Heterogeneity of consumption responses to income shocks in the presence of nonlinear persistence ✩

In this paper we use the enhanced consumption data in the Panel Survey of Income Dynamics (PSID) from 2005–2017 to explore the transmission of income shocks to consumption. We build on the nonlinear quantile framework introduced in Arellano et al. (2017). Our focus is on the estimation of consumption responses to persistent nonlinear income shocks in the presence of unobserved heterogeneity. To reliably estimate heterogeneous responses in our unbalanced panel, we develop Sequential Monte Carlo computational methods. We find substantial heterogeneity in consumption responses


Introduction
The empirical analysis of consumption and income dynamics has an important place in a number of key areas of economic research and policy design.A large literature aims at understanding income persistence, income inequality and income volatility, see Moffitt and Gottschalk (1993), Baker and Solon (2003) and references in Meghir and Pistaferri (2011).A parallel literature studies how income shocks impact consumption and savings decisions, see Hall and Mishkin (1982) and Blundell et al. (2008) among many other references.In this paper our goal is to empirically document the nature of consumption responses, with a particular focus on household heterogeneity and nonlinear persistence.
Economic models inform the empirical analysis of consumption and income.In a standard incomplete markets model of the life cycle, how much a household consumes in a given period is determined by the level of assets, the stage of the life cycle, as well as the income stream, see Deaton (1992) for a comprehensive review.Changes to income components ✩ We thank the editor Xiaohong Chen, an associate editor, two anonymous referees, and participants at the Whitney Newey Conference at MIT, and other venues for helpful comments.Arellano acknowledges research funding from the Ministerio de Economia y Competitividad, Grant ECO2016-79848-P.Blundell would like to thank the ESRC Centre for the Microeconomic Analysis of Public Policy (CPP) at the Institute for Fiscal Studies (ES/T014334/1) for support.Bonhomme acknowledges support from the National Science Foundation, Grant SES-1658920.with varying degrees of persistence, and for unobserved heterogeneity in consumption, are both important to accurately document these patterns quantitatively.Heterogeneity in consumer responses to income shocks matters for understanding the impact not only of fiscal policies but also of monetary policies which, as Auclert (2019) notes, can create large redistribution in favor of high MPC agents and be expansionary over and beyond the effect on real interest rates.
Our key finding is that consumption responses vary substantially with unobserved types.Our results clearly separate lower consumption types, who appear to follow the life-cycle patterns in consumption responses implied by standard models, from higher types, whose consumption responses to income shocks vary little with either assets levels or the stage of the life cycle.High-type households consistently have higher consumption levels, and relative to low-type households they have slightly higher incomes and levels of assets.For the younger low types, consumption responses to persistent income shocks exceed 0.50 while for older low types this falls below 0.20.Moreover, based on bootstrapped confidence intervals we conclude the difference between the two coefficients is significant at conventional levels.For the higher types, consumption responses are flatter across age and assets, and differences across age and assets are insignificant.These findings shed new light on the presence of heterogeneity in consumption behavior across households, on which there has been extensive micro-and macroeconomic research, see Alan et al. (2018), Crawley and Kuchler (2020), and references therein.
We examine several mechanisms that could lead to such heterogeneous consumption responses.First, the fact that high types consume more and hold more assets is difficult to reconcile with an explanation based on heterogeneity in preferences or discounting.Second, we estimate a specification that allows for latent heterogeneity in asset accumulation and find that the heterogeneity in consumption responses is virtually unaffected.Lastly, we link a subset of household heads in our sample (33%) to their parents, using the inter-generational linkages that the PSID provides.We find that high-type household heads have on average parents with higher consumption and income levels, suggesting that the heterogeneous responses that we find might in part reflect heterogeneity in access to other sources of insurance such as parental insurance.
We show the main results are robust to a number of specification changes.In particular, while we use disposable income in most of the analysis, we find similar patterns when using pre-tax labor income, with some quantitative differences.In addition, we find that including households where one member may not be working does not lead to major changes in our results.Lastly, we probe the robustness of our scalar individual effect modeling approach by allowing for a separate effect of education on consumption responses, in addition to the latent type.While the heterogeneity results remain qualitatively similar, the findings based on this specification allow us to discuss some limitations of our scalar individual effect modeling approach and to motivate future work.
The outline of the paper is as follows.In Section 2 we describe the sample and present motivating evidence on the nature of consumption responses.In Section 3 we provide a general description of the model, and in Section 4 we discuss implementation and present the computational methods we use.We then show our main empirical results in Section 5.In Section 6 we study possible mechanisms for those results.In Section 7 we show results based on extensions of our main model.We conclude in Section 8.An appendix describes implementation and provides additional empirical results.Lastly, replication codes are available in an onlinerepository.

Data
In this section we describe the PSID sample, and we provide preliminary motivating evidence about how consumption responds to income changes.

The PSID sample
We rely on the newly redesigned PSID, from 2005 to 2017.Since 1999, the PSID presents a unique combination of longitudinal data on income, consumption, and assets holdings for the US.Unlike the annual information available every year before 1997, after 1999 a new wave is only available every other year.Since 2005, the consumption information has been enhanced, with additional categories, see Li et al. (2010).The recent waves include food at home and away from home, gasoline, health, transportation, utilities, clothing, and leisure activities.Andreski et al. (2014) provide a detailed analysis of the post-2005 data and assess the new methodology developed by the PSID for collecting household expenditure data.The new survey methodology allows unfolding brackets as well as choice of time-frame for different consumption categories.They show that since 2005 the PSID has captured almost all expenditures measured in the cross-sectional Consumer Expenditure Survey (CE) and suggest the new measurement design is likely to improve on the accuracy of the expenditure data.For this reason, we expect the post-2005 PSID to provide more accurate information about household consumption patterns than the earlier period used in ABB.
Another difference with ABB is that we do not restrict the panel to be balanced.Following Blundell et al. (2016), we focus on a sample of household heads that participate in the labor market and are between 25 and 60 years old.Since we do not model labor supply, either at the extensive or intensive margin, in our baseline sample we focus on households where both adult members are working and present in at least two waves, and we keep their first spell of non-zero income observations.We refer to this baseline as the ''dual earners'' sample.However, in Section 7 we will also present results based on a broader sample that includes households where only one member is employed.Notes: PSID, 2005-2017.Means of variables, standard deviations in parentheses.Our baseline measure of consumption includes the following categories: food at home, food delivery, eating out, food stamps, clothing, gasoline, utilities, telephone bills, automobile insurance, parking, transport, education, childcare, institutional medical services, doctor services, prescriptions, health insurance, and trips and other recreation.
Our final dual earners sample contains 2,113 households and seven biennial waves from 2005-2017.In Table 1 we report some descriptive statistics about this sample.Food consumption, which was the only consumption item available in the PSID prior to the redesign of the data set, accounts for approximately one fourth of total non-durables consumption.Net disposable income is approximately 30% lower than pre-tax labor income.Since it is disposable income and not pre-tax income that should affect consumption decisions, we will focus on disposable income in most of the analysis.In Section 7 we will also present results using pre-tax labor income.
Table 1 also shows that total wealth tends to decrease around the 2008 recession, whereas income and especially consumption seem more stable over the period.See Krueger et al. (2015) for an analysis of consumption, income and wealth using the PSID with a focus on the great recession.In our analysis we will not focus on business cycle fluctuations, and we will attempt to remove calendar time effects in a prior partialling-out estimation step.
In Appendix Table D.1 we show additional statistics in order to describe the unbalanced structure of the panel sample.In the first column of that table we report statistics for households who are only observed for one wave, although we do not include these households in our main sample due to our focus on unobserved heterogeneity.More than half of households in our main sample are observed for at most three waves.For this reason, it will be important to account for the unbalancedness of the PSID in the modeling of income and consumption dynamics.
Following a common practice in the previous literature on income dynamics, we will work with residuals of logdisposable income on a set of demographics and time indicators.This partialling-out is meant to make household demographics as comparable to each other as possible, and to control for aggregate time effects.Specifically, we net out household size, year of birth, state indicators, number of kids, race of both adults, a higher education indicator for both adults interacted with age indicators, and a full set of age indicators interacted with year indicators.We similarly construct residuals of log-consumption and log-assets net of the same set of controls.Working with logarithms requires removing observations with zero or negative assets, which reduces the number of observations by approximately 200 households per year.In Appendix Table D.2 we report additional statistics for a sample which includes households with negative asset balances.

