Idiosyncratic Risk and the Business Cycle: A Likelihood Perspective

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1 Idiosyncratic Risk and the Business Cycle: A Likelihood Perspective Alisdair McKay Boston University April 214 Abstract This paper asks whether uninsurable idiosyncratic income risk affects aggregate consumption over the business cycle. I explore this question using an incomplete markets business cycle model in which the extent of idiosyncratic risk varies over the business cycle in line with recent empirical findings. I present a method for estimating general equilibrium incomplete markets models with rich household heterogeneity using likelihood-based methods. I use the estimated model to construct a counterfactual series for aggregate consumption under complete markets. The results show that time-varying idiosyncratic risk has substantial effects on the dynamics of aggregate consumption. Had markets been complete, the variance of aggregate consumption growth would have been up to 32% lower and the drop in consumption in the Great Recession up to 1.9 percentage points smaller. amckay@bu.edu. I would like to thank Susanto Basu, Simon Gilchrist, Jonathan Parker, Zhongjun Qu, and Ricardo Reis for helpful discussions and Carlos Ramos for excellent research assistance. I am also grateful to seminar participants at PSE/IMF Workshop on Advances in Numerical Methods, BU/BC Greenline Macro Meeting, FRB Chicago, MIT, Brown, ASSA 214, Columbia, UCL, and the Philadelphia Workshop on Macroeconomics.

2 1 Introduction Are idiosyncratic income risks important to the business cycle dynamics of aggregate quantities? Krusell and Smith (1998) provided a largely negative answer to this question. Their results show that the business cycle dynamics of an incomplete markets economy are generally very close to those of a representative agent economy. 1 I revisit this question with a focus on cyclical variation in uninsurable risks to an individual s long-term earnings potential. Recent empirical work using large panel datasets on individual earnings portray recessions as times when households face substantially larger downside risks to their earnings prospects. Moreover, these risks appear to have highly-persistent effects on household earnings. Davis and von Wachter (211) show that earnings losses from job-displacement are large, longlasting, and roughly twice as large when the displacement occurs in a recession as opposed to a expansion. The differential impact of displacement in a recession is evident even twenty years after the event occurred. Similarly, Guvenen et al. (213) show that the distribution of five-year earnings growth rates displays considerable pro-cyclical skewness meaning severe negative events are more likely in a recession. This type of cyclical risk has previously been estimated by Storesletten et al. (24) using PSID data. Those authors find the standard deviation of idiosyncratic labor income shocks is much larger during recessions than it is during expansions. The purpose of this paper is to investigate the implications of cyclical variation in idiosyncratic income risks for the business cycle dynamics of aggregate consumption. To do so, I develop a general equilibrium business cycle model with uninsurable idiosyncratic shocks to earnings. The idiosyncratic shock process is calibrated to match cyclical variation in income changes documented by Guvenen et al. (213). I estimate the model using aggregate data and recover the structural aggregate shocks that generated the observed time series for aggregate consumption, investment and labor market conditions. Feeding these shocks into an alternative model with a full set of insurance markets yields a counterfactual prediction 1 One exception in their paper is that the stochastic-β version of the model displays an increased correlation of consumption and output. 1

3 for consumption growth under complete markets. The results show that the combination of market incompleteness and cyclical idiosyncratic risks to long-term earnings prospects have a substantial effect on aggregate consumption dynamics. In the complete markets counterfactual, the variance of consumption growth is up to 32% lower than it is in the incomplete markets economy. The differences in consumption growth accumulate to substantial differences in the dynamics of aggregate consumption during recessions. For example the cumulative effect of uninsurable idiosyncratic risk during the Great Recession is such that aggregate consumption would have been up to 1.9 percentage points higher in the first quarter of 29 under complete markets. Methodologically this paper partially integrates two literatures: it presents a method by which incomplete markets business cycle models in the style of Krusell and Smith (1998) can be estimated using likelihood-based methods in the style of Sargent (1989) and Schorfheide (2) and others. The main idea behind the methodological contribution is to express the dynamics for aggregate variables implied by the incomplete markets model as a dynamic system in a small number of aggregate variables. That the aggregate dynamics of the incomplete markets model can very accurately represented in terms of a small number of aggregate variables was shown by Krusell and Smith (1998). I apply this idea to the statistical analysis of the model. Using this approach, one can apply the tools of statistical filtering to incomplete markets models to construct the likelihood function for the aggregate data and conduct inference on unobservable variables. This methodological approach is useful for the question at hand for three reasons. First, it allows me to investigate the contribution of idiosyncratic risk to particular episodes, such as the Great Recession, by backing out the structural shocks that generated the data. Second, it allows me to decompose the effect of idiosyncratic risk on aggregate dynamics according to its effect on the impulse response functions for the individual aggregate shocks. Finally, it allows me to estimate the persistence of the aggregate shock processes. As I explain, the persistence of the shocks is crucial because it determines the size of the wealth effect generated by a given innovation and therefore the size of the resulting change in consumption. 2

