Idiosyncratic risk and the dynamics of aggregate consumption: a likelihood-based perspective

Idiosyncratic risk and the dynamics of aggregate consumption: a likelihood-based perspective Alisdair McKay Boston University March 2013

Idiosyncratic risk and the business cycle How much and what types of insurance do households have access to? Answer is important to: Relevance of representative household assumption as a modeling paradigm. Aggregate and distributional consequences of aggregate shocks and policies. Potential welfare benefits of aggregate stabilization. 2 / 39

Evidence for market incompleteness 1. Consumption responds to predictable income changes, 2. Many studies find an MPC around 20% (Blinder, 1981; Poterba, 1988; Parker, 1999; Johnson et al., 2006; Parker et al., 2013) 3. Households report that they are borrowing constrained or respond to changes in borrowing limit (Jappelli, 1990; Gross and Souleles, 2002). 3 / 39

This paper How much can we learn about idiosyncratic risk from the behavior of aggregate consumption? Build an empirical version of Krusell and Smith (1998). Show how to construct a likelihood function for incomplete-markets business-cycle models. Evaluate the likelihood of aggregate data for different arrangements of household level risk sharing. 4 / 39

Joint test of model and insurance Can only view insurance in the data through the lens of the model. If the model s predictions are insensitive to insurance, then test has low power. Calibrate the model to match distribution of wealth and average MPC Much of household net worth is in illiquid assets, Illiquid wealth is not as useful for consumption smoothing...higher MPC. Model with an adjustment cost on household assets. Model predictions do depend on insurance arrangement. 5 / 39

Results Low insurance and full insurance fit the data similarly. Partial insurance is best fitting. 6 / 39

The model 7 / 39

Endowments Households endowed with employment status e {0, 1} skill level s S. Skill transition probabilities are fixed over time. Employment transition probabilities depend on skill and time. 8 / 39

Employment risk Variation over time: unemployment rate varies exogenously over time. Variation over skill group: Data show: Unemployment rate is decreasing with education. Driven by higher job-separation rates for low education. Job-finding rates are similar across groups. 9 / 39

Employment risk 2006 Asian 0.01 1976 1981 1986 1991 1996 2001 2006 Source: Bureau of Labor Statistics, Current Population Survey, and authors' calculations Montly Hazard (log scale) 0.08 (d) Inflow Rates by Education 2 0.04 <12 12 12-15 0.02 12-15 0.01 16+ 2006 0.01 1976 1981 1986 1991 1996 2001 2006 Source: Bureau of Labor Statistics, Current Population Survey, and authors' calculations Source: Elsby, Hobijn, Sahin (2010) 10 / 39

Employment risk 0.13 1976 1981 1986 1991 1996 2001 2006 Source: Bureau of Labor Statistics, Current Population Survey, and authors' calculations Monthly Hazard (log scale) 0.80 (d) Outflow Rates by Education 0.01 1976 Source: B Montly Hazard (log s 0.08 <12 12 0.04 0.40 0.02 16+ 12-15 0.01 0.20 1976 1981 1986 1991 1996 2001 2006 Source: Bureau of Labor Statistics, Current Population Survery, and authors' calculations 0.01 1976 Source: B Source: Elsby, Hobijn, Sahin (2010) 11 / 39

Employment risk Job-finding rate (same for all skill groups) with ɛ λ t N(0, σ λ ). λ t = ρ λ λ t 1 + ɛ λ t Job-separation rate for high skill ζ t = ρ ζ ζ t 1 + ɛ ζ t with ɛ ζ t N(0, σζ ). For lower-skill groups: ζt s = ζ t + ζ s ζ s = 0 for s = max{s}. 12 / 39

Preferences where U i = E 0 t=0 β t c1 χ Qt i,t e 1 χ with ɛ Q t N(0, σ Q ). Q t = (1 ρ Q ) Q + ρ Q Q t 1 + ɛ Q t Q t is an aggregate preference shock. 13 / 39

Technology with where ɛ A t N(0, σ A ). Y t = A t K α t L 1 α t log(a t ) = ρ A log(a t 1 ) + ɛ A t Capital accumulates according to K t+1 = (1 δ)k t + I t. 14 / 39

