 | 0.000260
DL23 | 0.000508
DL24 | -4.199E-05
DL2sum | -0.000322

DL31 | -0.00575
DL32 | 0.00582
DL33 | -0.000203
DL34 | -0.000272
DL3sum | -0.000413

#### MCE Equation

ZLHP is a weighted average of future growth rates of the target level of hours. The pattern of the weights is a function of the PAC adjustment coefficients in the estimated hours (LHP) equation. The sum of the weights equals one minus the sum of the coefficients on lagged growth of actual hours in the LHP equation.

zlhp = |
| 0.212 * Δ(log(qlhp))
| + -0.0413 * Δ(log(qlhpt+1))
| + 0.981 * zlhpt+1
| + -0.195 * zlhpt+2

### ZPI10: Expected cons. price infl., for RCCH, RRMET, and YHPNTN eqs. (10-yr mat.) (VAR exp.)

#### VAR Equation

ZPI10 is a weighted average of future PCE price inflation. The weights sum to one and decline geometrically at a rate based on the average duration of a ten-year bond. The equation shown below is the reduced-form representation of this expectational computation based on the core part of the expectations VAR.

zpi10 = |
| DL1(4){sum=0.074}picniat-1
| + DL2(4){sum=-0.15}rfft-1
| + 0.155[64809640708192.54]*rtrt-1
| + 0.926[189546297693701.4]*ptrt-1
| + DL3(4){sum=0.10}xgapt-1

Distributed Lag Coefficients

DL11 | 0.0436
DL12 | 0.0131
DL13 | 0.0155
DL14 | 0.00146
DL1sum | 0.0737

DL21 | -0.104
DL22 | -0.0143
DL23 | -0.0296
DL24 | -0.00704
DL2sum | -0.155

DL31 | 0.0603
DL32 | -0.0102
DL33 | 0.0255
DL34 | 0.0247
DL3sum | 0.100

#### MCE Equation

ZPI10 a weighted average of future PCE price inflation. The weights sum to one and decline geometrically at a rate based on the average duration of a ten-year bond.

zpi10 = |
| 0.0301 * picnia
| + 0.970 * zpi10t+1

### ZPI10F: Expected cons. price infl., for FPXR eq. (10-yr mat.) (VAR exp.)

#### VAR Equation

zpi10f | = zpi10

#### MCE Equation

zpi10f = |
| 0.0301 * picnia
| + 0.970 * zpi10ft+1

### ZPI5: Expected cons. price infl., for RCCD eq. (5-yr mat.) (VAR exp.)

#### VAR Equation

ZPI5 is a weighted average of future PCE price inflation. The weights sum to one and decline geometrically at a rate based on the average duration of a five-year bond. The equation shown below is the reduced-form representation of this expectational computation based on the core part of the expectations VAR.

zpi5 = |
| DL1(4){sum=0.13}picniat-1
| + DL2(4){sum=-0.23}rfft-1
| + 0.234[151449941153792.1]*rtrt-1
| + 0.871[275997302660806.2]*ptrt-1
| + DL3(4){sum=0.19}xgapt-1

Distributed Lag Coefficients

DL11 | 0.0818
DL12 | 0.0222
DL13 | 0.0250
DL14 | -9.007E-05
DL1sum | 0.129

DL21 | -0.146
DL22 | -0.0311
DL23 | -0.0295
DL24 | -0.0276
DL2sum | -0.234

DL31 | 0.174
DL32 | -0.0718
DL33 | 0.0407
DL34 | 0.0449
DL3sum | 0.188

#### MCE Equation

ZPI5 a weighted average of future PCE price inflation. The weights sum to one and decline geometrically at a rate based on the average duration of a five-year bond.

zpi5 = |
| 0.0548 * picnia
| + 0.945 * zpi5t+1

### ZPIB5: Expected output price infl., for RPD eq. (5-yr mat.) (VAR exp.)

#### VAR Equation

ZPI5 is a weighted average of future business sector price inflation. The weights sum to one and decline geometrically at a rate based on the average duration of a five-year bond. The equation shown below is the reduced-form representation of this expectational computation based on a small-scale VAR model.

zpib5 = |
| -1.265E-13[-7.68]
| + DL1(4){sum=0.13}picniat-1
| + DL2(4){sum=-0.34}rfft-1
| + 0.336[49264303130607.26]*rtrt-1
| + 0.700[115459764873131.9]*ptrt-1
| + DL3(4){sum=0.20}xgapt-1
| + DL4(4){sum=0.17}400*dlog(pxbt-1)

Distributed Lag Coefficients

DL11 | 0.0903
DL12 | 0.0317
DL13 | 0.0277
DL14 | -0.0151
DL1sum | 0.135

DL21 | -0.197
DL22 | -0.0596
DL23 | -0.0486
DL24 | -0.0307
DL2sum | -0.336

DL31 | 0.147
DL32 | -0.0389
DL33 | 0.0554
DL34 | 0.0401
DL3sum | 0.203

DL41 | 0.0783
DL42 | 0.0375
DL43 | 0.0161
DL44 | 0.0334
DL4sum | 0.165

#### MCE Equation

ZPI5 a weighted average of future business sector price inflation. The weights sum to one and decline geometrically at a rate based on the average duration of a five-year bond.

zpib5 = |
| 0.0548 * 400 * Δ(log(pxb))
| + 0.945 * zpib5t+1

### ZPIC30: Expected cons. price infl., for REQ eq. (30-yr mat.) (VAR exp.)

#### VAR Equation

ZPIC30 is a weighted average of future PCE price inflation. The weights sum to one and decline geometrically at a rate based on the average duration of a thirty-year bond. The equation shown below is the reduced-form representation of this expectational computation based on the core part of the expectations VAR.

zpic30 = |
| 1.044E-13[8.83]
| + 0.0402[30416237654149.4]*picnia
| + 0.00693[4776647021764.45]*picniat-1
| + 0.00819[5660807375569.69]*picniat-2
| + 0.000789[593507324804.068]*picniat-3
| + -0.0550[-24115027675954.86]*rff
| + -0.00756[-2412096360020.61]*rfft-1
| + -0.0158[-5029937955937.71]*rfft-2
| + -0.00358[-1556924824600.78]*rfft-3
| + 0.0820[16798496169736.65]*rtr
| + 0.944[214636286092698.4]*ptr
| + 0.0312[11342786247061.99]*xgap
| + -0.00476[-988190865805.51]*xgapt-1
| + 0.0135[2824557302251.42]*xgapt-2
| + 0.0130[4475369411695.43]*xgapt-3