A first look at consumption responses
We will analyze the PSID sample using a dynamic model of income, consumption, and assets holdings.The model is flexibly parameterized and it features various latent variables.Before describing how we specify the model and estimate it, here we provide preliminary motivating evidence about consumption and income, only using observed covariates and simple econometric methods.We highlight two features of the data in turn.Notes: The graphs show averages of the derivatives of quantile functions of log-consumption with respect to log-income, conditional on income, age and assets.In the left graph we report results for the bottom tercile (averaged over a fine grid of percentiles), in the right graph we report results for the top tercile.Estimates are based on quantile regressions of log-consumption on a second-order polynomial in log-income, age, and log-assets.The two horizontal axes show age and assets percentiles.
In Fig. 1, we show average derivatives of log-consumption with respect to log-income, controlling for age and logassets. 3The derivative effect is 0.45 on average, with a standard deviation of 0.07.In particular, wealthier and older households have a lower derivative (i.e., lower than 0.30), suggesting that they are relatively well insured against income shocks.In contrast, younger and less wealthy households have a higher derivative (i.e., higher than 0.50), suggesting less ability to insure.
In Fig. 2, we show quantile derivatives of log-consumption with respect to log-income.In the left graph, we average quantile derivatives over the bottom tercile, while in the right graph we report an average over the top tercile.We see that these quantile derivative coefficients tend to be somewhat higher at the bottom of the consumption distribution (0.48 on average) than at the top (0.43 on average).The main difference between the two graphs concerns the younger and less wealthy households, for whom the derivative drops from 0.60 to 0.40 when moving from the bottom tercile to the top tercile. 4his evidence is suggestive of the presence of heterogeneity in consumption responses and insurance.However, there are several reasons why it may be incomplete and quantitatively inaccurate.Standard consumption models imply that income components with varying degrees of persistence have a different impact on consumption.Hence, while in Fig. 1 we report derivatives with respect to observed income, in a model where log-income is the sum of a persistent and a transitory component, economically-relevant consumption derivatives should be computed with respect to the latent components of income.To do so, a dynamic model with latent variables is needed.The heterogeneity suggested by Fig. 2 is similarly ambiguous.Indeed, consumption quantiles are likely to reflect a combination of time-invariant household heterogeneity and time-varying shocks.Distinguishing the two requires estimating a dynamic panel data model that features latent heterogeneity explicitly.In the next two sections we describe such a model, and we explain how we estimate it using the PSID.

Consumption behavior
Our primary interest is to understand how shocks to income translate into consumption for different types of consumers.Consumers are allowed to differ along a number of dimensions, specifically according to their assets, the stage in their life cycle, observable characteristics, and unobserved heterogeneity.Our underlying framework is one where households act as single agents with access to a single risk-free asset.They receive income shocks each period and make consumption decisions subject to a period-to-period budget constraint.We assume all distributions are known to households, and there is no aggregate uncertainty.
In modeling the dynamic responses of consumption to earnings shocks, one strategy is to specify the functional form of the utility function and the distributions of the shocks, and to calibrate or estimate the model's parameters by comparing the model's predictions with the data, see Kaplan and Violante (2010) and references therein.Another strategy is to follow the partial insurance approach of Blundell et al. (2008) and linearize the Euler equation, with the help of the budget constraint.The approach we follow in this paper builds on the framework introduced in ABB.It differs from the earlier strategies as we directly estimate the consumption rule that comes from the optimization problem.In this approach the level of consumption is modeled as a function of beginning of period assets, income components, consumer characteristics and individual heterogeneity.The framework we develop here is a generalization of the main specification in ABB to allow for individual unobserved heterogeneity and a more flexible policy rule.The shape of the consumption function and its derivatives will depend on the distributions of beliefs about future incomes and characteristics.We are therefore able to document a rich set of derivative effects but, as our model does not separate the role of preferences from expectations, we cannot recover counterfactuals that involve a change in the income process.
In our approach, the income process is modeled using the framework of ABB which allows for nonlinear persistence.In this framework, log-income is decomposed into a predetermined life-cycle component and two latent stochastic factors that represent the level of persistent income and the level of transitory income.We consider an unbalanced panel of households, i = 1, . . ., N, in which household i is observed T i consecutive time periods.For any household i at time t we denote the persistent income component as η it and assume it follows a nonlinear first-order Markov process.The transitory income component ε it is assumed to be distributed independently across time and independent of the η ′ s.Log-income residuals are then y it = η it + ε it .The details of the income specification are developed in the next subsection.
Given beginning-of-period-t assets a it , and the realizations of the persistent and transitory income components η it and ε it , consumers make their consumption choices according to the policy rule where t i denotes the period when i enters the panel, c it is log-consumption for household i in period t, a it is log-assets, age it is the age of the household head in period t, and unobserved heterogeneity is given by the ''fixed effect'' ξ i . 5As mentioned above, both c it and a it are net of common effects of age and other demographics, and of time indicators.We also allow consumption choices to depend on transitory preference shocks ν it , with arbitrary dimension.
Our main goal is to estimate the empirical consumption response parameters ] . (2) Average derivative effects such as (2) can be identified without restricting the dimensionality of ν it , see Matzkin (2013) and references therein. 6Reporting features of estimates of the individual transmission parameters in the PSID will shed light on how much variation there is in consumption responses and insurance, over the life cycle and as a function of assets and income.Importantly, the dependence of the consumption function on the latent type ξ i will allow us to document individual heterogeneity in consumption responses.Exploring the relationship between φ it and ξ i is a main objective of this paper.
In order to estimate the consumption function g t in (1), one needs to recover the persistent and transitory income components η it and ε it , and the time-invariant consumption type ξ i , all of which are unobserved to the econometrician.
For this purpose, we will estimate a dynamic model of income and consumption with latent variables, following ABB.
Asset accumulation.Estimation of the consumption function g t requires taking a stand on the accumulation of assets.A simple case is when current assets only depend on lagged assets, income, and consumption, but not on the latent income components and heterogeneity separately.This would hold in a textbook asset accumulation rule with a constant riskfree interest rate, for example.Under the assumption that asset accumulation does not depend on the latent variables, one can estimate the consumption function consistently without having to model the assets process, in the spirit of partial likelihood estimation.We will use this approach in our main results.More generally, our approach can allow the latent income components and type heterogeneity to affect current assets, and we will report results based on such a specification as well, see Section 6.3.
Dispersion of consumption derivatives.Lastly, while we focus on recovering the average response parameters φ it , the distribution of the consumption derivatives conditional on (a it , η it , ε it , age it , ξ i ), is generally not identified unless ν it is scalar and has a monotone effect on g t .Yet, using an insight from Hoderlein and Mammen (2007), one can compute a lower bound on the variance of the consumption derivatives ∂c it ∂η , even though the variance itself is not identified.We make this point formally in Appendix C, and we will report empirical estimates of bounds on variances as a complement to our main average coefficients.

