Income Dynamics and Consumption Inequality: Nonlinear Persistence and Partial Insurance

Transcription

1 Income Dynamics and Consumption Inequality: Nonlinear Persistence and Partial Insurance Richard Blundell (UCL & IFS) Short Course, Northwestern University [Updated papers and references on my web page] November 217 Richard Blundell Income and Consumption Dynamics November / 1

2 The idea behind this research is to examine the transmission of income shocks to consumption: Richard Blundell Income and Consumption Dynamics November / 1

3 The idea behind this research is to examine the transmission of income shocks to consumption: > The overall objective is to model the links between income, earnings and consumption inequality - the distributional dynamics of inequality - Deaton and Paxson (1994), Blundell and Preston (1998), Krueger and Perri (25), Blundell, Pistaferri and Preston (28),..., Attanasio and Pistaferri (216),... > Recent work incorporates family labor supply and non-separabilities, see Nemmers Lecture and recent papers, Blundell, Pistaferri and Saporta, 216, 217. > I want to focus this lecture on nonlinear persistence and partial insurance. I will come back to family labor supply at the end. Richard Blundell Income and Consumption Dynamics November / 1

4 The idea behind this research is to examine the transmission of income shocks to consumption: > The overall objective is to model the links between income, earnings and consumption inequality - the distributional dynamics of inequality - Deaton and Paxson (1994), Blundell and Preston (1998), Krueger and Perri (25), Blundell, Pistaferri and Preston (28),..., Attanasio and Pistaferri (216),... > Recent work incorporates family labor supply and non-separabilities, see Nemmers Lecture and recent papers, Blundell, Pistaferri and Saporta, 216, 217. > I want to focus this lecture on nonlinear persistence and partial insurance. I will come back to family labor supply at the end. In particular, the aim is: 1. To consider alternative ways of modelling persistence, and 2. To explore the nonlinear nature of income shocks and the implications for consumption dynamics and inequality. Richard Blundell Income and Consumption Dynamics November / 1

5 The idea behind this research is to examine the transmission of income shocks to consumption: > The overall objective is to model the links between income, earnings and consumption inequality - the distributional dynamics of inequality - Deaton and Paxson (1994), Blundell and Preston (1998), Krueger and Perri (25), Blundell, Pistaferri and Preston (28),..., Attanasio and Pistaferri (216),... > Recent work incorporates family labor supply and non-separabilities, see Nemmers Lecture and recent papers, Blundell, Pistaferri and Saporta, 216, 217. > I want to focus this lecture on nonlinear persistence and partial insurance. I will come back to family labor supply at the end. In particular, the aim is: 1. To consider alternative ways of modelling persistence, and 2. To explore the nonlinear nature of income shocks and the implications for consumption dynamics and inequality. e.g. US Household Panel data and Norwegian Population Register data. Richard Blundell Income and Consumption Dynamics November / 1

6 New data on consumption and family income sources I. Administrative linked data: e.g. Norwegian population register. Linked registry databases with unique individual identifiers. Containing records for every Norwegian from 1967 to 214. Detailed demographic and socioeconomic information (market income, cash transfers). Recent links to real estate and assets; and to hours of work. New consumption measurements. Family identifiers allow to match spouses and children. see Blundell, Graber and Mogstad (215). Richard Blundell Income and Consumption Dynamics November / 1

7 New data on consumption and family income sources I. Administrative linked data: e.g. Norwegian population register. Linked registry databases with unique individual identifiers. Containing records for every Norwegian from 1967 to 214. Detailed demographic and socioeconomic information (market income, cash transfers). Recent links to real estate and assets; and to hours of work. New consumption measurements. Family identifiers allow to match spouses and children. see Blundell, Graber and Mogstad (215). II. Newly designed panel surveys: e.g. PSID since Collection of consumption and assets had a major revision in % of consumption expenditures, more since 24. The sum of food at home, food away from home, gasoline, health, transportation, utilities, clothing etc. Earnings and hours for all earners; Assets measured in each wave. see Blundell, Pistaferri and Saporta-Eksten (216). Richard Blundell Income and Consumption Dynamics November / 1

8 A prototypical canonical panel data model of (log) family (earned) income y it is: y it = η it + ε it, i = 1,..., N, t = 1,..., T. where y it is net of a systematic component, η it is a random walk with innovation v it, η it = η it 1 + v it, i = 1,..., N, t = 1,..., T. and ε it is a transitory shock. Richard Blundell Income and Consumption Dynamics November / 1

9 A prototypical canonical panel data model of (log) family (earned) income y it is: y it = η it + ε it, i = 1,..., N, t = 1,..., T. where y it is net of a systematic component, η it is a random walk with innovation v it, η it = η it 1 + v it, i = 1,..., N, t = 1,..., T. and ε it is a transitory shock. Consumption growth is then related to income shocks: c it = φ t v it + ψ t ε it + ν it, i = 1,..., N, t = 1,..., T. where c it is log total consumption net of a systematic component, > φ t is the transmission of persistence shocks v it, and > ψ t the transmission of transitory shocks; - the ν it are taste shocks, assumed to be independent across periods. Richard Blundell Income and Consumption Dynamics November / 1

10 Covariance Restrictions Baseline panel data model specification: Implying covariance restrictions: c it = φv it + ψε it + ν it, y it = v it + ε it, var( c it ) = φ 2 σ 2 v + ψ 2 σ 2 ε var( y it ) = σ 2 η + 2σ 2 ε cov( y it y it 1 ) = σ 2 ε cov( c it y it ) = φσ 2 v + ψσ 2 ε cov( c it 1 y it ) = ψσ 2 ε > For T>3, BPP include time(age) variation in the σ 2 and insurance parameters, > BPP allow for measurement error and extend to MA(1) transitory shocks, > BP develop these covariance restrictions for repeated cross-sections. Richard Blundell Income and Consumption Dynamics November / 1