4 This is not the first paper to estimate an incomplete markets model. Gourinchas and Parker (22) and Guvenen and Smith (21) estimate versions of the incomplete markets model focusing on cross-sectional variation in a stationary environment. Heathcote et al. (212) estimate an incomplete markets model that allows for time-varying wage and preference distributions. Their focus is on secular trends in inequality as opposed to business cycle fluctuations in aggregate quantities. The goal here is to estimate an incomplete markets business cycle model using aggregate time series data. The contribution of idiosyncratic risk to the dynamics of aggregate quantities has been studied previously and the most closely related contributions are Krusell and Smith (1998), Challe and Ragot (213), and Ravn and Sterk (213). This paper is primarily concerned with cyclical variation in highly-persistent earnings shocks as measured in the empirical papers mentioned above. By contrast, preceding work has incorporated cyclical variation in idiosyncratic risk through changes in the risk of unemployment, which is generally a short lived shock to earnings. 2 The highly-persistent earning shocks that I study are notable in that they alter the dynamics of aggregate consumption even when households are relatively far from the borrowing constraint. This is not true of employment shocks, which only have aggregate effects when the distribution of wealth is highly skewed (Krusell and Smith, 1998). Other implications of the cyclical variation in idiosyncratic risks to long-term earnings potential have been studied. For example, Storesletten et al. (27) investigate the asset pricing implications of this type of risk. The welfare cost of business cycles in the presence of these risks have been analyzed by Storesletten et al. (21); Krebs (23, 27). However, it does not appear that the consequences for the business cycle dynamics of aggregate quantities have been studied in the literature. The paper is organized as follows: Section 2 presents the model. Section 3 discusses the method to compute the likelihood function for incomplete markets models. Section 4 discusses the parameters that are estimated and the data used to estimate them. The 2 This distinction is not quite as stark as it might seem as Ravn and Sterk (213) allow for long-term unemployment spells. The calibration of the shock processes and cyclical variation in the risks is very different in that approach. 3

5 baseline results appear in Section 5 and Section 6 extends the model to generate more realistic heterogeneity in wealth holdings. Section 7 compares estimated parameter values from different model specifications. The paper concludes with Section 8. 2 Model I analyze a general equilibrium model with heterogeneous households and aggregate uncertainty. At the aggregate level, the model is similar to that of Krusell and Smith (1998). At the microeconomic level, I incorporate time-varying idiosyncratic risk with an income process similar to the one estimated by Guvenen et al. (213). 2.1 Preferences and endowments There is a unit continuum of households who survive from one period to the next with probability 1 ω. Each period, a mass ω of households is born leaving the total population unchanged. Households seek to maximize homothetic preferences ( t ) E ˆβ t c 1 γ e qτ t 1 γ, t τ= where c t is the household s consumption in period t. The economy grows at rate g and c t c t /e gt will denote consumption relative to trend. q t is a preference shock that follows q t = ρ q q t 1 + ɛ q,t. Throughout the paper, I will assume an aggregate shock ɛ z,t, for any z, is normally distributed with mean zero and standard deviation σ z. I use β ˆβ(1 ω) to denote the joint effect of pure time preference and discounting due to mortality risk. Households can be either employed (n = 1) or unemployed (n = ) and transition between these two states exogenously. Let λ and ζ be the job-finding and -separation rates, respectively. I assume that aggregate shocks occur at the start of a period and labor market 4

6 transitions occur under the new transition rates. The job-finding and -separation rates are exogenous and follow logistic AR(1) processes 1 λ t = 1 + e ˆλ t (1) ˆλ t = (1 ρ λ ) ˆλ + ρλˆλt 1 + ɛ λ,t (2) 1 ζ t = (3) 1 + e ˆζ t ˆζ t = (1 ρ ζ ) ˆζ + ρζ ˆζt 1 + ɛ ζ,t. (4) The unemployment rate, u, evolves according to u t = (1 λ t )u t 1 + ζ t (1 u t 1 ), (5) If employed, a household exogenously supplies e y efficiency units of labor, where y is the households individual efficiency. The cross-sectional dispersion in efficiency units could be due to differences in wage or due to differences in hours. For lack of a better term, I will refer to y as skill. This skill evolves according to y = θ + ξ, θ = θ + η, where ξ is a transitory income shock distributed N(µ ξ, σ ξ ). The permanent shock, η, is drawn from a time-varying distribution the tails of which vary over the business cycle in such a way to generate pro-cyclical skewness as shown by Guvenen et al. (213). Like Guvenen et al., I assume η is drawn from a mixture of normals. Specifically N(µ 1,t, σ η,1 ) with prob. 1 2p η N(µ 2,t, σ η,2 ) with prob. p N(µ 3,t, σ η,3 ) with prob. p. 5

7 µ 2,t and µ 3,t will control the tails of the distribution and I assume they are driven by a single driving process, x t : µ 2,t = µ 2 + µ 1 2x t (6) µ 3,t = µ 3 + µ 1 3x t. (7) This income process differs from the one estimated by Guvenen et al. in that it mixes three normals as opposed to two and the extent of idiosyncratic risk is a continuous function of the driving process x t as opposed to regime switching. Throughout this paper, I use the job separation rate as the cyclical indicator, x t = ζ t. This choice is motivated in part by the work of Davis and von Wachter (211) who show that job loss in mass layoff events is associated with large and long-lasting reductions in income and it is plausible that times of labor market turbulence lead to both job separations and more negative outcomes for long-term earnings. I show below that this assumption does fairly well in accounting for the cyclical variation in measured income risks reported by Guvenen et al.. The parameter µ 1,t controls the center of the distribution for η. I make µ 1,t a function of the other parameters of the distribution such that Ee θ = 1 in all periods. 3 Similarly, I choose the constant parameters of the distribution for ξ, µ ξ and σ ξ, such that E[e ξ ] = 1. As employment is independent of y, the aggregate labor input is L t = (1 u t ). It would be natural to assume that there is a correlation between shocks to skill and shocks to employment. I have experimented with this and found that it has little impact on the results because employment shocks are quite transitory. When a household dies, it is replaced by a household with no assets and θ normalized 3 This requires (1 2p)e µ1,t+σ2 η,1 /2 + pe µ2,t+σ2 η,2 /2 + pe µ3,t+σ2 η,3 /2 = 1. 6