Income and consumption
Our modeling of the income process closely follows ABB, with the main difference that we extend the model to an unbalanced panel.Specifically, let y it be the log-disposable income of household i in year t, net of common effects of age and other demographics, and time indicators.We specify the following persistent-transitory model where the persistent and transitory components η it and ε it , respectively, are zero-mean continuous latent variables given age.
We model the processes η it and ε it using their quantile representations.Let Q A (B, v) be a generic notation for the conditional quantile of A given B, evaluated at the percentile v in the unit interval.The quantile representation of A given B implies that A = Q A (B, V ), where V is standard uniform independent of B. 7  The persistent income component η it follows a nonlinear first-order Markov process with age-specific transitions; that is, 8 In order to model entry in the panel, we let the initial persistent latent component η i,t i depend on years of education and birth cohort of the household head, and on age at entry in the sample: (5) In turn, the transitory component ε it is assumed to be independent over time and independent of η is for all s with an age-specific distribution, Note that the income process is common across households.In this paper we do not attempt to model latent time-invariant heterogeneity in the income process beyond heterogeneity in initial conditions.However, we allow for an unobserved type that affects consumption and may be correlated with income.
Turning to consumption, we let the unobserved heterogeneity variable ξ i be correlated with birth cohort, education, and income; that is, we specify 7 For example, Q A (B, 0.50) is the conditional median of A given B, and Q A (B, 0.90) is the conditional 90th percentile of A given B. The fact that Here income i is a measure of the household's ''normal'' income.In our baseline specification we will take income i to be the average log-income over the period of observation.In addition, note that the age at entry in the panel does not affect ξ i given cohort, education, and income.Hence, ξ i is a time-invariant household characteristic that does not depend on when the household starts being recorded in the PSID, whereas the value of the initial persistent latent component in (5) depends on the stage of the life cycle the household was at when she entered the panel.
We then specify the log-consumption function as For the purpose of documenting consumption responses, it is important to know under which conditions estimating (8) allows one to learn about features of the household's consumption function g t in (1).Suppose that the transitory preference shocks ν it in (1) are i.i.d., independent of past assets and income components, age, and latent type ξ i .If in addition ν it are scalar and have a monotone impact on the consumption function g t , then g t will be identified based on (8), up to a nonlinear transformation of its last argument.Moreover, when the economic primitives are such that ν it are multidimensional or have a non-monotone impact on consumption, the conditional mean function of log-consumption implied by (1) will still be identified based on (8), even though the individual consumption function g t will not be identified in general.Indeed, under our assumptions we have In other words, using quantile methods to flexibly estimate the function Q c in (8), we will be able to consistently estimate our main target parameters, which are the average derivative quantities φ it .
Note that, under mild assumptions, the consumption response parameters in (2) are equal to the derivatives of the conditional mean of consumption given the state variables, However, η it , ε it and ξ i are unobserved in the data, so it is not enough to model the conditional mean to recover our key parameters φ it .ABB specified a nonlinear mean model with separable heterogeneity.A concern with their specification is that it might be too restrictive as a model of the conditional distribution of c it given (a it , η it , ε it , age it , ξ i ).In contrast, in this paper we employ a quantile specification to achieve a more flexible modeling of that conditional distribution.
In our baseline model where assets do not depend on the latent variables η it , ε it , and ξ i directly, a specification of the assets process is not needed.However, assuming that asset accumulation does not depend on the latent variables might be restrictive if, for example, assets returns are heterogeneous and the assets process is not independent of ξ i .For this reason, we will also estimate a model where we specify a reduced-form assets process as where in addition u a i,t+1 and u c i,t+1 are independent.In this model, we will specify initial assets holdings as To summarize the framework laid out in this section, we have described a model with three latent components.The time-invariant type ξ i is intended to capture household pre-sample-period observed and unobserved heterogeneity.The other two latent components enter the income process.The persistent component η it captures household heterogeneity that results from the accumulation of persistent shocks over time.Finally, independent transitory shocks ε it with an age-specific distribution combine with the persistent component and its profile to produce observed labor income.
The presence of the latent type ξ i as an argument of the consumption function may potentially reflect several mechanisms.For example, ξ i may indicate preference or discounting heterogeneity.Alternatively, it may capture heterogeneity in returns to assets.Yet another possible interpretation of ξ i is as additional resources that are available to the household but not observed in the data, such as consumption insurance provided by parents.We will examine the plausibility of these various mechanisms empirically in Section 6.We let the latent type ξ i correlate with income through the conditioning on income i in (7).In addition, although here we will use our most parsimonious specification as a baseline when reporting results, in an extension we will let ξ i enter asset accumulation directly, see Eqs. ( 9)-( 10).
The model thus features two levels of heterogeneity: (a) demographics and time effects, which we partial out linearly in an initial step, and (b) the latent type ξ i , which we include as part of our nonlinear model.We will study the possibility of an additional nonlinear impact of demographic heterogeneity in Section 7.4.

Estimation methodology and implementation
To specify and estimate the model, we closely follow ABB, with some differences.While in this section we focus on estimation and practical implementation, we note that given the similarity of the model's structure to that of ABB, nonparametric identification can be shown using the arguments they provide.Those arguments rely on insights from the literature on nonparametric instrumental variable models and nonlinear models with latent variables (see, among others, (Newey and Powell, 2003;Hu and Schennach, 2008;Wilhelm, 2015)).

Specification
Following ABB, we model all conditional quantile functions using linear quantile specifications at a grid of percentiles.
As an example, we model the conditional quantile function of the persistent latent component of income in (4) as where ϕ k are low-order products of Hermite polynomials in age and the lagged persistent latent component of income, and a η k (τ ) are piecewise-linear polynomial functions of τ .In practice we use a grid of 11 equidistant percentiles.In addition, following ABB we augment the model by specifying a η k (τ ) using an exponential modeling of the tails of the intercept coefficients.We use similar specifications for all the other Eqs.( 6)-( 10).We provide details in Appendix A.
A difference with ABB is that, while they modeled the nonlinear mean of log-consumption and assumed separable errors, here we flexibly estimate the entire conditional quantile function of log-consumption in (8) without imposing separability between u c it and the other determinants of consumption.This is important for estimating the average consumption derivative parameters φ it in the presence of latent variables, in a way which is robust to the presence of non-separabilities implied by the economic model.Fully nonlinear estimation of consumption quantiles has implications for the econometric specification of the model, given that the type ξ i is a latent variable.Indeed, note that ξ i and the conditional quantile function Q c are not separately nonparametrically identified, since it is always possible to take a transformation of ξ i , and to undo it in Q c . 9 In a general quantile model such as (8), we impose the following restriction: where a, η, ε, age are some fixed reference values for log-assets, persistent and transitory income components, and age.Imposing this restriction resolves the indeterminacy.10In this way, ξ i is measured in consumption units, which is meaningful when studying its distribution.In the implementation we set a, η, age to be the unconditional sample averages of log-assets, log-income and age, respectively, and we set ε to zero.