11 Linking Income Dynamics to Consumption Inequality More specifically, to account for the impact of income shocks on the evolution of consumption inequality we introduce transmission or partial insurance parameters, writing consumption growth as: Δ ln C it = γit + ΔZ it ϕ + φ t v it + ψ t ε it + ξ it φ t and ψ t provide the link between the consumption and income distributions - v it the permanent and ε it the transitory shock to income. Richard Blundell Income and Consumption Dynamics November / 1

12 Linking Income Dynamics to Consumption Inequality More specifically, to account for the impact of income shocks on the evolution of consumption inequality we introduce transmission or partial insurance parameters, writing consumption growth as: Δ ln C it = γit + ΔZ it ϕ + φ t v it + ψ t ε it + ξ it φ t and ψ t provide the link between the consumption and income distributions - v it the permanent and ε it the transitory shock to income. For a simple benchmark intertemporal consumption model for consumer of age t, BLP (213) show φ t = (1 π it ) and ψ t = (1 π it )γ Lt where Assets π it it Assets it + Human Wealth it and γ Lt is the annuity value of a temporary shock to income for an individual aged t retiring at age L. [Easily extend to ARMA processes for income.] Richard Blundell Income and Consumption Dynamics November / 1

13 This standard framework implies a set of extended covariance restrictions for panel data on consumption and income, allowing the insurance parameters and variances to depend on age and education turns out to be key (analysis of PSID and Norwegian data). can show (over-)identification and efficient estimation via nonlinear GMM, see Blundell, Preston and Pistaferri (AER, 28). Blundell, Pistaferri and Saporta (AER, 216) - develop the nonlinear GMM panel data estimator for wage shocks and family labor supply. Will return to this - if time. Also in main Nemmers lecture. Richard Blundell Income and Consumption Dynamics November / 1

14 This standard framework implies a set of extended covariance restrictions for panel data on consumption and income, allowing the insurance parameters and variances to depend on age and education turns out to be key (analysis of PSID and Norwegian data). can show (over-)identification and efficient estimation via nonlinear GMM, see Blundell, Preston and Pistaferri (AER, 28). Blundell, Pistaferri and Saporta (AER, 216) - develop the nonlinear GMM panel data estimator for wage shocks and family labor supply. Will return to this - if time. Also in main Nemmers lecture. Linearity of the income (or wage) process simplifies identification and estimation. However, by construction, it rules out the nonlinear transmission of shocks. Richard Blundell Income and Consumption Dynamics November / 1

15 Motivation The aim in this lecture is to step back and take a different tack - develop an alternative approach to modeling persistence in which the impact of past shocks on current incomes/earnings can be altered by the size and sign of new shocks. This new framework draws on a flurry of recent work on nonlinearity and heterogeneity in the dynamics of inequality and income risk (full references in Arellano, Blundell and Bonhomme, 217). The idea is to have a framework allows: unusual shocks to wipe out the memory of past shocks, and future persistence of a current shock to depend on the future shocks. Richard Blundell Income and Consumption Dynamics November / 1

16 Motivation The aim in this lecture is to step back and take a different tack - develop an alternative approach to modeling persistence in which the impact of past shocks on current incomes/earnings can be altered by the size and sign of new shocks. This new framework draws on a flurry of recent work on nonlinearity and heterogeneity in the dynamics of inequality and income risk (full references in Arellano, Blundell and Bonhomme, 217). The idea is to have a framework allows: unusual shocks to wipe out the memory of past shocks, and future persistence of a current shock to depend on the future shocks. We will see that the presence of unusual shocks matches the data and has a key impact consumption and saving over the life cycle. Richard Blundell Income and Consumption Dynamics November / 1

17 Background papers - Blundell, Pistaferri and Preston [BPP] Consumption inequality and partial insurance (AER, 28) - Blundell, Low and Preston [BLP] Decomposing changes in income risk using consumption data (QE, 213) - Blundell, Graber and Mogstad [BGM] Labor income dynamics and social insurance (JPubE, 215; 217) - Arellano, Blundell and Bonhomme [ABB] Earnings and consumption dynamics: a nonlinear framework (Ecta, 217) maybe finding time to look at family labor supply in: - Blundell, Pistaferri and Saporta-Eksten [BPS1/2] Consumption inequality and family labor supply (AER, 216; JPE, 217) -> on my website uctp39a/pub.html Richard Blundell Income and Consumption Dynamics November / 1

18 Nonlinear Persistence Consider a cohort of households, i = 1,..., N, and denote age as t. Let y it denote log-labor income, net of age dummies y it = η it + ε it, i = 1,..., N, t = 1,..., T. η it follows a general first-order Markov process (can be generalised). Denoting the τth conditional quantile of η it given η i,t 1 as Q t (η i,t 1, τ), we specify η it = Q t (η i,t 1, u it ), where (u it η i,t 1, η i,t 2,...) Uniform (, 1). ε it has zero mean, independent over time. The conditional quantile functions Q t (η i,t 1, u it ) and the marginal distributions F εt can all be age (t) specific. Richard Blundell Income and Consumption Dynamics November / 1

19 A measure of nonlinear persistence This framework allows for nonlinear dynamics of income. To see this, consider the following measure of persistence ρ t (η i,t 1, τ) = Q t(η i,t 1, τ). η ρ t (η i,t 1, τ) measures the persistence of η i,t 1 when, at age t, it is hit by a shock u it that has rank τ. Measures the persistence of histories. Allows a general form of conditional heteroscedasticity, skewness and kurtosis. Richard Blundell Income and Consumption Dynamics November / 1

