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Uncertainty Over Models and Data: The Rise and Fall of American Inflation1

Seth Pruitt2

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at http://www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at http://www.ssrn.com/.


Abstract:

An economic agent who is uncertain of her economic model learns, and this learning is sensitive to the presence of data measurement error. I investigate this idea in an existing framework that describes the Federal Reserve's role in U.S. inflation. This framework successfully fits the observed inflation to optimal policy, but fails to motivate the optimal policy by the perceived Philips curve trade-off between inflation and unemployment. I modify the framework to account for data uncertainty calibrated to the actual size of data revisions. The modified framework ameliorates the existing problems by adding sluggishness to the Federal Reserve's learning: the key point is that the data uncertainty is amplified by the nonlinearity induced by learning. Consequently there is an explanation for the rise and fall in inflation: the concurrent rise and fall in the perceived Philips curve trade-off.

Keywords: Data uncertainty, data revisions, real time data, optimal control, parameter uncertainty, learning, extended Kalman filter, Markov-chain Monte Carlo

JEL classification: E01, E58


1.  Introduction

A great deal of research has gone towards identifying the causes of the large swings in inflation in the United States between 1970 and 1985. One strand of literature advances the view that evolving government behavior had an important role in these events. Clarida, Gali, and Gertler (2000) and Boivin (2006), among others, provides evidence of time-varying U.S. monetary policy responses over different parts of the postwar era. Romer and Romer (1990) and Owyang and Ramey (2004) suggest that the changing response might be explained by changing Federal Reserve objectives. Alternatively, Sargent (1999) suggests that evolving Federal Reserve beliefs about the economic environment can explain the rise and fall of inflation.3

This last explanation relies on the idea that agents learn about their economic environment, as advocated by learning and expectations. In such a framework, agents' prediction errors update their beliefs, represented as the parameters to their own economic model. These prediction errors are based on the data actually observed, which may contain measurement error. The point of this paper is that agents' data uncertainty, engendered by measurement errors, affects the evolution of their beliefs. Similar to Brainard (1997), we see that this uncertainty tempers policy-makers' behavior: their learning is made more sluggish by data uncertainty.

To investigate the importance of this observation in practice, I modify the framework in Sargent, Williams, and Zha (2006) that itself extends Sargent (1999). The Federal Reserve optimally controls inflation in light of constant unemployment and inflation targets, but is unconvinced that its economic model is correct and hence changes its beliefs in response to new data. Thus the great American inflation is explained as optimal policy given changing estimates of the Federal Reserve's economic model, the Philips curve. However, those existing results suffer from three key problems. One, the Federal Reserve's unemployment rate forecasts, the basis for setting inflation, are very inaccurate, much more so than the Greenbook forecasts they should mimic. Second, the amount of estimated model uncertainty is very large, which undermines the plausibility that the Federal Reserve believed in its estimated Philips curve enough to use it as the basis for policy. Third, the framework explains the rise in inflation between 1973 and 1975, but does not give a good reason for the drastic fall in inflation between 1980 and 1984. The recent Carboni and Ellison (forthcoming) modifies Sargent, Williams, and Zha's (2006) optimal control problem to target actual Greenbook forecasts, which successfully addresses the first two problems.4 However, an explanation for the rise and fall of inflation remains missing. The results here address all three problems.

I modify the optimal control framework by assuming the Federal Reserve recognizes that data may be measured with error, the size of which I calibrate to existing evidence. In line with this, and following the general suggestion of Orphanides (2001), I differ from Sargent, Williams, and Zha (2006) and Carboni and Ellison (forthcoming) by fitting the model to real-time data. Since at least as far back as Zellner (1958) - who admonished readers to be "careful'' with "provisional'' data - the reality of data uncertainty has been clearly recognized.5 What has been less clear is the impact this data uncertainty may hold for the purposes of economic research, and this paper suggests that data uncertainty can have a sizeable effect in frameworks where agents are learning.6 Data uncertainty introduces sluggishness into Federal Reserve's learning process, and the ensuing framework is able to avoid the aforementioned problems. The Federal Reserve's model uncertainty drop significantly and the framework predicts unemployment forecasts that resemble Greenbook forecasts. Importantly, the results show a sharp drop in the perceived Philips trade-off between 1980 and 1984 that explains the concomitant fall of inflation.

The paper is organized as follows. Section 2 introduces the theory of the paper: Section 2.1 briefly analyzes the interaction of model uncertainty and data uncertainty, and Section 2.2 describes and extends Sargent, Williams, and Zha (2006) framework. Section 3 presents the estimation results both without- and including data uncertainty, and discusses how data uncertainty matters to this exercise. Section 4 concludes.

2.  Theory

I briefly discuss the meanings of the terms data uncertainty and model uncertainty and explain how they may be related. Then I introduce a basic optimal control framework following Sargent, Williams, and Zha (2006) that is meant to describe Federal Reserve behavior during the 1970s-1980s. Finally, I modify the framework to explicitly account for the Federal Reserve's data uncertainty.

2.1  Why Data Uncertainty Matters

Generally speaking, the simple point made in this section is that the estimated size of modeled random shocks is positively biased by other sources of variation that are unmodeled. Moreover, the agent's learning process is nonlinear in the latent variables (that is, latent parameters multiply latent economic variables), which amplifies bias.

Suppose an agent forms a forecast of an economic quantity $ y$. The agent's model maps data and parameters into this prediction. Model uncertainty is the situation where past predictions and realized data might change the agent's parameter vector going forward. Data uncertainty is the situation where data is measured with error, maybe but not necessarily observed after the fact. The main question here is, if the agent is uncertain both of the model and the data, what is the impact of the researcher ignoring the agent's data uncertainty?

The intuition is straightforward that when the agent's forecast is correct there is nothing to change about her beliefs (her model).7 When such is the case, the agent has nothing to learn since her model's performance cannot be improved. When the forecast error is nonzero, suppose the agent has an incentive to evaluate her model and learn from the error. At this point, the agent needs to understand why the forecast error is nonzero: to answer this, let us consider the forecast error decomposition, which is the machinery that answers the question why. In the case of linear forecasters, the forecast error decomposition is a function of the modeled errors' variances. To sketch out why the ignored data uncertainty biases up the researcher's estimate of the agent's model uncertainty, we turn to a simple example.

Let $ \mbs{\epsilon}$ be a normal error associated with the agent-predicted parameter vector $ \mbs{b}$, $ \mbs{\varepsilon}$ be a normal error associated with the agent-predicted data vector $ \mbs{x}$, $ \upsilon$ be a normal error, all errors be mean zero, and each error be uncorrelated with the others.8 Suppose that the true data generating process is $ y=(\mbs{x}+\mbs{\varepsilon})^{\prime}(\mbs{b}+\mbs{\epsilon})+\upsilon$ . The agent knows this specification of the data generating process and makes a linear forecast $ \hat{y}=\mbs{x}^{\prime}\mbs{b}$. Finally, the agent knows that $ \mvar\left( y-\hat{y}\right) $ is given by

$\displaystyle \sigma^{2} + 2\mbs{x}^{\prime}\mbs{P}\mbs{x} + \mbs{b}^{\prime}\mbs{Q}\mbs{b} + c $

where $ \mexp(\mbs{\epsilon\epsilon}^{\prime})=\mbs{P}$, $ \mexp(\mbs{\varepsilon\varepsilon}^{\prime})=\mbs{Q}$, and $ c>0$, and these three quantities are known to the agent.9

However the researcher ignores the agent's data uncertainty, which means he assumes that $ \mbs{\varepsilon}=\mbs{0}$. So the researcher decomposes the forecast error variance as $ \tilde{\sigma}^{2} + 2\mbs{x}^{\prime }\mbstop{P}{\tilde}\mbs{x}$ . But

$\displaystyle \tilde{\sigma}^{2} + \mbs{x}^{\prime}\mbstop{P}{\tilde}\mbs{x} = \sigma^{2} + \mbs{x}^{\prime}\mbs{Px} + \mbs{b}^{\prime}\mbs{Qb} + c , $

and it becomes clear that there will be a positive bias to his estimates due to the presence of the ignored $ \mbs{Q}$ and $ c$.