Estimation
To estimate the model we adapt the multi-step approach proposed by ABB to our setting.In a first step, we compute regression residuals of log-income, log-consumption, and log-assets on a set of controls, which includes demographics and time indicators, see Section 2 for the full list of controls.This allows us to construct the residualized variables y it , c it , and a it .
In a second step, we estimate the income process.To this end, we use a stochastic EM algorithm (Nielsen, 2000), which alternates between draws of the latent income components η it and ε it , and parameter updates based on the latent draws.
The updates are performed using quantile regressions, similarly to ABB.For example, to estimate the parameters a To generate the latent draws, we depart from ABB who relied on Metropolis Hastings, and use a Sequential Monte Carlo sampling method.We describe this method in the next subsection.The reason for using a different sampler compared to ABB is numerical stability.Indeed, the performance of Metropolis Hastings tends to deteriorate as the length of the panel and the number of households increase.In the longer and larger panel sample we use in this paper, Sequential Monte Carlo methods tend to be more robust to numerical issues such as initialization and seeding than Metropolis Hastings in our experience.A feature of Sequential Monte Carlo methods is that they take advantage of the Markovian structure of the model to improve performance relative to naive importance sampling.
In a third step, we estimate the consumption function, for given values of the parameters governing the income process.We perform this step using a similar strategy to the one we use for income.In this case also, we depart from ABB in the sampling step of the stochastic EM algorithm.However, the presence of the latent type ξ i further complicates implementation, since one needs to repeatedly draw ξ i together with the sequences of persistent and transitory components.To generate valid draws, we rely on the pseudo-marginal Markov Chain Monte Carlo algorithm proposed by Andrieu et al. (2010), which itself makes use of Sequential Monte Carlo sampling.We describe our implementation in the next subsection.
Quantile monotonicity.Given our quantile modeling, the parameters satisfy monotonicity restrictions (e.g., Chernozhukov et al. (2010). For example, in (11) In practice we do not enforce monotonicity in estimation.However, in each expectation step of the stochastic EM algorithm we draw from the likelihood implied by the estimated parameters.This ensures that we obtain posterior draws from a valid distribution of η's and ξ 's, irrespective of the lack of monotonicity of the quantile parameter estimates.To provide intuition in a simple setup, note that to draw η it according to model ( 11) one can compute, as in Machado and Mata (2005), where u η it are i.i.d.standard uniform.Although the estimatesâ η k (τ ) may not satisfy monotonicity restrictions, this approach produces ηit draws from a valid distribution function.In our setting we use this strategy to generate posterior draws of η's and ξ 's, see Appendix A for details.
Asymptotic distribution and inference.Under the assumption that the parametric model is correctly specified,12 averages of parameter draws are consistent and asymptotically normal with an asymptotic variance that can be estimated by bootstrap or analytical approximations, see Arellano and Bonhomme (2016) and ABB for details.We will report confidence bands computed using two versions of the bootstrap: a parametric bootstrap that relies on the model's structure for simulations, and a nonparametric bootstrap clustered at the household level.

Computational sampling techniques
Here we describe how we draw latent variables in every step of the stochastic EM algorithm.We present, in turn, the methods we use for the latent income components η it , ε it , and for the latent consumption type ξ i .In practice we run these simulation steps in parallel across households, which makes it easy to estimate the model on an unbalanced panel.
Income components: Sequential Monte Carlo.Estimating the income process requires solving a nonlinear filtering problem, where η i,t i , . . ., η i,t i +T i −1 are latent variables.To draw from their posterior distribution given the income data we use a Sequential Monte Carlo (SMC) approach, see Creal (2012) and Kantas et al. (2015) for surveys.
To describe the SMC approach, we focus on the problem of sampling η i,t i , . . ., η i,t i +T i −1 for a single household i from the posterior distribution f (η i,t i , . . ., η i,t i +T i −1 |y i,t i , . . ., y i,t i +T i −1 ).In practice we sample in parallel across households.With importance sampling, one might first sample directly from some proposal distribution π (η i,t i , . . ., η i,t i +T i −1 ), and then re-sample using importance sampling weights where ∝ is a proportionality symbol.However, finding a suitable proposal distribution in our flexible nonlinear model is challenging.Instead, we try and generate draws (also called ''particles'') sequentially.
At t = t i , we initialize S particles η (s) i,t i from a suitable proposal distribution π(η i,t i ).Re-sampling with weights gives S particles approximately distributed according to f (η i,t i |y i,t i ).At t = t i + 1, we now aim to approximate Since we already have S particles approximately distributed according to f (η i,t i |y i,t i ), we can simply use a second proposal distribution π (η i,t i +1 |η i,t i ) to extend these existing particles.Re-sampling with weights gives S particles approximately distributed according to f (η i,t i , η i,t i +1 |y i,t i , y i,t i +1 ).The process continues until we obtain S particles approximately distributed as f (η i,t i , . . ., η i,t i +T i −1 |y i,t i , . . ., y i,t i +T i −1 ). 13he choice of proposal distributions π is important for numerical performance.We found that a simple generalization of a linear permanent-transitory earnings model with Gaussian errors performed well.Specifically, we postulate the following model: where ε it and v η it are independent at all lags, and m is a Hermite polynomial.We re-estimate this model at each iteration of the stochastic EM algorithm, and then set π(η it |η i,t−1 ) to be the posterior distribution based on it.We provide details about the implementation of the SMC sampler in Appendix A. In addition, we provide a comparison of the SMC and Metropolis Hastings sampling methods in the ABB sample in Appendix B. We find that, while our SMC algorithm recovers similar estimates of nonlinear persistence to those reported in ABB, the SMC method is less sensitive to numerical instability than Metropolis Hastings.
Unobserved type in consumption: Particle Markov chain Monte Carlo.In order to incorporate unobserved heterogeneity ξ i , we embed the SMC sampler into a Particle Markov Chain Monte Carlo (PMCMC) algorithm, following Andrieu et al. (2010).
We use this method to estimate the parameters of the consumption process, after having estimated the parameters of the income process.
To outline the PMCMC approach, suppose we wish to sample ξ i , η i,t i , . . ., η i, , where w it = (y it , c it , a it ) is a vector of household i's observed income, consumption and assets at time t.In the PMCMC approach, SMC algorithms are used to generate efficient proposals to be used within a Metropolis Hastings algorithms.An important feature of these methods is that they only rely upon the availability of unbiased estimates of the marginal likelihood f (w i,t i , . . ., w i,t i +T i −1 |ξ i ), which are readily available as a by-product of the SMC algorithm.The use of unbiased estimates of a target distribution within a Metropolis Hastings algorithm can be viewed more generally as an example of a pseudo-marginal approach in which the resulting algorithms can be presented as bona fide Metropolis Hastings samplers whose marginal distribution is the target distribution of interest.We provide details about the implementation of the PMCMC sampler in Appendix A.

Main results
In this section we present the main empirical results on income and consumption, obtained using our baseline nonlinear model with unobserved heterogeneity.