20 A measure of nonlinear persistence This framework allows for nonlinear dynamics of income. To see this, consider the following measure of persistence ρ t (η i,t 1, τ) = Q t(η i,t 1, τ). η ρ t (η i,t 1, τ) measures the persistence of η i,t 1 when, at age t, it is hit by a shock u it that has rank τ. Measures the persistence of histories. Allows a general form of conditional heteroscedasticity, skewness and kurtosis. In the canonical model η it = η i,t 1 + v it, with v it independent over time and independent of past η s, η it = η i,t 1 + F 1 v t (u it ) ρ t (η i,t 1, τ) = 1 for all (η i,t 1, τ). Richard Blundell Income and Consumption Dynamics November / 1

21 A measure of nonlinear persistence This framework allows for nonlinear dynamics of income. To see this, consider the following measure of persistence ρ t (η i,t 1, τ) = Q t(η i,t 1, τ). η ρ t (η i,t 1, τ) measures the persistence of η i,t 1 when, at age t, it is hit by a shock u it that has rank τ. Measures the persistence of histories. Allows a general form of conditional heteroscedasticity, skewness and kurtosis. In the canonical model η it = η i,t 1 + v it, with v it independent over time and independent of past η s, η it = η i,t 1 + F 1 v t (u it ) ρ t (η i,t 1, τ) = 1 for all (η i,t 1, τ). But what is the evidence for such nonlinearities in persistence? Richard Blundell Income and Consumption Dynamics November / 1

22 Estimates of the average derivative of the conditional quantile function of y it given y i,t 1 with respect Richard to Blundell y i,t 1, using a grid Income ofand 11-quantiles Consumption Dynamics and a 3rd degree Hermite November polynomial / 1 Some motivating evidence: Quantile autoregressions of log-earnings PSID data Q y t y t 1 (y i,t 1,τ) y Norwegian administrative data persistence persistence percentile τ init percentile τ shock percentile τ init percentile τ shock.8 1 Note: Household labor earnings, Age 3-59, (US), (Norway).

23 Conditional skewness, Norwegian administrative data Family income Individual income skewness quantiles of distribution of yt-1 skewness quantiles of distribution of yt-1 Note: Skewness measured as a nonparametric estimate of Q yt y t 1 (y i,t 1,.9) + Q yt y t 1 (y i,t 1,.1) 2Q yt y t 1 (y i,t 1,.5). Q yt y t 1 (y i,t 1,.9) Q yt y t 1 (y i,t 1,.1) Age 3-59, years Richard Blundell Income and Consumption Dynamics November / 1

24 Layout Life-cycle model simulations and model specification Identification Data and estimation strategy Empirical results Richard Blundell Income and Consumption Dynamics November / 1

25 Life-cycle model: illustrative simulation Calibration based on Kaplan and Violante [KV] (21). Households enter the labor market at age 25, work until 6, and die with certainty at age 9. A single risk-free, one-period bond with return 1 + r ( r =.3), A t = (1 + r)a t 1 + Y t 1 C t 1. Log-earnings are ln Y t = κ t + η t + ε t, where κ t is a deterministic age profile. In period t agents know η t, ɛ t and their past values, but not η t+1 or ε t+1 (no advance information). Period-t optimization V t (A t, η t, ε t ) = max C t u(c t ) + βe t [ Vt+1 ( At+1, η t+1, ε t+1 )], where u( ) is CRRA (γ = 2), and β = 1/(1 + r).97. We compare the results for the canonical earnings process used by KV, with our nonlinear process. Richard Blundell Income and Consumption Dynamics November / 1

26 Simulation results Consumption (age 37) by decile of η t Average consumption over the life-cycle consumption consumption decile of t age Note: Blue is nonlinear earnings process, Green is canonical earnings process. Richard Blundell Income and Consumption Dynamics November / 1

27 An Empirical Consumption Rule Let c it and a it denote log-consumption and assets (beginning of period) net of age dummies. Our empirical specification is based on c it = g t (a it, η it, ε it, ν it ) t = 1,..., T, where ν it are independent across periods, and g t is a nonlinear, age-dependent function, monotone in ν it. ν it may be interpreted a taste shifter that increases marginal utility. We normalize its distribution to be standard uniform in each period. Richard Blundell Income and Consumption Dynamics November / 1

28 An Empirical Consumption Rule Let c it and a it denote log-consumption and assets (beginning of period) net of age dummies. Our empirical specification is based on c it = g t (a it, η it, ε it, ν it ) t = 1,..., T, where ν it are independent across periods, and g t is a nonlinear, age-dependent function, monotone in ν it. ν it may be interpreted a taste shifter that increases marginal utility. We normalize its distribution to be standard uniform in each period. > This consumption rule is consistent, in particular, with the standard life-cycle model on the earlier slide. > Can allow for individual heterogeneity, advance information and habits. Richard Blundell Income and Consumption Dynamics November / 1

29 Insurance coefficients With consumption specification given by c it = g t (a it, η it, ε it, ν it ), t = 1,..., T, consumption responses to η and ε are [ ] [ ] gt (a, η, ε, ν) gt (a, η, ε, ν) φ t (a, η, ε) = E, ψ η t (a, η, ε) = E. ε φ t (a, η, ε) and ψ t (a, η, ε) reflect the transmission of the persistent and transitory earnings components, respectively. Richard Blundell Income and Consumption Dynamics November / 1

30 Insurance coefficients With consumption specification given by c it = g t (a it, η it, ε it, ν it ), t = 1,..., T, consumption responses to η and ε are [ ] [ ] gt (a, η, ε, ν) gt (a, η, ε, ν) φ t (a, η, ε) = E, ψ η t (a, η, ε) = E. ε φ t (a, η, ε) and ψ t (a, η, ε) reflect the transmission of the persistent and transitory earnings components, respectively. The marginal effect of an earnings shock u on consumption is [ ) E g t (a, ] ( ) Qt (η, τ) Q t (η, u), ε, ν = φ u t a, Q t (η, τ), ε. u=τ u Richard Blundell Income and Consumption Dynamics November / 1

