Finance and Economics Discussion Series: 2014-59 Screen Reader version ♣

# Precautionary Volatility and Asset Prices

Andrew Y. Chen
Federal Reserve Board
August 12, 2014*

Abstract:

Many theories of asset prices assume time-varying uncertainty in order to generate time-varying risk premia. This paper generates time-varying uncertainty endogenously, through precautionary saving dynamics. Precautionary motives prescribe that, in bad times, next period's consumption should be very sensitive to news. This time-varying sensitivity results in time-varying consumption volatility. Production makes this channel visible, and external habit preferences amplify it. An estimated model featuring this channel quantitatively accounts for excess return and dividend predictability regressions. It also matches the first two moments of excess equity returns, the risk-free rate, and the second moments of consumption, output, and investment.
Keywords: Time-varying risk premia, time-varying volatility, habit, precautionary savings, the equity premium puzzle
JEL Classification: G12, E21, E30

# 1 Introduction

Time-varying uncertainty plays a central role in theories of aggregate asset prices. This feature is frequently required in order to capture the nature of stock market volatility: stock market fluctuations have little relationship with movements in expected cash flows but are closely linked with movements in expected excess returns (LeRoy & Porter (1981), Shiller (1981), Cochrane (2011)). Time-varying uncertainty addresses these facts by generating fluctuations in risk which move asset prices, while leaving average cash flows unchanged. This mechanism exists in a broad array of asset pricing models, though the details of the modeling differ. In long-run risk models, time-varying uncertainty comes from the assumption of time-varying consumption volatility (Bansal & Yaron (2004), Bansal et al. (2009)). In ambiguity aversion, it is the assumption of a time-varying magnitude of ambiguity (Sbuelz & Trojani (2008), Bianchi et al. (2012)). In disasters, it is the assumed time-varying severity or probability of disaster (Gabaix (2012), Wachter (2012)). In external habit models, it is the assumption of time-varying habit sensitivity (Campbell & Cochrane (1999)). These modeling approaches provide coherent and tractable descriptions of stock market volatility, but by assuming time-varying uncertainty, they leave unanswered the deeper question: why does uncertainty vary over time?1

This paper provides an answer to the question. I show that time-varying uncertainty is the natural result of a fundamental economic phenomenon: the desire for precautionary savings. The mechanism lies in the dynamics of this motive. Precautionary motives intensify in bad times, making investors hunker down and cut consumption. But in the following month, they may need to hunker down again. In fact, how much they will need to hunker down next month depends sharply on economic news. News that the bad times will get even worse would put them in austerity, leading to a sharp drop in consumption. News that the economy will recover relieves them of their concerns, and they can relax and sharply increase consumption. While they wait for this news, next month's consumption is very uncertain, risk is high, and asset prices are low. This volatility is absent in good times, when investors have a lot of wealth. High wealth shields investors from precautionary concerns and allows them to continue with their plans, regardless of economic news.

I formalize this "precautionary volatility" story in two ways. The first is a simple consumption-savings model which allows for proofs and propositions. The model features a precautionary savings motive, that is, the agent desires savings as protection against uncertainty. This motive leads to a concave consumption function (Carroll & Kimball (1996)), as depicted in Figure 1. Concave consumption means that in bad times, consumption is very sensitive to shocks (dashed lines), while in good times, there is little action (dotted lines). Countercyclical sensitivity then leads to time-varying consumption volatility, even in the presence of homoskedastic shocks. I also examine an amplification mechanism: external habit preferences à la Campbell & Cochrane (1999). Investors with external habit judge their consumption relative to an aggregate reference level. This perspective makes them feel poor, amplifying precautionary motives and the concavity of the consumption function (dash-dot line). These results are presented in a specific context for clarity, though the mechanism applies in a wide variety of settings. The model features a shock to wealth, though the underlying cause of the shock may be a shift in technology, a shift in monetary policy, or even an uncertainty shock in the sense of Bloom (2009). And though the model's precautionary motive comes from prudent preferences (Kimball (1990)), financial constraints would produce similar dynamics (Carroll (2001)). Additionally, analogous effects exist for firm policies under financial constraints (i.e. Almeida et al. (2004)).

#### Figure 1: Precautionary Savings and Time-Varying Volatility.

The bulk of the paper is devoted to the second formalization: an estimated general equilibrium model. The general equilibrium model is a real business cycle model with capital adjustment costs and external habit preferences. Fluctuations originate from a standard AR1 productivity shock. Despite the lack of a heteroskedastic driving process, the model produces time-varying consumption volatility and captures the nature of stock market volatility: the price-dividend ratio has little forecasting power for cash flow growth but is a good predictor of excess returns. This result is due to two key features: real investment and external habit. Real investment means that precautionary savings motives are reflected in aggregates, and thus the economy exhibits the precautionary volatility mechanism described above. External habit plays two roles. Habit amplifies precautionary volatility, and also gives investors strong concerns about risk which allow the model to address aggregate asset prices.

The model also provides a unified description of aggregate asset prices and business cycles. The model captures a long list of asset price facts: it matches the mean, volatility, and persistence of excess returns, the mean, volatility, and persistence of the risk-free rate, the volatility and persistence of the price-dividend ratio, as well as the previously mentioned excess return and dividend predictability regressions using the price-dividend ratio. And these data-like asset price dynamics originate from a data-like business cycle: the model matches the volatilities of output, consumption, and investment, as well as their autocorrelations and cross-correlations.

The preferences build off of the external habit preferences of Campbell & Cochrane (1999) but feature a critical deviation. While Campbell and Cochrane assume a time-varying "habit sensitivity," I make this preference parameter a constant. This changes the key economics. Constant sensitivity means that asset prices in my model are driven by time-varying consumption volatility, rather than a time-varying sensitivity of preferences. Indeed, the model's time-varying consumption volatility quantitatively mimics the Campbell-Cochrane habit sensitivity function. Thus, the model can be considered a way of endogenizing the Campbell-Cochrane mechanism. This description conflicts with the common wisdom that time-varying risk aversion, not time-varying volatility of habit, drives the external habit model. While the common wisdom provides an intuitive description of asset prices, it is not an entirely accurate description of the Campbell-Cochrane model. Indeed, Campbell & Cochrane (1999) explicitly state that time-varying sensitivity is required for time-varying risk premia (p. 211).

I provide empirical support for the precautionary volatility mechanism. GARCH and GJR-GARCH estimates of U.S. data show only mild evidence of time-varying consumption volatility, and the model is consistent with this fact. The mechanism predicts that low asset prices are linked to high consumption volatility, and regressions of consumption volatility on the price-dividend ratio show that this prediction is borne out by the data, consistent with previous work (Kandel & Stambaugh (1990), Lettau et al. (2008)). Additionally, regressions on simulated data show that the model does a good job of describing the magnitude of this relationship. The model is able to generate significant time-varying risk premia out of mild time-variation in consumption volatility for two reasons. The first is that habit preferences make investors very sensitive to changes in consumption volatility. As a result, a small change in consumption volatility leads to a large change in risk premia. The second is that consumption volatility is persistent in both the model and data. This persistence makes it difficult to detect time-varying volatility in only 50 years of quarterly data.

This paper belongs to the literature which studies the macroeconomic effects of uncertainty fluctuations. Uncertainty has been linked to a broad set of aggregate economic phenomena, including business cycles (Bloom (2009), Bloom et al. (2012), Gourio (2012)), aggregate stock prices (Kandel & Stambaugh (1990), Bansal & Yaron (2004), Lettau et al. (2008), Gabaix (2012)), and exchange rates (Lustig et al. (2011), Farhi & Gabaix (2013)). The bulk of these papers take uncertainty fluctuations as given and then investigate its consequences. This paper differs by proposing a theory of the origins of uncertainty fluctuations. A handful of papers do provide an origin theory. Uncertainty may rise in bad times because low production makes it hard to learn about economic fundamentals (Van Nieuwerburgh & Veldkamp (2006), Fajgelbaum et al. (2012)), because bad times are a good time for experimentation (Pastor & Veronesi (2011), Bachmann & Moscarini (2011)), or because bad shocks lead to doubts about the true model of the economy (Orlik & Veldkamp (2013)). A distinct channel arising from disaggregated volatility is proposed in Carvalho & Gabaix (2013) and Koren & Tenreyro (2013) who show that changes in the ability to diversify lead to changes in volatility. The precautionary volatility channel complements these theories. Indeed, since the vast majority of models contain precautionary savings motives (Carroll & Kimball (1996)), this channel is active in many of the above papers.

The quantitative model belongs to the literature on asset pricing models with habit formation. This literature shows that that habit formation can account for a wide variety of asset price facts, including aggregate stock markets (Campbell & Cochrane (1999)), term structure (Wachter (2006), Bekaert et al. (2010)), foreign exchange (Verdelhan (2010)), and option markets (Bekaert & Engstrom (2010)). The aforementioned papers are all endowment economies and so cannot address output and investment. In contrast, this paper shows that habit formation can be successfully extended into a production economy and account for asset prices as well as quantity fluctuations. Previous habit-production papers consider a short-lived, internal habit (Jermann (1998) and Boldrin et al. (2001)). In contrast, I consider a slow-moving, external habit, in the style of Campbell & Cochrane (1999). Without this modification, the model would produce a counterfactually volatile risk-free rate. Lettau & Uhlig (2000) do consider a slow-moving, external habit in a production economy, but they assume costless adjustment of the capital stock. In contrast, I model convex capital adjustment costs. Without such costs, RBC models imply a counterfactually smooth Tobin's Q (Boldrin et al. (1999)).

The paper proceeds as follows. Section 2 examines a simple consumption-savings model which illustrates the mechanism. Sections 3-6 estimates the general equilibrium model and demonstrates that the mechanism can quantitatively account for a long list of asset price and business cycle facts.

#
2 A Two-period Model

This section illustrates the precautionary volatility mechanism with a two-period model. The model favors simplicity over generality. It is a consumption-savings problem for a prudent investor with external habit who faces uncertainty in the return on savings. I prove two propositions and two corollaries. Proposition 1 and Corollary 1 show that consumption volatility is countercyclical. Proposition 2 and Corollary 2 show that habit amplifies this countercyclicality. Proofs are found in the Appendix. The consumption-savings problem can be formalized as a general equilibrium model with a linear technology (à la Constantinides (1990)), but the notation is simpler as a consumption-savings problem.

Consider a two-period-lived investor with an external habit stock , period utility

and . These preferences display precautionary savings motives because they are "prudent" in the sense of Kimball (1990). An alternative approach of modeling precautionary savings motives through financial constraints would produce similar effects. Habit is there to illustrate an amplification mechanism. For simplicity the investor has no time preference.

At date 0, the investor has wealth . Nothing occurs at date 0, but the date helps to serve as a reference point. Throughout this section, I describe a variable as `countercyclical' if it is negatively related to . At date 1, the investor has a habit level of , and he receives a wealth shock , making his wealth . He consumes , and saves the rest of his wealth in a risky asset. At date 2, the investor has a habit level of . The return on the risky asset is realized, and he consumes his remaining wealth .

The model comes down to finding the investor's optimal consumption at date 1. His optimal consumption policy solves

|  | (1)

Taking a Taylor of expansion of around , the volatility of is approximately

|  | (2)

That is, consumption volatility is proportional to the marginal propensity to consume (MPC). Intuitively, the MPC captures the responsiveness of consumption to shocks. So the higher the MPC, the more responsive, and thus the more volatile, is consumption. For the remainder of this section, I assume that is small enough so that equation (2) is a good approximation.