A function relating Var$ \mvar\left( y-\hat{y}\right) $ to $ \sigma,\mbs{P},$ and $ \mbs{Q}$ requires some knowledge about how these quantities are related. Writing the matrix \begin{displaymath}\left( \begin{array}[c]{cc} \mbs{P} & \mbs{0}\ \mbs{0} & \sigma^{2} \end{array}\right) = \delta\mbs{D}\end{displaymath} , assume that the researcher knows the true $ \mbs{D}$ (determining the correlation structure and relative proportions of the variances), but must estimate the scalar $ \delta$ (determining the scale of the variances). It is easy to see the researcher's estimate is related to the truth by

$\displaystyle \tilde{\delta}=\delta+\mbs{b}^{\prime}\mbs{Qb}+c. $

This algebra is directly related to frameworks that specify that economic agents are learning, and thus involve filtering. At each point in time the filter delivers conditional means (the data and parameter predictions) and conditional covariance matrices (the errors' covariance structure). The forecast is a function of the means, and the forecast error is decomposed by a gain matrix that is a function of the covariance matrices. Obviously, the gain matrix can only decompose the observed forecast error to sources that are modeled, and so those sources are inferred to have taken on a larger realization than they actually did.

The key idea is that this bias is a nonlinear function of the covariance matrix of the data errors, and so amplification of small (in terms of variance) data uncertainty is possible. It is a different matter to say whether or not this amplification happens in practice, and to investigate that possibility we turn to the next section.

2.2  The Federal Reserve's Optimal Control Problem

Why did inflation rise and fall so dramatically in the United States between 1973 and 1984? Sargent, Williams, and Zha (2006) reverse engineer Federal Reserve Philips curve beliefs that explain inflation as an optimal policy. The Federal Reserve's model evolves because we assume the Federal Reserve is learning about the Philips curve. In light of the previous section, it is worth asking if the exclusion of data uncertainty significantly affects the framework's predictions. We will see in Section 3 that the answer to this question is, yes. But first we describe Sargent, Williams, and Zha's (2006) framework, and then modify it to explicitly account for Federal Reserve data uncertainty.

The Federal Reserve has a dual mandate to keep both the unemployment rate and the inflation rate at target. Directly following Sargent, Williams, and Zha (2006), the Federal Reserve's objective function is written

$\displaystyle \min_{ \{x_{t-1+j}\}_{j=0}^{\infty}} \mathbb{E}_{t} \sum_{j=0}^{\infty}0.9936^{j} \Big( (\pi_{t+j} - 2)^{2} + (u_{t+j} - 1)^{2} \Big).$ (2.1)

Setting the inflation and unemployment targets to Sargent, Williams, and Zha's (2006) values delivers the attractive message that the Federal Reserve has always tried to lower inflation and unemployment while retaining the tractable quadratic objective function. The important message of this equation is that the Federal Reserve has always had the same inflation and unemployment targets.

To achieve this objective, the Federal Reserve controls the rate of inflation up to an exogenous shock. Therefore the annualized inflation rate $ \pi_{t}$ is

$\displaystyle \pi_{t} = x_{t-1} + \zeta_{1}\epsilon_{1t}$ (2.2)

where $ x_{t-1}$ is the part of inflation controllable by the Federal Reserve using information through time $ t-1$, and $ \epsilon_{1t} \sim iid(0,1)$ is an exogenous shock.

In order to understand the relationship between inflation (which can be somewhat directly controlled) and the unemployment rate (which cannot be directly controlled), the Federal Reserve uses a Philips curve model. However, at any point in time the Federal Reserve is uncertain that its estimated model is correct, and so is constantly learning and updating its model estimate. We accomplish this by assuming the parameters follow a random walk, which introduces model uncertainty (and a motivation for learning) following the basic idea of Cooley and Prescott (1976):

$\displaystyle u_{t}$ $\displaystyle = \boldsymbol{\alpha}^{\prime}_{t-1} \left( \begin{array}[c]{c} \pi_{t}\\ \pi_{t-1}\\ u_{t-1}\\ \pi_{t-2}\\ u_{t-2}\\ 1 \end{array} \right) + \zeta_{2} \epsilon_{2t} \equiv\boldsymbol{\alpha}^{\prime}_{t-1} \boldsymbol{\Phi}_{t} + \zeta_{2} \epsilon_{2t}$ (2.3)
$\displaystyle \boldsymbol{\alpha}_{t}$ $\displaystyle = \boldsymbol{\alpha}_{t-1} + \mbs{Z}^{-\frac{1} {2}}\boldsymbol{\epsilon}_{3t}$ (2.4)

Because the parameters follow a random walk whose steps are independent of everything else, the Federal Reserve's estimate of $ \boldsymbol{\alpha}_{t-1}$ is also its estimate of $ \boldsymbol{\alpha}_{t+j},\: \forall j \geq0$; thus we solve the problem for each period $ t$ using "anticipated utility'', following conquestamericaninflation and Sargent, Williams, and Zha (2006). Hence, the time $ t$ solution to the dynamic programming problem is found by using the Philips curve estimate $ \boldsymbol{\alpha}_{t-1\vert t-1}$ as the law of motion for all $ j\geq0$; the time $ t+1$ plugs in the estimate $ \boldsymbol{\alpha}_{t\vert t}$; and so forth.

Without Data Uncertainty  If we assume the Federal Reserve ignores data measurement error, and therefore has no data uncertainty, then the Federal Reserve's Philips curve estimates are the solution to a linear filtering problem:10

$\displaystyle \boldsymbol{a}_{t+1\vert t}$ $\displaystyle = \boldsymbol{a}_{t\vert t-1} + \frac{\boldsymbol{P} _{t\vert t-1}\boldsymbol{\Phi}_{t}(\tilde{u}_{t} - \boldsymbol{\Phi}^{\prime}_{t} \boldsymbol{a}_{t\vert t-1})}{\zeta_{2}^{2} + \boldsymbol{\Phi}^{\prime}_{t} \boldsymbol{P}_{t\vert t-1} \boldsymbol{\Phi}_{t}}$ (2.5)
$\displaystyle \boldsymbol{P}_{t+1\vert t}$ $\displaystyle = \boldsymbol{P}_{t\vert t-1} - \frac{\boldsymbol{P} _{t\vert t-1} \boldsymbol{\Phi}_{t} \boldsymbol{\Phi}^{\prime}_{t} \boldsymbol{P} _{t\vert t-1}}{\zeta_{2}^{2} + \boldsymbol{\Phi}^{\prime}_{t} \boldsymbol{P} _{t\vert t-1} \boldsymbol{\Phi}_{t}} + \boldsymbol{Z}$ (2.6)

where $ (\epsilon_{2t},\mbs{\epsilon}_{3t}^{\prime})^{\prime}\sim iid(\mbs{0},\mbs{I})$ . The information actually available to the Federal Reserve at time $ t$ is $ \{\tilde{u}_{t},\tilde{i}_{t},\tilde{u}_{t-1} ,\tilde{i}_{t-1},\ldots\}$ , which are the available real-time data on inflation and unemployment rates (cf. ?). When the Federal Reserve ignores measurement error, it assumes that the real-time data are the true values of the economic variables; that is, $ \tilde{u}_{t}=u_{t}$ and $ \tilde{i}_{t}=i_{t}$.