31 Earnings: identification For T = 3, Wilhelm (212) gives conditions under which the distribution of ε i2 is identified. In particular, completeness of the pdf s of (y i2 y i1 ) and (η i2 y i1 ). This requires η i1 and η i2 to be dependent. In this research we use this result to establish identification of the earnings model. Apply the result to each of the three-year sub-panels t {1, 2, 3} to t {T 2, T 1, T } The marginal distribution of ε it are identified for t {2, 3,..., T 1}. By independence the joint distribution of (ε i2, ε i3,..., ε i,t 1 ) is identified. By deconvolution the distribution of (η i2, η i3,..., η i,t 1 ) is identified. The distribution of ε i1, η i1, and ε it, η it are not identified in general. Richard Blundell Income and Consumption Dynamics November / 1

32 Consumption: assumptions u it and ε it are independent of past earnings shocks and past asset holding, for t 1, where η it = Q t (η i,t 1, u it ). We let η i1 and a i1 be arbitrarily dependent; this is important, because asset accumulation upon entry in the sample may be correlated with past persistent shocks. Denoting η t i = (η it, η i,t 1,..., η i1 ), we assume (in this talk) that: a it is independent of (ηi t 1, ai t 2, εi t 2 ) given (a i,t 1, c i,t 1, y i,t 1 ); consistent with the accumulation rule in the standard life-cycle model with one single risk-less asset. Richard Blundell Income and Consumption Dynamics November / 1

33 Consumption: initial assets Let y = (y 1,..., y T ). We have f (a 1 y) = = f (a 1 η 1, y)f (η 1 y)dη 1 f (a 1 η 1 )f (η 1 y)dη 1, where we have used that u it and ε it are independent of a i1. Note that f (η 1 y) is identified from the earnings process alone. If f (η 1 y) is complete, then f (a 1 η 1 ) is identified. Structure is as in the NPIV problem where η 1 is the endogenous regressor and y is the instrument. Richard Blundell Income and Consumption Dynamics November / 1

34 Consumption: first period We have f (c 1, a 1 y) f (c 1, a 1 η 1, y)f (η 1 y)dη 1 and given our assumptions f (c 1, a 1 y) = f (c 1 a 1, η 1, y 1 )f (a 1 η 1 )f (η 1 y)dη 1. f (a 1 η 1 ) can be treated as known. Provided we have completeness in (y 2,..., y T ) of f (η 1 y 1, y 2,..., y T ), then f (c 1 a 1, η 1, y 1 ), is identified. Intuition: y i2,..., y it are used as instruments for η i1. Subsequent periods discussed in ABB (217), briefly here... Richard Blundell Income and Consumption Dynamics November / 1

35 Consumption: subsequent periods We have f (a 2 c 1, a 1, y) = f (c 2 a 2, c 1, a 1, y) = f (a 2 c 1, a 1, η 1, y 1 )f (η 1 c 1, a 1, y)dη 1 f (c 2 a 2, η 2, y 2 )f (η 2 a 2, c 1, a 1, y)dη 2. By induction it can be shown that the joint density of η s, consumption, assets, and earnings is identified provided, for all t 1, the distributions of (η it ci t, at i, y i ) and (η it ci t 1, ai t, y i ) are complete in (ci t 1, ai t 1, yi t 1, y i,t+1,..., y it ). Intuition: lagged consumption and assets, as well as lags and leads of earnings, are used as instruments for η it. Richard Blundell Income and Consumption Dynamics November / 1

36 Identification: extensions Similar techniques can be used in the presence of advance information, e.g. c it = g t ( ait, η it, η i,t+1, ε it, ν it ), or consumption habits, e.g. c it = g t (c i,t 1, a it, η it, ε it, ν it ). also cases where the consumption rule depends on lagged η, or when η follows a second-order Markov process. (See Section 3 in ABB, 217). Richard Blundell Income and Consumption Dynamics November / 1

37 Identification: extensions Similar techniques can be used in the presence of advance information, e.g. c it = g t ( ait, η it, η i,t+1, ε it, ν it ), or consumption habits, e.g. c it = g t (c i,t 1, a it, η it, ε it, ν it ). also cases where the consumption rule depends on lagged η, or when η follows a second-order Markov process. (See Section 3 in ABB, 217). Households differ in their initial productivity η 1 and initial assets, the panel data provide opportunities to allow for additional, unobserved heterogeneity in earnings and consumption. For example: heterogeneity ξ i in discounting or preferences, or heterogeneity ξ i in the Markovian transitions of η it Richard Blundell Income and Consumption Dynamics November / 1

38 Extensions (cont.) Consumption rule with unobserved heterogeneity: c it = g t (a it, η it, ε it, ξ i, ν it ). We assume that u it and ε it, for t 1, are independent of (a i1, ξ i ). The distribution of (a i1, ξ i, η i1 ) is unrestricted. A combination of the above identification arguments and the main result of Hu and Schennach (8) identifies: the period-t consumption distribution f (c t a t, η t, y t, ξ), and the distribution of initial conditions f (η 1, ξ, a 1 ). Richard Blundell Income and Consumption Dynamics November / 1

39 Data: PSID (New) PSID , we use 6 waves (every other year), as in BPS. C it : Information on food expenditures, rents, health expenditures, utilities, car-related expenditures,... A it : Assets holdings are the sum of financial assets, real estate value, pension funds, and car value, net of mortgages and other debt. (Net worth). y it are residuals of log total pre-tax household labor earnings on a set of demographics. Note, c it and a it are residuals, using the same set of demographics as for earnings. cohort and calendar time dummies, family size and composition, education, race, and state dummies. As in BPS, we select married male heads aged between 25 and 59. In this talk we focus on a balanced sub-sample of N = 792 households. Richard Blundell Income and Consumption Dynamics November / 1