Unfortunately, even in a two-period consumption savings problem, the presence of precautionary savings motives typically precludes closed-form solutions (Carroll (2001)). However, using the methods from Carroll & Kimball (1996), I can still prove that the solution exhibits countercyclical volatility and that habit amplifies this cyclicality.2

Proposition 1 The date 1 MPC is decreasing in wealth, that is, .
The essence of the proof is that, as long as the date 2 habit is positive, the model falls into the broad class of models for which consumption is strictly concave (Carroll & Kimball (1996)). Carroll and Kimball do not show the strict concavity result for this model, but I show in Appendix 8.1 that their proofs can be extended to include this setting. The proof is rather technical, but, intuitively, the investor saves for precautionary reasons in the presence of uncertainty. As the agent becomes wealthier, the uncertainty becomes less relevant, and this motive declines, creating convex savings and concave consumption.
Corollary 1 The volatility of is countercyclical, that is, .
The proof is short and illustrates the role of the MPC, so I state it here. Proof. Since the MPC is positive and decreasing in wealth, the absolute value of the MPC is decreasing in wealth. Since the volatility of is proportional to the absolute value of the MPC (equation (2)), the volatility of is also decreasing in wealth.

Proposition 1 and its corollary capture "precautionary volatility," that is, the mechanism by which precautionary motives generate countercyclical volatility. Countercyclical consumption volatility stems from the effect of the date 1 wealth shock on the investor's desire for precautionary savings at date 1. A positive shock weakens this desire, decreasing savings and boosting consumption, while a negative shock encourages him to hunker down, with opposite effects. From the perspective of date 0, this uncertainty in the need for precautionary savings at date 1 causes additional consumption volatility. Since the need for precautionary savings intensifies at low levels of wealth, the effect on consumption volatility is countercyclical.

Mathematically, the precautionary volatility mechanism is manifested as a strictly concave consumption function. Strictly concave consumption is an implication of a broad class of models (Carroll & Kimball (1996)). Indeed, Posch (2011) shows that a standard real business cycle model generates time-varying risk premia. However, the magnitude in a standard model is too small to account for the data. The following proposition and corollary show that habit amplifies precautionary volatility.

Proposition 2 The date 1 habit increases the sensitivity of the MPC to wealth, that is,

To understand this result, note that an investor with high habit has become accustomed to a high standard of living. He judges consumption not by its absolute level, but by how much it exceeds this standard. As a result, he has a much stronger precautionary savings motive than a investor who is accustomed to living in poverty. The mechanism driving Proposition 1 becomes stronger, and the MPC becomes even more sensitive to wealth. Mathematically, an external and additive habit acts as a reduction in income, which reduces wealth and amplifies precautionary savings effects.

Corollary 2 As the date 1 habit increases, the volatility of becomes more countercyclical, that is, for any set of initial wealth ,

One can think of the set of initial wealth as the range of the business cycle states in the economy. Corollary 2 states that, across this business cycle, the range of consumption volatilities is increasing in habit. The proof simply applies the link between the MPC and consumption volatility (equation (2)) to Proposition 2. Because habit makes the MPC more countercyclical, habit also makes consumption volatility more countercyclical.

Figures 2 and 3 illustrate the propositions of this section. The figures show results from numerical solutions of the two-period model for different levels of habit. In all solutions, I set . This specification focuses on how habit "concavifies" the consumption function by eliminating intertemporal substitution effects which complicate the picture.

#### Figure 2: Two-period Model: Date 1 Consumption and MPC.

is date 1 consumption as a function of date 1 wealth. is its derivative (the MPC). The different lines show consumption for various levels of habit .

#### Figure 3: Two-period Model: Date 0 Consumption Volatility.

is the volatility of date 1 consumption, given that date 0 wealth is . The different lines show consumption volatility for various levels of habit .

Figure 2 plots date 1 consumption and MPC as a function of wealth. The left panel shows that, as long as habit is positive, consumption is strictly concave. The right panel shows that this concavity is reflected in an MPC which decreases in wealth (Proposition 1). Figure 2 also shows that habit intensifies this relationship (Proposition 2). As the line gets lighter, habit increases, and the slope of the MPC gets steeper. Intuitively, habit makes the investor feel poorer and intensifies his precautionary savings motives.

Figure 3 shows how these consumption policies are reflected in consumption volatility. As long as habit is positive, consumption volatility decreases in wealth (Corollary 1). This is the essence of precautionary volatility. Because precautionary savings motives prescribe a countercyclical MPC, and because consumption volatility is proportional to the MPC (equation (2)), consumption volatility is countercyclical. The figure also shows that habit amplifies this countercyclicality (Corollary 2). As the line gets lighter, habit increases, and the range that is spanned by consumption volatility increases.

The results of this section may come as a surprise since many models in the finance literature produce consumption policies which are linear in wealth. These linear consumption policies are the result of the fact that much of this literature is set in continuous time (Sundaresan (1989), Constantinides (1990)) or relies on log-linear approximations (Campbell (1994), Lettau (2003), Kaltenbrunner & Lochstoer (2010)). Continuous time allows the investor to make an infinite number of trades within any trading period. This instantaneous trading allows for consumption policies which, if applied in a discrete time setting, would imply a positive probability that marginal utility becomes infinite (Brandt (2009)). It is exactly fear of hitting this condition of infinite marginal utility which seems to drive concave consumption behavior (Attanasio (1999)). Log-linear approximations, on the other hand, implicitly assume that consumption policies are (log) linear in state variables. The non-linearities introduced by combining power utility functions with linear budget constraints are important for generating non-linear consumption (Posch (2011)).

The results may also appear to conflict with the common intuition that habit encourages smooth consumption. This intuition is straightforward in internal habit models, where an increase in consumption has a direct effect of lowering utility in later periods (Sundaresan (1989)). In an external habit model, however, the investor by assumption does not take into account this indirect effect. Prices may encourage smooth consumption via general equilibrium effects, but the habit itself acts very similarly to a reduction in income which increases the marginal propensity to consume (Carroll & Kimball (1996)).

#
3 The General Equilibrium Production Model

The previous section illustrates the intuition behind the precautionary volatility mechanism, but a quantitative evaluation requires a richer model. This section presents a richer model. The remainder of the paper examines the predictions of the quantitative model.

The model sticks as close to the standard real business cycle model as possible. The only features are external habit preferences and quadratic capital adjustment costs. External habit serves as an amplification mechanism. Capital adjustment costs are required for a volatile Tobin's Q. There is a representative household and representative firm. Time is discrete, the horizon is infinite, and markets are complete.

For the remainder of the paper, I denote log variables with lowercase, i.e. .

## 3.1 Representative Household

There is a continuum of identical households with external habit preferences. Each household chooses its asset holdings to maximize

| (3)

Where , the level of habit, is taken as external to the household. Habit is important for both amplifying the precautionary volatility mechanism and for providing strong concerns about risk which have the chance at being consistent with asset price facts. For simplicity, the household does not value leisure and is endowed with a unit of labor.

I specify the evolution of habit using surplus consumption, rather than the level of habit itself. That is, define surplus consumption as

| (4)

where the hats denote aggregates. This approach leads to a simple and symmetric stochastic discount factor and allows for some clean asset pricing analysis. It also eases comparison with the existing literature on external habit (Campbell & Cochrane (1999), Wachter (2006), among others).

Surplus consumption evolves according to an autoregressive process

| (5)

where is a constant. Endowment economy external habit models specify as a decreasing function of surplus consumption. This assumption builds in a countercyclical volatility of marginal utility which is essential for addressing excess market volatility. The model does not require this assumed countercyclicality because, as we will see, production and precautionary motives endogenously generate countercyclical consumption volatility. For comparability with the literature, I fix at the Campbell & Cochrane (1999) steady state value

| (6)

This modification causes the issue that habit may move negatively with consumption. However, that habit is still a geometric average of the history of consumption (Appendix 8.4).

The non-linear habit specification is not critical to the economic mechanism, but the persistent AR1 specification is. Persistent habit is important for capturing the persistence in aggregate asset prices such as the price-dividend ratio. This strong persistence contrasts with previous habit models in production economies (Jermann (1998) and Boldrin et al. (2001)) which assume that habit depends only on last quarter's consumption.

The external nature of habit and the surplus consumption specification produce a simple stochastic discount factor

|  | (7)

The stochastic discount factor has the traditional consumption growth term , but habit adds a surplus consumption term . This SDF is the same as that in Campbell & Cochrane (1999) but is very different from the Epstein-Zin-habit model of Dew-Becker (2011). Dew-Becker's SDF also has two components, but the two components are those generated by Epstein & Zin (1989) preferences: a consumption growth component and a term related to the return on wealth. In Dew-Becker's model, habit enters the SDF by affecting the curvature (power parameter) of the return on wealth component. As a result, the mechanisms driving Dew-Becker's model are very different from those in this paper.

## 3.2 Representative firm

The production side of the economy is standard. The only feature is quadratic capital adjustment costs. There is a unit measure of identical firms which produce consumption using capital and labor . Production is given by

| (8)

where is capital's share of output and is productivity.

The Cobb-Douglas specification of production, combined with the fact that the household does not value leisure implies that wages are equal to the marginal product of labor

| (9)

Productivity follows the standard AR(1) process

|  | (10)

Where is a standard normal i.i.d. shock. This is the only source of uncertainty in the model, and by assumption it is homoskedastic. is chosen so that the non-stochastic steady state capital stock is approximately one. The choice of does not have a material effect on the results, but keeping the steady state capital stock near one helps with the computation.

Capital accumulates according to the usual capital accumulation rule

| (11)

and firms face a convex capital adjustment cost

| (12)

This formulation punishes the firm for deviating from the non-stochastic steady state investment rate of . I assume that the adjustment costs are a pure loss. They do not represent payments to labor. Adjustment costs are included because production economies produce a counterfactually smooth Tobin's Q unless one includes an investment friction.

The firm's objective is standard

| (13)

It chooses investment, capital, and labor to maximize future dividends, discounted with the household's stochastic discount factor.

## 3.3 Recursive Competitive Equilibrium

Market clearing is standard:

| (14)

Due to the consumption externality, the welfare theorems do not hold and the equilibrium cannot be easily described by a social planner's problem. Thus, I define equilibrium competitively. The aggregate state variables are aggregate capital , surplus consumption , and productivity .

Definition 1 Equilibrium is a firm decision rule for investment , cum-dividend value function , law of motion for aggregate consumption , and a law of motion of aggregate capital such that

(i)
  Firms optimize: and solve

  |  | (15)

  subject to capital accumulation (11), competitive wages (9), the productivity process (10), the habit process (5), the SDF is equal to the household's IMRS (7), , and where is the standard normal CDF.
(ii)
  Markets clear and aggregates are consistent with individual behavior:

  |  | (16)
  |  | (17)

Since households and firms are identical, in equilibrium, , and . Thus, in what follows, I drop the hats.

## 3.4 Solution Method

Capturing the economics of this model requires a global and non-linear solution method. This is important for both external habit (Campbell & Cochrane (1999)) and precautionary savings (Carroll (1997)). The result is a challenging numerical problem which I discuss in this section.

### 3.4.1 Projection Method

I solve the model using a projection method (Judd (1992)). Specifically, I represent the law of motion for capital using cubic splines and then use Broyden's method (a quasi-Newton algorithm) to find cubic spline coefficients which satisfy the firm's Euler equation. The solution program makes extensive use of the Miranda & Fackler (2001) CompEcon toolbox.