Income persistence
We start by reporting the results on nonlinear income persistence.In the left graph of Fig. 3 we show the derivative of the conditional quantile function of log-income given lagged log-income and age, with respect to lagged log-income.Formally, we compute an estimate of where Q y is the conditional quantile function of log-income given lagged log-income and age, and average it with respect to age.The nonlinear persistence parameters ρ y (y, age, τ ) can be interpreted as heterogeneous autoregressive coefficients, which may depend on both the income level y and the income shock τ . 14We plot the derivative as a function of lagged log-income (which we refer to as ''initial income'') and of the innovation in the quantile model (which we refer to as ''income shock'').
The results show that most households, for most shocks, have current disposable incomes that are quite persistent, with a derivative coefficient that is above 0.80.However, households with low initial income and high income shocks have incomes that are substantially less persistent, with a coefficient as low as 0.40.Likewise, persistence is also low for households with high initial income and low income shocks, with a coefficient of a similar magnitude.These nonlinear persistence estimates are closely related to those found by ABB on a smaller balanced sample drawn from the earlier pre-recession years of the PSID.ρ y (y, age, τ ) averaged over age.The right graph shows quantile derivatives of the persistent latent component η it with respect to η it−1 , ρ η (η, age, τ ) averaged over age, in a model estimated using sequential Monte Carlo with a stochastic EM algorithm.In this case, the two horizontal axes show percentiles of η it−1 (''initial income'') and conditional percentiles of η it given η it−1 (''income shock''), respectively.
In the right graph of Fig. 3 we show nonlinear income persistence, but now for the persistent latent component η it .
That is, we show where Q η is the conditional quantile function of η it given η i,t−1 and age, see (4).We plot the derivative as a function of η i,t−1 (''initial income'') and the innovation in the quantile model (''income shock''). 15We see that average persistence is higher than for the case of log disposable income -it is 0.92 in the right graph, versus 0.78 in the left graph -due to the removal of the transitory income component.For households with high values of initial persistent income and high shocks, persistence is close to unity, and similarly for households with low initial persistent income and low shocks. 16he nonlinear pattern for the persistent latent component η it is qualitatively similar to the one for log-income, although it is quantitatively less pronounced.These nonlinear persistence patterns are rather precisely estimated, see the parametric bootstrap 95% confidence bands in Appendix .1,we see that, while nonlinearities are somewhat more salient in our larger and more recent sample compared to the balanced sample used in ABB, the persistence patterns in both cases are comparable.17

Average consumption responses to income shocks
The main goal of the paper is to study heterogeneity in consumption responses to unexpected changes in income.That is, the way income shocks are transmitted into consumption which underpins the degree of ''partial insurance'' achieved by the household.In this subsection, and the next, we document several key features of household partial insurance, which we measure using the household-and-time-varying transmission coefficients given by the average derivative effects (2) introduced in Section 3.1.The transmission coefficient φ it quantifies the change in consumption induced by an exogenous marginal change in the persistent latent component of income.
In Fig. 4 we start by showing how the mean of the estimated transmission parameters φ it varies with assets levels and over the life cycle.We compare four specifications.The ''models without filtering'' in the upper panel correspond to specifications without transitory component ε it , so the derivative on the right-hand side of (2) is taken with respect to log current disposable income y it instead of the persistent latent component η it .The ''models with filtering'' in the lower panel allow for a separate role of η it and ε it .For both models with and without filtering, we distinguish two specifications with and without unobserved heterogeneity ξ i , in the left and right columns, respectively.Fig. 4 shows that all specifications agree quite well qualitatively.In particular, the association between consumption and income or its persistent latent component is weaker for older and wealthier households.At the same time, there are important quantitative differences between the four specifications.We find that allowing for unobserved heterogeneity ξ i tends to dampen the consumption impacts of income shocks, the difference being particularly noticeable for the models without filtering where average responses decrease from 0.40 to 0.14.The impact of heterogeneity can be explained by the fact that, according to our estimates, ξ i is positively correlated with income, see Section 6.In contrast, allowing for a transitory income component tends to increase consumption responses to income shocks, as is typically the case in estimates that correct for measurement error bias.As a result, in our main model with unobserved heterogeneity and a transitory component, the lower right hand graph shows an estimated average response parameter of 0.33.There are strong differences by assets and age too, with the estimated average transmission coefficient dropping towards 0.10 for older and wealthier households, while for younger households the estimated mean transmission rises to around 0.40. 18  Comparison with ABB.It is informative to compare the average responses in Fig. 4 to the results obtained by ABB.In a model without heterogeneity but with a transitory component, ABB found an average transmission coefficient of 0.38.This is lower than the responses in Fig. 4(c), which are 0.54 on average. 19As we previously noted, the period of observation, the sample of households, and the income measure used in ABB all differ from the ones we focus on in the current paper.In particular, ABB focus on labor income as opposed to disposable income.Our estimates of consumption responses based on labor income are substantially lower than the responses based on disposable income shown in Fig. 4, see Section 7.
Test of homogeneity.By comparing average response coefficients in models with and without household-specific heterogeneity, one can assess whether the data supports an homogeneous model without latent types.To do this, in Appendix