40 Empirical specification: income The quantile function of η it given η i,t 1 is specified as Q t (η t 1, τ) = Q(η t 1, age t, τ) = K a Q k (τ)ϕ k (η t 1, age t), k= where ϕ k, k =, 1,..., K, are polynomials (Hermite). In addition, the quantile functions of ε it and η i1 are Q ε (age t, τ) = Q η1 (age 1, τ) = K ak ε (τ)ϕ k (age t), k= K a η 1 k (τ)ϕ k (age 1). k= Richard Blundell Income and Consumption Dynamics November / 1

41 Empirical specification: consumption We specify the (log) consumption function as: g t (a t, η t, ε t, τ) = g(a t, η t, ε t, age t, τ) = K b g k ϕ k (a t, η t, ε t, age t ) + b g (τ) k=1 additivity in the taste shifters, though not essential, is convenient given the sample size. In addition, the conditional quantiles of a i1 given η i1 and age i1 are Q (a) (η 1, age 1, τ) = K bk(τ) ϕ a k (η 1, age 1 ). k= Richard Blundell Income and Consumption Dynamics November / 1

42 Implementation choices Model a Q k (τ) as piecewise-linear interpolating splines (Wei and Carroll, 29) on a grid < τ 1 < τ 2 <... < τ L < 1, convenient as the likelihood function is available in closed form. We extend the specification of the intercept coefficient a Q (τ) on (, τ 1] and [τ L, 1) using a parametric model: exponential (λ). In practice, for the PSID data, we take L = 11 and τ l = l/l + 1. ϕ k and ϕ k are low-dimensional tensor products of Hermite polynomials. We set b (τ) = α + σφ 1 (τ), where (α, σ) are to be estimated. Richard Blundell Income and Consumption Dynamics November / 1

43 Estimation algorithm The first estimation step recovers estimates of the income parameters θ. The second step recovers estimates of the consumption parameters μ, given a previous estimate of θ. Our choice of a sequential estimation strategy, rather than joint estimation of (θ, μ), is motivated by the fact that θ is identified from the income process alone. Richard Blundell Income and Consumption Dynamics November / 1

44 Model s restrictions: income Let θ be the income-related parameters with true values θ. Let ρ τ (u) = u(τ 1{u }) denote the check function of quantile regression, and let a Q kl denote the value of aq kl = aq k (τ l) evaluated at the true θ. The model implies ( ) a Q l,., aq K l = argmin E (a Q l,.,aq K l) [ ρτl ( K η it a Q kl ϕ k (η i,t 1, age it) k= with additional restrictions involving the other parameters in θ. In the above, f i denotes the posterior density of (η i1,..., η it ) given the income data f i (η T i ; θ) = f (η T i y T i, age T i ; θ). ) f i (η T i ; θ)d Richard Blundell Income and Consumption Dynamics November / 1

45 Model s restrictions: consumption Letting μ (true value μ) be the consumption-related parameters, the model implies ( α, b1, g., b g ) K = argmin E ( K 2 c it α b g (α,b g k ϕ k (a it, η it, y it η it, age it )) g i ( 1,.,bg K ) k=1 and σ 2 = E ( c it α K k=1 b g k ϕ k (a it, η it, y it η it, age it )) 2 g i (η T i ; θ, μ)dη T i with additional restrictions involving the other parameters in μ. Here g i denotes the posterior density of (η i1,..., η it ) given the earnings, consumption, and asset data, g i (η T i ; θ, μ) = f (η T i c T i, a T i, y T i, age T i ; θ, μ). Richard Blundell Income and Consumption Dynamics November / 1

46 Overview of estimation A compact notation for the restrictions implied by the income model is [ ] θ = argmin E R(y i, η; θ)f i (η; θ)dη. θ We use a stochastic EM algorithm (in a non-likelihood setup). Starting with θ () we iterate on s=,1,... the following two steps until convergence of the Markov Chain: 1. Stochastic E-step: draw η (m) i = (η (m) i1,..., η(m) ) for m = 1,..., M from it f i ( ; θ (s) ). ABB use a random-walk Metropolis-Hastings sampler. 2. M-step: update θ (s+1) = argmin θ N M i=1 m=1 R(y i, η (m) i ; θ). Richard Blundell Income and Consumption Dynamics November / 1

47 Overview of estimation (cont.) As the likelihood function is available in closed form, the E-step is straightforward. The M-step consists of a number of ordinary regressions and quantile regressions, such as ( ) N T M min ρ (a Q τl η (m) K it a Q kl l,...,aq K l) ϕ k (η(m) i,t 1, age it), l = 1,..., L. i=1 t=2 m=1 k= We compute θ as an average of θ (s) across S iterations. We estimate θ and μ sequentially. Richard Blundell Income and Consumption Dynamics November / 1

48 Statistical properties Nielsen (2) studies the properties of this algorithm in a likelihood case. He provides conditions for the Markov Chain θ (s) to be ergodic (for a fixed sample size). He also shows that N autoregressive process as N tends to infinity. ( ) θ (s) θ converges to a Gaussian Arellano and Bonhomme [AB] (215) adapt Nielsen s arguments to derive the form of the asymptotic variance in a non-likelihood case. AB also study consistency as K (number of polynomial terms) and L (number of knots) tend to infinity with N. Richard Blundell Income and Consumption Dynamics November / 1

49 Empirical results Richard Blundell Income and Consumption Dynamics November / 1

50 Nonlinear persistence of η it (PSID): ρ t (η i,t 1, τ) = Q η t η t 1 (η i,t 1,τ) η persistence percentile τ init percentile τ shock Note: Estimates of the average derivative of the conditional quantile function of η it on η i,t 1 with respect to η i,t 1, evaluated at percentile τ shock and at a value of η i,t 1 that corresponds to the τ init percentile of the distribution of η i,t 1. Evaluated at mean age in the sample. Richard Blundell Income and Consumption Dynamics November / 1

