Nonlinear Persistence and Partial Insurance: Income and Consumption Dynamics in the PSID†

AEA Papers and Proceedings 2018, 108: 281–286 https://doi.org/10.1257/pandp.20181049 Nonlinear Persistence and Partial Insurance: Income and Consumption Dynamics in the PSID† By Manuel Arellano, Richard Blundell, and Stephane Bonhomme* In this paper we highlight the important role income dynamics using the extensive population the PSID has played in our understanding of register data from Norway. income dynamics and consumption insurance, Exploiting the enhanced consumption and see for example, Krueger and Perri 2006 ; asset data in recent waves of the PSID, we show Blundell, Pistaferri, and Preston 2008 ,( hence)- that nonlinear persistence has key implications forth, BPP ; and Guvenen and (Smith) 2014 . for consumption insurance. The approach is In the (partial) insurance approach, transmission( ) used to provide new empirical measures of parameters are specified that link “shocks” to partial insurance in which the transmission of income with consumption growth. These trans- income shocks to consumption varies systemat- mission parameters can change across time ically with assets, the level of the shock, and the and may differ across individuals reflecting the history of past shocks. degree of “insurance” available. They encom- pass self-insurance through simple credit mar- I. Earnings and Consumption Dynamics kets as well as other mechanisms used to smooth consumption. A prototypical “canonical” panel data model We explore the nonlinear nature of income of log family earned income yit is shocks and describe a new quantile-based panel ( ) ( ) data framework for income dynamics, devel- yit it it , i 1, , N, t 1, , T, oped in Arellano, Blundell, and Bonhomme = η + ε = … = … ABB 2017 . In this approach the persistence of where y is net of a systematic component, is ( ) it ηit past income shocks is allowed to vary accord- a random walk with innovation vit , ing to the size and sign of the current shock. We find that the model provides a good match it it 1 vit , i 1, , N, t 1, , T, with data on family earnings and on individual η = η − + = … = … wages from the PSID. We confirm the results on and it is a transitory shock. Thereε is good economic reasoning behind this decomposition: persistent shocks to earned * Arellano: CEMFI, Madrid email: [email protected] ; income are more difficult to insure, especially Blundell: University College London,( Gower Street, London) for young families with low assets. How families email: [email protected] ; Bonhomme: University of ( ) cope with persistent shocks is the main focus of Chicago, Chicago email: [email protected] We this research. Short-run fluctuations will matter would like to thank (David Johnson and Karen Dynan for). com- ments. Luigi Pistaferri and Itay Saporta-Eksten for insights too, of course, especially for households with low assets or low access to liquid assets . on the PSID data, Magne Mogstad and Michael Graber for ( ) the estimations and insights using the Norwegian popula- In the partial insurance framework, consumption register data as part of the project on “Labour Income tion growth is related to income shocks, Dynamics and the Insurance from Taxes, Transfers and the Family.” Arellano acknowledges research funding from the Ministerio de Economía y Competitividad, Grant ECO2016- c v , 79848-P. Blundell would like to thank the ESRC Centre CPP Δ it = ϕt it + ψt εit + νit at IFS and the ERC MicroConLab project for financial assis- tance. Bonhomme acknowledges support from the European i 1, , N, t 1, , T, Research Council ERC grant agreement no. 263107. = … = … † Go to https://doi.org/10.1257/pandp.20181049/ to visit the article page for additional materials and author disclo- where cit cit 1 ci , t−1 , cit is log consump- Δ = − − sure statement s . tion net of a systematic component, is the ( ) ϕt 281 282 AEA PAPERS AND PROCEEDINGS MAY 2018 transmission of persistent shocks v , and the a general first-order Markov process.1 Denoting it ψt transmission of transitory shocks; the it are the th conditional quantile of it given i , t 1 as ν τ η η − taste shocks, assumed to be independent across Qt i , t 1 , , we specify periods. BPP show identification and efficient (η − τ) estimation using GMM. They also allow for it Qt i,t 1 , it , measurement error and extend to MA 1 tran- η = ( η − Τ ) ( ) sitory shocks. where uit i,t 1 , i,t 2 , Uniform 0, 1 , The parameters and link the evolution and has( zero| η −mean,η − independent…) ~ over (time.) ϕt ψt εit of consumption inequality to income inequality. The conditional quantile functions Qt i , t 1 , uit (η − ) They indicate the degree of partial insurance, and the marginal distributions F t can all be age and will differ by age, assets, and human capital. specific. ε For example, using a linearized approximation This framework allows for quite general non- to a simple