Including Data Uncertainty  On the other hand, there is evidence that data, particularly real-time, is subject to measurement error. Macroeconomic data get revised (cf. Croushore and Stark (2001)), and data collection agencies document and publicly analyze collection errors. It is reasonable to suppose a well-informed policy-maker, such as the Federal Reserve, associates some uncertainty to the observed real-time data.

This modification makes latent both the true values of economic variables and the true values of economic model parameters. Therefore the state-space will be nonlinear in the state variables due to the Philips curve, where latent data multiply latent parameters. This nonlinearity in the transition equation for the optimal control problem spreads also to the optimal policy rule equation delivering the Federal Reserve's inflation control variable $ x_{t-1}$ (which, due to the unpredictability of $ \epsilon_{1t}$, is also the forecast of $ \pi_{t}$). I choose to put both of these nonlinear equations in the state equation and leave the observation equation linear.11 The interesting state equations are

$\displaystyle \pi_{t}$ $\displaystyle = x_{t-1}(\boldsymbol{\alpha}_{t-1},\pi_{t-1},u_{t-1},\pi _{t-2},u_{t-2}) + \zeta_{1} \epsilon_{1t}$ (2.7)
$\displaystyle u_{t}$ $\displaystyle = \boldsymbol{\alpha}^{\prime}_{t-1} \boldsymbol{\Phi}_{t} + \zeta_{2} \epsilon_{2t}$ (2.8)
$\displaystyle \boldsymbol{\alpha}_{t}$ $\displaystyle = \boldsymbol{\alpha}_{t-1} + \mbs{Z}^{-\frac{1} {2}}\boldsymbol{\epsilon}_{3t}.$ (2.9)

(2.7), (2.8), and (2.9) repeat (2.2), (2.3), and (2.4), respectively, and $ x_{t-1}$ is written explicitly as the optimal policy function depending on latent state variables.

Turning to the observation equations, we have

$\displaystyle \tilde{\pi}_{t} = \pi_{t} + \zeta_{4}\varepsilon_{4 t}$ (2.10)
$\displaystyle \tilde{u}_{t} = u_{t} + \zeta_{5}\varepsilon_{5t}.$ (2.11)

where \begin{displaymath}\left( \begin{array}[c]{c} \varepsilon_{4 t}\ \varepsilon_{5t} \end{array}\right) \sim iid\left( \left[ \begin{array}[c]{c} \mu_{4}\ \mu_{5} \end{array}\right] , \left[ \begin{array}[c]{cc} \zeta_{4}^{2} & 0\ 0 & \zeta_{5}^{2} \end{array}\right] \right) \end{displaymath} , a distribution that is calibrated to evidence presented in Section 3.3.

3.  Empirical Results

In this section we estimate the two frameworks described above. The results without data uncertainty in Section 3.2 are similar to the Sargent, Williams, and Zha's (2006) results. I describe three problems that emerge from these results and propose to address them by modifying the model so as to allow the Federal Reserve to explicitly account for data measurement error. Then Section 3.3 calibrates the amount of data uncertainty to existing evidence and presents estimated results that ameliorates the three fore mentioned problems. The final section discusses how the problems were ameliorated.

3.1  Estimation

The Extended Kalman filter approximates the state space model using a Taylor-expansion about the linear prediction of the state, as suggested by Anderson and Moore (1979) and following Tanizaki (1996). I have found little difference in practice between the first-order and second-order expansions and use the former to computational simplicity. In the interest of exposition, discussion of the Extended Kalman filter and the likelihood is put in Appendix A.I.

The parameter estimated is

$\displaystyle \boldsymbol{\Psi} \equiv\left( \zeta_{1}^{-1},\zeta_{2}^{-1},\text{vech} \left( \text{Chol}\left( \boldsymbol{Z}\right) \right) ^{\prime },\text{vech}\left( \text{Chol}\left( \boldsymbol{P}_{1\vert}\right) \right) ^{\prime},\right) ^{\prime} $

where Chol$ (\cdot)$ is the Cholesky factor of positive definite matrix.12 Because of $ \boldsymbol{\Psi}$'s large dimension, I follow Sargent, Williams, and Zha (2006) and use a Bayesian empirical method discussed in Appendix A.III: a Markov-Chain Monte Carlo algorithm using the Metropolis-Hastings algorithm with random walk proposals to draw from the posterior distribution13

$\displaystyle p(\boldsymbol{\Psi}\vert\mathcal{Y}_{T}) \propto\mathcal{L}(\mathcal{Y} _{T}\vert\boldsymbol{\Psi})p(\boldsymbol{\Psi})$ (3.1)

where $ p(\boldsymbol{\Psi})$ is the prior and $ \mathcal{L}(\mathcal{Y} _{T}\vert\boldsymbol{\Psi})$ is the likelihood (in Appendix A.I). From the simulated posterior distribution I report medians as my point estimates and quantiles as probability intervals for the parameters. Following Sargent, Williams, and Zha (2006), $ \boldsymbol{a} _{1\vert}$ is set to a regression estimate from presample data.

Here we must note a distinction between the framework with and without data uncertainty. Without data uncertainty, the parameters $ \zeta_{2} ,\boldsymbol{Z},$ and $ \boldsymbol{P}_{1\vert}$ can be scaled together with no effect on the likelihood - there is an unidentification problem. Sargent, Williams, and Zha (2006) note this problem and overcome it by assuming that $ \zeta_{2}$ is one-tenth the size of the standard deviation of a structural equation for the unemployment rate that they additionally estimate.14 Because this structural equation has nothing to do with the Federal Reserve's belief-generating mechanism, I refrain from specifying it at all, just as Carboni and Ellison (forthcoming) refrain during the analysis of their third and fourth sections.

Table 1: Parameter Estimates without Data Uncertainty - $ \boldsymbol{Z}$: Standard Deviations and Correlations

0.2959 -0.9854 0.1539 0.8839 -0.1088 -0.2848
  0.2796 -0.0762 -0.8792 0.1675 0.3918
    0.1611 0.3821 0.8009 0.7949
      0.1901 0.2922 -0.0387
        0.3224 0.7471
          5.0793

$   			\zeta_{1}$:0.21, (0.20, 0.22)

$ \zeta_{2}$: 0.23

Notes on table 1: median of posterior distribution. $ \zeta_1$ is the standard deviation of $ \epsilon_{1t}$, the additive shock to the Federal Reserve's inflation control; $ \zeta_2$ is the standard deviation of the additive shock in the Federal Reserve's Philips curve, and is imposed as the value estimated in the model including data uncertainty (Table 2). 95% probability intervals in parentheses. The bottom array is comprised as follows: the main diagonal are the square roots of the main diagonal of $ \mmm{Z}$; the off-diagonal elements are the correlations derived from $ \mmm{Z}$; $ \mmm{Z}$ is the covariance matrix of the $ \mmm{\epsilon}_{3t}$ shock to the time-varying Philips curve parameters $ \mmm{\alpha}_t$. The vector $ \mmm{\Phi}_t = (\pi_t, \pi_{t-1}, u_{t-1}, \pi_{t-2}, u_{t-2}, 1)'$ multiplies $ \mmm{\alpha}_t$.