Insurance y i,t = [e i,t + b u (1 e i,t )]s 1 bs i,t e j,t (1 τ)w t s j,t dj. [ej,t + b u (1 e j,t )]s 1 bs j,t dj }{{}}{{} income share aggregate income If b u = b s = 0: y i,t = e i,t (1 τ)w t s i,t. If b u = b s = 1: y i,t = e j,t (1 τ)w t s j,t dj. 15 / 39

Asset market Households savings fund aggregate capital stock and government debt. Government debt pays same return as capital. Households cannot borrow. Wealthy hand-to-mouth Motivated by Kaplan and Violante (2012) model of illiquid assets. Simpler version here: quadratic adjustment cost on household asset position ( a ) a 2 a Γ(a, a) = γ 2 a 16 / 39

Asset market Households savings fund aggregate capital stock and government debt. Government debt pays same return as capital. Households cannot borrow. Wealthy hand-to-mouth Motivated by Kaplan and Violante (2012) model of illiquid assets. Simpler version here: quadratic adjustment cost on household asset position Γ(a, a) = γ 2 ( a ) a 2 a a+1 17 / 39

Taxes and government spending Why include taxes and spending? G affects C through aggregate resource constraint. T affects C through disposable income. Ĝ t = ρ G Ĝ t 1 + ɛ G t with ɛ G t N(0, σ G ). ˆT t = ρ T ˆTt 1 + φ Y Ŷ t + φ B ˆBt + ɛ T t with ɛ T t N(0, σ T ). τ = T wl. B = (1 + r)b + G T. 18 / 39

State variables Individual: a, e, s Aggregate: Θ = exogenous: Q, A, λ, ζ, G, T endogenous: B distribution of households over (a, e, s) Φ(a, e, s) 19 / 39

Household problem { V (a, e, s; Θ) = max u(c) + βe Q E [ V (a, e, s ; Θ ) ]} a,c a = R(Θ)a + y(e, s; Θ) c Γ(a, a) 20 / 39

Firm problem and equilibrium Firm problem: representative firm rents capital and labor in spot markets. Equilibrium definition: Similar in spirit to Krusell and Smith (1998). Capital market clearing condition: K + B = h(a, e, s, Θ)Φ(da, e, s). e,s 21 / 39

Methods: challenges Solving: Solving model requires forecasting the distribution of wealth. Many shocks and aggregate states. Filtering: To construct likelihood function, must filter data through model to uncover shocks. Dynamics of the economy depend on how shocks affect distribution of wealth. 22 / 39

Methods: solving Solve the model using technique of Reiter (2009a): Create a discrete representation of the economy Represent savings policy functions with splines, Represent distribution of wealth with a histogram. An equilibrium of the model is a solution to F (X t 1, X t, η t, ɛ t ) = 0 Euler equation evaluated at finite set of points, Equations for evolution of wealth histogram. Linearize and solve with standard techniques. End result: backward-looking state-space system. 23 / 39

Methods: model reduction and filtering System is very large (almost 4K variables) Most variables are tracking distribution of wealth and are highly co-linear. Model reduction: 1. Solve first, reduce second (Antoulas, 2005; Matlab Control System Toolbox) 2. Reduce first, solve second (Reiter, 2009b). After model reduction, system has about 100 variables. Linear system with Gaussian shocks: apply Kalman filter to calculate likelihood of data. 24 / 39

Empirical strategy Four groups of parameters: 1. Steady state, unrelated to risk and insurance (e.g. capital share) 2. Steady state, related to risk and insurance, (e.g. skill shocks) 3. Dynamics, shock persistence and fiscal adjustment rule 4. Dynamics, shock variances 25 / 39

Empirical strategy Two reference models: ICM = low insurance, Rep. Agent = full insurance. Four groups of parameters: 1. Steady state, unrelated to risk and insurance (e.g. capital share) calibrated for both models 2. Steady state, related to risk and insurance, (e.g. skill shocks) calibrated 3. Dynamics, shock persistence and fiscal adjustment rule estimated/calibrated 4. Dynamics, shock variances estimated computationally feasible due to certainty equivalence w.r.t. aggregate shocks 26 / 39