#### MCE Equation

ZPIC30 a weighted average of future PCE price inflation. The weights sum to one and decline geometrically at a rate based on the average duration of a thirty-year bond.

zpic30 = |
| 0.0141 * picnia
| + 0.986 * zpic30t+1

### ZPIC58: Expected 4-qtr consumer price inflation (8 qtrs. in the future) (VAR exp.)

#### VAR Equation

ZPIC58 is the expected four-quarter rate of PCE price inflation measured eight quarters in the future. The equation shown below is the reduced-form representation of this expectational computation based on the core part of the expectations VAR.

zpic58 = |
| DL1(4){sum=0.46}picnia
| + DL2(4){sum=-0.18}rff
| + 0.176[109427301097439.8]*rtr
| + 0.543[166146844821212.5]*ptr
| + DL3(4){sum=0.20}xgap

Distributed Lag Coefficients

DL11 | 0.359
DL12 | 0.0540
DL13 | 0.0604
DL14 | -0.0170
DL1sum | 0.457

DL21 | 0.0552
DL22 | -0.115
DL23 | -0.00495
DL24 | -0.110
DL2sum | -0.176

DL31 | 0.162
DL32 | -0.175
DL33 | 0.0837
DL34 | 0.128
DL3sum | 0.199

#### MCE Equation

ZPIC58 is the expected four-quarter rate of PCE price inflation measured eight quarters in the future.

zpic58 | = pic4t+8

### ZPICXFE: Expected value of picxfe in the next quarter (VAR exp.)

#### VAR Equation

The equation shown below for the expectation of core PCE price inflation one quarter in the future is the reduced-form representation of this expectation in the small linear model used to jointly estimate the PICXFE and PIECI equations.

zpicxfe = |
| ( 0.381 * picxfet-1 + 0.00113 * picxfet-2 + 0.00146 * picxfet-3 + 0.00226 * picxfet-4)
| + ( 0.0461 * piecit-1 + 0.0339 * piecit-2 + 0.0229 * piecit-3 + 0.0112 * piecit-4)
| + ( -0.0140 * rfft-1 + 0.00112 * rfft-2 + 0.00760 * rfft-3 + -0.00299 * rfft-4)
| + ( 0.0470 * xgap2t-1 + -0.0278 * xgap2t-2 + -0.00506 * xgap2t-3 + -0.00225 * xgap2t-4)
| + 0.00828 * rtrt-1
| + 0.500 * ptrt-1
| + 11.938 * log(qpcniat-1/pcniat-1)
| + 6.844E-05 * log(qplt-1/plt-1)
| + -0.114 * (hlprdtt-1 - 400*huqpctt-1)
| + ( -0.00384 * (lurt-1 - lurnatt-1) + -0.000695 * (lurt-2 - lurnatt-2))

#### MCE Equation

zpicxfe | = picxfet+1

### ZPIECI: Expected value of pieci in the next quarter (VAR exp.)

#### VAR Equation

The equation shown below for the expectation of the rate of growth of ECI hourly compensation one quarter in the future is the reduced-form representation of this expectation in the small linear model used to jointly estimate the PICXFE and PIECI equations.

zpieci = |
| ( -0.0260 * picxfet-1 + 0.00320 * picxfet-2 + 0.00403 * picxfet-3 + 0.00650 * picxfet-4)
| + ( 0.202 * piecit-1 + 0.196 * piecit-2 + 0.196 * piecit-3 + 0.0247 * piecit-4)
| + ( -0.0329 * rfft-1 + 0.00136 * rfft-2 + 0.0230 * rfft-3 + -0.00862 * rfft-4)
| + ( 0.149 * xgap2t-1 + -0.0777 * xgap2t-2 + -0.0138 * xgap2t-3 + -0.00648 * xgap2t-4)
| + 0.0172 * rtrt-1
| + 0.393 * ptrt-1
| + -4.495 * log(qpcniat-1/pcniat-1)
| + 0.000155 * log(qplt-1/plt-1)
| + 0.381 * (hlprdtt-1 - 400*huqpctt-1)
| + ( -0.0172 * (lurt-1 - lurnatt-1) + -0.00416 * (lurt-2 - lurnatt-2))

#### MCE Equation

zpieci | = piecit+1

### ZRFF10: Expected federal funds rate, for RG10 eq. (10-yr mat.) (VAR exp.)

#### VAR Equation

ZRFF10 is a weighted average of future federal funds rates. The weights sum to one and decline geometrically at a rate based on the average duration of a ten-year bond. The equation shown below is the reduced-form representation of this expectational computation based on the core part of the expectations VAR.

zrff10 = |
| 2.379E-13[10.004]
| + DL1(4){sum=-0.036}picnia
| + DL2(4){sum=0.27}rff
| + 0.731[74486915458169.02]*rtr
| + 0.0365[4124786896497.41]*ptr
| + DL3(4){sum=0.043}xgap

Distributed Lag Coefficients

DL11 | 0.000122
DL12 | -0.00918
DL13 | -0.0159
DL14 | -0.0115
DL1sum | -0.0365

DL21 | 0.221
DL22 | -0.0158
DL23 | 0.130
DL24 | -0.0661
DL2sum | 0.269

DL31 | 0.361
DL32 | -0.273
DL33 | -0.0378
DL34 | -0.00708
DL3sum | 0.0433

#### MCE Equation

ZRFF10 is a weighted average of future federal funds rates. The weights sum to one and decline geometrically at a rate based on the average duration of a ten-year bond.

zrff10 = |
| 0.0301 * rff
| + 0.970 * zrff10t+1

### ZRFF30: Expected federal funds rate, for RG30 eq. (30-yr mat.) (VAR exp.)

#### VAR Equation

ZRFF30 is a weighted average of future federal funds rates. The weights sum to one and decline geometrically at a rate based on the average duration of a thirty-year bond. The equation shown below is the reduced-form representation of this expectational computation based on the core part of the expectations VAR.