Instead, we note that in the framework including data uncertainty $ \zeta_{2}$ is separately identified from $ \boldsymbol{Z}$ and $ \boldsymbol{P}_{1\vert}$. This identification follows from the fact that changes in $ \boldsymbol{Z}$ and $ \boldsymbol{P}_{1\vert}$ affect the optimal control policy rule while changes in $ \zeta_{2}$ have no such effect. The inflation equation (2.7) will therefore respond differently to $ \boldsymbol{Z}$ and $ \boldsymbol{P}_{1\vert}$ than to $ \zeta_{2}$, and therefore the likelihood will respond differently, and thus $ \boldsymbol{Z}$ and $ \boldsymbol{P}_{1\vert}$ are separately identified. Of course, since the effect of $ \boldsymbol{P}_{1\vert}$ dies out rapidly, the evolution of beliefs is almost solely directed by the estimate of $ \boldsymbol{Z}$, which allows for identification between $ \boldsymbol{P}_{1\vert}$ and $ \boldsymbol{Z}$. Therefore, I impose that the value for $ \zeta_{2}$ in the framework without data uncertainty is equal to the estimate from the framework including data uncertainty. Moreover, I place the estimates of $ \mbs{P}_{1\vert}$ in Appendix A.I because this parameter is far less important to the story and point of the paper than are $ \mbs{V}$, $ \zeta_{1}$, and $ \zeta_{2}$.

3.2  Without Data Uncertainty

Data on inflation and the civilian unemployment rate for ages 16 and older comes from the ALFRED, a real-time data archive established by the St. Louis Federal Reserve Bank. Table 1 displays the estimates for the framework without data uncertainty as the posterior median of 700,000 MCMC iterations from two separate runs of 400,000 with different initial conditions where the first 50,000 of each run is burned.15 The estimates in Table 1, and the following figures, are qualitatively similar to those of Sargent, Williams, and Zha (2006). However, they are not identical since I use real-time data and calibrate $ \zeta_{2}$ to a different value.

Figure 1: Real-Time Inflation vs. Federal Reserve Control, Without Data Uncertainty

Figure 1: CPI inflation and model predicted Fed inflation control. NBER recessions shaded. The prediction errors are quite small.

Figure 1 shows the predicted inflation control choices. Figure 1 shows the Federal Reserve choosing to set inflation high in the two high-inflation episodes of the mid 1970s and early 1980s. With $ \zeta_{1}$ estimated at about 1/5 the Federal Reserve believes that it has rather tight control of inflation.

The Philips curve beliefs $ a_{t-1\vert t-1}$ are used to forecast the next month's unemployment rate for any inflation control setting. According to the model, the Federal Reserve sets inflation with this forecast in mind. Therefore an important aspect of the model-predicted Federal Reserve beliefs are what they deliver in terms of unemployment forecasts: these are plotted in Figure 2. These forecasts have a bias of -2.13 percentage points and a root mean squared error of 2.71 percentage points. Both the bias and the volatility are due to the estimated Philips curve parameters. A volatile estimated evolution for the natural rate of unemployment, plotted in Figure 3, accounts for the large fluctuations.16 Meanwhile, the bias is due to a persistent and large estimated Philips curve trade-off, a point to which I return in Section 3.4.17

Figure 2: Real-Time Unemployment vs. Federal Reserve Forecasts, Without Data Uncertainty

Figure 2: Step-ahead unemployment forecasts come from the Philips curve using the Fed ’s inflation setting and actual unemployment and inflation data. NBER recessions shaded. The predicted forecast errors are quite large.

Notes: Step-Ahead unemployment forecasts come from Phillips curve (2.3) using the Federal Reserve's inflation setting and real-time unemployment and inflation data. NBER recessions shaded.

Figure 3: Natural Rate of Unemployment, Without Data Uncertainty

Figure 3: The rate is quite volatile, attaining values of 60% in the mid 1970s, and dropping to 10% by the late 1980s.

Notes: Estimated natural rate of unemployment from the Philips curve (2.3) using the Federal Reserve's inflation setting and real-time unemployment and inflation data. NBER recessions shaded.

These unemployment rate forecasts should explain actual Federal Reserve unemployment rate forecasts because this framework explains high inflation as Federal Reserve policy that attempts to bring down unemployment. Therefore, it is informative to compare the predicted forecasts to actual Greenbook forecasts over this time span.18 A simple test of similarity between these forecasts is Diebold and Mariano's (1995). Their statistic $ S_{1}$ is a two-sided test of the null hypothesis that the predicted unemployment forecasts have accuracy equal to the Greenbook forecasts and $ S_{1}$ has an asymptotic standard normal distribution. Here $ \vert\hat{S}_{1}\vert = 2.7834$, so we can reject the null of equal forecasting accuracy at the 99% level. This suggests that the predicted forecasts are unlike the Greenbook forecasts they should mimic.

The large unemployment forecast errors drive up the estimated standard deviation of the time-varying Philips curve. Notice that the estimated $ Z^{(6,6)}$ implies that the Federal Reserve believes that every month the constant parameter is very volatile. Since this parameter drives the natural rate of unemployment, this result can be interpreted as saying the Federal Reserve believed there was on any given month about a 30% chance that the natural rate of unemployment would jump by 5 percentage points!19 This is borne out in the volatility of the Federal Reserve's estimated natural rate of inflation plotted in Figure 3. In addition, the estimated natural rate of unemployment shoots up to unreasonable values during the early 1970s as inflation rose, the Federal Reserve tried to exploit a large perceived Philips trade-off, but unemployment continued to rise. In other words, the Federal Reserve has several reasons to disbelieve this Philips curve model.

These results cast doubt on the economic story that the Federal Reserve's Philips curve beliefs motivated high inflation. The estimates imply the Federal Reserve perceived its Philips curve model as unstable, and the model delivers implausible estimates of the natural rate of unemployment and poor forecasts. Given the large estimate of $ \boldsymbol{Z}$ and $ \zeta_{2}$ and the poor forecasting performance, it seems implausible that the Federal Reserve would have tried to exploit a Philips curve trade-off by letting inflation achieve the heights it did. Moreover, even if the Federal Reserve did believe its estimated Philips curve enough to use it as a basis for policy, we will see below that this framework's estimated Philips curve trade-off does not explain inflation's rise and fall.

3.3  Including Data Uncertainty

I propose to modify the framework by allowing the Federal Reserve to explicitly account for measurement error in the data it observes, and argue that so doing is natural for two reasons. First and most importantly, we have data on measurement errors. Much macroeconomic data is revised and this issue is common knowledge in policy and forecasting discussions (e.g. Cunningham, Jeffery, Keptanios, and Labhard (2007), Pesaran and Timmermann (2005). Secondly, Section 2.1's analysis suggested that unmodeled data uncertainty could increase the estimated volatility of an agent's beliefs, which apparently is a problem for the optimal control framework as it now stands.