Model comparison Compare different insurance arrangements using Bayesian model comparison. Pr(M i y) = Pr(y M i )P i J j=1 Pr(y M j)p j where Pr(y M i ) = Pr(y θ i, M i ) Pr(θ i M i )dθ i. Taking the ratio for models i, j θ Pr(M i y) Pr(M j y) = Pr(y M i) P i. Pr(y M j ) P j }{{} Bayes factor 27 / 39

Empirical strategy Fix all parameters except shock variances and insurance. Vary insurance parameters and repeat. Report marginal likelihood of the data (i.e. integrate across shock variances). Full estimation is not possible: each likelihood evaluation requires approx. 45 min to solve and compress the model. Recalibrate the steady state? Yes: model should be consistent with steady state. No: the results become difficult to interpret if too many parameters are changing. Also, representative agent model cannot be consistent with cross-sectional heterogeneity. Compromise: recalibrate β to match capital-output ratio as insurance changes, leave other parameters at their baseline ICM levels. 28 / 39

Data Y = GDP per capita I = Investment and durable consumption per capita c i di = Non-durable and service consumption per capita T = tax receipts per capita Labor market data. Aggregate quantities are deflated by GDP deflator and linearly detrended. 29 / 39

Modeling aggregate hours Complication: aggregate hours worked reflect i) hours per worker, ii) labor force participation rate, iii) unemployment rate. In representative agent model, all that matters is total effective hours (other aspects of distribution are irrelevant). With incomplete markets, distribution matters. Incorporating i) iii) to model would complicate matters substantially. Stark assumption: all fluctuations in aggregate hours are unemployment (involuntary and binary). Insurance system will smooth out these shocks. 30 / 39

Labor market data Since 1960, aggregate hours per capita peaked in 1999:IV. Define full employment as [labor force participation] 1999:IV [hours per worker] 1999:IV Define the unemployment rate as [full employment] [Aggregate hours per capita] t [full employment] 31 / 39

Labor market data: λ vs. ζ Fluctuations in unemployment can reflect the job-finding rate or the job-separation rate. Short-term unemployed (less than 15 weeks) driven primarily by job-separation. Long-term unemployed (15 weeks or more) driven primarily by job-finding rate. Use data on mix of short-term and long-term unemployed in total unemployment pool to identify movements in λ and ζ. 32 / 39

Unemployment rates 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 1966 1971 1976 1981 1986 1991 1996 2001 2006 2011 Short- term Long- term Total Reported by BLS 33 / 39

Choosing parameters Risk aversion = 2. Capital share = 30% (all taxes fall on labor). Three skill groups: skill risk to match distribution of wealth Castaneda et al. (2003), Domeij and Heathcote (2004), Heathcote (2005). Depreciation = 1.5% per quarter γ to match average MPC of 20%. Blinder (1981), Poterba (1988), Parker (1999), Johnson et al. (2006), Parker et al. (2013). 34 / 39

Unemployment-skill correlation Job-separation rate differs by skill level. The low- and high-skill groups are small (3.4% of population). Impose u low skill ū model = u <high school ū data u high skill ū model = u college ū data 35 / 39

Shock persistence parameters Consider two cases: Posterior mode of full insurance specification. Low-persistence case: ρ x = 0.9 x. 36 / 39

Impulse response functions consumption 0.6 0.5 0.4 0.3 0.2 0.1 0 Q preference bs = 0, bu = 0.3 Full insurance 0.1 0 5 10 15 20 0.15 0.1 0.05 A technology 0 0 5 10 15 20 0.07 0.06 0.05 0.04 0.03 0.02 0.01 λ job-finding 0 0 5 10 15 20 0 0.02 0.04 0.06 0.08 0.1 0.12 ζ job-separation 0.14 0 5 10 15 20 0 0.05 0.1 0.15 G exog. spending 0.2 0 5 10 15 20 0.05 0 0.05 0.1 T taxes 0.15 0 5 10 15 20 Plots show log change in response to one standard deviation shock. 37 / 39

Model comparison results Marginal log likelihoods b u 0.3 1 1 1 b s 0 0 0.5 1 High persistence -1295-1309 -1286-1295 Low persistence -1341-1340 -1337-1346 38 / 39

Future directions Why is partial insurance the best fitting? Idiosyncratic risk and the Great Recession: what would the recession have looked like if there were more/less insurance agains idiosyncratic risk? What role for the adjustment costs? 39 / 39