zrff30 = |
| 1.711E-13[12.13]
| + DL1(4){sum=-0.019}picnia
| + DL2(4){sum=0.14}rff
| + 0.856[146971930365859.7]*rtr
| + 0.0195[3712076118807.21]*ptr
| + DL3(4){sum=0.023}xgap

Distributed Lag Coefficients

DL11 | 0.000181
DL12 | -0.00492
DL13 | -0.00851
DL14 | -0.00622
DL1sum | -0.0195

DL21 | 0.119
DL22 | -0.00855
DL23 | 0.0699
DL24 | -0.0356
DL2sum | 0.144

DL31 | 0.194
DL32 | -0.147
DL33 | -0.0202
DL34 | -0.00373
DL3sum | 0.0231

#### MCE Equation

ZRFF30 is a weighted average of future federal funds rates. The weights sum to one and decline geometrically at a rate based on the average duration of a thirty-year bond.

zrff30 = |
| 0.0141 * rff
| + 0.986 * zrff30t+1

### ZRFF5: Expected federal funds rate, for RG5 eq. (5-yr mat.) (VAR exp.)

#### VAR Equation

ZRFF5 is a weighted average of future federal funds rates. The weights sum to one and decline geometrically at a rate based on the average duration of a five-year bond. The equation shown below is the reduced-form representation of this expectational computation based on the core part of the expectations VAR.

zrff5 = |
| 2.601E-13[12.011]
| + DL1(4){sum=-0.051}picnia
| + DL2(4){sum=0.43}rff
| + 0.570[63832198886307.72]*rtr
| + 0.0508[6309318921722.55]*ptr
| + DL3(4){sum=0.094}xgap

Distributed Lag Coefficients

DL11 | 0.00624
DL12 | -0.0132
DL13 | -0.0248
DL14 | -0.0191
DL1sum | -0.0508

DL21 | 0.356
DL22 | -0.0297
DL23 | 0.214
DL24 | -0.111
DL2sum | 0.430

DL31 | 0.615
DL32 | -0.453
DL33 | -0.0589
DL34 | -0.00912
DL3sum | 0.0943

#### MCE Equation

ZRFF5 is a weighted average of future federal funds rates. The weights sum to one and decline geometrically at a rate based on the average duration of a five-year bond.

zrff5 = |
| 0.0548 * rff
| + 0.945 * zrff5t+1

### ZYH: Expected level of real after-tax household income, for QEC eq. (VAR exp.)

#### VAR Equation

Permanent household income (ZYH) is the scaled discounted stream of future household income (YH). The discount factor is 25 percent per year. The scaling factor is chosen so that ZYH and YH have the same value when YH is expected to grow steadily at 3 percent per year. The equation shown below is the reduced-form representation of this expectational computation based on a small-scale VAR used in the estimation of the main consumption (ECO) equation.

log(zyh) = |
| DL1(4){sum=0.0019}picnia
| + DL2(4){sum=-0.0024}rff
| + DL3(4){sum=0.0030}xgap2
| + -0.00185[-150057050145.010]*ptr
| + 0.00243[176351782948.98]*rtr
| + DL4(4){sum=0.0044}yhgap
| + 0.1000[280285385735409.3]*log(zyhst*xgdpt)

Distributed Lag Coefficients

DL11 | 8.403E-05
DL12 | 0.000702
DL13 | 0.000599
DL14 | 0.000469
DL1sum | 0.00185

DL21 | -0.00211
DL22 | 0.000166
DL23 | -0.000778
DL24 | 0.000291
DL2sum | -0.00243

DL31 | 0.000906
DL32 | 0.00269
DL33 | 0.000682
DL34 | -0.00126
DL3sum | 0.00302

DL41 | 0.00416
DL42 | 0.000750
DL43 | -0.000192
DL44 | -0.000360
DL4sum | 0.00436

#### MCE Equation

Permanent household income (ZYH) is the scaled discounted stream of future household income (YH). The discount factor is 25 percent per year. The scaling factor is chosen so that ZYH and YH have the same value when YH is expected to grow steadily at 3 percent per year.

zyh = |
| (1 - (0.750^.25)*exp(0.008)) * yh
| + (0.750^.25) * zyht+1

### ZYHP: Expected level of real after-tax property income, for QEC eq. (VAR exp.)

#### VAR Equation

Permanent household property income (ZYHP) is the scaled discounted stream of future household property income (YHP). The discount factor is 25 percent per year. The scaling factor is chosen so that ZYHP and YHP have the same value when YHP is expected to grow steadily at 3 percent per year. The equation shown below is the reduced-form representation of this expectational computation based on a small-scale VAR used in the estimation of the main consumption (ECO) equation.

log(zyhp) = |
| DL1(4){sum=0.0033}picnia
| + DL2(4){sum=-0.0031}rff
| + DL3(4){sum=0.0016}xgap2
| + -0.00331[-183774665668.85]*ptr
| + 0.00310[166602195390.43]*rtr
| + DL4(4){sum=0.0036}yhgap
| + DL5(4){sum=0.0018}yhpgap
| + 0.1000[186145509558884.1]*log(zyhpst*zyhst*xgdpt)

Distributed Lag Coefficients

DL11 | 0.000864
DL12 | 0.00104
DL13 | 0.000941
DL14 | 0.000459
DL1sum | 0.00331

DL21 | -0.00154
DL22 | -0.000641
DL23 | -0.000877
DL24 | -4.075E-05
DL2sum | -0.00310

DL31 | -0.00204
DL32 | 0.00308
DL33 | 0.00153
DL34 | -0.000979
DL3sum | 0.00160

DL41 | 0.00422
DL42 | 0.000131
DL43 | 5.342E-05
DL44 | -0.000835
DL4sum | 0.00357

DL51 | 0.00225
DL52 | 0.000201
DL53 | -0.000427
DL54 | -0.000252
DL5sum | 0.00177

#### MCE Equation

Permanent household property income (ZYHP) is the scaled discounted stream of future household property income (YHP). The discount factor is 25 percent per year. The scaling factor is chosen so that ZYHP and YHP have the same value when YHP is expected to grow steadily at 3 percent per year.