I introduce data uncertainty through the measurement errors $ (\varepsilon _{4t},\varepsilon_{5t})^{\prime}$ in (2.10)-(2.11) whose distribution I calibrate. To do this, I look at the revision between the first-reported value of the data and the recent vintage (circa 2008). Making the assumption - ubiquitous in macroeconomics - that the recent data vintage is more accurate than the real-time data, I define the measurement error as the observed revision. These revisions' statistical properties (c.f. Aruoba (2005) calibrate the measurement error stochastic process:

\begin{displaymath} \left( \begin{array}[c]{c} \mu_{4}\ \mu_{5} \end{array}\right) = \left( \begin{array}[c]{c} \-0.005\ 0 \end{array}\right) , \left( \begin{array}[c]{cc} \zeta_{4}^{2} & 0\ 0 & \zeta_{5}^{2} \end{array}\right) = \left( \begin{array}[c]{cc} 0.106^{2} & 0\ 0 & 0.112^{2} \end{array}\right) . \end{displaymath}

Obviously, the amount of data uncertainty is small.

Table 2 reports estimates for the framework including data uncertainty from the posterior median of 700,000 MCMC iterations from two separate runs of 400,000 with different initial conditions where the first 50,000 of each run is burned. Figure 4 shows that the Federal Reserve's inflation control explains the rise and fall of American inflation. Once again the estimate of $ \zeta_{1}$ implies the Federal Reserve believes its inflation control is quite good. Turning now to the Federal Reserve's unemployment rate forecasts in Figure 4, we find a far different picture than in the framework without data uncertainty. The Federal Reserve's forecasts are considerably more accurate than before, with insignificant bias and a RMSE of 0.30 percentage points.20

Again, seeing as the model forecasts are intended to predict the Federal Reserve's actual forecasts, we can directly compare them to Greenbook unemployment rate forecasts. Again using Diebold and Mariano's (1995) test we find $ \vert\hat{S}_{1}\vert = 0.4858$ and we accept the hypothesis that the model forecasts and Greenbook forecasts are equally accurate: by this measure the predicted unemployment rate forecasts are statistically indistinguishable from actual Greenbook forecasts.

Table 2: Parameter Estimates, Including Data Uncertainty - $ \boldsymbol{Z}$: Standard Deviations and Correlations

0.0106 0.9665 -0.9746 -0.8234 0.9520 -0.1226
  0.0211 -0.9894 -0.9395 0.9541 0.0005
    0.0566 0.8950 -0.9870 0.1251
      0.0128 -0.8334 -0.2346
        0.0475 -0.2685
          0.2351
           

$ \zeta_{1}$: 0.22, (0.20, 0.25)

$ \zeta_{2}$: 0.23, (0.21, 0.26)

Notes: median of posterior distribution. $ \zeta_1$ is the standard deviation of $ \epsilon_{1t}$, the additive shock to the Federal Reserve's inflation control; $ \zeta_2$ is the standard deviation of the additive shock in the Federal Reserve's Philips curve. 95% probability intervals in parentheses. The bottom array is comprised as follows: the main diagonal are the square roots of the main diagonal of $ \mmm{Z}$; the off-diagonal elements are the correlations derived from $ \mmm{Z}$; $ \mmm{Z}$ is the covariance matrix of the $ \mmm{\epsilon}_{3t}$ shock to the time-varying Philips curve parameters $ \mmm{\alpha}_t$. The vector $ \mmm{\Phi}_t = (\pi_t, \pi_{t-1}, u_{t-1}, \pi_{t-2}, u_{t-2}, 1)'$ multiplies $ \mmm{\alpha}_t$.

Figure 4: Real-Time CPI Inflation vs. Federal Reserve Control, Including Data Uncertainty

Figure 4: CPI inflation and model predicted Fed inflation control. NBER recessions shaded. The prediction errors are quite small.

Notes: Real-time CPI inflation and model predicted Federal Reserve inflation control. NBER recessions shaded.

The estimate of $ \boldsymbol{Z}$ in Table 2 is much smaller than before. For instance, the estimated $ Z^{(6,6)}$ implies that the Philips curve's intercept has a monthly shock with a standard deviation of about 20 unemployment rate basis points as opposed to the 500 basis points we saw before. Roughly speaking, the shocks are smaller by a factor of 8 for inflation parameters, a factor of 5 for unemployment parameters, and a factor of 20 for the constant parameter. Now the estimated natural rate of unemployment, plotted in Figure 6, is much less volatile and does not shoot off to the ethereal levels it attain before, although still it elevates implausibly high.

Figure 5: Real-Time Unemployment vs. Federal Reserve Forecast, Including Data Uncertainty

Figure 5: Step-ahead unemployment forecasts come from the Philips curve using the Fed’s inflation setting and actual unemployment and inflation data. NBER recessions shaded. The predicted forecast errors are quite small.

Notes: Step-ahead unemployment forecasts come from the Philips curve (2.3) using the Federal Reserve's inflation setting and real-time unemployment and inflation data. NBER recessions shaded.

Figure 6: Natural Rate of Unemployment, Including Data Uncertainty

Figure 6: The rate is smooth, attaining values of 35% around 1981, then falling to 5% by the late 1980s.

Notes: Estimated natural rate of unemployment from the Philips curve (2.3) using the Federal Reserve's inflation setting and estimated values of past values of the unemployment and inflation rates. NBER recessions shaded.

Figure 7: Evolution of the Philips Curve Trade-Off

Figure 7: Top panel: time series of Philips curve inflation response estimates from model without data uncertainty. Bottom panel: time series of Philips curve inflation response estimates from model including data uncertainty. NBER recessions are shaded. The top panel shows a sharp negative decrease around 1973, and then steady trend upward crossing 0 around 1990. The bottom panel shows a trade-off that closely mirrors the low-frequency movement of inflation.

Notes: Top Panel: From framework without data uncertainty, sum of Philips curve inflation coefficient estimates. Bottom Panel: From framework including data uncertainty, sum of Philips curve inflation coefficient estiamtes. NBER recessions are shaded.

Let us turn now to the predicted evolution of the Federal Reserve's beliefs about the Philips curve trade-off, which is the sum of the coefficients on inflation in the Philips curve. As seen in the top panel of Figure 7, the trade-off in the framework without data uncertainty experiences a large jump between 1973 and 1975 which explains the great rise in inflation over those years. Thereafter the trade-off stays high, with no sharp activity around the disinflation of the early 1980s. But this does not bear out the main story, which is that the evolution of the Philips curve trade-off led to the rise and fall of inflation. In the framework without data uncertainty, the dramatic fall of inflation from 14.4 to 2.2% between 1980 and 1984 occurs without any sharp change in the Federal Reserve's beliefs.21

On the other hand, consider the trade-off in the framework including data uncertainty, depicted in the bottom panel of Figure 7. Here the we predict a drastic drop in the Philips curve trade-off starting around 1980. As this perceived trade-off diminishes, inflation falls by than 80% off its peak going into 1984.22 Thus, the framework including data uncertainty describes a consistent connection between the Federal Reserve's inflation control and the Federal Reserve's beliefs about the Philips curve trade-off.

3.4  Discussion

How can such a small amount of data uncertainty change the results? There are two parts to the answer. Firstly, introducing measurement error into the learning framework creates a nonlinearity (recall Section 2.1) that amplifies the modest amount of data uncertainty into a largely biased estimate of model uncertainty. Secondly, the evident data uncertainty is actually a fair bit larger than the Federal Reserve's model uncertainty (particularly the variances of the parameters on inflation and unemployment in the Philips curve) to begin with.

By ignoring the Federal Reserve's data uncertainty, we overestimate the size of the shocks to the Philips curve coefficients on inflation and unemployment. Therefore, these parameters shift a lot from period to period in response to the large unemployment rate forecast errors; hence, there are shifts in the optimal policy rule's dependence on past inflation and unemployment. But inflation over this period is rather persistent: so each period ceteris paribus the current optimal policy should be somewhat near last period's policy in order to fit the data. So the constant in the Philips curve adjusts to offset the change in the policy rule engendered by the shifting Philips curve coefficients on inflation and unemployment.