51 Nonlinear persistence of y it Qyt yt 1 (yi,t 1,τ) PSID panel data y Nonlinear model persistence persistence percentile τ init percentile τ shock percentile τ init percentile τ shock Note: Estimates of the average derivative of the conditional quantile function of y it given y i,t 1 with respect to y i,t 1, evaluated at percentile τ shock and at a value of y i,t 1 that corresponds to the τ init percentile of the dist. of y i,t 1. Richard Blundell Income and Consumption Dynamics November / 1

52 Nonlinear persistence of y it Qyt yt 1 (yi,t 1,τ) Norwegian register data y Nonlinear model persistence persistence percentile τ init percentile τ shock percentile τ init percentile τ shock Note: Estimates of the average derivative of the conditional quantile function of y it given y i,t 1 with respect to y i,t 1, evaluated at percentile τ shock and at a value of y i,t 1 that corresponds to the τ init percentile of the dist. of y i,t 1. Richard Blundell Income and Consumption Dynamics November / 1

53 Figure: Densities of persistent and transitory earnings components (PSID) (a) Persistent component η it (b) Transitory component ε it density.8.6 density η component ε component Note: Nonparametric estimates of densities based on simulated data according to the nonlinear model, using a Gaussian kernel. Richard Blundell Income and Consumption Dynamics November / 1

54 Nonlinear persistence of y it (cont.) PSID data Q y t y t 1 (y i,t 1,τ) y Canonical model persistence persistence percentile τ init percentile τ shock percentile τ init percentile τ shock Note: Estimates of the average derivative of the conditional quantile function of y it given y i,t 1 with respect to y i,t 1, evaluated at percentile τ shock and at a value of y i,t 1 that corresponds to the τ init percentile of the dist. of y i,t 1. Richard Blundell Income and Consumption Dynamics November / 1

55 Nonlinear persistence, 95% confidence bands (a) Earnings, PSID data (b) Earnings, nonlinear model persistence persistence percentile τ init percentile τ shock percentile τ init percentile τ shock Note: Pointwise 95% confidence bands. Parametric bootstrap, 5 replications. Richard Blundell Income and Consumption Dynamics November / 1

56 Figure: Conditional skewness of log-earnings residuals and η component (a) Log-earnings residuals y it (b) Persistent component η it conditional skewness conditional skewness percentile y i,t-1 percentile η i,t-1 Note: Conditional skewness sk (y, τ) and sk(η, τ), for τ = 11/12. Log-earnings residuals (left) and η component (right). The x-axis shows the conditioning variable, the y-axis shows the corresponding value of the conditional skewness measure. Bootstrap confidence intervals in the Appendix. Richard Blundell Income and Consumption Dynamics November / 1

57 Consumption response to η it, by assets and age φ t (a) = E [ ] gt (a,η it,ε it,ν it ) η, nonlinear model.8 consumption response percentile τage percentile τ assets Note: Estimates of the average consumption response φ t (a) to variations in η it, evaluated at τ assets and τ age. Richard Blundell Income and Consumption Dynamics November / 1

58 Consumption responses to y it, by assets and age [ E ] yit E (c it a it = a, y it = y, age it = age) y PSID data Nonlinear model consumption response percentile τage percentile τ assets 1 consumption response percentile τage percentile τ assets 1 Note: Estimates of the average derivative of the conditional mean of c it given y it, a it & age it with respect to y it, evaluated at values of a it & age it corresponding to their τ assets & τ age percentiles, and averaged over the values of y it. Richard Blundell Income and Consumption Dynamics November / 1

59 Figure: Household heterogeneity in earnings (a) Nonlinear persistence of η it (b) Conditional skewness of η it persistence percentile τ init percentile τ shock conditional skewness percentile η i,t-1 Notes: (a) Estimates of the average derivative of the conditional quantile function of η it on η i,t 1 with respect to η i,t 1, based on estimates from the nonlinear earnings model with an additive household-specific effect. (b) Conditional skewness sk (η, τ), for τ = 11/12, based on the same model. Richard Blundell Income and Consumption Dynamics November / 1

60 Consumption response to η it, by assets and age, household heterogeneity [ ] gt (a,η φ t (a) = E it,ε it,ξ i,ν it ) η, nonlinear model consumption response percentile τage percentile τ assets 1 Note: Estimates of the average consumption response φ t (a) to variations in η it, evaluated at τ assets and τ age. Richard Blundell Income and Consumption Dynamics November / 1

61 Model s simulation Simulate life-cycle earnings and consumption after a shock to the persistent earnings component (at age 37). We report the difference between: Households that are hit by a bad shock (τ shock =.1) or by a good shock (τ shock =.9). Households that are hit by a median shock τ =.5. Age-specific averages across 1, simulations. At age 35 all households have the same persistent component (percentile τ init ). Richard Blundell Income and Consumption Dynamics November / 1

62 Impulse responses, earnings Bad shock: τ shock =.1 τ init =.1 τ init =.5 τ init = log-earnings log-earnings log-earnings age age age Good shock: τ shock =.9 τ init =.1 τ init =.5 τ init = log-earnings log-earnings log-earnings age age age Richard Blundell Income and Consumption Dynamics November / 1

63 Impulse responses, consumption Bad shock: τ shock =.1 τ init =.1 τ init =.5 τ init =.9 log-consumption log-consumption log-consumption age age age Good shock: τ shock =.9 τ init =.1 τ init =.5 τ init = log-consumption.1.5 log-consumption.1.5 log-consumption age age age Richard Blundell Income and Consumption Dynamics November / 1

64 Impulse responses, consumption, household heterogeneity Bad shock: τ shock =.1 τ init =.1 τ init =.5 τ init =.9 log-consumption log-consumption log-consumption age age age Good shock: τ shock =.9 τ init =.1 τ init =.5 τ init = log-consumption.1.5 log-consumption.1.5 log-consumption age age age Richard Blundell Income and Consumption Dynamics November / 1