8 to. The mortality risk allows for a stationary cross-sectional distribution of productivity along with permanent shocks to θ. The unemployment rate among newborn households is the same as prevails in the surviving population at that date. This paper investigates the implications of these cyclical idiosyncratic risks. A related question concerns the source of these risks. While some progress has been made here (Huckfeldt, 214), it is still not well understood. The perspective that I adopt here is that the income shocks are uninsurable and their realizations are unknown before they occur. 2.2 Technology and markets A composite good is produced out of capital and labor according to Ȳ t = A t Kα t ( e gt L t ) 1 α, (8) where g is the trend growth rate. Throughout the paper, I normalize variables by trend productivity. Defining K t K t /e gt and Y t Ȳt/e gt, Eq. (8) becomes Y t = A t K α t L 1 α t, TFP, A t, evolves according to log A t = ρ A log A t 1 + ɛ A,t. Capital depreciates at rate δ and evolves according to C t + e g K t+1 = Y t + (1 δ)k t. 7

9 The factors of production are rented from the households each period at prices that satisfy the representative firm s static profit maximization problem W t = (1 α)a t K α t L α t (9) R t = αa t Kt α 1 Lt 1 α + 1 δ. (1) Here R is the return on capital. Households save in the form of annuities and the return to surviving households is R t R t /(1 ω). I assume that savings must be non-negative due to borrowing constraints. Other than self-insurance, the only mechanism for risk-sharing is an unemployment insurance scheme. The pre-insurance labor income of a household with characteristics (θ, ξ, n) is W e θ+ξ n. I assume that all households in the economy share unemployment risk by pooling their incomes and then allocating after-transfer incomes such that a household with characteristics (θ, ξ, n) receives l(θ, ξ, n, W, u) W e θ+ξ [n + b u 1 u (1 n)] 1 u + b u u, (11) where the parameter b u controls the degree of insurance against unemployment shocks. The final term in this expression plays the role of a tax that balances the unemployment insurance budget period by period. 2.3 The household s decision problem The individual state variables of the household s decision problem are its cash on hand, call it a, its permanent skill, θ, and its employment status. The aggregate states are S {A, ˆλ, ˆζ, q, Γ}, where Γ is the distribution of households over the state space from which one can calculate both K and L. The household s decision variables are consumption and endof-period savings, k. k is normalized by the level of trend productivity in the next period 8

10 so the current resource cost of saving k is e g k. The household s decision problem is then 4 { V (a, n, θ, S) = max e q(s) u(a e g k ) + e q(s) e g(1 γ) βe ɛ,ξ,η,n [V k (a, n, θ, S )] } subject to a = R(S )k + l(θ + η, ξ, n, W (S ), u(s )), the law of motion Γ = H Γ (S, ˆλ, ˆζ ) and the other equations that govern the dynamics of the aggregate state. The household s Euler equation and budget constraint are c γ = e γg βe [e ] q R c γ c + e g k = Rk + l(θ, ξ, n, W, u). Analysis of the model can be simplified by normalizing a household s consumption and income by it s permanent income e θ leading to [ ( c ) ( ) ] γ = e γg βe e q γη c R γ e θ e θ c k + eg eθ e = R k θ e + l(, ξ, n, W, u) a θ e. θ Making use of this normalization allows one to reduce the size of the state space by only keeping track of a household s assets relative to its permanent income. A permanent-transitory income process and finite lifetimes are necessary to achieve this simplicity while maintaining a finite variance of income in the cross-section and this approach has been used previously by Carroll et al. (213). While permanent income can be eliminated as a state of the house- 4 Due to economic growth, the household s discount factor is modified by the term e g(1 γ). If c t is consumption relative to trend, then total consumption is c t e gt. The utility function then becomes E t= β t (c te gt ) 1 γ 1 γ = E t= (βe g(1 γ)) t c 1 γ t 1 γ. 9

11 hold s problem, the risk to permanent income still affects consumption decisions through the η term in the Euler equation and through its effect on the denominator of normalized cash on hand. 2.4 Equilibrium Let f be the optimal decision rule for k in the household s problem. Aggregate savings are K = n f (Rk + l(θ, ξ, n, W, u), n, S) Φ(dξ)Γ(dk, dθ, n), k θ ξ where Φ is the distribution of ξ. Given a set of exogenous stochastic processes for A, ˆλ, ˆζ, and q, a recursive competitive equilibrium consists of the law of motion for the distribution, H Γ, household value function, V, and policy rule, f, and pricing functions W and R. In an equilibrium, V and f are optimal for the household s problem, R = R/(1 ω) and W satisfy (9)-(1), and H Γ is induced by f and the idiosyncratic income process. 3 Calculating the likelihood function The literature has developed several algorithms to solve incomplete markets models with aggregate uncertainty. 5 The difficulty in solving this class of models is that the distribution of wealth is an infinite-dimensional endogenous state variable. It may appear that the inherit complexity of the state of this type of model makes computing the likelihood function a challenging problem. However, the same methods that reduce the state of the system in solving the model apply to the statistical filtering problem and make it straightforward to calculate the likelihood. 5 The most widely used algorithm is the Krusell and Smith (1998) algorithm. See also den Haan (21). 1