These measurement errors work differently than do the shocks $ \epsilon _{1t},\epsilon_{2t}$ appearing in the inflation control and Philips curve equation: they make the values of past inflation and unemployment latent. Therefore, their presence helps to explain forecast errors for several periods because past (true) values of inflation and unemployment enter the forecast rule. Therefore the Federal Reserve is more sluggish to change its Philips curve estimate on the basis of the observed forecast errors, inferring that some of this forecast error may be due to measurement error in past values of inflation and unemployment on which the forecast is based. The important characteristic of data uncertainty is that these past economic variables remain latent to the agent for some time.

These results highlight two main points in favor of explicitly including data uncertainty in economic frameworks that specify agents' model uncertainty. Firstly, the nonlinearity of a framework with learning amplifies small amounts of data uncertainty. By explicitly accounting for the data uncertainty, we obtain an intuitive message that is similar to Brainard's (1967) point that a policy-maker's model uncertainty leads to an optimal "dampening'' of her policies: here the point is that a policy-maker's data uncertainty leads to an optimal "dampening'' of her learning. As we have just seen, an economic researcher may make substantially different inferences depending on whether or not data uncertainty is accounted for, even if the amount of uncertainty they would ignore is small.

Second and perhaps more importantly, we are able to discipline this "dampening'' with the facts. We have evidence, one form of which are data revisions, to point to when discussing data uncertainty and ascertaining how large it may be. The role of data uncertainty can be disciplined by observables, and this makes it a potentially constructive part of learning frameworks. In fact, this quality positively distinguishes the concept of data uncertainty from the concept of model uncertainty, which has little if any such evidence to discipline its role.23

4.  Conclusion

This paper analyzes the effects of data uncertainty in frameworks with agents who learn. Ignoring data uncertainty can bias estimates of agents' model uncertainty. I investigate the importance of this point by extending the framework of Sargent, Williams, and Zha (2006) and showing that the explicit modeling of data uncertainty remedies some well-known issues with that paper's results. The mechanism by which data uncertainty matters is through introducing sluggishness into the Federal Reserve's learning. Once this is the case, the framework predicts that the inflation of the 1970s and 1980s can be explained by evolving beliefs about the Philips curve trade-off between inflation and unemployment.


References

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A  Appendix

Sargent, Williams, and Zha (2006).

A.I  Extended Kalman Filter and Likelihood

Extended Kalman Filter  We first discuss the state space model more generally, and then relate this algebra to the notation used in the paper. Let $ \mathbb{E}(\boldsymbol{\beta }_{t}\vert\mathcal{Y}_{s}) \equiv\boldsymbol{b}_{t\vert s}$ and $ \mathbb{V}ar\left( \boldsymbol{\beta}_{t}\vert\mathcal{Y}_{s}\right) \equiv\Sigma_{t\vert s}$ for $ \mathcal{Y}_{s} \equiv\sigma(\boldsymbol{y}_{s}, \boldsymbol{y}_{s-1} ,\ldots)$ . The "expansion" about $ (\boldsymbol{\beta}_{t},\boldsymbol{\varepsilon}_{t}) = (\boldsymbol{b} _{t\vert t-1},0)$ is in fact exact:

$\displaystyle \boldsymbol{h}_{t}(\boldsymbol{\beta}_{t},\boldsymbol{\varepsilon}_{t}) = \boldsymbol{H} \boldsymbol{b}_{t\vert t-1} + \boldsymbol{H}(\boldsymbol{\beta} _{t}-\boldsymbol{b}_{t\vert t-1}) + \boldsymbol{\varepsilon}_{t}$ (A.1)

The "expansion" about $ (\boldsymbol{\beta} _{t-1},\boldsymbol{\eta}_{t}) = (\boldsymbol{b}_{t-1\vert t-1},0)$ is approximate:

$\displaystyle \boldsymbol{g}_{t}(\boldsymbol{\beta}_{t-1},\boldsymbol{\eta}_{t}) \approx\boldsymbol{g}_{t\vert t-1} + \boldsymbol{T}_{t\vert t-1}(\boldsymbol{\beta}_{t} - \boldsymbol{b}_{t-1\vert t-1}) + \boldsymbol{R}_{t\vert t-1} \boldsymbol{\eta} _{t}$ (A.2)

where

$\displaystyle \boldsymbol{g}_{t\vert t-1}$ $\displaystyle = \boldsymbol{g}_{t}(\boldsymbol{b}_{t-1\vert t-1},0)$    
$\displaystyle \boldsymbol{T}_{t\vert t-1}$ $\displaystyle = \frac{\partial\boldsymbol{g}_{t} (\boldsymbol{\beta}_{t-1},\boldsymbol{\eta}_{t})}{\partial\boldsymbol{\beta }_{t-1}^{\prime} }\bigg\vert _{(\boldsymbol{b}_{t-1\vert t-1},0)}$    
$\displaystyle \boldsymbol{R}_{t\vert t-1}$ $\displaystyle = \frac{\partial\boldsymbol{g}_{t} (\boldsymbol{\beta}_{t-1},\boldsymbol{\eta}_{t})}{\partial\boldsymbol{\eta }_{t}^{\prime} }\bigg\vert _{(\boldsymbol{b}_{t-1\vert t-1},0)}$    

I motivate the derivation of optimal prediction and updating for the approximating system (A.1) and (A.2) by assuming Gaussian shocks, as in Howrey (1987), Watson and Engle (1983), and Harvey (1989).24 In this case, the relevant conditional expectations have the known forms given below. In particular, I assume

$\displaystyle \boldsymbol{\eta}_{t} \sim iid \: \mathcal{N}(\boldsymbol{0},\boldsymbol{Q}), \quad\quad\boldsymbol{\varepsilon}_{t} \sim iid \: \mathcal{N}(\boldsymbol{0} ,\boldsymbol{N}), \quad\quad\boldsymbol{\eta}_{t} \bot\boldsymbol{\varepsilon }_{\tau}, \forall t,\tau $

The Extended Kalman Filtering equations are

$\displaystyle \boldsymbol{b}_{t\vert t-1}$ $\displaystyle = \boldsymbol{g}_{t\vert t-1}$  
$\displaystyle \boldsymbol{\Sigma}_{t\vert t-1}$ $\displaystyle = \boldsymbol{T}_{t\vert t-1} \boldsymbol{\Sigma }_{t-1\vert t-1} \boldsymbol{T}_{t\vert t-1}^{\prime}+ \boldsymbol{R}_{t\vert t-1} \boldsymbol{Q} \boldsymbol{R}_{t\vert t-1}^{\prime}$    
$\displaystyle \boldsymbol{y}_{t\vert t-1}$ $\displaystyle = \boldsymbol{H} \boldsymbol{b}_{t\vert t-1}$  
$\displaystyle \boldsymbol{F}_{t\vert t-1}$ $\displaystyle = \boldsymbol{H} \boldsymbol{\Sigma}_{t\vert t-1} \boldsymbol{H}^{\prime}+ \boldsymbol{N}$ (A.3)
$\displaystyle \boldsymbol{M}_{t\vert t-1}$ $\displaystyle = \boldsymbol{H} \boldsymbol{\Sigma}_{t\vert t-1}$    
$\displaystyle \boldsymbol{K}_{t}$ $\displaystyle = \boldsymbol{M}_{t\vert t-1}^{\prime}\boldsymbol{F} _{t\vert t-1}^{-1}$    
$\displaystyle \boldsymbol{\Sigma}_{t\vert t}$ $\displaystyle = \boldsymbol{\Sigma}_{t\vert t-1} - \boldsymbol{K} _{t} \boldsymbol{F}_{t\vert t-1} \boldsymbol{K}_{t}^{\prime}$    
$\displaystyle \boldsymbol{b}_{t\vert t}$ $\displaystyle = \boldsymbol{b}_{t\vert t-1} + \boldsymbol{K}_{t} (\boldsymbol{y}_{t} - \boldsymbol{y}_{t\vert t-1} )$  

Conditional on the data $ \mathcal{Y}_{T}$, parameters $ \{\boldsymbol{G} ,\boldsymbol{H},\boldsymbol{Q},\boldsymbol{N}\}$ , and initial conditions $ \{\boldsymbol{b}_{1\vert},\boldsymbol{\Sigma}_{1\vert}\}$, the sequences of left hand side variables (A.3)-(A.4) are found by matrix multiplication.