65 Impulse responses, consumption, linear assets rule Nonlinear model τ init =.1 (a) τ shock =.1 (b) τ shock =.9.15 log-consumption log-consumption age age τ init =.9 (e) τ shock =.1 (f) τ shock =.9.15 log-consumption log-consumption age age Note: Linear assets accumulation rule. Assets are constrained to be non-negative. Richard Blundell Income and Consumption Dynamics November / 1

66 Impulse responses: canonical earnings and linear consumption model Earnings τ shock =.1 τ shock = log-earnings log-earnings age age Consumption τ shock =.1 τ shock =.9.15 log-consumption log-consumption age age Richard Blundell Income and Consumption Dynamics November / 1

67 Impulse responses, by age and initial assets Earnings τ init =.9, τ shock =.1 τ init =.1, τ shock =.9 Young Old Young Old log-earnings -.2 log-earnings -.2 log-earnings.2 log-earnings age age age age Consumption τ init =.9, τ shock =.1 τ init =.1, τ shock =.9 Young Old Young Old.2.2 log-consumption log-consumption log-consumption log-consumption age age age age Richard Blundell Income and Consumption Dynamics November / 1

68 Summary New framework to shed new light on the nonlinear transmission of income shocks to consumption and the nature of insurance to income shocks. Richard Blundell Income and Consumption Dynamics November / 1

69 Summary New framework to shed new light on the nonlinear transmission of income shocks to consumption and the nature of insurance to income shocks. A Markovian permanent-transitory model of household income, which reveals asymmetric persistence of unusual shocks in the PSID and in large administrative registers. An age-dependent nonlinear consumption rule that is a function of assets, permanent income and transitory income. Richard Blundell Income and Consumption Dynamics November / 1

70 Summary New framework to shed new light on the nonlinear transmission of income shocks to consumption and the nature of insurance to income shocks. A Markovian permanent-transitory model of household income, which reveals asymmetric persistence of unusual shocks in the PSID and in large administrative registers. An age-dependent nonlinear consumption rule that is a function of assets, permanent income and transitory income. Provide conditions for nonparametric identification: explain how a simulation-based sequential QR method is feasible. This framework leads to new empirical measures of the degree of partial insurance and the link between income and consumption inequality. But what about looking inside the family labour income measure...? Richard Blundell Income and Consumption Dynamics November / 1

71 A role for family labour supply? Families have the possibility of insuring consumption on many margins. Distinguish four separate mechanisms: Richard Blundell Income and Consumption Dynamics November / 1

72 A role for family labour supply? Families have the possibility of insuring consumption on many margins. Distinguish four separate mechanisms: 1. Labor supply of other family members, 2. Non-linear taxes and welfare, 3. Self-insurance (i.e., savings through the direct use of net assets), 4. Other informal mechanisms and networks... - Then examine each step in the distributional dynamics from wages to consumption: wages->earnings->family earnings->net income->consumption->wealth. Richard Blundell Income and Consumption Dynamics November / 1

73 A role for family labour supply? BPS use data on wage, consumption, income, labor supply and assets from the PSID. As described in the Nemmers Lecture, BPS show that family labor supply can be a key mechanism for insuring unexpected shocks - especially for younger families and for those with limited access to assets, - a strong added-worker effect as a response to a permanent shock. * Find an important role for unusual shocks and nonlinear persistence in the wages... Richard Blundell Income and Consumption Dynamics November / 1

74 Measured Nonlinear Persistence in the Male Wage Data: PSID persistence percentile τ init percentile τ shock 1 Notes: Log male wages, Age (US). Estimates of the average derivative of the conditional quantile function. Source: Arellano, Blundell and Bonhomme (217b). Richard Blundell Income and Consumption Dynamics November / 1

75 Simulated Nonlinear Persistence in the Male Wage Data: PSID persistence percentile τ init percentile τ shock 1 Notes: Log male wages, Age (US). Simulation of the average derivative of the conditional quantile function. Source: Arellano, Blundell and Bonhomme (217b). Richard Blundell Income and Consumption Dynamics November / 1

76 Family labour supply, time-use and consumption smoothing Recent research (BPS2) combines data on wage, consumption, income, labor supply, assets and time-use from the PSID, ATUS and CEX. Time-use data from ATUS is used to unpack what s going on in terms of family time allocation responses to to male and female wage shocks -> results uncover a tension between the desire of spouses to spend leisure time with each other, and the specialization in care of children. -> the presence of young children is found to give rise to Frisch substitutability of hours between spouses, with a switch to Frisch complements as children age and leave home. Richard Blundell Income and Consumption Dynamics November / 1

77 Family labour supply, time-use and consumption smoothing Recent research (BPS2) combines data on wage, consumption, income, labor supply, assets and time-use from the PSID, ATUS and CEX. Time-use data from ATUS is used to unpack what s going on in terms of family time allocation responses to to male and female wage shocks -> results uncover a tension between the desire of spouses to spend leisure time with each other, and the specialization in care of children. -> the presence of young children is found to give rise to Frisch substitutability of hours between spouses, with a switch to Frisch complements as children age and leave home. The strong added-worker effect from a response to an adverse permanent shock to his earnings is found to induce a fall in mother s time-use with young children, especially for low-educated with low assets. -> Details of the family labour supply, time-use and consumption smoothing model and results at the end of these lecture slides. Richard Blundell Income and Consumption Dynamics November / 1

78 Next steps 1 Study the implications for child outcomes, currently linking to CDS. 2 Separate housing equity and allow a role for local labour markets. 3 Include firm to firm transitions and lay-offs. 4 Include experience/human capital => as in BDMS (Ecta 216). 5 Health and other types of (partially insured) shocks (HRS, ELSA). 6 Estimate on the full population (Norwegian) register data. 7 and more... Richard Blundell Income and Consumption Dynamics November / 1