12 3.1 State reduction and filtering A statistical filtering problem involves recursive application of a prediction step and an updating step. Let S t be the unobserved state at date t, which evolves according to S t = F (S t 1, ɛ t ) where F is possibly non-linear and ɛ is i.d.d. and possibly non-gaussian. The data vector Z t is generated from a possibly non-linear measurement equation Z t = G(S t, ν t ), where ν is an i.i.d. possibly non-gaussian measurement error. Some prior belief about the state is given by Pr(S ). The prediction step is then Pr(S t Z 1:t 1 ) = Pr(S t S t 1 ) Pr(S t 1 Z 1:t 1 )ds t 1, where Z 1:t 1 is the data observed up to date t 1. The updating step is Pr(S t Z 1:t ) = Pr(Z t S t ) Pr(S t Z 1:t 1 ) Pr(Z t Z 1:t 1 ) where Pr(Z t Z 1:t 1 ) = Pr(Z t S t ) Pr(S t Z 1:t 1 )ds t is the normalizing constant and also the likelihood of the data at date t conditional on the preceding data. In these equations, the probability distributions depend on the functions F and G and on the properties of the shocks ɛ t and ν t, which in turn depend on the vector of parameters to be estimated, call it Θ. The log likelihood of the data Z 1:T is then given by T log Pr(Z t Z 1:t 1 ; Θ). t=1 11

13 For the model presented in Section 2, the state S t includes the distribution of wealth and so one cannot immediately apply standard filtering methods to the problem at hand. Under two conditions, however, we can replace the state S t with a low-dimensional state vector S t and proceed with standard techniques. The first condition is that the probability distribution Pr(Z t S t ) can be approximated with Pr(Z t S t ). In the problem at hand, one does not need to know the distribution of wealth in order to compute aggregate output and consumption. Suppose L t, K t and K t+1 are given, 6 one can then solve for C t and Y t from Y t = A t K α t L 1 α t C t = Y t + (1 δ)k t e g K t+1. The probability distribution of current-period aggregate data only depends on the evolution of K and L = 1 u and not on Γ. Therefore in the updating step it is not necessary to include Γ in the state as it is sufficient to include K and u. If the observable data include disaggregated quantities, such as percentiles of the distribution of wealth or the number of borrowing constrained households, one may need to include more information in S t. Incorporating data of this kind would be a very useful direction for future research. The second condition that the model must satisfy is that the distribution Pr( S t S t 1 ) can be approximated by Pr( S t S t 1 ) so it is not needed to keep track of the full state for the prediction step. This is the same idea that underlies all algorithms to solve incomplete markets models with aggregate uncertainty: it is impossible to track the entire distribution of idiosyncratic state variables so these algorithms find ways of replacing that distribution with some lower-dimensional representation. Most famously, Krusell and Smith (1998) showed that future values of the capital stock can be forecast very well using only the first moment of the distribution. So again, S contains K and not Γ. 6 As K t+1 is determined at date t, it can be included in S t and S t. 12

14 3.2 Application To solve the model I use the Krusell-Smith algorithm in which the distribution of wealth is summarized by a small number of moments, in particular the first moment, K. The dynamics of the capital stock can then be expressed in terms of the law of motion h log K = h(a, ˆλ, ˆζ, q, û, log K), where the distribution of wealth is replaced by log K and the inverse-logistic transformation of the unemployment rate, û. As usual, the solution algorithm solves for h by making a guess for h, solving the household s problem, simulating a population of households, and updating the guess for h. I assume that h is linear in its arguments, which in this application yields highly accurate forecasts of the future capital stock. Where the algorithm deviates from the approach taken in Krusell and Smith (1998) is in the approach to solving the household s decision problem for a given h. I approximate the household s decision problem with a function that is non-linear with respect to idiosyncratic states, but linear in aggregate states. 7 The reason for adopting this approach as opposed to a fully non-linear solution method as in Krusell and Smith (1998) is to improve the speed of the computations as I will need to solve the model repeatedly during estimation. Empirical DSGE models typically include a large number of persistent shocks that result in a considerable number of aggregate state variables making fully non-linear solutions computationally costly. As the resulting approximate policy rule is non-linear in the individual s state variables I am able to capture the precautionary savings motive that arises out of idiosyncratic risk. Precaution against aggregate shocks, however, will not be captured by this solution method. 8 7 This part of the method is closely related to the work of Reiter (29) and Ravn and Sterk (212). Reiter s methods has two components: linearizing the household decision rules with respect to aggregate shocks and summarizing the distribution of wealth with a histogram. I only make use of the first of these components. Ravn and Sterk (212) use a version of the Krusell and Smith (1998) algorithm in which the household decision rules are linear with respect to aggregate states. 8 For a given function h, the household s decision rule f displays the certainty equivalence property with respect to the variance of aggregate shocks. However, these variances affect the dynamics of the distribution of wealth and lead to different solutions for h. Therefore, the full solution of the model does not display certainty equivalence. 13