Table A.1.a: Remaining Parameter Estimates: Framework Without Data Uncertainty - P1|0: Standard Deviations and Correlations

0.3273 0.9875 0.9187 -0.9953 -0.6626 -0.9959
  0.4412 0.9237 -0.9978 -0.6624 -0.9972
    0.0748 -0.9182 -0.9134 -0.9191
      0.7743

0.6545

0.9996
        0.0424 0.6515
          0.3111

Table A.1.b: Remaining Parameter Estimates: Framework Including Data Uncertainty - P1|0: Standard Deviations and Correlations

0.3273 0.9875 0.9187 -0.9953 -0.6626 -0.9959
  0.4412 0.9237 -0.9978 -0.6624 -0.9972
    0.0748 -0.9182 -0.9134 -0.9191
      0.7743 0.6545 0.9996
        0.0424 0.6515
          0.3111

Notes: The arrays are comprised as follows: The main diagonal is the square roots of the main diagonal of P1|0; the off-diagonal elements are the correlations derived from P1|0; P1|0 is the Federal Reserve's initial step-ahead uncertainty over the initial Philips curve parameter estimate a1|0.

Referring back to the paper's notation,

$\displaystyle \mbs{\beta}_{t}$ $\displaystyle = (\mbs{\alpha}_{t-1}^{\prime},\pi_{t},u_{t},\pi _{t-1},u_{t-1})^{\prime}$
$\displaystyle \mbs{\Sigma}_{t}$ $\displaystyle =$   blkdiag$\displaystyle (\mbs{P}_{t},\zeta_{4},\zeta_{5} ,\zeta_{4},\zeta_{5})$
$\displaystyle \mbs{Q}$ $\displaystyle =$   blkdiag$\displaystyle (\mbs{V},\zeta_{1},\zeta_{2},0,0)$
$\displaystyle \mbs{N}$ $\displaystyle =$   blkdiag$\displaystyle (\zeta_{4},\zeta_{5})$

Likelihood  The likelihood is

$\displaystyle \mathcal{L}(\mathcal{Y}_{T}\vert\boldsymbol{\Psi}) = \prod^{T}_{t=1} \left\vert \boldsymbol{F}_{t\vert t-1} \right\vert ^{-1/2} \exp\left\{ -\frac{1}{2} (\boldsymbol{y} _{t} - \boldsymbol{y}_{t\vert t-1})^{\prime}\boldsymbol{F} _{t\vert t-1}^{-1} (\boldsymbol{y}_{t} - \boldsymbol{y}_{t\vert t-1})\right\} $

where the $ \boldsymbol{F}_{t\vert t-1}$ come from (A.3).

The estimates of $ \mbs{P}_{1\vert}$ from both models are displayed in Table A.1.

A.II  Greenbook Forecasts

The main issue with comparing the model predicted unemployment forecasts to Greenbook forecasts is the difference in the frequency of observation. The model forecasts the monthly unemployment rate one month into the future. The Greenbook forecasts quarterly unemployment rates and are released without rigid frequency. For example, there are Greenbook forecasts published monthly through the 1970s, but into the 1980s and 1990s these forecasts are published almost at a bimonthly frequency. I take the following steps to make the comparison.

First, I form a quarterly unemployment rate series as the average of unemployment rate for the three underlying months. It is against these series that the forecasts produce forecast errors.

Second, I form a quarterly model-forecast series as the average of the step-ahead forecasts for the three underlying months. That is, the model's quarterly unemployment forecast for quarter $ q$ composed of months $ m_{1},m_{2},m_{3}$ is

$\displaystyle \frac{1}{3}(u_{m_{1}\vert m_{1}-1} + u_{m_{2}\vert m_{1}} + u_{m_{3}\vert m_{2}}) $

where $ u_{j\vert j-1}$ is the forecast made at time $ j-1$ pertaining to time $ j$

Third, I form the quarterly Greenbook-forecast series as an average of all the forecasts made the month before or anytime during a quarter. That is, the Greenbook quarterly forecast for $ q$ composed of months $ m_{1},m_{2},m_{3}$ and immediately preceded by month $ m_{0}$ is

$\displaystyle \frac{1}{n_{obs}}(gb_{m_{0}} + gb_{m_{1}} + gb_{m_{2}} + gb_{m_{3}}) $

where $ gb_{j}$ is the Greenbook forecast for quarter $ q$ published in month $ j$. It should be noted that all four of these forecasts do not exist for every quarter, in which case only those observed are summed and $ n_{obs}$ adjusts to however many forecasts are observed.

The Diebold and Mariano (1995) statistic $ S_{1}$ takes forecast error series $ \{e_{it}\}$ and $ \{e_{jt}\}$

$\displaystyle S_{1} = \frac{\overline{d}}{\sqrt{\frac{2\pi\widehat{f_{d}(0)}}{T}}} $

where

$\displaystyle \overline{d} = \frac{1}{T}\sum_{t=1}^{T} \left( e_{it}-e_{jt}\right) $

and I take $ \widehat{f_{d}(0)}$ to be Andrews (1991) quadratic-spectral HAC estimator. The errors under consideration run from 1970 through 1995 so that $ T=104$.

A.III  MCMC Implementation and Robustness

Priors  The prior for $ \boldsymbol{\Psi}$ is multivariate normal with a non-zero mean and a diagonal covariance matrix - so equivalently, the priors for each parameter are independent normals. The exact specifications are listed below where $ \boldsymbol{\varphi} \equiv\left( \text{vech}\left( \text{Chol} \left( \boldsymbol{Z}\right) \right) ^{\prime},\text{vech}\left( \text{Chol}\left( \boldsymbol{P}_{1\vert}\right) \right) \right) ^{\prime}$ following the notation of Sargent, Williams, and Zha (2006):

Convergence of the MCMC  To address the convergence of the MCMC algorithm to its posterior distribution, I computed the number of iterations required to estimate the 0.025 quantile with a precision of 0.02 and probability level of 0.950 using the method of rafteryle92. For each chain (with different initial conditions) the max of these across $ \boldsymbol{\Psi}$ was below the $ 3.5E5$ iterations taken from each chain, suggesting that mixing the two chains (after burn-in) yields satisfactory precision.