benchmark intertemporal consump- linear dynamics of income, allowing a general tion model, Blundell, Low, and Preston 2013 form of conditional heteroscedasticity, skew- show ( ) ness, and kurtosis. To see this, consider the following measure of persistence: t 1 it and t 1 it Lt , Q , ϕ = ( − π ) ψ = ( − π )γ ___________t i, t 1 t i , t 1 , ∂ (η − τ) , _________________ Assetsit ρ (η − τ) = where it and Lt is the ∂ η π ≈ Assetsit Human Wealth it γ annuity value of a+ temporary shock to income for an individual aged t retiring at age L. which measures the persistence of i , t 1 when, at age t, it is hit by a shock u thatη has− rank . For the PSID, estimates of 1 it typically it τ average at around 0.82. BPP (estimate− π ) a partial This measures the persistence of histories, which insurance coefficient of 0.642 0.09 . This may depend on both the level of earnings it 1 ϕ ( ) and the percentile of the shock . Below(η −we) represents excess insurance relative to 1 it . (τ) They document higher values for samples( − withπ )- show strong evidence for such nonlinearities in out college education, for older cohorts, and for persistence. low wealth samples. This linearized partial insurance framework III. An Empirical Model for Consumption provides key insights on the distributional dynamics of income and consumption. However, it rules To motivate the specification of consumption out the nonlinear transmission of shocks and we use a standard life-cycle incomplete markets restricts interactions in consumption responses. model. Let cit and ait denote log-consumption and assets beginning of period net of age dum- mies. We model( consumption in) levels and leave II. Nonlinear Persistence the nonlinear rule flexible. Our empirical specification is based on The aim in the new work on nonlinear persistence is to step back from the standard panel c g a , , , , t 1, , T, data model of income dynamics and take a dif- it = t( it ηit εit νit) = … ferent track: develop an alternative approach in which the impact of past shocks can be altered where it are independent across periods, and gt by the size and sign of new shocks. The frame- is a nonlinear,ν age-dependent function, mono- work allows “unusual” shocks to wipe out the tone in it ; it may be interpreted a taste shifter memory of past shocks. Additionally the future that increasesν ν marginal utility. This consumption persistence of a current shock will depend on rule is consistent, in particular, with the standard future shocks. We will see that the presence of life-cycle model. ABB derive conditions under “unusual” shocks matches the data well and has which g is nonparametrically identified. a key impact on consumption and saving deci- sions over the life cycle. In this framework we maintain the factor 1 The first-order Markov assumption can be generalized to structure for log income but allow to follow Markov p , with any fixedp although this requires larger T . ηit ( ) ( ) VOL. 108 NONLINEAR PERSISTENCE AND PARTIAL INSURANCE 283 The consumption responses to and are demographics as for earnings. Following BPS, η ε we select married male heads aged between gt a, , , 25 and 59. We focus on a subsample of 792 a, , E ___________∂ ( η ε ν) , ϕt ( η ε) = households. [ ∂ η ] The conditional quantile function for the permanent income factor it , given i , t 1 , is gt a, , , η η − a, , E ∂___________( η ε ν) , specified as ψt ( η ε) = [ ] ∂ ε Qt t 1 , Q t 1 , ag et , (η − τ) = (η − τ) where a, , and a, , reflect the trans- K ϕt ( η ε) ψt ( η ε) Q mission of the persistent and transitory earnings a k k t 1 , ag et , − = ∑k 0 (τ)φ (η ) components, respectively. They generalize the = partial insurance coefficients of BPP. Similar techniques can be used in the pres- where k , k 0, 1, . , K , are polynomials ence of advance information, consumption Hermiteφ . We =proceed similarly for , etc. The ( ) εit habits, and in cases where the consumption consumption log function, g at , t , t , ag et , is rule depends on lagged , or when follows specified as a( flexible) polynomial( η inε assets,) per- a second-order Markov process, η see ηSection 3 manent income factor, the transitory shock and in ABB. The framework allows for additional, age, gt is additive in it . unobserved heterogeneity in earnings and con- Estimation takes placeν in two steps, see ABB sumption. Households can also differ in their for details. The first step recovers estimates of initial productivity 1 and initial assets. the income parameters. The second step recov- η ers estimates of the consumption parameters, IV. Data and Estimation given an estimate of the income parameters. The estimation algorithm alternates between draws The PSID went through a redesign in the late of latent variables from candidate posteriors and 1990s, introducing new consumption and asset quantile regressions using those draws, see also modules.

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