15 To calculate the likelihood function for aggregate data, suppose one has solved for h as described above. We can then summarize the aggregate dynamics of the economy with the following system of equations log K = h(a, ˆλ, ˆζ, q, û, log K) C + K e g = Y + (1 δ)k Y = AK α L 1 α L = 1 u u = (1 λ )u + ζ (1 u) and the exogenous processes for ˆλ, ˆζ, and A. The system can be reduced to a small number of variables and equations because the function h( ) summarizes the aggregate effect of household heterogeneity. I then take a linear approximation to this system and express the result in state-space form. Under the assumption of Gaussian aggregate shocks, I can then use the Kalman filter to compute the likelihood. Non-linear or non-gaussian models could be analyzed using the methods of Fernández-Villaverde and Rubio-Ramírez (27) but I do not pursue this here. Appendix B discusses the implementation of the methods in more detail and discusses the accuracy of the solution including the quality of the approximation K = h(a, ˆλ, ˆζ, q, û, log K). 4 Parameterization The model parameters can be grouped into three categories (see Table 1). First, there are parameters that relate to preferences and technologies, which I calibrate to values on the balanced growth path. Second, there are the parameters of the income process. I select these parameters to match the data moments reported by Guvenen et al. (213) using a procedure described in Section 4.2. Finally, there are the parameters of the shock processes, which I estimate from the time series data described in Section 4.3. Prior distributions for 14

16 these parameters are listed in Table 1 although these do not exert a strong influence on the results. The persistence of aggregate shocks is important to the cyclical behavior of aggregate consumption. More persistent shocks generate larger wealth effects and therefore larger consumption responses on impact. More persistent shocks also generate more persistent fluctuations in endogenous variables. Both of these effects are important to assessing the contribution of a given shock to aggregate consumption dynamics. 9 Therefore it is important to choose these persistence parameters carefully. 4.1 Calibrated parameters Some of the parameter values are standard in line with Cooley and Prescott (1995): I set the coefficient of relative risk aversion to 2, and set the household s discount factor to match an annual capital-output ratio of The capital share is set to.36 and the quarterly depreciation rate is set to.2. The average labor market transition rates are set so that the steady state rates of short-term and long-term unemployment match the average values in the data described in Section 4.3. Similarly, I set the trend growth rate to the mean growth of income and consumption in the data to be described. Lastly, I set the mortality rate to.1 quarterly to generate an expected working of life of 25 years. I set the unemployment insurance parameter b u to.3 for a 3% replacement rate, which is in line with replacement rates for the United States reported by Martin (1996). 4.2 Parameters for the income process The income process I use is motivated by the work of Guvenen et al. (213). Those authors estimate a related income process with a mixture of two normals the coefficients of which take different values in expansions and recessions. To choose parameters of the income process, I use a simulated method of moments approach similar to Guvenen et al.. I simulate quarterly 9 Cogley and Nason (1995) and King and Rebelo (1999) discuss the importance of the persistence parameters in more detail. 15

17 Symbol Parameter Value Target/Prior Panel A. Calibrated γ Risk aversion 2 λ Avg. job finding rate.679 Mean long-term unemployment. ζ Avg. high-skill job separation rate.37 Mean short-term unemployment. b u Unemployment insurance.3 3% replacement rate. ω Mortality risk.1 25-year expected working life. β Discount factor.985 Capital-output ratio = α Capital share.36 δ Depreciation rate.2 g Growth rate Mean ( Y + C)/2 Panel B. Income process parameters µ ξ Mean transitory skill.283 See text σ ξ Std. dev. transitory skill See text µ 2 Mean of mixture component.2963 See text µ 1 2 Cyclicality of mixture component.146 See text µ 3 Mean of mixture component.3153 See text µ 1 3 Cyclicality of mixture component.1534 See text σ η,1 Std. dev. of mixture component See text σ η,2 Std. dev. of mixture component 1.45 See text σ η,3 Std. dev. of mixture component 1.45 See text p Weight on mixture components.526 See text Panel C. Estimated ρ A Autoregressive coefficient of A.9797 Beta: mn. =.5, var. =.4. ρ λ Autoregressive coefficient of λ.8955 Beta: mn. =.5, var. =.4. ρ ζ Autoregressive coefficient of ζ.8467 Beta: mn. =.5, var. =.4. ρ q Autoregressive coefficient of q.9714 Beta: mn. =.5, var. =.4. σ A Standard deviation of ɛ A.71 Inv. Gam.: mn. = 1, var. = 4. σ λ Standard deviation of ɛ λ 11.6 Inv. Gam.: mn. = 1, var. = 4. σ ζ Standard deviation of ɛ ζ 6.56 Inv. Gam.: mn. = 1, var. = 4. σ q Standard deviation of ɛ q.41 Inv. Gam.: mn. = 1, var. = 4. Table 1: Parameter values, targets and priors. Estimated parameter values correspond to the general equilibrium economy. Standard deviations are 1. 16