The projection method is important for two reasons. The first is that, due to the consumption externality, the welfare theorems do not hold and the model cannot be easily solved by value function iteration. Projection avoids this problem by working with equilibrium conditions from the recursive competitive equilibrium. The second is that projection produces a global and non-linear solution. These properties are important for asset pricing models in general (Cochrane (2008b)), but are particularly important for capturing the mechanisms in this model. The precautionary savings channel which drives countercyclical risk premia are related to the third derivative of the utility function and are not captured by traditional linearizations. Moreover, precautionary effects are particularly pronounced when the investor is threatened with infinite marginal utility (Attanasio (1999)), and fully capturing this effect requires a global solution.

### 3.4.2 Homotopy Method

Projection methods require a good initial guess of the spline coefficients. There is no guarantee that a general non-linear equation solver will converge and the high dimensionality of the problem tends to make the solvers unstable. The standard approach is to use the real business cycle model as an initial guess. Unfortunately, with external habit preferences, the real business cycle model is a poor initial guess.

To overcome this issue, I use a homotopy method. Specifically, I modify the firm's problem so that it discounts future profits using the SDF

Note that corresponds to a model with no habit, and corresponds to the full model. I begin by solving the model for , and then slowly increase , using the coefficients from the previous as the initial guess for the current .

This homotopy algorithm is very computationally intensive. To aid in the speed of computation, I discretize the productivity process using the Rouwenhorst method.

### 3.4.3 Solving for the Evolution of Surplus Consumption

Another difficult issue which arises with external habit preferences is that they result in a state variable which is not predetermined and is endogenous. Surplus consumption tomorrow is not known today and it depends on consumption tomorrow, which is endogenous. This makes solving for the surplus consumption process rather difficult.

To see this clearly, it helps to write the evolution of surplus consumption (5) as functions of state variables

|  | (18)

Where is tomorrow's discrete productivity state. Since is predetermined, is a function of four variables . Note that surplus consumption tomorrow appears on both sides of this equation. Thus determining surplus consumption tomorrow requires solving this non-linear equation. I once again use Broyden's method to solve this equation. This calculation is very computationally intensive since it must be done at every collocation node for every potential productivity shock within every iteration of the big Broyden's method which is solving for the coefficients of the law of motion of capital.

For further details about the solution method, see Appendix 8.5.

#
4 Simulated Method of Moments Estimation

In order to capture economic magnitudes, I need to give the parameters numerical values. Most papers in this literature use an informal calibration procedure. I choose a more rigorous estimation method. I estimate the model using simulated method of moments (SMM), the simulated version of Hansen (1982)'s generalized method of moments (GMM). SMM differs from traditional GMM in that it uses simulation to compute model moments rather than closed-form expressions, which may be unavailable for moments of interest. Duffie & Singleton (1993) derive adjustments to GMM formulas and additional regularity conditions required by the use of simulation.

## 4.1 Data

The estimation uses post-war (1948-2011) data from the CRSP, BEA, and BLS. Some authors argue for using the longest sample available when evaluating consumption-based asset pricing models (e.g. Bansal et al. (2009)). However, in a production economy I must also address the data on aggregate investment and output. The nature of this data is significantly affected by using pre-war data. For example, the correlation between investment growth and output growth have a high correlation of 0.73 in the post-war sample, but have a mild correlation of 0.23 for the sample 1929-2011. This consideration leads me to target only post-war data.

All variables are real and per capita. Consumption is real per capita non-durable goods and services consumption. This measure excludes volatile consumer durables such as automobiles, which produce a smooth consumption flow over the life of the durable. Aggregate equity is represented by the CRSP value-weighted index, adjusted for inflation with the consumer price index. Dividends are calculated with the assumption that all dividends are reinvested in the stock market. This method of aggregation preserves the Campbell & Shiller (1988) present value identity. The risk-free rate is a forecast of the inflation-adjusted 90-day T-bill return using the previous year's inflation rate and the nominal 90-day yield (following Beeler & Campbell (2009)). No adjustments are made for financial leverage. Further details regarding the data are found in the Appendix 8.2.

## 4.2 Estimation Method and Predefined Parameters

Because SMM is computationally intensive, I set the more traditional parameters outside of the estimation. I set the depreciation rate to match the mean, growth-adjusted, investment rate. I set the capital share parameter to match the capital share of output implied by constant returns to scale. The persistence of productivity shocks matches the persistence of the Solow residual with a fixed labor input. In an external habit model, the utility curvature and the steady state surplus consumption ratio jointly control risk aversion. As a result, it is difficult to identify these parameters separately. For ease of comparison with the literature, is set at Campbell & Cochrane (1999)'s value.3

The remaining five parameters are estimated by SMM. Explicitly, let represent the five parameters as a vector and represent a vector of target data moments. SMM estimates the parameters by minimizing the distance between data moments and simulated moments:

| (19)

where is the model-simulated counterpart to and is a weighting matrix chosen by the econometrician. I use one-stage, exactly-identified GMM, that is, I target five data moments which are economically informative about the five parameter values. This approach has the advantage of transparency. Seeing that the empirical targets are equal to the model moments immediately verifies that the minimization algorithm is successful. Under exact identification, a consistent estimator for the asymptotic variance of the estimated parameter values is

| (20)

where is the number of simulations used to calculate model moments, is the derivative of the simulated moments with respect to and is a consistent estimate of the asymptotic variance of the estimated data moments. Intuitively, we have a good estimate if the moments are informative about the parameter values ( is large), or if we have a precise estimate of the moments ( is small).4

I optimize using Levenberg-Marquardt, a variant of Newton's method. I choose this method rather than the more commonly-used simulated annealing algorithm for two reasons. The first is that the moment function does not display an extreme number of local minima, which is where simulated annealing has an advantage. With a relatively smooth objective function, a method which uses derivative information is much more efficient. Another advantage of LM is robustness. Simulated annealing tends to be sensitive to the choice of the optimization parameters (Press et al. (1992)). Additional details regarding the estimation method can be found in Appendix 8.3.

## 4.3 Parameter Estimates and Moment Targets

Table 1 summarizes the estimation. It shows estimated parameter values, standard errors, and targeted moments. For convenience, the predefined parameters are also shown, with standard errors omitted. The model is quarterly and all parameter values are quarterly. To eliminate seasonality in dividends, both the U.S. and simulated data are aggregated to the annual level. All moments are annual.

#### Table 1: Parameter Estimates and Moment Targets

Parameter | Estimate | SE | Empirical Target | Target Value
Time Preference | 0.971 | (0.009) | Mean 90-Day T-bill Return (%) | 0.98
Persistence of Surplus Consumption | 0.963 | (0.006) | Volatility of Excess Return (%) of CRSP Index | 16.07
Steady State Surplus Consumption | 0.063 | (0.010) | Mean Sharpe Ratio of CRSP Index | 0.48
Utility Curvature | 2 |  | (Chosen to Match Campbell-Cochrane (1999) |
Adjustment Cost Parameter (Quadratic) | 75.00 | (10.65) | Relative Volatility of Consumption to GDP | 0.47
Volatility of Productivity | 0.014 | (0.001) | Volatility of GDP | 0.015
Persistence of Productivity | 0.979 |  | Persistence of Solow Residual | 0.979
Capital Share | 0.35 |  | Capital's Share of GDP | 0.35
Depreciation Rate | 0.02 |  | Mean Investment Rate (Growth-Adjusted) | 0.02

The model is quarterly, and all parameter values are quarterly. Empirical figures are annual. Standard errors are Newey-West with 10 lags and shown only for estimated parameters. The sample period is 1948-2011. Consumption is non-durable goods and services consumption. GDP and consumption are logged and HP-filtered with a smoothing parameter of 6.25. Further details are found in Appendices 8.2 and 8.3.

Preference parameters are identified with asset prices. Because time-preference is reflected in the risk-free rate, I choose the mean 90-Day T-bill return as a target moment. The resulting is rather low because the model features a non-trivial precautionary savings motive, and a low level of patience helps counteract that motive. Steady state surplus consumption controls the magnitude of habit, and, in effect, the degree of risk aversion in the model. I thus choose the mean Sharpe ratio of the CRSP index as a target moment. The resulting value is close to the values used in the external habit literature (Campbell & Cochrane (1999), Wachter (2006), Santos & Veronesi (2010)). The persistence of surplus consumption has a strong effect on the volatility of the market return, and so I choose the volatility of the excess return on the CRSP index as a target. The resulting value of indicates a very persistent habit process, which is also consistent with values used in the external habit literature.

Technological parameters are identified with moments of the real economy. The volatility of productivity targets the volatility of HP-filtered log GDP of 0.014. Since the data is annual, I use the annual smoothing parameter of 6.25 advocated by Ravn & Uhlig (2002). The quadratic adjustment cost parameter targets the relative volatility of consumption to GDP (also HP-filtered). This estimated value results in mean adjustment costs as a percentage of output of less that 1%.

#
5 Matching Asset Price and Business Cycle Moments

This section contains the main quantitative results. It shows that precautionary volatility channel, when amplified by external habit, provides a quantitative description of time-varying risk premia. The model also matches numerous other moments, and provides a unified description of asset prices and business cycles. I make no adjustments to account for un-modeled leverage or payout policy. The price of equity is simply the present value of dividends from the representative firm, discounted with the household's SDF.

For quick navigation, this section does not discuss mechanisms. Readers focused on mechanisms may want to skip to Section 6.

## 5.1 Unconditional Asset Price Moments

Table 2 shows that the model produces a nice fit for all of the basic moments of asset prices. As intended by the estimation, it hits three of these moments exactly: the mean risk-free rate, the Sharpe Ratio, and the volatility of excess returns. Producing this large equity volatility is a difficult task in production economies (Gourio (2010), Kaltenbrunner & Lochstoer (2010), Croce (2010)).

#### Table 2: Unconditional Asset Price Moments

Identifying Moments

Moment | Data, 1948-2011 | Model mean | 5% | 50% | 95%
| 0.98 | 0.98 | -2.72 | 0.28 | 6.83
| 0.48 | 0.48 | 0.29 | 0.48 | 0.68
| 16.07 | 16.07 | 10.77 | 15.71 | 22.53

Untargeted Moments

Moment | Data, 1948-2011 | Model mean | 5% | 50% | 95%
(%) | 6.47 | 6.77 | 5.03 | 6.96 | 7.88
AC1 | -0.03 | -0.07 | -0.28 | -0.07 | 0.16
| 2.24 | 2.96 | 0.55 | 1.88 | 8.60
AC1 | 0.56 | 0.88 | 0.69 | 0.91 | 0.99
| 3.42 | 2.67 | 2.09 | 2.66 | 3.27
| 0.43 | 0.46 | 0.26 | 0.44 | 0.76
AC1 | 0.95 | 0.90 | 0.79 | 0.91 | 0.96
|  | 2.03 | 0.93 | 2.10 | 2.92
|  | 1.42 | 0.62 | 1.37 | 2.44
AC1 |  | 0.85 | 0.70 | 0.86 | 0.95

Figures are annual. No adjustments are made to account for financial leverage. The model columns show means and percentiles across simulations of the same length as the empirical sample. , , and are the logs of returns, prices, and dividends from the CRSP value-weighted index. Capital letters show levels rather than logs. is a forecast of the ex-post real return on 90-day T-bills following Beeler & Campbell (2009). , , and AC1 represent the sample mean, standard deviation and first-order autocorrelation. Further details are found in Appendix 8.2.