Metropolis Algorithm  An important part of the MCMC algorithm sampling from the posterior in a reasonable number of iterations is the covariance matrix of the proposal random step in the Metropolis algorithm. The Metropolis algorithm is

1.  Given $ \boldsymbol{\Psi}^{\text{previous}}$, propose a new value

$\displaystyle \boldsymbol{\Psi}^{\text{proposal}} = \boldsymbol{\Psi}^{\text{previous}} + \boldsymbol{\xi} $

where $ \boldsymbol{\xi}$ is normal with mean zero and covariance matrix $ c\boldsymbol{\Sigma}_{\xi}$

2.  Compute

$\displaystyle q = \min\left\{ \frac{p\left( \boldsymbol{\Psi}^{\text{proposal}} \vert\mathcal{Y}_{T}\right) }{p\left( \boldsymbol{\Psi}^{\text{previous} }\vert\mathcal{Y}_{T}\right) },1 \right\} $

3.  Randomly draw $ w \sim U(0,1)$

4.  If $ w\leq q$, accept $ \boldsymbol{\Psi}^{\text{proposal}}$ as current draw; otherwise, set $ \boldsymbol{\Psi}^{\text{previous}}$ as the current draw

Given the manner in which all parameters affect the optimal policy, I arrived at this proposal covariance matrix $ \boldsymbol{\Sigma}_{\xi}$ by doing the following. Using the covariance matrix for $ \boldsymbol{\varphi}$ numerically solved for as described in Sargent, Williams, and Zha (2006)'s Appendix D and the prior covariance terms for all other elements of $ \boldsymbol{\Psi}$ given above, the MCMC was started. For tens of thousands of iterations based on one initial condition, I considered only elements of the MCMC chain where a proposal had been accepted. From these chain elements I calculated the sample covariance matrix of the successful proposal shocks and set $ \boldsymbol{\Sigma}_{\xi}$ equal to this. I tried different initial conditions and took the weighted average of the Cholesky factors of these sample covariance matrices. The tuning parameter $ c$ was adjusted to achieve an acceptance rate of around 25-35% during the first 20,000 iterations: after this, it was unadjusted, as continual chain-dependent adjustment of Metropolis step-size can negate the ergodicity upon which MCMC methods are based (see Robert and Casella (2004)).


Footnotes

1.  First version: February 2007. I thank Jim Hamilton for his insightful advice. Thanks also to Alina Barnett, Gray Calhoun, Martin Ellison, Margie Flavin, Nir Jaimovich, Garey Ramey, Valerie Ramey, Chris Sims, Allan Timmermann, Harald Uhlig, Chris Wignall, and Tao Zha for helpful discussions; to an anonymous referee for extensive comments; and to seminar participants at UCSD, Georgetown, HEC Montreal, Texas A&M, Penn State, UCSC, Cambridge University Learning and Macroeconomic Policy Conference, the Philadelphia, Kansas City, and Atlanta Federal Reserve Banks, and the Federal Reserve Board. Return to text

2.  Division of International Finance, mail to: seth.j.pruitt@frb.gov. This paper reflects the views and opinions of the author and does not reflect on the views or opinions of the Federal Reserve Board, the Federal Reserve System, or their respective staffs. Return to text

3.  Orphanides and Williams (2006) and Sims (2006) also argue that policy is sensitive to model uncertainty, and Nason and Smith (forthcoming) finds instability in the Philips curve relationship since the 1950s. Return to text

4.  In contrast, this paper produces unemployment forecasts that are statistically indistinguishable from the Greenbook forecasts without using those Greenbook forecasts as data. Return to text

5.  In fact, data uncertainty was recognized by Burns and Mitchell, who revised their business cycle indicators as data revisions came in, and considered many macroeconomic variables for that reason. Return to text

6.  Recent related work is in Aruoba (2004) who analyzes the welfare effects of more accurate data collection, and Aruoba (2005) who analyzed the statistical features of data revision processes. Return to text

7.  This naturally follows from usual assumptions: the agent has an economic loss function, the agent's prediction is rational with respect to that loss function, and the loss function is minimized when the prediction error is zero. See Elliott, Komunjer, and Timmerman (2005) for more extensive discussion. Return to text

8.  That is $ \mexp(\mbs{\epsilon\varepsilon}^{\prime})=\mbs{0}$, etc. Return to text

9.  The value $ c$ is the sum of expectations of fourth-order products of the elements of $ \mbs{\epsilon}$ and $ \mbs{\varepsilon}$. Kan's (2008) Proposition 1 ensures that this $ c$ is positive, as well as that expectation of the terms involving $ \upsilon$ drop out. Return to text

10.  Given initial conditions $ \boldsymbol{a}_{1\vert}$ and $ \boldsymbol{P}_{1\vert}$Return to text

11.  One could rewrite the state space in other equivalent ways. Return to text

12.  I estimate the reciprocals of $ \zeta_{1},\zeta_{2}$ because it is easy to draw them as normals and avoid nonnegativity constraints. Return to text

13.  I must use a accept/reject simulation technique because, due to the effects of $ \boldsymbol{\Psi}$ on the whole sequence of forecasts, the form of (3.1) is not known. Further details are in Appendix A.III. Return to text

14.  This has the effect of lowering the estimated values for $ \boldsymbol{Z}$ and $ \boldsymbol{P}_{1\vert}$. We will see later that high values of $ \boldsymbol{Z}$ imply a difficult economic story (namely that the Federal Reserve's model of the economy was extremely unstable), and so to some extent the calibration of $ \zeta_{2}$ to one-tenth (as opposed to, say, one-half or two times) is an assumption that drives favorable results for Sargent, Williams, and Zha (2006). Return to text

15.  Sargent, Williams, and Zha's (2006) results come from a sequence of 50,000 draws with an unspecified burn-in interval. Return to text

16.  The natural rate at time $ t$ is the steady-state rate of unemployment with inflation set at target, defined as $ \left( \alpha^{(6)}_{t-1\vert t-1}+2[\alpha^{(1)}_{t-1\vert t-1}+\alpha^{(2)} _{t-1\vert t-1}+\alpha^{(4)}_{t-1\vert t-1}]\right) / \left( 1-\alpha^{(3)} _{t-1\vert t-1}-\alpha^{(5)}_{t-1\vert t-1}\right) $ , following Sargent, Williams, and Zha (2006) and Carboni and Ellison (forthcoming). The natural rate essentially follows the estimated constant parameter: a plot of the constant parameter time series is available from the author upon request. Return to text

17.  See Figure 7  Return to text

18.  Appendix A.II discusses these forecasts and provides more details on the test statistic. Return to text

19.  The probability that a normal random variable is at least one standard deviation away from its mean is around 30%. Return to text

20.  The other measurement equation forecasts are pictured in Appendix A.III. Return to text

21.  Sargent, Williams, and Zha (2006) and Carboni and Ellison's (forthcoming) estimated trade-offs are similar to each other but a little different from the one here. Theirs' experiences the large jump between 1973 and 1975, but no sharp fall between 1980 and 1984. Return to text

22.  One manner of seeing this is to compute the coherence between the Philips curve trade-off and inflation at low-frequencies. This coherence is $ -0.75$ for fluctuations with a period between 1 and 10 years. For the model without uncertainty this coherence is $ -0.10$Return to text

23.  This last point is disputable. Research on disagreement between policy-makers, and related evidence, may be useful in this regard (c.f. Romer (2009)). Return to text

24.  Technically, I must assume that the vector $ \boldsymbol{\eta}_{t}$ appearing in the first-order expansion iis Gaussian; assuming a Gaussian $ \boldsymbol{\eta}_{t}$ for the general nonlinear case does not assure that the shock in the first-order expansion would be Gaussian. Return to text



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