18 data from the model, including unemployment spells and mortality risk. I then aggregate the quarterly data to annual observations and compute 1-year, 3-year, and 5-year log income changes. I then choose the parameters of the income process to minimize the distance between the moments of simulated income change distributions to the annual data reported in Guvenen et al.. 1 Among the moments that I ask the model to match are the widths of the right and left tails as measured by the difference between the 9 th percentile and the median and the 1 th percentile and the median. These differences can be combined to form Kelley s skewness Kelley s skewness = (P 9 P 5) (P 5 P 1), P 9 P 1 which is Guvenen et al. s preferred measure of skewness because it is less sensitive to extreme observations than the third moment. Figure 1 plots Kelley s skewness for 5-year income changes for the model and the Guvenen et al. data. The model does a good job of replicating the amplitude of the fluctuations in skewness but the timing of these fluctuations appears to lag the data. Appendix A further explains the procedure. The left panel Figure 2 shows the density of η with x =. Most of the mass is concentrated near zero, but there is some probability of both large positive and negative values. The right panel of the figure omits the mass in the center to focus on the behavior of the tails. It shows a hypothetical recession with ζ one standard deviation below its mean and an expansion with ζ one standard deviation above its mean. While the tails are quite volatile over the cycle, the central mass is relatively stable, which results in the pro-cyclical skewness. 4.3 Time series data and estimation After fixing the parameters in the first two panels of Table tab:params, I estimate the aggregate shock processes by searching for the posterior mode. I use data on short-term unemployment, long-term unemployment, consumption of non-durables and services, consumption of durables and investment in fixed capital. I relate the data series to model 1 I impose the restriction σ η,2 = σ η,3. 17

19 .2 Kelley s skewness Figure 1: Keylley s skewness of 5-year income changes: model (solid line) and data (dashed line). Data are from Guvenen et al. (213). The value for year t shows the skewness of income changes from year t to t Full distribution.25 Tails 2.2 Recession Boom density η η Figure 2: Density of distribution for η. The right-hand panel omits the central component of the mixture for clarity. 18

20 variables as follows consumption of non-durables and services = C t investment plus consumption of durables = Y t C t short-term unemployment = ζ t (1 u t 1 ) long-term unemployment = (1 λ t )u t 1 I will refer to the first two data series as consumption and investment, respectively, and each is transformed as 1 log( ). I use quarterly data from 1966:I to 213:IV. In the context of incomplete markets, the choice of data on labor market conditions is more complicated than it is under complete markets because the distribution of labor income is now relevant to the dynamics of aggregate consumption. I ask the model to match the contribution of aggregate hours to fluctuations in aggregate labor income, but to avoid complicating the model I abstract from labor force participation and fluctuations in hours per worker. I use 1 H t / H as the empirical observation of the unemployment rate, where H t is aggregate hours per capita and H is aggregate hours at full employment. To define a fullemployment level of aggregate hours per capita I look to 1999:IV as the quarter since 196 with the largest value of aggregate hours per capita. I then define full employment hours per capita as the labor force participation rate in 1999:IV times the index of average weekly hours in that quarter. The unemployment rate alone does not convey information about the duration of unemployment spells. I therefore use data on the composition of the unemployed pool by splitting the pool into those with fewer than 15 weeks and at least 15 weeks of unemployment. As the model period is one quarter, I take less than 15 weeks to be individuals who are unemployed, but were employed in the previous quarter. These data define the shares of short-term and long-term unemployment in total unemployment. I use these shares to split the constructed unemployment rate into short-term and long-term unemployment pools. Figure 3 shows the 19

21 Short- term Long- term Total Reported by BLS Figure 3: Constructed short-term and long-term unemployment data. constructed unemployment rate and its decomposition by duration. From this figure, one can see there are some low-frequency movements in the unemployment rates, which are partly attributable to demographic factors. To remove these low-frequency trends, I de-trend the unemployment series using an HP filter with smoothing parameter 1, following Shimer (25). 5 The aggregate effects of time-varying risk I assess the contribution of time-varying idiosyncratic risk through counterfactual experiments. The first counterfactual is a representative agent economy in which a full set of insurance markets allow idiosyncratic risks to be fully shared among households. In addition, I assume a dynastic structure such that upon death a household is replaced by a child 2

22 5 x 1 3 A x 1 4 λ x 1 3 ζ 2 x 1 3 q IM CM Figure 4: Impulse responses of aggregate consumption to aggregate shocks. IM = incomplete markets, CM = complete markets. Shocks are scaled to one standard deviation. Units are log deviation from steady state. who inherits the parent s capital holdings and the parent values the child as an extension of themselves. Finally, I assume a capital subsidy financed by a lump sum tax such that the steady state capital stock is the same as in the incomplete markets economy. 11 In this counterfactual, aggregate consumption is determined by a relatively standard Euler equation [ C γ = e γg ˆβ(1 ] + τ K )E e q R C γ. I call this counterfactual the complete markets case. The second counterfactual is simply the incomplete markets economy with a time-invariant idiosyncratic risk process (µ 1 2 = µ 1 3 = ). I call this counterfactual the constant risk case. Figure 4 shows the impulse response of aggregate consumption to the four aggregate shocks for both the incomplete markets and complete markets cases. In the top two panels, 11 This requires ˆβ(1 + τ K ) =