The model also captures many asset market features beyond those used in the identification. The model produces a risk-free rate volatility of 2.96% which is close to the data estimate of 2.24%. This moment is difficult to match in habit models, which tend to produce an excessively volatile risk-free rate (i.e. 11% in Jermann (1998) and 25% in Boldrin et al. (1999)). This low risk-free rate volatility is reflected in a reasonable term premium. The model produces a mean excess return on 10-year bonds of 2%, indicating that the model's equity premium is distinct from the term premium, as in the data. The volatility of the log price-dividend ratio is 0.46, which is close to the data value of 0.43. This moment has been difficult to capture even in endowment economies (Bansal et al. (2007), Bansal et al. (2009)).

Additionally, the model generates data-like persistence. As in the data, the excess market return is mildly negatively autocorrelated, and the risk-free rate and price-dividend ratio are highly positively autocorrelated. The model somewhat overstates the persistence of the risk-free rate, but this deviation contains some uncertainty because risk-free rate is not directly observable.

##
5.2 Excess Return and Dividend Predictability

The previous section shows that the model generates a volatile, data-like stock market. This large volatility does not mean that the model captures the nature of stock market fluctuations, however. The data show that these fluctuations are driven by time-varying risk premia: stock price movements have little relationship with movements in expected cash flows, but are closely linked to movements in expected excess returns (LeRoy & Porter (1981), Shiller (1981), Campbell & Shiller (1988), Cochrane (1992)). In this section, I consider dividends as cash flows, but other notions of cash flow (i.e. consumption or profits) produce similar results.

#### Table 3: Predicting Dividends and Excess Returns with the Price-Dividend Ratio

Panel A: Predicting dividend growth

Estimate | L | Data, 1948-2011 | Model, Mean | 5% | 50% | 95%
| 1 | -0.03 | -0.00 | -0.04 | -0.00 | 0.03
| 3 | 0.01 | 0.07 | 0.00 | 0.06 | 0.18
| 5 | 0.03 | 0.13 | 0.02 | 0.12 | 0.30
SE( ) | 1 | 0.03 | 0.03 | 0.01 | 0.02 | 0.04
SE( ) | 3 | 0.07 | 0.05 | 0.03 | 0.05 | 0.09
SE( ) | 5 | 0.09 | 0.07 | 0.03 | 0.06 | 0.11
| 1 | 0.01 | 0.00 | 0.00 | 0.00 | 0.02
| 3 | 0.00 | 0.04 | 0.00 | 0.03 | 0.13
| 5 | 0.00 | 0.10 | 0.01 | 0.09 | 0.26

Panel B: Predicting excess returns

Estimate | L | Data, 1948-2010 | Model, Mean | 5% | 50% | 95%
| 1 | -0.12 | -0.12 | -0.23 | -0.11 | -0.04
| 3 | -0.27 | -0.30 | -0.55 | -0.29 | -0.12
| 5 | -0.40 | -0.44 | -0.78 | -0.44 | -0.17
SE( ) | 1 | 0.05 | 0.05 | 0.02 | 0.05 | 0.08
SE( ) | 3 | 0.08 | 0.09 | 0.04 | 0.08 | 0.13
SE( ) | 5 | 0.12 | 0.10 | 0.05 | 0.10 | 0.17
| 1 | 0.09 | 0.10 | 0.02 | 0.09 | 0.19
| 3 | 0.19 | 0.26 | 0.07 | 0.26 | 0.45
| 5 | 0.26 | 0.37 | 0.11 | 0.38 | 0.59

Figures are annual. The model columns show means and percentiles across simulations of the same length as the empirical sample. , , and are the log-returns, prices, and dividends from the CRSP value-weighted index. is a forecast of the ex-post real return on 90-day T-bills following Beeler & Campbell (2009). Standard errors are Newey-West with lags. Further details are found in Appendix 8.2.

Table 3 shows that the model captures key elements of time-varying risk premia. Panel A shows regressions of future dividend growth on the log price-dividend ratio at various horizons. The model predicts no relationship between the price-dividend ratio and future dividend growth at the one-year horizon, as is seen in the data. As in the data, this predictability increases with the horizon, but the coefficients remain small and statistically insignificant. The predictability is somewhat overstated at the 5-year horizon, but Section 6.6 shows that this issue can be fixed with a richer dividend process which accounts for deviations between neoclassical firm cash flows and dividends.

If the price-dividend ratio does not forecast dividends, then it must forecast returns (Campbell & Shiller (1988), Cochrane (2008a)). Panel B shows the ability of the model to capture this flip side of dividend predictability. It shows regressions of future excess returns on the log price-dividend ratio. Regression coefficients, standard errors, and 's, are close to the data for all forecasting horizons. The coefficients on the price-dividend ratio are negative, and they are both economically and statistically significant. At the one-year horizon, the model exactly matches the data coefficient of -0.12. To understand this economically, recall that the volatility of the log price-dividend ratio is roughly 0.40 in both the model and the data. This means that a one standard deviation rise in the price-dividend ratio predicts a huge 4% reduction in the equity premium over the next year. This forecasting power increases with the forecasting horizon, reaching 's of roughly 30% at the 5-year horizon in both model and data.

Overall, the model captures both sides of predictability. Asset price fluctuations have little relationship with fluctuations in future dividends, but are tightly linked to fluctuations in future excess returns.

## 5.3 Unconditional Business Cycle Moments

The previous sections show that the model produces a good description of numerous asset price moments. Table 4 shows that these asset price moments come with data-like fluctuations in the real economy. The model hits the volatility of output and relative volatility of consumption to output, as intended by the estimation. Investment is more volatile than output, as in the data. Like the data, the model moments display strong co-movement between output, consumption, and investment. Output, consumption, and investment are highly persistent, and are nearly as persistent as the data. First-differenced log consumption has a low volatility. In both the model and data this volatility is about 1% per year. Lastly, the table also shows that average adjustment costs are small at less than 1% of output.

#### Table 4: Basic Business Cycle Moments

Identifying Moments

Moment | Data, 1948-2011 | Model, mean | 5% | 50% | 95%
(%) | 1.50 | 1.50 | 1.11 | 1.49 | 1.93
| 0.47 | 0.47 | 0.40 | 0.47 | 0.55

Untargeted Moments

| 2.46 | 3.43 | 3.09 | 3.44 | 3.76
| 0.83 | 0.99 | 0.98 | 0.99 | 1.00
| 0.85 | 1.00 | 0.99 | 1.00 | 1.00
AC1 | 0.12 | 0.21 | 0.01 | 0.21 | 0.38
AC1 | 0.32 | 0.21 | 0.01 | 0.22 | 0.39
AC1 | 0.27 | 0.20 | 0.01 | 0.21 | 0.38
(%) | 1.11 | 1.41 | 0.97 | 1.38 | 1.90
Adj Cost |  | 0.54 | 0.21 | 0.49 | 1.04
Adj Cost |  | 3.21 | 1.13 | 2.70 | 6.88

Figures are annual. The model columns show means and percentiles across simulations of the same length as the empirical sample. is log GDP, is log nondurable and services consumption, and is the log of fixed investment plus durable goods plus government investment. The subscript indicates that the moment is calculated from HP-filtered data with a smoothing parameter of 6.25. indicates first-differences. Further details are found in the Appendix 8.2.

#
6 Inspecting the Mechanism

I have shown that the model provides a unified description of numerous asset price and business cycle moments, including time-varying risk premia. The remainder of the paper is devoted to explaining how.

Most of the new economics can be seen by examining time-varying risk premia. The model contains the key features of the simple model of Section 2, suggesting that precautionary volatility is the underlying mechanism. However, the model is closely related to Campbell & Cochrane (1999), and so a closer look is informative. Section 6.1 shows that time-varying risk premia come from time-varying consumption volatility. Section 6.2 compares this mechanism with Campbell & Cochrane (1999). Section 6.3 shows that the model's time-varying consumption volatility is consistent with the data.

The other asset pricing results are examined later in this section. The large and volatile equity premium is analyzed in Section 6.4. The low and smooth risk-free rate is examined in Section 6.5. Section 6.6 investigates cash flow dynamics and alternative dividend processes. Section 6.7 ends the inspection by examining comparative statics.

##
6.1 Time-varying Risk Premia and Precautionary Volatility

The model addresses time-varying risk premia through the precautionary volatility mechanism described in Section 2. To demonstrate this, I'll work backward, starting with time-varying risk premia and ending with precautionary motives.

The model produces fluctuations in risk by producing time-varying consumption volatility. This can be seen in Figure 4, which shows scatterplots of asset prices against consumption volatility produced by model simulations. Consumption volatility varies over time, and both the price-dividend ratio and equity premium are nicely linked to consumption volatility. The price-dividend ratio declines in consumption volatility, while the equity premium increases.

#### Figure 4: Time-Varying Consumption Volatility and Asset Prices.

The figures show scatterplots from model simulations. Consumption volatility and the equity premium are calculated using the model's laws of motion. All values are annualized.

This result has a very simple intuition. Times of high consumption volatility are risky times with large risk premia and low asset prices. But a little formalism provides additional insight and builds confidence in the mechanism. A log-normal approximation of the SDF shows that the conditional maximum Sharpe ratio is5

| (21)

The conditional maximum Sharpe ratio is the conditional volatility of consumption growth, multiplied by preference parameters: is the utility curvature (3), and is the conditional volatility of the habit process (5). This expression shows that consumption volatility is critical. Since consumption volatility is the sole term on the RHS which can vary over time, it must be the driver of time-varying Sharpe ratios.

Time-varying consumption volatility in turn is the result of the precautionary volatility channel described in the simple model (Section 2). To see this, note that the model is driven by a single, homoskedastic productivity shock (equation (10)). Moreover, the conditional volatility of the habit process is assumed to be constant (equation (5)). Thus, there are no exogenous drivers of time-varying volatility. On the other hand, the model features the same preferences as the simple model. These preferences produce precautionary savings motives which lead to strictly concave consumption. This shape means that consumption has time-varying volatility, even though the underlying shock is homoskedastic. External habit amplifies this channel and produces the quantitative result seen in Figure 4. Unlike previous external habit models, this model features real investment, which allows the precautionary volatility channel to be visible in equilibrium quantities.

##
6.2 Endogenizing the Campbell-Cochrane Mechanism

Though the model features external habit preferences, the mechanism is distinct from the existing literature. Existing external habit models drive time-varying risk premia through a time-varying preference parameter, i.e. the Campbell & Cochrane (1999) function. My model drives time-varying risk premia through time-varying consumption volatility. In this section, I show that these two channels are economically distinct, but have similar quantitative properties. Thus, my model can be considered a way of endogenizing the Campbell & Cochrane (1999) mechanism.

To compare the channels, it helps to examine the maximum Sharpe ratio from Campbell & Cochrane (1999)

| (22)

where

| (23)

The volatility of consumption growth is assumed to be constant. So the only way the model can generate time-varying Sharpe ratios is through the preference parameter . This is a direct contrast to my model, where time-varying Sharpe ratios must come from consumption volatility (equation (21)). is the "habit sensitivity function," that is, it controls how sensitive habit is to shocks. Existing models assume that this sensitivity declines in surplus consumption (equation (23)). This assumption builds in a time-varying volatility of marginal utility and thus a time-varying maximum Sharpe ratio.