Heterogeneity in consumption responses to income shocks
We have already seen that the introduction of unobserved heterogeneity has a systematic effect on the estimated average response of consumption to changes in income.We hypothesize that there are also systematic differences in responses across consumers that differ according to unobserved heterogeneity.To examine this, we study how consumption responses differ among households that are at the same point in the life cycle and have the same level of assets.For this purpose, we show how the transmission coefficients φ it vary by quantiles of the unobserved type ξ i , in addition to showing how they vary with assets levels and over the life cycle.
In Fig. 5 we show transmission parameters as a function of assets and age, for five different percentiles of ξ i , and we also show the average across ξ i values.The results show clear evidence of household heterogeneity in consumption responses to income shocks.Consider the 10th percentile of ξ i , in the top left graph.For these ''low consumption type'' households, average transmission is 0.36, yet the magnitude of the transmission coefficient varies substantially with age and assets.Indeed, while younger and less wealthy households have transmission coefficients of close to 0.60, the coefficient is as low as 0.10 for older and wealthier households.This pattern is qualitatively consistent with the implications of a standard lifecycle model of consumption and saving behavior in which persistent shocks are harder to self-insure for young consumers and for those consumers with low levels of net assets.
This ''life-cycle consistent'' pattern of responses is maintained through to the median type, albeit less pronounced.As we move to the higher consumer types, a pattern that is much less sensitive to assets and age appears.Consider the 90th percentile of ξ i , in the bottom right graph of Fig. 5.For these high-type households, the transmission coefficients are 0.29 on average, hence lower than the coefficients of the low-type households.In addition, the variation of the transmission coefficients with assets and age is less pronounced than for the low types.Indeed, while coefficients are approximately 0.15 for the older and wealthier households, the young and less wealthy households have coefficients that do not exceed 0.40.These patterns for the high-types are less in accordance with the forces at play in conventional life-cycle models of the individual household.
In order to provide measures of uncertainty associated with our main results, we rely on the bootstrap.We report results based on a parametric bootstrap approach, where we use the model to simulate bootstrapped data sets given parameter estimates.In Appendix Fig. D.7 we report pointwise 95% bands for the transmission parameters of Fig. 5.We see that our estimates are rather precise.As a complement to the parametric bootstrap, in Appendix Fig. D.8 we report pointwise 95% bands based on the nonparametric bootstrap clustered at the household level.Precision is lower in this case, which is not surprising, since, relative to the clustered nonparametric bootstrap, the parametric bootstrap exploit our modeling of the time-series dependence.
As a summary measure of the salient dimensions of heterogeneity that we find, in the top panel of Table 2 we report estimates of average transmission parameters for various categories of households: high and low types, corresponding to ξ i being at the 90th percentile or the 10th percentile, young/low assets for whom age and assets are below the median, and old/high assets for whom age and assets are above the median.In the bottom panel we repeat the exercise for high types corresponding to ξ i being at the 75th percentile and low types corresponding to ξ i being at the 25th percentile.
Alongside point estimates, we report 95% confidence intervals based on the parametric bootstrap.
18 The relative magnitudes of the nonlinear estimates in Fig. 4 are reminiscent of the situation in a linear model with a mismeasured persistent regressor and fixed effects, where the (positive) fixed effects bias and the (negative) measurement error bias tend to offset each other, while only accounting for fixed effects exacerbates the measurement error bias (Griliches and Hausman, 1986).19 The consumption responses in a model without heterogeneity in ABB can be found in their Fig.5(c).In addition, ABB also reported average responses based on a model with unobserved heterogeneity, albeit using a different specification for the consumption rule.They found lower responses in this case, amounting to 0.32 on average, see Figure S24(b) in the supplementary appendix of ABB.We find that, while for high consumption types at the 90th percentile the transmission of income shocks is only 0.09 higher for young/low assets households and insignificant at conventional levels, for low types at the 10th percentile the  average response coefficient is 0.27 higher for the young and low assets and significant at the 5% level.This supports our main conclusion regarding the fact that the behavior of low types appears to be consistent with a standard life-cycle model of consumption and saving, yet the behavior of high types appears less consistent with the mechanisms of the model.In addition, the cross-type difference 0.09 − 0.27 = −0.18 between these two estimates, which is akin to a difference-in-differences estimate, is significant at the 5% level. 20he results in this section, based on a dynamic model with latent income components and unobserved heterogeneity, provide evidence for the presence of heterogeneous types of consumers, confirming what Fig. 2 suggested.In the next section, we develop the implications of these results for life-cycle patterns of consumption and savings, and we examine various possible mechanisms for the patterns in transmission parameters displayed in Fig. 5.
Dispersion of consumption responses around their means φ it .While our main focus is on the average consumption response parameters φ it , there may be dispersion around those averages.In Appendix C we show how to compute an upper bound on the share of variance in responses ∂c it ∂η explained by the means φ it , obtained by calculating a lower bound on the variance of ∂c it ∂η conditional on (a it , η it , ε it , age it , ξ i ).The reason why only bounds are available is because transitory preference shocks ν it , which may generate additional heterogeneity in responses beyond the mean transmission parameters φ it , may be multi-dimensional.We report estimates of the upper bounds on the variance shares in Appendix Fig. D.9.We find high variance shares, in many cases higher than 80%, suggesting that the φ it parameters capture a large part of the heterogeneity in responses (although we note that, since those are upper bounds, this evidence does not strictly speaking rule out the presence of substantial additional heterogeneity).

Candidate mechanisms to explain the heterogeneity
In this section we study various mechanisms that might potentially explain the type heterogeneity that we find.

Three candidate mechanisms
Informed by standard models of consumption and saving decisions, which guide our empirical analysis, we can outline three candidate mechanisms to explain the heterogeneous types that we document.
A first possible explanation is heterogeneity in preferences and discounting.There is a long history of incorporating discount rate heterogeneity to help explain lifetime wealth accumulation, for example Krusell and Smith (1998) and different ages and percentiles of ξ i (10%, 25%, Median, 75%, 90%).The dashed lines show the age-specific unconditional 10th and 90th percentiles for each outcome measure.Hendricks (2007).Everything else equal, individuals with higher marginal utility of consumption will consume more, and hold fewer assets.Individuals with higher discount factors will delay consumption relative to those with lower discount factors, and hold more assets.This type of heterogeneity should lead to high-type households consuming more and holding fewer assets.We examine this hypothesis by showing how consumption and assets profiles depend on the latent type.
A second candidate explanation is heterogeneity in returns to assets.The rate of return is a key determinant of consumption choice in standard models, so the types we find might in fact reflect heterogeneity in those returns across households.Fagereng et al. (2016) find evidence of individual heterogeneity in returns to wealth using administrative records from Norway.We examine this heterogeneity in the PSID by estimating an extension of the model with heterogeneity in the asset accumulation rule (see Eqs. ( 9)-( 10)), and by empirically documenting the form of this rule.
A third candidate explanation is heterogeneity in access to external resources, such as parental insurance.Individuals with access to other forms of insurance would be expected to consume more, for a comparable level of income and assets.Altonji et al. (1992), Hayashi et al. (1996) and, more recently, Charles et al. (2014) and Attanasio et al. (2019), use the generational links in the PSID to document a significant role for parents and family networks in providing additional insurance.To probe this hypothesis, we link the household heads in the PSID to their parents, and study how the latent types relate to parental income, wealth, and consumption.

Life-cycle profiles
As a step towards examining the plausibility of a preference and discounting channel, we show the life-cycle profiles implied by our dynamic model, for various percentiles of the unobserved heterogeneity ξ i .In the top panel of Fig. 6 we show consumption profiles, in logs. 21We see that consumption levels are monotone in the types.This is partly a result of our restriction in (12), which implies monotonicity at the reference age.In addition, comparing the dispersion of the solid lines (which correspond to the ξ i percentiles) with the dashed lines (which correspond to 10th and 90th unconditional percentiles of log-consumption), we see that type heterogeneity explains a large part of the overall variation in log-consumption.Our results imply that ξ i accounts for 25% of the variance of log-consumption. 22n the bottom panel of Fig. 6 we show the profiles of assets and income, in logs.In the left graph we see that, similarly to consumption, assets are monotone in types.This suggests that, while high-type households consume more than low types, they also hold more assets.However, the variation in types explains a relatively small share of the overall variation in log-assets.Note that, while the restriction in ( 12) imposes that log-consumption increases with the type ξ i at particular covariates values, nothing in our approach restricts log-assets to be monotone in the type.Quantitatively, we find that ξ i accounts for 3% of the variance of log-assets.In the right graph we show the results for the persistent latent component of income.We see the same monotone behavior in the type as for consumption and assets.Our results imply that ξ i accounts for 4% of the variance of the persistent latent component of log-income. 23We have already seen that the correlation between the latent type and income is sufficient to generate sizable differences between specifications with and without latent heterogeneity, see Fig. 4. Overall, our results show that high-type households consume more, hold more assets, and have higher income.Quantitatively, individual types mainly differ in their consumption profiles.While these findings do not rule out that differences in preferences and discounting may be present in the data, they are difficult to reconcile with this channel being the main driver of the heterogeneity in consumption responses that we find.

Heterogeneity in consumption and assets
We next assess the role of heterogeneity in assets returns as an explanation for type heterogeneity.For this purpose, we estimate a specification where asset accumulation depends on the latent type ξ i , see Eqs. ( 9)-( 10).The results based on this specification are similar to the baseline ones for both income and consumption.In Appendix Fig. D.12 we show the type heterogeneity in consumption responses to variation in the persistent latent component of income, and find overall very similar responses to the ones based on a specification without assets heterogeneity.In Appendix Figs.D.13 and D.14 we report estimates of assets responses, by type, in this generalized specification that allows the latent type to enter the asset accumulation rule.We find that the association between lagged assets and current assets conditional on lagged income and consumption increases with the latent type, and that assets responses are higher for the young, decrease with the level of lagged assets, and increase with the type ξ i , especially for older households.
Overall, the results based on the extended specification with latent heterogeneity in assets and consumption suggest that returns to assets are indeed heterogeneous across households in the data.However, allowing the heterogeneity to enter asset accumulation does not materially affect the conclusions regarding the heterogeneity in consumption responses.