23 .2 x 1 3 ζ with constant risk case IM CM Constant Risk Figure 5: Impulse responses of aggregate consumption to ζ shock. IM = incomplete markets, CM = complete markets, CR = constant risk. Shocks are scaled to one standard deviation. Units are log deviation from steady state. the response of consumption to shocks to TFP (A) and the job-finding rate (λ) is slightly larger in the incomplete markets economy than in the complete markets economy. The key difference between the two economies comes in the response to the shock to ζ, which leads to both an increase in the job-separation rate and negative skewness in the distribution of skill shocks. Finally, the response to a preference shock (q) is very similar in the two models. Figure 5 shows that the differences in the response to a ζ shock in the incomplete and complete markets economies are primarily driven by the time-varying idiosyncratic risk process. Specifically, the figure shows the constant risk case in addition to the two others. One can see that the the incomplete markets model with a constant level of idiosyncratic risk is quite similar to the complete markets case. Table 2 shows the variance of aggregate consumption growth and its correlation with aggregate income growth, short-term unemployment and long-term unemployment. For the three model economies, these moments are generated without preference shocks. One can see that time-varying idiosyncratic risk makes consumption growth more volatile and more correlated with short-term unemployment. The estimated model can be used to recover the structural shocks that generated the data. To recover the structural shocks I use the Kalman smoother. Feeding these shocks 22

24 Data IM CM CR Variance Corr. w/ Y Corr. w/ short-term u Corr. w/ long-term u Table 2: Moments of consumption growth. IM = incomplete markets model. CM = complete markets model. CR = IM with constant idiosyncratic risk. A λ ζ q Figure 6: Decomposition of contribution of idiosyncratic risk to aggregate consumption growth. Each series is the contribution of the given shock to aggregate consumption growth in the incomplete markets model less the complete markets model. Standard deviations listed in the figures. 23

25 one at a time into the incomplete markets model produces a time series for the contribution of the given shock s contribution to aggregate consumption. Given the linear structure of the empirical model, summing across the four shocks nearly 12 reproduces the data. same shocks can be fed into the complete markets model to generate a counterfactual contribution of each shock. Figure 6 shows the difference between the generated series for the incomplete and complete markets economies. Each panel shows the standard deviation of the difference in the predicted series from that two models. Three of the four panels show almost no difference between the complete markets and incomplete markets economies. The exception is the ζ shock where the impact of idiosyncratic risk on aggregate consumption has a standard deviation of.13. The contribution of incomplete markets to consumption growth is particularly evident during the recessions in 1975, the early 198 s and the Great Recession where the incomplete markets model generates larger drops in consumption. To find a counterfactual prediction for total consumption growth, I could feed the smoothed structural shocks that were recovered from the incomplete markets model into the complete markets model. As the model is linear, an equivalent calculation is to sum the series in the four panels of Figure 6 and then subtract the result from the consumption growth data. This alternative method for arriving at the same counterfactual is useful because it makes clear which shocks are contributing to the difference between the counterfactual and the data. As there is not a clear structural interpretation of the preference shock and I choose to omit it s contribution altogether. Therefore, I sum over the remaining three panels of Figure 6 and subtract the result from the observed data to arrive at the counterfactual. The variance of the counterfactual consumption growth series under complete markets is.235 as opposed to the observed variance of.271 so consumption growth would have been 13% less volatile had markets been complete. The real benefit of this counterfactual time series is that it can be used to investigate the contribution of idiosyncratic risk to particular episodes. The left panel of Figure 7 shows the cumulative consumption growth 12 The data also depends on the past shocks through the state variables. The procedure described sets these to zero before the sample begins so there is a small discrepancy at the start of the sample that fades quickly. The 24

26 4 6 3 Complete markets 5 percentage points 2 1 Data percentage points Complete markets 1 Data Figure 7: Cumulative consumption growth. Fraction of wealth held by Quintile Gini Data Baseline / Table 3: Distribution of wealth. The 8/2 model includes preference heterogeneity. Data are from the 27 Survey of Consumer Finances as reported by Diaz-Gimenez et al. (211). Model distributions refer to the stationary equilibrium around which the economy fluctuates. during the Great Recession starting from 27:III. By 29:I, the difference in aggregate consumption is.71 percentage points. The right panel of the figure shows the cumulative consumption growth in the early 198 s starting from 1979:IV. By 1982:I the difference in aggregate consumption is 1.4 percentage points. The differences in Figure 6 appear small, but they accumulate to more substantial differences in Figure 7. 6 More wealth inequality The first two rows of Table 6 show the distribution of wealth for the data and the baseline incomplete markets economy above. The baseline model does not generate the high concentration of wealth holdings found in the data. In this section, I modify the model to generate 25

27 additional concentration in wealth holdings. To do so, I incorporate preference heterogeneity in the spirit of Krusell and Smith s stochastic-β economy. I assume 8% of the population of households are impatient and face incomplete markets as described above. The remaining 2% are more patient, more skilled and pool their idiosyncratic income shocks as in the complete markets counterfactual. Specifically, the budget constraint of a patient household is c + e g k = Rk + χ ν W L, where ν =.2 is the mass of patient households and χ is the earnings share of patient households. The assumption that these households do not face idiosyncratic risk does not have a large effect on the results because if they did face idiosyncratic labor income risks they would be very well self insured by virtue of being patient. At the same time, this assumption is very useful from a computational standpoint because it does not increase the number of non-linear household decision rules that I need to compute. The budget constraint of an impatient household is c + e g k = Rk + 1 χ l(θ, ξ, n, W, u). 1 ν These assumptions introduce two new parameters: the relative patience of the two groups and the relative skills, χ. I choose these parameters to match the wealth share and earnings share of the patient households to the top 2% of the distribution of wealth. The resulting distribution of wealth appears in the bottom row of Table 6. The 8/2 model generates more wealth in the bottom quintiles than we observe in the data, which is partly due to the tight borrowing limit k. I re-estimate the model and perform the counterfactual decomposition from the previous section. 13 As the 8/2 model includes preference heterogeneity, the complete markets counterfactual is somewhat different from the one above. In this section the counterfactual is that all households are patient and fully share risks. The steady state interest rate of the 8/2 model is determined by the Euler equation of the patient households so it is not 13 The estimated parameter values appear in Table 4. 26