The above analysis also shows that the common intuition regarding the external habit mechanism is misplaced. The common intuition is that time-varying risk premia are driven by time-varying risk aversion. But time-varying risk aversion is not sufficient for generating time-varying Sharpe ratios. To see this, note that a constant still results in time-varying surplus consumption (see equation (5)). Since risk aversion is approximately , a constant still produces time-varying risk aversion. But equation (22) shows that, even though risk aversion is time-varying, a constant implies a constant maximum Sharpe ratio. It is time-varying habit sensitivity , not time-varying risk aversion, which drives the traditional external habit model. The critical role of the sensitivity function is acknowledged in Campbell & Cochrane (1999). This is not to say that the common intuition cannot be modeled. Dew-Becker (2011) uses Epstein-Zin preferences to create a habit model which captures the common intuition.

The fact that, in traditional external habit models, time-varying habit sensitivity drives time-varying risk leaves open many questions. How do we interpret a time-varying habit sensitivity? Where does this time-varying sensitivity come from? This model can help address these questions since it replaces the time-varying habit sensitivity function with time-varying consumption volatility, which has a very simple interpretation. And time-varying consumption volatility originates from the precautionary savings motives, that is, bad times make investors unsure of how much they can consume next period.

Indeed, the model's time-varying consumption volatility does a good job of mimicking the habit sensitivity function of Campbell and Cochrane. To demonstrate this I construct an `implied consumption volatility' from the Campbell-Cochrane sensitivity function by equating the two expressions for the maximum Sharpe ratio (equations (21) and (22)). I call this implied consumption volatility because one can replace the Campbell-Cochrane sensitivity function with this implied consumption volatility and get similar quantitative results. Explicitly,

Implied Consumption Volatility | (24)

Implied consumption volatility is simply the habit sensitivity , rescaled by its steady state value , and multiplied by the unconditional volatility .

Figure 5 compares the Campbell-Cochrane implied consumption volatility to the consumption volatility of my model. It shows scatterplots of the volatilities against surplus consumption. Both are countercyclical, that is, they decline in surplus consumption. Moreover, for much of the plot, implied consumption volatility runs right through the middle of the cloud of dots representing consumption volatility. The two channels are both qualitatively and quantitatively similar. Thus, this production economy could be considered a method for endogenizing the external habit mechanism.

#### Figure 5: Consumption Volatility and the Campbell-Cochrane .

Dots represent consumption volatility from model simulations. The dashed line represents Campbell-Cochrane implied consumption volatility (equation (24)). Figures are annualized.

##
6.3 Empirical Support for the Precautionary Volatility Mechanism

I have shown that the precautionary volatility channel generates time-varying consumption volatility, and that this time-varying consumption volatility produces time-varying risk premia. This section provides direct empirical support for this mechanism. I show that the mechanism generates mild time-variation in consumption volatility, as is seen in the data. I also show that, in the data, consumption volatility has a negative relationship with asset prices. The model captures both the sign and the magnitude of this relationship.

I use a number of measures of conditional consumption volatility. All measures are constructed by first fitting an AR(1) model to log consumption growth to remove an expected growth component:

| (25)

I then either estimate GARCH-type models on the residual or use the mean absolute residual as a non-parametric measure of conditional volatility. I use quarterly data because it is difficult to detect time-varying volatility with the post-war annual sample of only 50 observations. This procedure follows Bansal et al. (2005), Bansal et al. (2009) and Beeler & Campbell (2009).

Table 5 compares GARCH estimates of consumption volatility from the data and model. Panel A shows results from a GARCH(1,1) process

| (26)

The data columns show modest time-variation in consumption volatility. Consistent with Bansal et al. (2005), the ARCH parameter just makes it to the 95% significance level if one uses non-robust standard errors. However, using standard errors which are robust to the assumption of non-normal shocks (Bollerslev & Wooldridge (1992)), one cannot reject the hypothesis of no time-varying volatility. This modest time-variation in volatility is consistent with the model columns. The model's mean estimate of the ARCH parameter is close to zero, and 5% of simulations produce a value of 0. Supposing that ARCH effects do exist, the data produce GARCH parameter estimates of indicating that consumption volatility is persistent. The model captures this persistence, producing a GARCH parameter of 0.76. Indeed, this high persistence of consumption volatility may be the reason why time-varying volatility is hard to detect in the data.

#### Table 5: GARCH Estimates of Time-Varying Consumption Volatility

Panel A: GARCH

Data: 1948Q1-2011Q4 | Estimate | SE | Robust SE | Model, mean | 5% | 50% | 95%
| 1.39E-06 | 8.51E-07 | 1.27E-06 | 9.69E-06 | 2.94E-07 | 2.65E-06 | 4.42E-05
| 0.14 | 0.08 | 0.13 | 0.03 | 0.00 | 0.03 | 0.09
| 0.79 | 0.09 | 0.15 | 0.76 | 0.00 | 0.91 | 0.99

Panel B: GJR-GARCH

Data: 1948Q1-2011Q4 | Estimate | SE | Robust SE | Model, mean | 5% | 50% | 95%
| 1.97E-06 | 1.25E-06 | 2.29E-06 | 1.09E-05 | 3.19E-07 | 2.00E-06 | 4.70E-05
| 0.07 | 0.07 | 0.08 | 0.01 | 0.00 | 0.00 | 0.06
| 0.76 | 0.10 | 0.19 | 0.73 | 0.00 | 0.93 | 0.98
| 0.14 | 0.11 | 0.20 | 0.05 | 0.00 | 0.06 | 0.16

Data and figures are quarterly. This table shows measures GARCH estimates of the residual from an AR(1) model of consumption growth

GARCH estimation is done by quasi maximum likelihood. Robust standard errors use the Bollerslev-Wooldridge method. The model columns show means and percentiles across simulations of the same length as the empirical sample. Further details are found in Appendex 8.2.

Panel B compares estimates of a GJR-GARCH(1,1,1) model (Glosten et al. (1993))

GJR-GARCH introduces an additional term which allows negative shocks to have a larger effect on volatility, as predicted by the analysis of Section 2. Though the data show significant sampling uncertainty, the point estimates are consistent with the intuition. The asymmetric GJR-GARCH parameter is much larger than the symmetric ARCH in both the model and the data. In terms of magnitudes, the model's parameter estimates are all smaller than the data's. Consistent with Panel A, the model reproduces the modest time-variation in consumption volatility, as seen in the data.

There are two reasons why the model can generate substantial fluctuations in risk premia out of modest GARCH effects. The first is that habit preferences amplify fluctuations in consumption volatility. This can be seen in the surplus consumption process (5). Surplus consumption is basically an AR1 process, where the shocks are consumption growth and the conditional volatility is a free parameter. The calibration chooses a large which provides substantial amplification. Thus small movements in the volatility of consumption growth produce large fluctuations in the volatility of marginal utility. The second is that consumption volatility generated by the model is very persistent. This persistent consumption volatility makes it hard to measure time-varying volatility using only 50 years of quarterly data.

A key prediction of the precautionary volatility channel is that consumption volatility is high when asset prices are low. Table 6 shows that this prediction is supported by the data. The table shows regressions of various proxies for consumption volatility on the price-dividend ratio. The data column shows that, using all proxies, consumption volatility and the price-dividend ratio are negatively related. A high price-dividend ratio indicates a safe time of low volatility. The model replicates this pattern. The large standard errors in Table 5 suggest that comparing magnitudes should be done with caution, but with that in mind, the model coefficients are of similar magnitude to those from the data.

#### Table 6: Regressions of Consumption Volatility on the Price-Dividend Ratio

Panel A: GARCH estimates of consumption volatility

Estimate | Data: 1948Q1-2011Q4 | Model: mean | 5% | 50% | 95%
GARCH | -0.43 | -0.19 | -0.50 | -0.15 | 0.00
GARCH SE | 0.14 | 0.05 | 0.00 | 0.05 | 0.13
GARCH | 0.37 | 0.23 | -0.01 | 0.18 | 0.61
GJR-GARCH | -0.43 | -0.32 | -0.75 | -0.32 | 0.00
GJR-GARCH SE | 0.16 | 0.07 | 0.00 | 0.06 | 0.16
GJR-GARCH | 0.31 | 0.33 | -0.01 | 0.34 | 0.77

Panel B: Non-parametric estimates of consumption volatility

cvol

Estimate | L (qtr) | Data: 1948Q1-2011Q4 | Model: mean | 5% | 50% | 95%
| 4 | -0.66 | -0.21 | -0.01 | -0.22 | -0.66
| 12 | -0.59 | -0.25 | 0.00 | -0.24 | -0.68
| 20 | -0.53 | -0.25 | 0.00 | -0.24 | -0.61
SE | 4 | 0.16 | 0.25 | 0.12 | 0.23 | 0.45
SE | 12 | 0.12 | 0.19 | 0.08 | 0.17 | 0.36
SE | 20 | 0.09 | 0.14 | 0.06 | 0.13 | 0.29
| 4 | 0.22 | 0.02 | 0.00 | 0.01 | 0.06
| 12 | 0.36 | 0.04 | 0.00 | 0.02 | 0.13
| 20 | 0.44 | 0.07 | 0.00 | 0.04 | 0.23

This table shows regressions of the form

cvol |

where cvol is a measure of conditional consumption volatility, is the log equity price, and is the log dividend. To generate cvol , first an AR(1) model (25) is run on log consumption growth. Panel A estimates either a GARCH(1,1) or GJR-GARCH(1,1,1) model on the residuals. Panel B uses a non-parametric measure: cvol , where is the residual from the AR(1) model. Consumption data is quarterly and price-dividend ratio data is annual, which results in some abuse of notation. The model columns show means and percentiles across simulations of the same length as the empirical sample. Further details are found in Appendix 8.2.

##
6.4 The Large and Volatile Equity Premium

The high volatility of the equity premium comes from high capital adjustment costs and the low elasticity of intertemporal substitution (EIS) of habit preferences. High capital adjustment costs mean that productivity shocks are absorbed by asset prices rather than investment. This channel is not new (Jermann (1998), Kogan (2004), Jermann (2010), Kogan & Papanikolaou (2012)) and so I provide only a brief discussion. The role of the low EIS is that a low EIS pins down high adjustment costs via estimation and general equilibrium effects. The importance of the EIS is less well understood and so I focus on it.

The link between equity volatility and adjustment costs can be seen in the investment return - stock return identity (Cochrane (1991), Restoy & Rockinger (1994)). Since the model has a homogenous production technology, this identity means we can express the stock return as

| (27)

The stock return is related to the marginal product of capital , the investment rate , and the adjustment cost parameter . This equality holds state-by-state, which means that the volatilities of the two sides are equal.

This identity shows the difficulty of matching equity volatility in a production economy. The marginal product of capital and the investment rate have a very low volatility, less than 2% per year. On the other hand, the equity return has a huge volatility of about 20% per year. The only way to reconcile these two sides, then, is with capital adjustment costs. Adjustments costs increase the curvature of the RHS. Jensen's inequality then implies that adjustment costs increase the volatility of the investment return. The large discrepancy between the equity volatility and the volatility of the investment rate imply a very large adjustment cost parameter.

The role of the EIS comes through general equilibrium effects and standard empirical restrictions. General equilibrium means that both the EIS and capital adjustment costs affect consumption volatility. The EIS channel works through the SDF. Since the firm uses the investor's IMRS as an SDF, the firm is rewarded for providing consumption which matches the investor's preferences. A low EIS results in a strong incentive to produce smooth dividends, and through market clearing, smooth consumption. Capital adjustment costs also affect consumption volatility through market clearing. Large adjustment costs encourage the firm to keep investment smooth in the face of shocks. Since shocks must be absorbed by either investment or dividends, this encourages volatile dividends, and, through market clearing, volatile consumption.