zyhp = |
| (1 - (0.750^.25)*exp(0.008)) * yhp
| + (0.750^.25) * zyhpt+1

### ZYHPST: Expected trend share of property income in household income

log(zyhpst) = |
| log(zyhpstt-1) + 0.05*yhpgapt-1/100

### ZYHST: Expected trend ratio of household income to GDP

log(zyhst) = |
| log(zyhstt-1) + 0.05*yhgapt-1/100

### ZYHT: Expected level of real transfer income, for QEC eq. (VAR exp.)

#### VAR Equation

Permanent household transfer income (ZYHT) is the scaled discounted stream of future household transfer income (YHT). The discount factor is 25 percent per year. The scaling factor is chosen so that ZYHT and YHT have the same value when YHT is expected to grow steadily at 3 percent per year. The equation shown below is the reduced-form representation of this expectational computation based on a small-scale VAR used in the estimation of the main consumption (ECO) equation.

log(zyht) = |
| DL1(4){sum=0.00075}picnia
| + DL2(4){sum=-0.0027}rff
| + DL3(4){sum=0.0077}xgap2
| + -0.000746[-129368288338.99]*ptr
| + 0.00274[437547186771.30]*rtr
| + DL4(4){sum=0.0036}yhgap
| + DL5(4){sum=0.0027}yhtgap
| + 0.1000[489814668175101.9]*log(zyhtst*zyhst*xgdpt)

Distributed Lag Coefficients

DL11 | -0.000335
DL12 | 0.000473
DL13 | 0.000280
DL14 | 0.000328
DL1sum | 0.000746

DL21 | -0.00250
DL22 | 0.000886
DL23 | -0.00121
DL24 | 8.524E-05
DL2sum | -0.00274

DL31 | 0.00217
DL32 | 0.00313
DL33 | 0.00211
DL34 | 0.000248
DL3sum | 0.00767

DL41 | 0.00271
DL42 | 0.000679
DL43 | 0.000309
DL44 | -6.935E-05
DL4sum | 0.00363

DL51 | 0.00184
DL52 | 0.000510
DL53 | 9.471E-05
DL54 | 0.000243
DL5sum | 0.00268

#### MCE Equation

Permanent household transfer income (ZYHT) is the scaled discounted stream of future household transfer income (YHT). The discount factor is 25 percent per year. The scaling factor is chosen so that ZYHT and YHT have the same value when YHT is expected to grow steadily at 3 percent per year.

zyht = |
| (1 - (0.750^.25)*exp(0.008)) * yht
| + (0.750^.25) * zyhtt+1

### ZYHTST: Expected trend share of transfer income in household income

log(zyhtst) = |
| log(zyhtstt-1) + 0.05*yhtgapt-1/100

### ZYNID: Expected rate of growth of target real dividends, for YNIDN eq. (VAR exp.)

#### VAR Equation

The weighted average growth rate of expected future target dividends, ZYNID, is computed from forecasts of the small-scale VAR model used in the estimation of the dividends (YNIDN) PAC equation. The forward weights are those implied by the estimated PAC coefficients. The equation shown below is the reduced-form representation of this expectational computation.

zynid = |
| 5.207E-16[6.91]
| + DL1(4){sum=0.00070}picniat-1
| + DL2(4){sum=-0.0012}rfft-1
| + 0.00117[48662776893950.1]*rtrt-1
| + -0.000705[-33097352629009.33]*ptrt-1
| + DL3(4){sum=-0.0014}xgapt-1
| + DL4(4){sum=0.023}dlog(qynidnt-1/pxbt-1)
| + 1.186[172844532426645.2]*hggdptt-1/400

Distributed Lag Coefficients

DL11 | -0.000102
DL12 | 0.000349
DL13 | 0.000252
DL14 | 0.000206
DL1sum | 0.000705

DL21 | 0.000353
DL22 | -0.000912
DL23 | -0.000833
DL24 | 0.000222
DL2sum | -0.00117

DL31 | -0.00510
DL32 | 0.00166
DL33 | 0.000635
DL34 | 0.00140
DL3sum | -0.00141

DL41 | 0.00252
DL42 | 0.0155
DL43 | 0.0124
DL44 | -0.00697
DL4sum | 0.0235

#### MCE Equation

ZYNID is a weighted average of future growth rates of the target level of dividends. The pattern of the weights is a function of the PAC adjustment coefficients in the estimated dividend (YNIDN) equation. The sum of the weights equals one minus the sum of the coefficients on lagged growth of actual dividends in the YNIDN equation.

zynid = |
| 0.130 * Δ(log((qynidn / pxb)))
| + 0.0262 * Δ(log(qynidnt+1 / pxbt+1))
| + 0.670 * zynidt+1
| + 0.201 * zynidt+2