28 A λ ζ q Figure 8: Decomposition of contribution of idiosyncratic risk to aggregate consumption growth for the 8/2 model. Each series is the contribution of the given shock to aggregate consumption growth in the incomplete markets model less the complete markets model. Standard deviations listed in the figures. affected by changes in the share of these households in the population. 14 Figure 8 shows the contribution of idiosyncratic risk associated with the four aggregate shocks. With more agents near the borrowing constraint, consumption becomes more sensitive to TFP shocks and to the ζ shock. In total, the variance of consumption growth is 32% lower in the counterfactual complete market series. Precautionary saving behavior generates a consumption function that is concave in wealth (Carroll and Kimball, 1996). An increase in ζ reduces the expected lifetime income of a household by changing the probability of unemployment in the future. Due to the concavity of the consumption function, this wealth effect has larger effects on the consumption of lowwealth households. However, it is only at very low asset levels that this effect is quantitatively important. This can be seen in Figure 9 which shows the change in consumption in response to a one standard deviation positive shock to ζ across different levels of wealth. The lightcolored line corresponds to the constant risk case. Notice that is only very near zero cash 14 See McKay and Reis (213) for further discussion of the determination of the steady state interest rate in a related model. 27

29 2 x 1 3 units of consumption normalized cash on hand Figure 9: Percentage change in consumption on impact in response to a standard deviation increase in ζ for an employed household with the given level of cash on hand relative to permanent income. 8/2 model (dark line) and 8/2 model with constant risk (light line). on hand that the drop in consumption is amplified. At higher levels of assets the concavity of the consumption function is obscured by general equilibrium effects. There is, however, an additional source of curvature when risk is time-varying. When the increase in ζ brings about additional uninsurable risk the precautionary savings motive changes and more so at low wealth levels. Households with low wealth are poorly self insured and so the change in risk has a particularly strong effect on their consumption. This can be seen in the dark line in Figure 9 which corresponds to the baseline 8/2 model. The consumption response to this shock now depends strongly on the household s asset position. As a result of this curvature, the distribution of wealth makes a substantial difference to the overall impact of the shock. Notice that at high wealth levels consumption actually increases due to general equilibrium effects. The contributions of idiosyncratic risk to aggregate consumption growth shown in Figure 8 accumulate to substantial differences in consumption during recessions. The left panel of Figure 1 shows the cumulative effect during the Great Recession. By 29:I, aggregate consumption would have been 1.93 percentage points higher under complete markets. Similarly, 28

30 5 7 4 Complete markets percentage points 2 1 Data percentage points Complete markets 1 Data Figure 1: Cumulative consumption growth for the 8/2 model. during the early 198s shown in the right panel, aggregate consumption would have been 2.24 percentage points higher by 1982:I. 7 Idiosyncratic risk and parameter estimates The previous two sections treat the complete markets model as a counterfactual and the structural parameters are the same in the two models. I now switch to a perspective where the complete markets model is a potential data generating process and re-estimate the parameters of the model to provide the best fit to the data. I document the extent to which the estimated structural parameters are sensitive to the inclusion of microeconomic heterogeneity in the model. The value of doing so is nicely illustrated by the work of Chang et al. (213) who simulate data from an incomplete markets model and estimate an approximating representative agent model using that data. They find that the estimated structural parameters of the representative agent model are not stable across policy regimes due to the misspecification of the micro-foundations. The methods presented here make it possible to estimate the incomplete markets model itself and therefore check sensitivity of the estimates to household heterogeneity directly using actual data. Table 4 shows the estimated parameter values from the posterior mode for both the complete and incomplete markets models. The primary difference between the two sets of parameters is the persistence of the job-separation shock, ρ ζ. With a larger ρ ζ, a given 29

31 IM CM 8/2 ρ A ρ λ ρ ζ ρ q σ A σ λ σ ζ σ q Table 4: Estimated parameter values. Standard deviations are 1. innovation to ζ will have a larger wealth effect on consumption. For a given value of ρ ζ, the incomplete markets model makes consumption more sensitive to innovations to ζ because of the extra effect coming from the increase in idiosyncratic risk, which leads to an extra precautionary motive for saving. Suppose that fitting the data well requires a certain level of co-movement between short-term unemployment and consumption growth. To achieve that co-movement, the incomplete markets model requires less persistence in the shock processes. Figure 11 illustrates the effect of the persistence parameters on the magnitude of the consumption response to a ζ shock by plotting the impulse responses implied by the baseline parameters and that implied by the complete markets posterior mode. Table 4 also shows the estimated parameter values from the posterior mode of the 8/2 model where a similar pattern emerges. In the 8/2 model, the TFP shock and ζ shocks are less persistent than in the estimates from the incomplete markets model. This fits with the intuition above as aggregate consumption is more sensitive to these shocks when there is more heterogeneity in wealth holdings. 8 Conclusion This paper began by asking whether idiosyncratic risks are relevant for the dynamics of aggregate quantities. The results show that time-varying risks to long-term earnings potential can generate substantial movements in aggregate consumption. In a version of the model 3