Since both the EIS and capital adjustment cost affect consumption volatility, the ubiquitous requirement that asset pricing models should fit consumption data means that these two elements must match each other. A low EIS requires high capital adjustment costs, and vice versa. In this model, the large Sharpe ratio pins down a low EIS. Given the low EIS, the volatility of consumption pins down a high adjustment cost. Note that the adjustment cost is not pinned down by equity volatility, and so equity volatility forms an overidentified restriction which is satisfied by the model.

This EIS-adjustment cost link exists in long-run risk and disaster models, but it tends to hurt rather than help their asset pricing results. The vast majority of long-run risk and disaster papers do not calibrate the EIS. However, the EIS must be larger than one in both models in order to match certain qualitative facts, and the common wisdom is that the larger the EIS, the better the asset pricing results. For long-run risk models, this is required to qualitatively match the fact that consumption volatility and the price-dividend ratio are negatively related (Bansal & Yaron (2004)). In disaster models, this is required to match the intuitive notion that a rise in disaster risk lowers investment and increases excess returns (Gourio (2010)). This large EIS then implies low adjustment costs and counterfactually low equity volatility (Kaltenbrunner & Lochstoer (2010), Gourio (2010)).

Given that equity volatility is high, the large equity premium is the consequence of a large Sharpe ratio. The large Sharpe ratio is simply due to the fact that habit preferences offer an additional degree of freedom for the modeler. Habit results in an additional term in the SDF (7). The modeler can choose the volatility of this term to be high by adjusting the preference parameter in equation (5).

Economically, can be interpreted as the `moodiness' of the economy. The surplus consumption process says that if consumption growth goes up by 1%, surplus consumption gets boosted by %. The calibration chooses , meaning that changes in `mood' are responsible for the vast majority of changes in marginal utility. Checking this magnitude by introspection is, of course, a dangerous activity. But Section 5 shows that it is consistent with numerous overidentifying restrictions regarding asset prices.

##
6.5 The Low and Smooth Risk-Free Rate

The low risk-free rate is simply the result of time preference. Time preference is effectively a free parameter which allows me to hit the low risk-free rate in the data. Economically speaking, this results in an intuitive time preference of annually, that is, consumption one year from today is worth 90% of consumption today.

The smooth risk-free rate comes from an interplay of intertemporal substitution and precautionary savings effects. This channel has a simple intuition. In bad times, people want to borrow in order to consume today. But in bad times, the economy is particularly volatile, and the desire for precautionary savings prevents them from borrowing. This channel is similar to that of Campbell & Cochrane (1999), but with endogenous consumption volatility driving precautionary savings effects rather than the exogenous habit sensitivity function. Indeed, Section 6.1 shows that the model does a good job of mimicking the Campbell-Cochrane sensitivity function and so the smoothness of the risk-free rate should not be surprising.

These intuitions are fleshed out by examining the log-normal approximation of the risk-free rate6

|  | (28)

The 1st term reflects time preference. Time preference has a small impact on other unconditional moments, and so it is essentially a free parameter which one can use to fit the low mean risk-free rate in the data.

The 2nd and 3rd terms are due to intertemporal substitution and tend to create excessive volatility in habit models (Jermann (1998), Boldrin et al. (2001)). They reflect the Friedman (1957) permanent income hypothesis. In bad times, investors want to borrow from the future in order to consume today. This motive pushes down the price on the risk-free bond and pushes up the risk-free rate, leading to a countercyclical effect on a risk-free rate. Habit models imply a very strong consumption smoothing motive which makes this channel very volatile.

This model has the volatile intertemporal substitution effect typical of habit models, but it counters this effect with a precautionary savings effect. The precautionary savings effect runs through the 4th term, which is decreasing in the conditional volatility of consumption. Intuitively, in bad times, high consumption volatility creates a desire for savings. Investors buy bonds, pushing up the price and down the risk-free rate. This channel creates a procyclical effect on the risk-free rate, which helps counteract the countercyclical effect of the intertemporal substitution channel.

The previous discussion shows that, qualitatively, precautionary savings effects help counteract intertemporal substitution effects. Whether the quantitative effect is enough generate a smooth risk-free rate is another question. Figure 6 examines the quantitative efffect. It plots the risk-free rate decomposition (28) against surplus consumption for various levels of productivity. The solid red lines represent intertemporal substitution effects. The dashed blue lines represent the precautionary savings effects. The two effects are near mirror-images of each other, showing that, quantitatively, the channels balance each other quite nicely.

#### Figure 6: Decomposition of the Risk-Free Rate.

`Intertemporal Substitution' represents the 2nd and 3rd terms of equation (28). `Precautionary Saving' represents the last term. Computed from model's laws of motion. Risk-free rate is in annualized %. Capital in all panels is fixed at the mean capital stock.

A nice feature of the smoothing effects in this model is that they arise endogenously. This contrasts with the risk-free rate smoothing effects of Campbell & Cochrane (1999), which are the result of a parameter choice. To see this, it helps to return to approximation of the risk-free rate (28). This expression shows that, if one allows the preference parameter to vary over time, then one can control the magnitude of the precautionary savings effect. Indeed, Campbell & Cochrane (1999) do just that, and they choose the magnitude of the channel to exactly cancel out the intertemporal substitution effect. This model has no such freedom. The magnitude of this channel comes through the amount of countercyclicality in consumption volatility, which is the result of general equilibrium effects of investors' preferences for precautionary savings on firms' production decisions.

##
6.6 Cash Flow Dynamics and Alternative Dividend Processes

The model has a very simple notion of dividends. Dividends are just the profits (net of investment) of a representative neoclassical firm (see equation (13)). This formulation has the advantage of transparency, but it abstracts from a number of issues which affect dividends in the data. For example, the model abstracts from debt and payout policy, as well as labor market frictions. The literature has shown that these abstractions lead to some counterfactual behavior of dividends (Rouwenhorst (1995), Jermann (1998), Kaltenbrunner & Lochstoer (2010)), and my model is not exempted from this issue. Extending the model to incorporate richer financial and production elements would help the model fit dividend dynamics (i.e. Jermann & Quadrini (2012), Kuehn et al. (2012)), at the expense of complexity.

Would the main results still hold under a richer and more data-like dividend process? Table 7 suggests that the answer is yes. The table shows basic properties of dividends and excess returns for alternative dividend processes. The alternative processes specify dividends as stochastically related to consumption, which allows one to capture basic empirical dividend moments. This procedure follows the endowment economy literature (Bansal & Yaron (2004), Campbell & Cochrane (1999)). The table shows that, regardless of the dividend process, the model still produces a large and volatile equity premium.

#### Table 7: Alternative Dividend Processes: Basic Statistics

| US 1948-2011 | Neoclassical Dividend | Calibrated Dividend | Consumption Claim
(%) | 13.11 | 6.21 | 13.10 | 1.20
| 0.17 | -0.68 | 0.18 | 1.00
(%) | 6.47 | 6.77 | 8.01 | 6.86
(%) | 16.07 | 16.07 | 20.69 | 17.71
AC1 | -0.03 | -0.07 | -0.06 | -0.06

Figures are annual. `Neoclassical Dividend' is a claim on the dividends from the firm (see equation 13). `Calibrated Dividend' is a claim on the dividend process (29) calibrated to match U.S. data. `Consumption Claim' is a claim on consumption. is dividend growth, is consumption growth, and is excess returns of the CRSP index. Dividend growth is computed using no reinvestment. Details of the data are found in Appendix 8.2.

The `neoclassical dividend' column shows results from the baseline model. Consistent with previous studies, this simple technology leads to some counterfactual dividend statistics. Dividends are too smooth compared to the data, and dividend growth is negatively correlated with consumption, while the data show a mild positive correlation. Despite the negative correlation with consumption growth, a claim on the neoclassical dividend produces a large and volatile equity premium. One might think that, due to this negative correlation, this claim would serve as a hedge for consumption risk and offer low returns. However, the vast majority of the claim's value comes from its continuation value, not the next year's dividends. A negative shock leads the firm to cut investment and boost dividends temporarily, but this negative shock is a bad sign for a long future of dividends. This result of a large risk premium with negative consumption-dividend correlation is consistent with results from previous habit models (Jermann (1998)).

The `calibrated dividend' column presents results from a dividend process which matches some basic dividend data moments. The dividend process is a simple function of consumption

| (29)

where and are parameters and is a standard normal shock. This approach captures some key statistical features of dividend growth and follows Campbell & Cochrane (1999) and Bansal & Yaron (2004), among others. Specifically, I choose and to hit the correlation between consumption and dividend growth, and the volatility of dividend growth.

Table 7 shows the calibrated dividend process matches the volatility of dividend growth and its correlation with consumption growth, as intended. It also generates a data-like equity premium. Excess returns on this claim are large on average, volatile, and mildly negatively autocorrelated. This large risk premium exists despite the low correlation between dividend and consumption growth. Once again, the vast majority of this asset's value comes from its continuation value, not the next year's dividends. The `consumption claim' column shows further robustness of the large and volatile equity premium. A claim on consumption also produces data-like stock returns.

Figures are annual. `Neoclassical Dividend' is a claim on the dividends from the firm (see equation 13). `Calibrated Dividend' is a claim on the dividend process (29) calibrated to match U.S. data. `Consumption Claim' is a claim on consumption. is dividend growth, is the log price-dividend ratio, and is excess returns. Standard errors are Newey-West with lags. Details of the data are found in Appendix 8.2.

#### Table 8: Alternative Dividend Processes: Predictive Regressions

Panel A: Predicting dividend growth

Estimate | L | US 1948-2011 | Neoclassical Dividend | Exogenous Dividend | Consumption Claim
| 1 | -0.03 | -0.00 | 0.03 | -0.04
| 3 | 0.01 | 0.07 | 0.04 | -0.02
| 5 | 0.03 | 0.13 | 0.05 | -0.01
SE( ) | 1 | 0.03 | 0.03 | 0.04 | 0.03
SE( ) | 3 | 0.07 | 0.05 | 0.06 | 0.04
SE( ) | 5 | 0.09 | 0.07 | 0.06 | 0.04
| 1 | 0.01 | 0.00 | 0.01 | 0.01
| 3 | 0.00 | 0.04 | 0.02 | 0.01
| 5 | 0.00 | 0.10 | 0.02 | 0.01

Panel B: Predicting excess returns

| L | US 1948-2011 | Neoclassical Dividend | Exogenous Dividend | Consumption Claim
| 1 | -0.12 | -0.12 | -0.15 | -0.15
| 3 | -0.27 | -0.30 | -0.38 | -0.38
| 5 | -0.40 | -0.44 | -0.56 | -0.55
SE( ) | 1 | 0.05 | 0.05 | 0.07 | 0.07
SE( ) | 3 | 0.08 | 0.09 | 0.12 | 0.12
SE( ) | 5 | 0.12 | 0.10 | 0.15 | 0.15
| 1 | 0.09 | 0.10 | 0.08 | 0.08
| 3 | 0.19 | 0.26 | 0.20 | 0.21
| 5 | 0.26 | 0.37 | 0.29 | 0.30

Table 8 shows that alternative dividend processes help the model better fit the evidence of time-varying excess returns. It shows regressions of future dividend growth and future excess returns on today's price-dividend ratio, using alternative dividend processes. As seen in Section 5, the benchmark model (`neoclassical dividend' column) captures the lack of dividend predictability at shorter horizons, although the amount of predictability is somewhat overstated at the 5-year horizon. The calibrated dividend column shows that this mild overstatement is fixed with a more realistic dividend process. The calibrated dividend claim produces essentially zero dividend predictability, with a maximum of 0.02 at the 5-year horizon. This result is an intuitive extension of the consumption claim results. As seen in the `consumption claim' column, consumption growth is unpredictable at all horizons. Since the calibrated dividend claim is essentially a levered claim on consumption, it inherits this lack of predictability. This lack of dividend predictability shows up in excess return forecasts. Both the calibrated dividend claim and consumption claim produce strongly predictable excess returns.