Varable Directory
Name | Definition | Sector
adjlegrt | Potential government employment ratio (relative to business) |
d79a | Dummy, post-1979 indicator |
d8095 | Dummy, 1980-1995 indicator |
d83 | Dummy, post-1983 indicator |
d87 | Dummy, post-1986 indicator |
ddockm | Dock strike dummy, import equation |
ddockx | Dock strike dummy, export equation |
delrff | Federal funds rate, first diff | Financial Sector
deuc | EUC switch: 1 for including EUC, 0 for not including |
dfmprr | Dummy, Foreign monetary policy switch: Exogenous real interest rate |
dfpdbt | Fiscal policy switch: 1 for debt ratio stabilization |
dfpex | Fiscal policy switch: 1 for exogenous personal income trend tax rates |
dfpsrp | Fiscal policy switch: 1 for surplus ratio stabilization |
dglprd | Switch to control for long-run productivity growth in the government sector |
dmpalt | Monetary policy switch: MA rule |
dmpex | Monetary policy switch: exogenous federal funds rate |
dmpgen | Monetary policy switch: Generalized reaction function |
dmpintay | Monetary policy switch: inertial taylor rule |
dmprr | Monetary policy switch: exogenous real federal funds rate |
dmpstb | Stabilization switch: 0 for standard applications, 1 for stochastic simulations |
dmptay | Monetary policy switch: Taylor's reaction function |
dmptlr | Monetary policy switch: Taylor's reaction function with unemployment gap |
dmptlur | Monetary policy indicator for unemployment threshold | Financial Sector
dmptmax | Monetary policy indicator for both thresholds | Financial Sector
dmptpi | Monetary policy indicator for inflation threshold | Financial Sector
dmptr | Monetary policy indicator for policy rule thresholds | Financial Sector
dmptrsh | Monetary policy threshold switch: 0 for no threshold, 1 for threshold |
dpadj | Price inflation aggregation adjustment | Wages and Prices
dpgap | Price inflation aggregation discrepancy | Wages and Prices
drstar | RSTAR updating switch: 1 is on, 0 is off |
ebfi | Business Fixed Investment, cw 2012$ | Business Expenditures
ebfin | Investment in equipment, current $ | Business Expenditures
ec | Consumption, cw 2012$ (FRB/US definition) | Household Expenditures
ecd | Consumer expenditures on durable goods, cw 2012$ | Household Expenditures
ech | Consumer expenditures on housing services, cw 2012$ | Household Expenditures
ecnia | Personal consumption expenditures, cw 2012$ (NIPA definition) | Household Expenditures
ecnian | Personal consumption expenditures, current $ (NIPA definition) | Household Expenditures
eco | Consumer expenditures on non-durable goods and non-housing services, cw 2012$ | Household Expenditures
egfe | Federal Government expenditures, CW 2012$ | Government
egfen | Federal Government expenditures, current $ | Government
egfet | Federal Government expenditures, CW 2012$, Trend | Government
egfl | Federal government employee compensation, cw 2012$ | Government
egfln | Federal government employee compensation, current $ | Government
egflt | Federal government employee compensation, cw 2012$, trend | Government
egse | S&L Government expenditures, CW 2012$ | Government
egsen | S&L Government expenditures, current $ | Government
egset | S&L Government expenditures, CW 2012$, Trend | Government
egsl | S&L government employee compensation, cw 2012$ | Government
egsln | S&L government employee compensation, current $ | Government
egslt | S&L government employee compensation, cw 2012$, trend | Government
eh | Residential investment expenditures, cw 2012$ | Household Expenditures
ehn | Residential investment expenditures | Household Expenditures
ei | Change in private inventories, cw 2012$ | Business Expenditures
ein | Change in business inventories, current $ | Business Expenditures
em | Imports of goods and services, cw 2012$ | Foreign Trade
emn | Imports of goods and services, current $ | Foreign Trade
emo | Imports of goods and services ex. petroleum, cw 2012$ | Foreign Trade
emon | Imports of goods and services ex. petroleum | Foreign Trade
emp | Petroleum imports, cw 2012$ | Foreign Trade
empn | Petroleum imports, current $ | Foreign Trade
emptrt | Oil imports to GDP ratio (Trend) |
ex | Exports of goods and services, cw 2012$ | Foreign Trade
exn | Exports of goods and services, current $ | Foreign Trade
fcbn | US current account balance, current $ | Foreign Trade
fcbrn | US current account balance residual, current $ | Foreign Trade
fgdp | Foreign aggregate GDP (world, bilateral export weights) | Foreign Activity
fgdpt | Foreign aggregate GDP (world, bilateral export weights), trend | Foreign Activity
fnicn | Gross stock of claims of US residents on the rest of the world, current $ | Foreign Trade
fniln | Gross stock of liabilities of US residents to the rest of the world, current $ | Foreign Trade
fnin | Net stock of claims of US residents on the rest of the world, current $ | Foreign Trade
fnirn | Net stock of claims of US residents on the rest of the world, residual | Foreign Trade
fpc | Foreign aggregate consumer price (G39, import/export trade weights) | Foreign Activity
fpi10 | Foreign consumer price inflation (G10) | Foreign Activity
fpi10t | Foreign consumer price inflation, trend (G10) | Foreign Activity
fpic | Foreign consumer price inflation (G39, bilateral export trade weights) | Foreign Activity
fpitrg | Foreign target consumer price inflation (G10) |
fpx | Nominal exchange rate (G39, import/export trade weights) | Foreign Activity
fpxr | Real exchange rate (G39, import/export trade weights) | Foreign Activity
fpxrr | Real exchange rate residual | Foreign Activity
fpxrrt | Real exchange rate residual, trend |
frl10 | Foreign long-term interest rate (G10) | Foreign Activity
frs10 | Foreign short-term interest rate (G10) | Foreign Activity
frstar | Equilibrium real short-term interest rate used in foreign Taylor rule | Foreign Activity
ftcin | Corporate taxes paid to rest of world, current $ | Foreign Trade
fxgap | Foreign output gap (world, bilateral export weights) | Foreign Activity
fynicn | Gross investment income received from the rest of the world, current $ | Foreign Trade
fyniln | Gross investment income paid to the rest of the world, current $ | Foreign Trade
fynin | Net investment income received from the rest of the world, current $ | Foreign Trade
gfdbtn | Federal government debt stock, current $ | Government
gfdbtnp | Federal government debt stock held by the public, current $ | Government
gfdrt | Federal government target debt-to-GDP ratio |
gfexpn | Government payments | Government
gfintn | Federal government net interest payments, current $ | Government
gfrecn | Government receipts and residual | Government