Heterogeneity in parental insurance
A third candidate mechanism is heterogeneity in access to other forms of insurance, such as parental insurance.In order to examine the plausibility of this mechanism, we take advantage of the inter-generational linkages available in the PSID to match households to their parents.This aspect makes the PSID uniquely suited to study income and consumption dynamics in the presence of links across generations.Specifically, we start by matching the heads of each household to those households headed by a parent of the head.If matches to the household head are not available, we alternatively try and match the spouse of each household to those households headed by a parent of the spouse.In our baseline sample we are able to successfully match approximately 33% of households.
Given this matched panel dataset, we then regress posterior means of the types ξ i on various parental outcomes, such as consumption, income, and assets.In Table 3 we report the results of various specifications with different sets of controls.For robustness, in addition to the results for all households (in the top panel) we also report results for household heads who are less than 45 years old (in the bottom panel).We find that parental income and consumption correlate positively with the mean type, although the correlation with assets is insignificant from zero at conventional levels.When including all parental variables together, parental consumption remains significantly positively correlated with the type.This suggests that, indeed, the latent type ξ i may partly reflect heterogeneous access to parental insurance.This interpretation is further supported by the monotonicity of assets in the type documented in Fig. 6.
However, these results are purely indicative and we leave it to future work to assess whether this channel is quantitatively important.

Other results and extensions
In this section we report results based on extensions of the model and other robustness checks.

Impulse responses
We start by reporting impulse responses implied by the model's estimates.In Fig. 7 we estimate the impact of a shock to the persistent latent component of income, η it , at age 34.The figure is divided into three parts.In the upper part, we report the difference between the average persistent latent component of income for households hit by the shock and the average persistent latent component of income for households hit by a ''median'' shock, i.e., corresponding to the 50th percentile of η it conditional on η i,t−1 .To highlight the heterogeneity in impulse responses, we show results for various percentiles of the latent type distribution.In the middle and bottom parts of the figure we proceed similarly for log-consumption and log-assets, respectively, instead of the income component.
Within each part of the figure, we show impulse responses for various values of initial income and the shock.In the left, middle and right columns we consider households who are at the 10th, 50th and 90th percentile of the distribution of the persistent income component at age 32, respectively.In the top (respectively, bottom) subpanels, we show the results for a shock at the 10th (respectively, 90th) percentile of the distribution of shocks.Hence, top subpanels correspond to negative income shocks, whereas bottom subpanels correspond to positive income shocks.
Focusing first on the upper part of Fig. 7, and moving across columns, we observe that negative shocks tend to have a stronger impact for those on higher income, and that positive shocks have a stronger impact for those on lower income.This illustrates the nonlinear persistence in the income process documented in ABB.In addition, the fact that all lines corresponding to different values of the latent type ξ i reflects our assumption that the income process does not depend on ξ i .
Moving then to the middle part of Fig. 7, we also observe nonlinearities in consumption responses, although those are stronger for the negative income shocks than for the positive ones.In addition, the differences between lines reflect the heterogeneity between types.In particular, low types with higher income tend to respond more strongly to negative shocks than other types.To further illustrate this heterogeneity, in Appendix Fig. D.15 we show how consumption levels evolve, on impact, after an income shock.
Lastly, focusing on the bottom part of Fig. 7 we see only moderate differences in assets evolution after a shock depending on the initial income level.In Appendix Fig. D.16 we show impulse responses based on the model that allows for heterogeneity in both assets and consumption, see Eqs. ( 9)-( 10) and the results discussed in Section 6.3.The responses to a shock to the persistent latent component of income are overall similar to the ones based on the model without heterogeneity in the asset accumulation rule.

Robustness to the complexity of the quantile model used in estimation
The complexity of our empirical specification is controlled in part by the number of knots at which we evaluate the quantiles of the variables in the model (i.e., the income components, consumption, and the latent type).Our estimates of the functions, such as a η k (τ ) in ( 11), interpolate between those τ values.Hence, a large number of knots can approximate any continuous quantile function well, while a small number of knots may provide a worse approximation.However, in estimation one faces the usual bias/variance trade-off, and the impact of the number of knots on the estimates is a priori unclear.To probe the sensitivity of our main results to the number of knots, we report average consumption derivatives based on 19 knots in Appendix Fig. D.17.By comparison, our baseline results were obtained using 11 knots (see Fig. 5).Overall the two sets of estimates agree very well.

Robustness to income definition and sample restriction
Next, we probe the robustness of our results to changes in income definition and sample restriction.While our main results rely on using disposable, post-tax income, in Appendix Fig. D.18 we report results on nonlinear income persistence based on pre-tax labor income.In Appendix Fig. D.19 we report the corresponding results for heterogeneity in consumption responses.The findings suggest a higher degree of nonlinearity in income persistence, and a higher degree of consumption insurance, compared to the results based on disposable income.This is not surprising, as the non-proportionality in the tax system can be interpreted as a source of insurance to households.Moreover, since the results in ABB were based on labor income, these findings help explain the quantitative differences between the results in ABB and the ones we report in this paper when relying on disposable income.
Another important feature of our sample is the restriction to dual earner households.While this restriction is motivated by the goal to abstract from extensive labor supply decisions, it also results in a smaller and potentially more insured sample.We have estimated our model on a larger sample that also includes single earners, where the second member of the household is not working. 24

Additional dimensions of heterogeneity
Our specification of the consumption function flexibly allows for heterogeneity in income, assets, age effects, and the effect of the latent type, see Eq. ( 1).However, it is possible that the effects of additional observed and unobserved factors might matter for consumption insurance.For example, differences in education and birth cohorts might be associated with different consumption responses to income shocks.In Appendix Fig. D.23 we show that neither education nor cohort are strongly associated with the latent type ξ i .Yet, it is theoretically possible that they enter the consumption function, and interact with income components in meaningful ways, even though our modeling approach rules out this possibility.
To tentatively explore this question, in Appendix Fig. D.24 we report consumption responses to income shocks, by type ξ i , in a specification that also controls for a fully interacted education indicator.Since we do not re-estimate the model with latent variables, we view this exercise as indicative.We see that the consumption responses across types are qualitatively similar to the baseline ones, yet those responses appear somewhat muted.This motivates future work extending our framework to allow for multiple observed and unobserved sources of heterogeneity in consumption insurance and income processes across households.