The apparent lack of cash flow predictability is due to the uninformativeness of asset prices, rather than a fundamental lack of predictability in the economy itself. This effect of conditioning information is seen in Figure 7, which shows plots of expected consumption growth computed using the model's laws of motion, that is, conditioning on all information available in the economy. The left panel shows consumption growth against the price-dividend ratio from the benchmark `neoclassical' dividend process. The panel shows a cloud of consumption growth observations, and no relationship with asset prices. But the significant vertical dispersion of the cloud indicates that consumption growth is very predictable: it's just that asset prices are not very informative about consumption growth. Note that neither the price-dividend ratio nor the consumption-wealth ratio predicts consumption growth (see the `consumption claim' column of Table 8). On the other hand, the model's state variables are informative. The middle and right panels show that consumption growth varies systematically with capital, surplus consumption, and productivity. Consumption growth is decreasing in capital and productivity, but increasing in surplus consumption. These relationships are more complicated than these panels suggest however, since the state variables are correlated in simulations.

#### Figure 7: Predictability of Consumption Growth: Effect of the Conditioning Set.

Figures are annualized and computed from the model's laws of motion. Price-dividend ratio is for a claim on the firm's dividends. The left panel shows scatterplots from model simulations. The middle and right panels show consumption growth as a function of state variables.

##
6.7 Comparative Statics

The quantitative results come from estimated parameter values, but as with all econometric methods, the point estimates can be sensitive to choices of the econometrician. This section investigates the effect of changing the parameter values. The comparative statics also confirm the intuition developed earlier in the paper.

Table 9 shows key moments from these comparative statics exercises. Each column examines moments generated by models where only one of the parameter values is changed from the estimation described by Table 1. Three different parameter changes are examined: lower persistence of habit, weaker steady state habit, and lower capital adjustment costs. The magnitude of the perturbations are chosen to be the smallest change that produces a clearly recognizable deviation from the estimated results. This approach helps isolate the direct effect of changing a parameter value from its interaction with other elements of the model.

#### Table 9: Comparative Statics

Identifying Moments

Moment | Data: 1948-2011 | Estimated | Lower persistence of habit | Weaker steady state habit | Lower adjustment costs
| 0.98 | 0.98 | 3.31 | 8.25 | 4.66
| 0.48 | 0.48 | 0.39 | 0.31 | 0.38
| 16.07 | 16.07 | 24.22 | 14.77 | 8.92
| 0.47 | 0.47 | 0.43 | 0.64 | 0.39
(%) | 1.50 | 1.50 | 1.63 | 1.53 | 1.49

Untargeted Moments

Moment | Data: 1948-2011 | Estimated | Lower persistence of habit | Weaker steady state habit | Lower adjustment costs
(%) | 6.47 | 6.77 | 7.46 | 3.84 | 3.11
AC1 | -0.03 | -0.07 | -0.12 | -0.05 | -0.04
| 2.24 | 2.96 | 9.20 | 5.20 | 2.30
AC1 | 0.56 | 0.88 | 0.66 | 0.86 | 0.93
| 3.42 | 2.67 | 2.23 | 2.15 | 2.63
| 0.43 | 0.46 | 0.39 | 0.33 | 0.39
AC1 | 0.95 | 0.90 | 0.78 | 0.89 | 0.92
, 1-year | 0.08 | 0.10 | 0.07 | 0.04 | 0.06
, 3-year | 0.19 | 0.26 | 0.14 | 0.10 | 0.17
, 5-year | 0.26 | 0.37 | 0.17 | 0.15 | 0.25

Figures are annual. `Estimated' represents parameter values from Table 1. All other columns use the estimated values but with one parameter changed. `Lower persistence of habit' sets . `Weaker steady state habit' sets . `Lower adjustment costs' sets . Details of the data are found in Appendix 8.2.

### 6.7.1 Lower persistence of habit

The third column of Table 9 examines a model where the persistence of habit is lowered from the estimated value of 0.963 to 0.800. This parameter is important because the strong persistence of habit is one way in which this model deviates from previous models with habit and production. The high estimated persistence means that habit today depends on a very long history of consumption. In contrast, habit in Jermann (1998) and Boldrin et al. (2001) depends only on the last quarter's consumption.

The table shows the high persistence is critical. In the lower persistence model, the volatility of the risk free rate triples from the estimated value of 2.96% to 9.20%, bringing the model in line with the high risk-free rate volatility of Jermann (1998) and Boldrin et al. (2001). Figure 8 explains why. The figure plots the risk-free rate decomposition of equation (28) for the low persistence of habit model. The figure shows that the intertemporal substitution effect is still countercyclical and the precautionary saving effect is still procyclical. However, the two channels no longer cancel each other out quantitatively. The intertemporal substitution effect is extremely countercyclical and overwhelms precautionary saving effects. This change can traced to the lower persistence of habit. A low persistence of habit means that habit will strongly mean revert tomorrow. This means that, in bad times, there is a pronounced desire to borrow in order to consume today.

#### Figure 8: Decomposition of the Risk-Free Rate: low persistence of habit.

. All other parameter values are from the estimation (Table 1). `Intertemporal Substitution' represents the 2nd and 3rd terms of equation (28). `Precautionary Saving' represents the last term. Computed from model's laws of motion. Risk-free rate is in annualized %. Capital in all panels is fixed at the mean capital stock.

Another consequence of faster mean reversion is that precautionary saving effects become weaker. Faster mean reversion means that habit will recover quickly from bad times, and thus there is less countercyclical uncertainty about savings. This weakening of precautionary savings effects is reflected in a reduction in the amount of time-varying risk premia. The third column of Table 9 shows that the amount of time-variation in risk premia falls. 's from regressions of future excess returns on the log price-dividend ratio drop significantly.

The persistence of habit has a strong effect on many other asset price moments. The parameter is identified with the volatility of excess returns, and, as expected, the low persistence model strongly overpredicts this volatility. Another effect is that the persistence of the risk-free rate and the price-dividend ratio drops. Intuitively, the persistence of preferences is reflected in the persistence of asset prices.

### 6.7.2 Weaker steady state habit

The fourth column of Table 9 raises steady state surplus consumption from the estimated value of 0.063 to 0.120. This comparative static brings the model closer to CRRA utility. Roughly speaking, is an economy with no habit, so this model is still very far from the standard model.

Weakening steady state habit lowers the volatility of marginal utility of the household (see equation (21)), and thus has the direct effect of lowering the Sharpe ratio from the estimated value of 0.48 to 0.31. This reduced Sharpe ratio has the obvious effect of reducing the equity premium from the estimated value of 6.77% to 3.84%. The Sharpe ratio does not drop as much as one might expect, however, since the volatility of consumption increases. This increase in consumption volatility is due to a weaker general equilibrium consumption smoothing effect (see Section 6.4).

Weakening steady state habit has the additional effect of weakening precautionary savings effects. With weaker habit, there is less need for precautionary savings, and the risk-free rate jumps nearly ten-fold. This weakening of precautionary savings effects can be seen in the return forecasting regressions too. The 's from regressions of future returns on the price-dividend ratio all drop by more than half.

### 6.7.3 Lower adjustment costs

The final column of Table 9 examines the effect of lowering capital adjustment costs. The adjustment cost parameter is lowered from the estimated value of 75.00 to 40.00. This comparative static helps compare my model with Lettau & Uhlig (2000), who also examine external habit in a production economy. My model deviates from theirs in two important ways. The first is that I include capital adjustment costs, while Lettau and Uhlig assume costless adjustment. The second is that I use a non-linear, global solution method, while Lettau and Uhlig linearize the model around the non-stochastic steady state. This comparative static shows that capital adjustment costs make a large impact on the model results and that lowering the costs brings my model closer to Lettau & Uhlig (2000).

The table shows that lowering adjustment costs reduces the relative volatility of consumption to output falls from the estimated value of 0.47 to 0.39. This is consistent with the intuition from Section 6.4. Low adjustment costs reduce the incentive for smooth investment and encourages volatile consumption.

This decrease in consumption risk is reflected in a lower Sharpe ratio. The equity premium drops by half. With lower consumption risk, the precautionary motive is weakened and the risk-free rate rises by a factor of five. This decrease in the precautionary motive is also seen in lower 's from return forecasting regressions.

#
7 Conclusion

Time-varying uncertainty plays a key role in many theories of aggregate asset prices. Previous papers take time-varying uncertainty as exogenous. This paper shows that time-varying uncertainty is the natural result of a fundamental economic motive: the desire for precautionary savings. These motives make investors very sensitive to shocks in bad times, and lead to countercyclical consumption volatility. This mechanism, when amplified by external habit preferences, quantitatively accounts for the empirical evidence of time-varying risk premia. The model also provides a parsimonious description of asset prices and the real economy. The estimated model matches a long list of facts about the aggregate stock market, safe government bonds, as well as consumption, output, and investment.

# 8 Appendix

##
8.1 Proofs for the Two-period Model

Proof of Proposition 1 A change of variables shows that this model is equivalent to a standard consumption-savings problem. Shift consumption by assigning . Then the date 1 consumption rule can be written as

| (30)

where solves a simple consumption-savings problem with wealth and certain future income of :

| (31)

and for ease of notation I suppress the subscript 2 on . It turns out that for , random, and CRRA utility, is strictly convex in . That is, with CRRA utility, rate of return randomness, and the introduction of any (even constant) future income is a sufficient condition for generating strict convexity of the consumption function. This is not one of the sufficient conditions shown in Carroll & Kimball (1996), so I will show that it is sufficient in what follows.

The FOC of the shifted problem (31) is

| (32)

Where, for convenience, I've defined the function

Taking of the FOC and rearranging gives

Take another , do some serious algebra, and we get an expression for the convexity of :

| (33)

The first bracket is negative simply because and . To show that the second bracket is (strictly) positive, first note that, due to the CRRA specification, . I will now show that, due to the non-zero future income , . This is an extension of Carroll & Kimball (1996)'s Lemma 4.

Proving requires the following technical Lemma.

Lemma 1 Let for be symmetric matrices with the following properties:

- the diagonals of each are positive
- the off-diagonals of each are all negative
- for every ,

Then for each pair there is some constant such that

Proof. I will first show this for the case where . I'll then use the results to prove the general case. For ease of notation assign and . With some algebra, one can show that

| (34)

A few facts will let us simplify this expression dramatically. First , so those terms all drop out. Then note that, since , we have and , and we can rewrite

and so the last term in equation (34) also drops out. Thus equation (34) implies that , or

| (35)

where is the conjectured constant of proportionality. We just need to show that . To show this, plug and into (35) and we have

This completes the case.

To show the general case, first note that if and satisfy the requirements of the lemma, then also satisfies those requirements. Thus I can apply the results to the general case, where one matrix is and the other matrix is . Moreover, note that if there is a such that , then there is such that . Apply this to all and get the desired result .

Now, back to proving the proposition. I want to show that . Suppose, for contradiction, that . I can write this expression using the determinant of a matrix by defining Where, for ease of notation, . This expression can now be written compactly as

Note that is the weighted sum of many component matricies , and that due to the CRRA specification of , the determinant of each component matrix is zero. Thus is positive semidefinite, so . But our assumption for contradiction says is negative semidefinite, and so it must be that .