79 Additional slides Richard Blundell Income and Consumption Dynamics November / 1

80 Identification when T = 3: Wilhelm (12) We work in L 2 -spaces relative to suitable distributions. Let g(y 2, y 3 ) such that there exists a s(y 2 ) such that E [g(y 2, Y 3 ) Y 1 ] = E [s(y 2 ) Y 1 ]. Under completeness of Y 2 Y 1, s( ) is unique. By conditional independence, E [E (g(y 2, Y 3 ) η 2 ) Y 1 ]. Under completeness of η 2 Y 1, it follows that E [g(y 2, Y 3 ) η 2 ] = E [s(y 2 ) η 2 ]. Richard Blundell Income and Consumption Dynamics November / 1

81 The case T = 3 (cont.) Wilhelm (12) considers the functions g 1 (Y 3 ) = 1{Y 3 y 3 }, and g 2 (Y 2, Y 3 ) = Y 2 1{Y 3 y 3 }, for a given value y 3. This yields E [1{Y 3 y 3 } η 2 ] G (η 2 ) = E [s 1 (Y 2 ) η 2 ] E [Y 2 1{Y 3 y 3 } η 2 ] = η 2 G (η 2 ) = E [s 2 (Y 2 ) η 2 ]. Hence, taking Fourier transforms (i.e., F(h)(u) = h(x)e i ux dx), F(G )(u) = F(s 1 )(u)ψ ε2 ( u) i 1 df(g )(u)/du = F(s 2 )(u)ψ ε2 ( u), where ψ ε2 (u) = F(f ε2 )(u) is the characteristic function of ε 2, and i = 1. Richard Blundell Income and Consumption Dynamics November / 1

82 The case T = 3 (cont.) This yields the following first-order differential equation [ ] df(s1 )( u) F 2 (u)du = i F(s 2 )( u) ψ du ε2 (u). In addition, ψ ε2 () = 1. This ODE can be solved in closed form for ψ ε2 ( ), provided that F(s 1 )(u) = for all u (which is another injectivity condition). As a result, the distribution of ε 2, and the distribution of Y 3 given η 2, are both nonparametrically identified. Richard Blundell Income and Consumption Dynamics November / 1

83 Descriptive statistics PSID (means) Earnings 85,1 93,984 1,281 16, ,39 122,98 Consumption 3,182 35,846 39,843 47,636 52,175 5,583 Assets 266, , , ,91 51,59 46,262 Notes: Balanced subsample from PSID, N = 749, T = 6. Compared to BPS (12), households in our balanced sample have higher assets, and to a less extent higher earnings and consumption. Richard Blundell Income and Consumption Dynamics November / 1

84 Consumption response, two-period model CRRA utility. The Euler equation is (assuming β(1 + r) = 1) C γ 1 = E 1 [((1 + r)a 2 + Y 2 ) γ], where γ > is risk aversion and we have used the budget constraint A 3 = (1 + r)a 2 + Y 2 C 2 =. Let X 1 = (1 + r)a 1 + Y 1, R = (1 + r)x 1 + E 1 (Y 2 ), and Y 2 = E 1 (Y 2 ) + σw. Expanding as σ we obtain C 1 (1 + r)x 1 + E 1 (Y 2 ) 2 + r }{{} certainty equivalent γ + 1 2R E 1(W 2 ) }{{} precautionary-variance (2 + r)(γ + 1)(γ + 2) + 6R 2 E 1 (W 3 ). }{{} precautionary-skewness Richard Blundell Income and Consumption Dynamics November / 1

85 Nonlinear persistence, 95% confidence bands (a) Earnings, PSID data (b) Earnings, nonlinear model persistence persistence percentile τ init percentile τ shock percentile τ init percentile τ shock (c) Persistent component η it, nonlinear model persistence percentile τ init percentile τ shock Richard Blundell Income and Consumption Dynamics November / 1

86 Conditional skewness of log-earnings residuals and η component, 95% confidence bands (a) Log-earnings residuals y it (b) Persistent component η it conditional skewness conditional skewness percentile y i,t-1 percentile η i,t-1 Note: Pointwise 95% confidence bands. Parametric bootstrap, 5 replications. Richard Blundell Income and Consumption Dynamics November / 1

87 Nonlinear persistence of η it (Norwegian register data): ρ t (η i,t 1, τ) = Q η t η t 1 (η i,t 1,τ) η persistence percentile τ init percentile τ shock Note: Estimates of the average derivative of the conditional quantile function of η it on η i,t 1 with respect to η i,t 1, evaluated at percentile τ shock and at a value of η i,t 1 that corresponds to the τ init percentile of the distribution of η i,t 1. Evaluated at mean age in the sample. Richard Blundell Income and Consumption Dynamics November / 1

88 Family Labour Supply, Time-Use and Consumption Smoothing Modelling Slides () NOVEMBER 26, / 39

89 SOME RELATED LITERATURE Added worker effect: Lundberg (1985), Hyslop (21), Stephens (22), Attanasio, Low and Sanchez-Marcos (25), Juhn and Potter (27), Haan and Prowse (215), Blundell, Pistaferri and Saporta-Eksten (216), Autor, Kostol, Mogstad and Setzler (217),... Time use, time spent with children: Ghez and Becker (1975), Becker (1976), Aguiar and Hurst (27, 213), Guryan, Hurst and Kearney (28), Ramey and Ramey (21), Browning, Chiappori and Weiss (214), Del Boca, Flinn and Wiswall (214),... Consumption insurance: Hall and Mishkin (1982), Blundell and Preston (1998), Krueger and Perri (26), Guvenen (27), Blundell, Pistaferri and Preston (28), Heathcote, Storesletten and Violante (28), Kaufmann and Pistaferri (29), Kaplan and Violante (21), Guvenen and Smith (213), Heathcote, Storesletten and Violante (214), Arellano, Blundell and Bonhomme (217)... () NOVEMBER 26, / 39