gfsrpn | Federal government budget surplus, current $ | Government
gfsrt | Federal government target surplus-to-GDP ratio |
gtn | Federal government net transfer payments, current $ | Government
gtr | Government net transfer payments, deflated by PGDP | Government
gtrd | Deviation of ratio of government transfers to GDP from trend ratio | Government
gtrt | Government, trend ratio of transfer payments to GDP |
hgemp | Petroleum imports, cw 2012$, trend growth rate | Foreign Trade
hggdp | Growth rate of GDP, cw 2012$ (annual rate) | Aggregate Output Identities
hggdpt | Trend growth rate of XGDP, cw 2012$ (annual rate) | Aggregate Output Identities
hgpbfir | Trend growth rate of price of business investment (relative to PXB) | Aggregate Output Identities
hgpcdr | Trend growth rate of price of consumer durable goods (relative to PCNIA) |
hgpkir | Trend growth rate of PKIR | Wages and Prices
hgynid | Growth rate of real after-tax corporate profits | Expectations
hks | Growth rate of KS, cw 2012$ (compound annual rate) | Business Expenditures
hksr | Residual growth of capital services |
hlept | Trend growth rate of LEP (annual rate) | Labor Market
hlprdt | Trend growth rate of output per hour | Labor Market
hmfpt | Trend growth rate of multifactor productivity | Aggregate Output Identities
hqlfpr | Drift component of change in QLFPR | Labor Market
hqlww | Trend growth rate of workweek | Labor Market
huqpct | Drift term in stochastic component of trend ratio of PCNIA to PXP | Wages and Prices
huxb | Drift term in UXBT | Aggregate Output Identities
hxbt | Trend rate of growth of XB , cw 2012$ (annual rate) | Aggregate Output Identities
hxbtr | Residual Factor (Trend rate of growth of XB) | Aggregate Output Identities
jccan | Consumption of fixed capital, current $ | Nominal Income
jkcd | Consumption of fixed capital, consumer durables | Household Expenditures
jrbfi | Depreciation rate, business investment |
jrcd | Depreciation rate, consumer durables |
jrh | Depreciation rate, housing |
kbfi | Capital stock - BFI, 2012$ | Business Expenditures
kcd | Stock of consumer durables, cw 2012$ | Household Expenditures
kh | Stock of residential structures, cw 2012$ | Household Expenditures
ki | Stock of private inventories, cw 2012$ | Business Expenditures
kps | Capital stock - nonresidential structures, 2012$ | Business Expenditures
ks | Capital services, 2012$ | Business Expenditures
lefr | Federal Government employment ratio |
leg | Government civilian employment ex. gov. enterprise | Government
leh | Civilian employment (break adjusted) | Labor Market
leo | Difference between household and business sector payroll employment, less government employment | Labor Market
lep | Employment in business sector (employee and self-employed) | Labor Market
leppot | Potential employment in business sector | Labor Market
leuc | Emergency unemployment compensation (EUC) |
lf | Civilian labor force (break adjusted) | Labor Market
lfpr | Labor force participation rate | Labor Market
lhp | Aggregate labor hours, business sector (employee and self-employed) | Labor Market
lprdt | Trend labor productivity | Labor Market
lqualt | Labor quality, trend level |
lur | Civilian unemployment rate (break adjusted) | Labor Market
lurnat | Natural rate of unemployment | Labor Market
lurtrsh | Unemployment threshold |
lww | Workweek, business sector (employee and self-employed) | Labor Market
mei | Multiplicative discrepancy for the difference between XGDI and XGDO | Financial Sector
mep | Multiplicative discrepancy for the difference between XGDP and XGDO | Financial Sector
mfpt | Multifactor productivity, trend level | Aggregate Output Identities
n16 | Noninstitutional population, aged 16 and over (break adjusted) |
pbfir | Price level of BFI compared to PXP | Wages and Prices
pcdr | Price index for consumer durables, cw (relative to to PCNIA) | Wages and Prices
pcer | Price index for personal consumption expenditures on energy (relative to PCXFE) | Wages and Prices
pcfr | Price index for personal consumption expenditures on food (relative to PCXFE) | Wages and Prices
pcfrt | Real PCE price of food, trend |
pchr | Price index for housing services, cw (relative to to PCNIA) | Wages and Prices
pcnia | Price index for personal consumption expenditures, cw (NIPA definition) | Wages and Prices
pcor | Price index for non-durable goods and non-housing services, cw (relative to to PCNIA | Wages and Prices
pcpi | Consumer price index,total | Wages and Prices
pcpix | Consumer price index,excluding food and energy | Wages and Prices
pcstar | Target consumption price level (used in RFFGEN policy rule) |
pcxfe | Price index for personal consumption expendits ex. food and energy, cw (NIPA definit | Wages and Prices
pegfr | Price index for federal government expenditures, CW (relative to PXP) | Government
pegsr | Price index for S&L government expenditures, CW (relative to PXP) | Government
pgdp | Price index for GDP, cw | Wages and Prices
pgfl | Price index for federal government employee compensation, cw | Wages and Prices
pgsl | Price index for S&L government employee compensation, cw | Wages and Prices
phouse | Loan Performance House Price Index | Financial Sector
phr | Price index for residential investment, cw (relative to PXP) | Wages and Prices
pic4 | Four-quarter percent change in PCE prices | Wages and Prices
picnia | Inflation rate, personal consumption expenditures, cw | Wages and Prices
picx4 | Four-quarter percent change core in PCE prices | Wages and Prices
picxfe | Inflation rate, personal consumption expenditures, ex. food and energy, cw | Wages and Prices
pieci | Annualized rate of growth of EI hourly compensation | Wages and Prices
pigdp | Inflation rate, GDP, cw | Wages and Prices
pipl | Rate of growth of PL | Wages and Prices
pipxnc | Inflation rate, price of adjusted final sales excluding consumption (annual rate) | Wages and Prices
pitarg | Target rate of consumption price inflation (used in policy reaction functions) |
pitrsh | Inflation threshold |
pkbfir | Ratio of price of BFI stock (KBFI) to PXP | Wages and Prices
pkir | Price index for stock of inventories, cw (relative to PXP) |
pl | Compensation per hour, business | Wages and Prices
pmo | Price index for imports ex. petroleum, cw | Wages and Prices
pmp | Price index for petroleum imports | Wages and Prices
poil | Price of imported oil ($ per barrel) | Wages and Prices
poilr | Price of imported oil, relative to price index for bus. sector output | Wages and Prices
poilrt | Price of imported oil, relative to price index for bus. sector output, trend |
ptr | 10-year expected PCE price inflation (Survey of Professional Forecasters) | Expectations
