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

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, ϕ = ( − π ) ψ = ( − π )γ ______Qt i, t 1, t i, t 1, ∂ (η − τ) , ______Assetsit ρ (η − τ) = where it and Lt is the ∂ η π ≈ Assetsit Human Wealthit 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 dummies. 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, aget, (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, aget , − = ∑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, aget , 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. Since 1999 it collects some 70 per- Arellano and Bonhomme 2016 . cent of non-durable consumption expenditures, ( ) and more than 90 percent since 2005. We use V. Empirical Results the sum of food at home, food away from home, gasoline, health, transportation, utilities, etc. We Figure 1 provides our initial evidence for non- also make use of the more detailed asset data, see linear income dynamics. It presents estimates of Blundell, Pistaferri, and Saporta-Eksten 2016 . the average derivative of the conditional quantile ( ) For comparison we use family earnings data from function of y it given yi, t 1 with respect to yi, t 1 administrative records in the Norwegian popula- for both the PSID in panel− A, and the Norwegian− tion registers, see Blundell, Graber, and Mogstad register data in panel B. These are evaluated at 2015 .2 percentiles of the shock and at a value of ( ) τshock The results we present on the PSID use data yi, t 1 that corresponds to the init percentile of − τ from the 1999–2009 surveys. Assets holdings the distribution of y i, t 1. are the sum of financial assets, real estate value, The estimates in −Figure 1 display distinct pension funds, and car value, net of mortgages and systematic nonlinearity. The persistence of and other debt. Income yit are residuals of log income shocks is much lower for large negative total pretax household labor earnings on a set positive shocks for high low initial incomes. of demographics—cohort and calendar time The( results) for the PSID are( confirmed) in panel dummies, family size and composition, educa- B for the Norwegian data. tion, race, and state dummies. Log consump- Turning to the income model, Figure 2, tion cit is also a residual, using the same set of panel A provides estimates of the average derivative of the conditional quantile function of the persistent income factor it on i, t 1 with respect η η − 2 to i, t 1 , evaluated at percentile shock and at a The Norwegian results are part of the project on “Labor η − τ Income Dynamics and the Insurance from Taxes, Transfers value of i, t 1 that corresponds to the init per- η − τ and the Family.” See ABB Appendix C. centile of the distribution of i, t 1. The estimates η − 284 AEA PAPERS AND PROCEEDINGS MAY 2018

Panel lo earnins in the PSID data Panel Persistent component η

Persistence Persistence 1 0.8 1 1 0.8 1 0.6 0.8 0.8 0.4 0.6 0.6 0.4 0.4 0.6 0.2 0.2 0.4 Percentile 0 0 Percentile Percentile 0.2 0.2 Percentile τ τ τ 0 0 τ

Panel lo earnins in the Noreian Panel arnins nonlinear model administratie data

Persistence Persistence 1 1 0.8 1 0.8 1 0.6 0.8 0.6 0.4 0.6 0.8 0.4 0.4 0.6 0.2 0.2 0.4 Percentile 0 0 Percentile 0.2 τ τ 0 0.2 Percentile 0 Percentile τ τ Figure 1. Quantile Autoregressions Figure 2. Nonlinear Persistence

Notes: Residuals yit of log pretax household labor earnings, Age 25–60 1999–2009 United States , Age 25–60 2005– Notes: PSID data. Panel A shows estimates of the average ( ) derivative of the conditional quantile function of it on i,t 1 2006 Norway . Estimates of the average derivative of the η η − ( ) with respect to i,t 1, based on estimates from the nonlinear conditional quantile function of y it given y i,t 1 with respect to η − − earnings model. Panel B is based on data simulated accord- yi,t 1. Quantile functions are specified as third-order Hermite polynomials.− ing to our nonlinear earnings model with parameters set to Source: Arellano, Blundell, and Bonhomme 2017 their estimated values. ( ) Source: Arellano, Blundell, and Bonhomme 2017 ( ) are evaluated at mean age in the sample. Panel B and ag e corresponding to their and it τassets τage in Figure 2 is based on data simulated according percentiles, and averaged over the values of yit . to our nonlinear earnings model with parame- It shows consumption responses vary ters set to their estimated values. It shows a close systematically with age and assets and in a way accordance with the persistence in the PSID that accords with standard life-cycle theory. It income data, see panel A of Figure 1. also shows an agreement between the consump- Moving to the estimated consumption model, tion model and data. Figure 3 displays the average derivative of the Finally, we provide preliminary evidence that conditional mean of cit given yit , ait , and ageit the nonlinear persistence we have uncovered in with respect to yit , evaluated at values of ait family earnings data is also evident in hourly VOL. 108 NONLINEAR PERSISTENCE AND PARTIAL INSURANCE 285

Panel PSID data y assets and ae Panel aes PSID

Persistence Consumption response 0 1 0.2 0.8 1 1 0.4 0.6 0.8 0.8 0.4 0.6 0.6 0.6 0.2 0.4 0.4 0.2 0.8 Percentile 0 0 1 0.2 τ Percentile Percentile 0 Percentile τ τ τ

Panel Nonlinear model y assets and ae Panel Persistent component η

Persistence Consumption response 1 0 1 0.8 1 0.2 0.8 0.6 0.8 0.4 0.6 0.4 0.6 0.4 0.6 0.4 0.2 0.2 0.2 Percentile 0 0 Percentile Percentile 0.8 Percentile τ τ τ 1 0 τ

Figure 3. Consumption Responses to y it Figure 4. Nonlinear Persistence, Hourly Wages

Note: Estimates from PSID of the average derivative of the Notes: Log male hourly wages, Age 30–60 PSID 1999–2009 conditional mean of c given y , a , and age with respect United States . Estimates of the average derivative of the it it it it ( ) to yit , evaluated at values of a it and ageit corresponding to conditional quantile function. their and percentiles, and averaged over the val- Source: Authors’ calculations τassets τage ues of yit . Source: Arellano, Blundell, and Bonhomme 2017 ( ) VI. Summary and Conclusions wage data. Figure 4, panel A, presents estimates of nonlinear persistence in PSID male hourly We have outlined a new framework for earnings. Panel B provides the estimates of the income dynamics and the nonlinear transmis- average derivative of the conditional quantile sion of income shocks to consumption. We have function of the component from the nonlin- exploited important new measurements for con- ear model. These results showη an important role sumption and assets in the PSID. We have also for unusual shocks and nonlinear persistence shown the complementarities between the use of in hourly wage data, suggesting nonlinear per- “big” administrative data, e.g., Norwegian reg- sistence may be a key feature for life-cycle mod- isters, and purpose-designed panel surveys, like els of family labor supply. the PSID. 286 AEA PAPERS AND PROCEEDINGS MAY 2018

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