Now I use Lemma 1. The lemma states that if , all of the component matrices must be proportional to one another. This means that for any states and the ratio of the diagonal terms of the corresponding matrices is equal, that is, for any and ,

which is a contradiction, since is random. Note that the presence of a nonzero income is critical because otherwise, I could not move from the fourth line to the fifth line in the equations above.

Therefore, , and by equation (33), is strictly concave in , and by equation (30), .

Proof of Proposition 2 I first show that the transformed consumption function satisfies . To show this, note that is linear, but for any , is strictly concave. Thus we can sign the derivative

Assuming that is continuous, this means that there is some such that for any , .

Since is HD1, I can take derivatives to show that

Now note that I am free to choose , so this means, for any , . But by homogeneity, , and thus the third derivative with respect to wealth of the transformed consumption function is negative (as long as ).

Now to finish proving the proposition. Using the transformed consumption function (31) I can relate the desired derivative to the third derivative of the consumption function. First take twice:

Then take

Proof of Corollary 2 Let and be the minimum and maximum of , respectively.

Where the 2nd line uses equation (2), the third line uses .

Then take

##
8.2 Data Details

The data span 1948 thru 2011. Macroeconomic quantities are from the BEA NIPA and BEA fixed asset tables. All data is real and per-capita. Output is simply GDP. Consumption is nondurable consumption plus services consumption. Investment and capital are fixed investment plus durable consumption plus government investment. All quantities are constructed by dividing nominal figures (Table 1.1.5 or Table 3.9.5) by the appropriate price index (Table 1.1.4 or Table 3.9.4), and then dividing by the population (from Table 2.1).

Most studies exclude government investment from their definition of investment, citing the fact that their models do not include government. However, this logic would also imply that one should remove government purchases from GDP, which is not typically done. I choose to keep government purchases in GDP and include government investment in investment as this approach preserves the idea that the data come from a single general equilibrium system. Moreover, this approach is closer to the spirit of Cooley & Prescott (1995).7 Choice of the empirical definition of investment does not have a significant effect on any of the results.

Asset price data is taken from CRSP. The equity is the CRSP value-weighted index. Price-dividend ratios are computed annually using the stock market reinvestment assumption. As discussed in Cochrane (2011), this aggregation method preserves the Campbell-Shiller identity, which is useful for identifying the source of asset price fluctuations. CRSP returns are deflated using the CPI, also obtained from CRSP. The risk-free rate is computed using a forecast of the ex-post real return of the 90-day T-bill following Beeler & Campbell (2009). Following Beeler & Campbell (2009), the ex-post real return is calculated by deflating the 90-day nominal T-bill yield using seasonally adjusted CPI from the BLS. The forecast is constructed by regressing next quarter's ex-post real return on today's nominal 90-day yield and the mean inflation rate over the previous year.

##
8.3 Simulated Method of Moments Details

This section spells out details of the SMM method. I use a different notation than the brief discussion in the text in order to be more precise.

### 8.3.1 Econometric Details

I transform the parameters so that I do not need to be concerned about corner solutions. For example, rather than estimate , I estimate logit . After the estimation is done, I find the point estimates of the original parameters by inverting the transformation, and the standard errors by the delta method.

To be explicit, let be the vector of original parameters and by the transformed parameters. Then

|  | (36)
|  | (37)

so the asymptotic variance of the original parameters can be estimated by

| (38)

I also use the delta method on the moment errors, because some of the "moments" I use are not moments in the traditional sense (i.e. the Sharpe Ratio) but are more formally described as transformations of moments. That is, let the GMM objective be

where are some moment errors.

where

| is a set of moments from simulation s |
| transforms the moments into some other statistic |

and is the total number of simulations. This function has no effect on the GMM formula for the asymptotic variance

where

but it does have an effect on the estimator of the variance of the moment errors .

That is, we simply adjust the Duffie & Singleton (1993) estimator by applying the delta method to the spectral density. To see how this works, note that the moment error can be approximated by

Thus we just need to adjust the traditional Newey-West estimator by .

### 8.3.2 Numerical Details

To calculate the moment errors, I simulate the model 1000 times ( ). The initial state is set to be the median state for a long simulation close to the estimated parameter values. Derivatives are computed with a two-sided finite difference.

I optimize using Levenberg-Marquardt (LM). I choose this method rather than the more commonly used simulated annealing method for two reasons. The first is that the moment function does not display an extreme number of local minima, which is where simulated annealing has an advantage. With a relatively smooth objective function, a method which uses derivative information is much more efficient. Derivative information is particularly helpful in the GMM setting with small residuals, since the Gauss-Newton method provides a quick positive semi definite approximation of the Hessian. Speed is important as the estimation takes around 24 hours with a very good guess. Another advantage of LM is robustness. Simulated annealing tends to be very sensitive to the choice of the annealing schedule (Press et al. (1992).

I begin with an LM parameter of 1000, and use a simple algorithm for adjusting the parameter: if the new function value is a good enough improvement, I decrease the LM parameter by a factor of 10. Otherwise, I set it back to 1000. "Good enough" is judged by the difference between the improvement in the objective and the predicted improvement according to a quadratic model.

Since I use exact identification, the maximum moment error provides a clean convergence criterion. I consider the algorithm converged if the maximum moment error is less than 1.0E-4.

##
8.4 The Interpretation of Habit

Here I describe how deviating from Campbell & Cochrane (1999)'s specification of raises some questions regarding the interpretation of habit in the model. The key issue is that the constant of my model can make habit decrease in response to an increase in consumption. This violates some traditional notions of habit.

This issue can be illustrated by taking the derivative of log habit with respect to log consumption :

Thus if is large enough, will be negative.

The preferences of this paper still preserve the standard notion of habit in that is a geometric average of previous consumption. This can be seen by following the analysis of Campbell & Cochrane (1999) found in Campbell (2003). I can log-linearize the log surplus consumption ratio around the steady state:

| (39)

Plugging this into the definition of the habit process (5), I find the link between habit and historical consumption

| (Constants) | (40)
| (Constants) | (41)

This informal demonstration is verified by simulated data. In the simulated data, habit is highly correlated with consumption. The contemporaneous correlation is 0.978 and the correlation with lagged consumption is 0.983. Habit growth and consumption growth are moderately correlated. The correlation between and is 0.411.

That habit should move non-negatively with consumption everywhere is not required if one entertains a very slow-moving, historical average of consumption as responsible for our current reference point for consumption. Moreover, Campbell & Cochrane (1999)'s specification is also vulnerable to this this issue. Ljungqvist & Uhlig (2009) show that while habit moves positively with small movements in consumption, it can move negatively with large movements.

The issue illustrated in this section is related to Campbell & Cochrane (1999)'s three requirements on . They require (i) the risk-free rate is constant, (ii) habit is predetermined at the steady state surplus consumption, and (iii) habit is predetermined near the steady state. The first assumption is not critical for making habit move non-negatively with consumption. In my model, (ii) is satisfied, but (iii) is not. (iii), in combination with Campbell & Cochrane (1999)'s specification for results in for all .

##
8.5 Solution Method Details

Euler Equation To be explicit about the firm's Euler equation, let be the transition matrix for the discretized productivity process. The firm's problem is to find capital policy to solve

subject to

The FOC for investment and the envelope condition are:

which together produce the Euler equation

Impose the fact that the household does not value leisure and consistency, and we have

|  | (42)

Where

|  | (43)

and the evolution of surplus consumption satisfies

| (44)

The projection algorithm looks for cubic spline coefficients which solve equations (42), (43), and (44).

Solving for asset prices To find asset prices, the firm's Bellman equation, with optimal values plugged in, is:

In equilibrium, , , , and , so

Let's define . The above equation suggests that can be found by repeatedly applying the above equation as an operator (using the law of motion for capital ).

### 8.5.1 Approximations

I approximate the autoregressive process for productivity with a 13 point Markov Chain using the Rouwenhorst method. I approximate the law of motion for capital in the and directions using a two-dimensional cubic spline. The spline is of 6th degree in the direction and 14th degree in the direction. The spline breakpoints are log-spaced in both the and directions. I find that increasing the degree to the 14th in the direction has no material impact on the quantitative results.

In a projection method, one must define what it means to satisfy the Euler equation. I use the collocation method, which specifies that the Euler equation should hold exactly at a set of points (collocation nodes) in the and domain. I choose these nodes to be the standard nodes for splines using knot averaging. I consider the algorithm converged if, across these collocation nodes, the maximum absolute Euler equation residual, expressed as , is less than 1.0E-8. I search for spline coefficients which satisfy this condition by using Broyden's method.

## 8.6 Adjustment Costs and Accounting

Many papers specify adjustment costs in the following manner:

(e.g. Jermann (1998), Gourio (2009), Kaltenbrunner & Lochstoer (2010), Guvenen (2009), among others). This formulation preserves the traditional Cobb-Douglas formulation of output:

This formulation, however, deviates from the standard accounting treatment of investment and capital. The standard treatment specifies that end-of-period capital is beginning-of-period capital plus investment less depreciation. With geometric depreciation, this translates into the standard, adjustment-cost-free formulation of capital accumulation:

I choose to preserve this accounting identity. As a result, capital adjustment costs are pushed into output:

Adj Cost |

Fortunately, both choices result in the same capital and consumption allocations. For example, Gourio (2009) uses

From these expressions, we have capital evolution and consumption

which is identical to my formulation with

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#### Footnotes

* I am indebted to Aubhik Khan, Xiaoji Lin, René Stulz, Julia Thomas, and especially Lu Zhang for their training, advice, and inspiration. I would also like to thank John Cochrane, Urban Jermann, Pok Sang Lam, and Stijn Van Nieuwerburgh for comments which have significantly improved the paper, as well as seminar participants at the University of Arizona, Federal Reserve Board, and the Ohio State University. This paper was previously circulated under the titles "External Habit in a Production Economy," and "Prudential Uncertainty Causes Time-Varying Risk Premiums." The views expressed herein are those of the author and do not necessarily reflect the position of the Board of Governors of the Federal Reserve or the Federal Reserve System. Return to Text
1. Early versions of these models often do not feature time-varying volatility (e.g. Bansal & Yaron (2004) model I, Rietz (1988), Barro (2006), Constantinides (1990)), though later versions add this feature in order to address time-varying risk premia. Return to Text
2. I am grateful to Pok-Sang Lam for teaching me his version of the Carroll & Kimball (1996) proof. Return to Text
3. Campbell & Cochrane (1999) choose to fit the Sharpe ratio and to produce a constant risk-free rate, but that cannot be done here because I set as a constant in order eliminate exogenous time-varying volatility. Return to Text
4. The moments I use are not moments in the strict sense (i.e. the Sharpe ratio), and I estimate transformations of the parameters in order to avoid corner solutions. This leads to slightly more complicated formulas than those presented here. These details can be found in the Appendix 8.3. Return to Text
5. The log-SDF is and the habit process is . Plug the habit process into the log-SDF, then assume that the SDF is log-normal, and we have . Return to Text
6. If is normal,

Var |

Then just plug in the log SDF and habit process . Return to Text
7. Cooley & Prescott (1995) say that "Our economy is very abstract: it contains no government sector, no household production sector, no foreign sector and no explicit treatment of inventories. Accordingly, the model economy's capital stock, , includes capital used in all of these sectors plus the stock of inventories. Similar, output, , includes the output produced by all of this capital." Return to Text
♣ This version is optimized for use by screen readers. Descriptions for all mathematical expressions are provided in LaTex format. A printable pdf version is available. Return to Text