pwstar | Equilibrium business price markup |
pxb | Price index for business output | Wages and Prices
pxnc | Price of adjusted final sales excluding consumption | Wages and Prices
pxp | Price index for final sales plus imports less gov. labor | Wages and Prices
pxr | Price index for exports, cw (relative to PXP) | Wages and Prices
qebfi | Desired level of investment in business investment | Business Expenditures
qec | Desired level of consumption (FRBUS definition) | Household Expenditures
qecd | Target level of consumption of durable goods, trending component | Household Expenditures
qeco | Desired level of consumption of nondurable goods and nonhousing services | Household Expenditures
qeh | Target level of residential investment | Household Expenditures
qkir | Desired Inventory Sales Ratio | Business Expenditures
qleor | Desired ratio of employment discrepancy to the labor force |
qlf | Desired level of civilian labor force | Labor Market
qlfpr | Trend labor force participation rate | Labor Market
qlhp | Desired level of business labor hours | Labor Market
qlww | Trend workweek, business sector (employee and self-employed) | Labor Market
qpcnia | Desired level of consumption price | Wages and Prices
qpl | Desired level of compensation per hour, trending component | Wages and Prices
qpmo | Random walk component of non-oil import prices | Wages and Prices
qpxb | Desired price level of private output ex. energy, housing, and farm | Wages and Prices
qpxnc | Desired level of nonconsumption price | Wages and Prices
qpxp | Desired price level of adjusted final sales | Wages and Prices
qynidn | Desired level of dividends | Nominal Income
rbbb | S&P BBB corporate bond rate | Financial Sector
rbbbp | S&P BBB corporate bond rate, risk/term premium | Financial Sector
rbfi | After-tax real financial cost of capital for business investment | Business Expenditures
rcar | New car loan rate at finance companies | Financial Sector
rccd | Cost of capital for consumer durables | Household Expenditures
rcch | Cost of capital for residential investment | Household Expenditures
rcgain | Rate of capital gain on the non-equity portion of household wealth | Financial Sector
req | Real expected rate of return on equity | Financial Sector
reqp | Real expected rate of return on equity, premium component | Financial Sector
rff | Federal funds rate | Financial Sector
rffalt | Value of eff. federal funds rate given by estimated policy rule | Financial Sector
rfffix | Federal funds rate given by fixed, pre-determined funds rate path |
rffgen | Value of eff. federal funds rate given by the generalized reaction function | Financial Sector
rffintay | Value of eff. federal funds rate given by the inertial Taylor rule | Financial Sector
rffmin | Minimum nominal funds rate (set at 0 to impose zero lower bound) |
rffrule | Federal funds rate | Financial Sector
rfftay | Value of eff. federal funds rate given by the Taylor rule with output gap | Financial Sector
rfftlr | Value of eff. federal funds rate given by the Taylor rule with unemployment gap | Financial Sector
rfnict | Residual in FNICN equation |
rfrs10 | Real foreign short-term interest rate |
rfynic | Average yield earned on gross claims of US residents on the rest of the world | Foreign Trade
rfynil | Average yield earned on liabilities of US residents on the rest of the world | Foreign Trade
rg10 | 10-year Treasury bond rate | Financial Sector
rg10p | 10-year Treasury bond rate, term premium | Financial Sector
rg30 | 30-year Treasury bond rate | Financial Sector
rg30p | 30-year Treasury bond rate, term premium | Financial Sector
rg5 | 5-year Treasury note rate | Financial Sector
rg5p | 5-year Treasury note rate. term premium | Financial Sector
rgfint | Average rate of interest on existing federal debt | Financial Sector
rgw | Approximate average rate of interest on new federal debt | Financial Sector
rme | Interest rate on conventional mortgages | Financial Sector
rrff | Real federal funds rate | Financial Sector
rrfix | Real federal funds rate given by fixed, pre-determined real funds rate path |
rrtr | Expected long-run real federal funds rate | Expectations
rspnia | Personal saving rate | Nominal Income
rstar | Equilibrium real federal funds rate (for monetary policy reaction functions) | Financial Sector
rtb | 3-month Treasury bill rate | Financial Sector
rtbfi | User cost of capital for business investment | Business Expenditures
rtinv | User cost of capital for inventories | Business Expenditures
rtr | Expected federal funds rate in the long run (Blue Chip) | Expectations
t47 | Time trend, begins in 1947q1 (0 before) |
tapddp | Proportion of investment tax credit deducted from depr. base |
tapdt | Investment tax credit rate for equipment |
tcin | Government corporate income tax accruals, current $ | Government
tdpv | Present value of depreciation allowances, BFI |
tpn | Government personal income tax and non-tax receipts, current $ | Government
trci | Average government corporate income tax rate | Government
trcit | Government corporate income tax rate, trend |
trfcim | Marginal federal corporate income tax rate |
trfpm | Marginal federal personal income tax rate (at twice median family income) |
tritc | Investment tax credit for business investment |
trp | Average government tax rate for personal income tax and non-tax receipts | Government
trpt | Average government tax rate for personal income tax, trend | Government
trptx | Average tax rate for personal income tax, trend, policy setting |
trspp | Marginal S&L tax rate on personal property |
tryh | Average tax rate on household income | Government
uces | Energy share of nominal consumption expenditures |
ucfs | Food share of nominal consumption expenditures |
uemot | Trend in ratio of EMON to XGDEN |
ufcbr | Multiplicative factor in FCBRN identity |
ufnir | Multiplicative factor in FNIRN identity |
uftcin | Multiplicative factor in FTCIN identity |
ugfdbt | Multiplicative factor in GFDBTN identity |
ugfdbtp | Federal government debt ratio |
ugfsrp | Federal Government budget surplus, residual | Government
uleg | Multiplicative factor for government civilian employment | Government
upcpi | Multiplicative factor in PCPI identity |
upcpix | Multiplicative factor in PCPIX identity |
upgfl | Multiplicative factor in PGFL identity |
upgsl | Multiplicative factor in PGSL identity |
upkbfir | Multiplicative factor in PKBFIR identity |
upmp | Multiplicative factor in PMP identity |
upxb | Multiplicative factor in PXB identity |
uqpct | Stochastic component of trend ratio of PCNIA to PXP | Wages and Prices
uvbfi | Multiplicative factor in VBFI identity |
uxbt | Stochastic component of trend ratio of XGDPT to XBT | Aggregate Output Identities
uyd | Multiplicative factor in YDN identity |
uyhibn | Multiplicative factor (Consumer interest payments to business) |
uyhln | Multiplicative factor in YHLN identity |
uyhptn | Multiplicative factor in YHPTN identity |
uyhsn | Multiplicative factor in personal saving identity (accounts for transfers to foreign |
uyhtn | Multiplicative factor in YHTN identity |
uyl | Multiplicative factor in YLN identity |
uyni | Multiplicative factor in YNIN identity |
uynicpnr | Corporate profits, residual | Nominal Income
uyp | Multiplicative factor in YPN identity |
vbfi | Desired investment-output ratio | Business Expenditures
wpo | Household property wealth ex. stock market, real | Financial Sector
wpon | Household property wealth ex. stock market, current $ | Financial Sector
wps | Household stock market wealth, real | Financial Sector
wpsn | Household stock market wealth, current $ | Financial Sector
xb | Business output (BEA definition), cw 2012$ | Aggregate Output Identities
xbn | Business output (BEA definition), current $ | Nominal Income
xbo | Business output, adjusted for measurement error, cw 2012$ | Aggregate Output Identities
xbt | Potential business output, cw 2012$ | Aggregate Output Identities
xbtr | Residual Factor (Potential business output) | Aggregate Output Identities
xfs | Final sales of gross domestic product, cw 2012$ | Aggregate Output Identities
xfsn | Final sales of gross domestic product, current $ | Nominal Income
xgap | Output gap for business plus oil imports (100*log(actual/potential) | Aggregate Output Identities
xgap2 | Output gap for GDP (100*log(actual/potential) | Aggregate Output Identities
xgdi | Gross domestic income, cw 2012$ | Aggregate Output Identities
xgdin | Gross domestic income, current $ | Nominal Income
xgdo | Gross domestic product, adjusted for measurement error, cw 2012$ | Aggregate Output Identities
xgdp | GDP, cw 2012$ | Aggregate Output Identities
xgdpn | GDP, current $ | Nominal Income
xgdpt | Potential GDP, cw 2012$ | Aggregate Output Identities
xgdptn | Potential GDP, current $ | Aggregate Output Identities
xp | Final sales plus imports less government labor, cw 2012$ | Aggregate Output Identities
xpn | Final sales plus imports less government labor, current $ | Nominal Income
ydn | Disposable income | Nominal Income
yh | Income, household, total (real after-tax) | Nominal Income
yhgap | Income, household, total, ratio to XGDP, cyclical component (real after-tax) | Nominal Income
yhibn | Consumer interest payments to business | Nominal Income
yhl | Income, household, labor compensation (real after-tax) | Nominal Income
yhln | Income, household, labor compensation | Nominal Income
yhp | Income, household, property (real after-tax) | Nominal Income
yhpcd | Imputed income of the stock of consumer durables, 2012$ | Household Expenditures
yhpgap | Income, household, property, ratio to YH, cyclical component (real after-tax) | Nominal Income
yhpntn | Income, household, property, non-taxable component | Nominal Income
yhpshr | Income, household, property, ratio to YH (real after-tax) | Nominal Income
yhptn | Income, household, property, taxable component | Nominal Income
yhshr | Income, household, total, ratio to XGDP (real after-tax) | Nominal Income
yhsn | Personal saving | Nominal Income
yht | Income, household, transfer (real after-tax), net basis | Nominal Income
yhtgap | Income, household, transfer, ratio to YH, cyclical component (real after-tax) | Nominal Income
yhtn | Income, household, transfer payments. net basis | Nominal Income
yhtshr | Income, household, transfer, ratio to YH (real after-tax) | Nominal Income
ykbfin | Income from stock of BFI | Nominal Income
ykin | Income from stock of inventories | Nominal Income
ymsdn | Microsoft one-time dividend payout in 2004Q4 |
ynicpn | Corporate profits (national income component) | Nominal Income
ynidn | Dividends (national income component) | Nominal Income
yniln | Labor income (national income component) | Nominal Income
ynin | National income | Nominal Income
ynirn | Net interest, rental and proprietors' incomes (national income components) | Government
ypn | Personal income | Nominal Income
zdivgr | Expected growth rate of real dividends, for WPSN eq. (VAR exp.) | Expectations
zebfi | Expected growth rate of business output EBFI (VAR exp.) | Expectations
zecd | Expected growth rate of target durable consumption, for ECD eq. (VAR exp.) | Expectations
zeco | Expected growth rate of target nondurables and nonhousing services, for ECO eq (VAR | Expectations
zeh | Expected growth rate of target residential investment, for EH eq. (VAR exp.) | Expectations
zgap05 | Expected output gap, for RG5 eq. (VAR exp.) | Expectations
zgap10 | Expected output gap, for RG10 eq. (VAR exp.) | Expectations
zgap30 | Expected output gap, for RG30 eq. (VAR exp.) | Expectations
zgapc2 | Expected output gap, for ECD eq. (VAR exp.) | Expectations
zlhp | Expected growth rate of target aggregate hours (VAR exp.) | Expectations
zpi10 | Expected cons. price infl., for RCCH, RRMET, and YHPNTN eqs. (10-yr mat.) (VAR exp.) | Expectations
zpi10f | Expected cons. price infl., for FPXR eq. (10-yr mat.) (VAR exp.) | Expectations
zpi5 | Expected cons. price infl., for RCCD eq. (5-yr mat.) (VAR exp.) | Expectations
zpib5 | Expected output price infl., for RPD eq. (5-yr mat.) (VAR exp.) | Expectations
zpic30 | Expected cons. price infl., for REQ eq. (30-yr mat.) (VAR exp.) | Expectations
zpic58 | Expected 4-qtr consumer price inflation (8 qtrs. in the future) (VAR exp.) | Expectations
zpicxfe | Expected value of picxfe in the next quarter (VAR exp.) | Expectations
zpieci | Expected value of pieci in the next quarter (VAR exp.) | Expectations
zrff10 | Expected federal funds rate, for RG10 eq. (10-yr mat.) (VAR exp.) | Expectations
zrff30 | Expected federal funds rate, for RG30 eq. (30-yr mat.) (VAR exp.) | Expectations
zrff5 | Expected federal funds rate, for RG5 eq. (5-yr mat.) (VAR exp.) | Expectations
zyh | Expected level of real after-tax household income, for QEC eq. (VAR exp.) | Expectations
zyhp | Expected level of real after-tax property income, for QEC eq. (VAR exp.) | Expectations
zyhpst | Expected trend share of property income in household income | Expectations
zyhst | Expected trend ratio of household income to GDP | Expectations
zyht | Expected level of real transfer income, for QEC eq. (VAR exp.) | Expectations
zyhtst | Expected trend share of transfer income in household income | Expectations
zynid | Expected rate of growth of target real dividends, for YNIDN eq. (VAR exp.) | Expectations