Conclusion
The motivation for this research has been to better understand nonlinear income dynamics and heterogeneous consumption responses to changes in income.In this paper we have developed methods that build on and extend Arellano et al. (2017), and we have applied them to a larger and more comprehensive sample from the PSID which includes a richer set of consumption categories.We have developed computational tools to better handle larger and more complex models, including in settings with unbalanced panels, within a nonlinear quantile-based latent variables framework.These new data and tools allow us to go beyond confirming the presence of nonlinear income and consumption dynamics, and to document rich heterogeneity in consumption responses across households.
Our results point to consumption responses to income shocks that vary substantially with unobserved types.We distinguish lower types, who appear to follow the life-cycle patterns in consumption responses implied by standard models, from higher types, whose consumption responses to income shocks vary little with either assets levels or the stage of the life cycle.High-type households consistently have higher consumption levels and, relative to low-type households, have slightly higher incomes and levels of assets.For the younger low types, consumption responses to persistent income shocks are close to 0.60 while for older low types this falls to 0.10.For the higher types, consumption responses are flatter across age and assets.
We examined alternative mechanisms that could lead to such heterogeneous consumption responses.The fact that high types both consume more and hold more assets is difficult to reconcile with an explanation based on heterogeneity in preferences or discounting.We also argue that it is difficult to align with a specification that allows for latent heterogeneity in asset accumulation, finding that the heterogeneity in consumption responses is virtually unaffected by this extension.To explore a third mechanism, parental insurance, we used the inter-generational linkages in the PSID to link a subset of household heads in our sample to their parents.We found that high-type household heads have on average parents with higher consumption and income levels, suggesting that the heterogeneous responses might in part reflect heterogeneity in access to other sources of insurance such as parental insurance.
Our findings motivate further work on two fronts.First, whilst we have examined several mechanisms and found a correlation between the latent types and parental consumption, we lack a quantitative understanding of how these and other factors shape the household differences in consumption responses and insurance.Second, although we have leveraged a single-latent-factor model to maintain tractability in the presence of heterogeneous responses, generalizing the model to account for other sources of heterogeneity is an important next step.In particular, it would be valuable to extend the model to allow for time-invariant heterogeneity in income, in addition to the latent consumption type.

A.1. Empirical specification
Earnings components.Let ϕ k , for k = 0, 1, . .., denote a dictionary of functions, with ϕ 0 = 1.In practice we use low-order products of Hermite polynomials for ϕ k .We specify, for t ∈ {t i + 1, . . ., t i + T i − 1}, the conditional quantile function of η it given η i,t−1 and age it as in (11).We specify the quantile function of ε it (for t = 1, . . ., T ) given age it , and that of η i1 given age at the start of the period age i1 , in a similar way.Specifically, we set with outcome-specific choices for K and ϕ k .
Consumption type.To specify the latent type we set Consumption rule.To specify the consumption process we set To fix the scale of the function we impose that which translates into linear restrictions on the parameters ∫ 1 0 a c k (τ )dτ .Assets evolution.For initial assets we set For assets evolution we set Implementation.We base our implementation on ABB, and model the functions a k (τ ) as piecewise-linear interpolating , contained in the unit interval.We extend the specification of the intercept coefficient a 0 on (0, τ 1 ] and [τ L , 1) using a Laplace model indexed by λ − (for the left tail) and λ + (for the right tail).All a k for k ≥ 1 are constant on [0, τ 1 ] and [τ L , 1], respectively.We denote a kℓ = a k (τ ℓ ).In practice, we take L = 11 and τ ℓ = ℓ/(L + 1).We use tensor products of Hermite polynomials for ϕ k , each component of the product taking as argument a standardized variable.

A.2. Estimation
Overview of the estimation strategy.We start by estimating the earnings parameters.Next, we recover estimates of the consumption, assets, and type parameters, given the previous earnings estimates.
Parameters.We collect all parameters governing the income process into a vector θ, given by θ = ) .
Likewise, we collect all parameters governing the consumption process into a vector µ, given by µ = ) .
We estimate θ and µ sequentially.
Model's restrictions.Let ρ τ (u) = u(τ − 1{u ≤ 0}) denote the ''check'' function of quantile regression.Consider the parameters of Q η ; that is, the a η kℓ and the corresponding Laplace parameters λ η .The true values of a where f i is the posterior distribution of the (η i,t i , . . ., η i,t i +T i −1 ) given the data and the true parameter values.In turn, the true values of λ η satisfy ] , with an analogous formula for the upper tail parameter λ η + .The model implies related restrictions on all the other quantile and tail parameters in θ and µ.
Likelihood function.The likelihood function is, letting z i = (cohort i , educ i ) and where notice we have imposed the assumption that ξ i is independent of (y Similarly to ABB, the likelihood function is available in closed form.For example, we have )]
Note that the likelihood function is non-negative by construction.In particular, drawing from the posterior density of η automatically produces rearrangement of the various quantile curves (Chernozhukov et al., 2010).
Estimation algorithm.Like in ABB, starting from initial parameter values, we iterate between two steps.
In the stochastic E-step, we draw M values η (m) i = (η (m) i,t i , . . ., η (m) i,t i +T i −1 ) and ξ (m) i from their posterior distribution.In practice we take M = 1.
In the M-step, we estimate parameters by solving empirical counterparts of the population restrictions.This involves running multiple quantile regressions in order to estimate the a kℓ parameters, and estimating the λ parameters which are available in closed form.
Solving the indeterminacy in consumption.To impose the restriction (12), which solves the indeterminacy in the relationship between consumption and the latent type, we proceed as follows.At the start of every M-step, given draws η (m) i and ξ (m) i , we regress c it on polynomials in a it , η (m) it , ε (m) it = y it − η (m) it , age it , and ξ (m) i , using the same polynomial specification as in the quantile model for log-consumption.Letting ĉit denote the predicted value at (a, η, ε, age, ξ (m) i ), we then reset ĉit ↦ → ξ (m)

i
.

Stochastic E-step (income estimation).
The target for a given household i is the posterior distribution f (η i,t i , . . ., η i,t i +T i −1 |y i,t i , . . ., y i,t i +T i −1 ).
At t = t i , we initialize S particles η (s) i,t i from the following proposal distribution π: where the β ε k , σ 2 η 1 and σ 2 ε are parameters estimated by running OLS counterparts to the M-step quantile regressions (in the previous stochastic EM iteration), and c ≥ 1 is a constant (we take c = 2).Time t = t i re-sampling weights are then given by where π is the normal density with mean µ i and variance σ 2 .These weights, which are available in closed form, are used to re-sample particles with replacement from the set of particles η (s) ) 2 exceeds some threshold (see below).This simple adaptive rule avoids degeneracy of the particles.After re-sampling we reset w (s) i,t i = 1 S .Otherwise we keep all the existing particles and weights.
At t = t i + r > t i , we use the following proposal distribution, again denoted as π, to generate new draws to append to the existing set of particles: where again the β ε k , σ 2 η and σ 2 ε are parameters estimated by running OLS counterparts to the M-step quantile regressions.The re-sampling weights are given by w (s) i,t i +r ∝ w (s) respectively, where ξ * i is a draw from a random walk proposal.We make use of the same proposal distributions π as in the income estimation.
In the very first iteration of the stochastic EM algorithm we initialize the Metropolis Hastings chains using random draws from the following proposal: i,t i +r .The unbiasedness of these marginal likelihood estimates implies that the resulting algorithm can be represented as a bona fide Metropolis Hastings algorithm yielding the desired target as its marginal.
Pseudo-code of the stochastic EM algorithm.A short pseudo-code for the algorithm we use is presented in Algorithm A1.
end for 10: Keep the last values ξ K i and (η K i,t i , ..., η K i,t i +T i −1 ). 11:

M-Step:
12: Estimate the quantile parameters by quantile regressions given the draws ξ K i and (η K i,t i , ..., η K i,t i +T i −1 ), as explained in Appendix A. Estimate the Laplace tail parameters.

13:
Update the parameters of the proposal distribution, as explained in Appendix A. 14: end for Practical issues: number of particles and threshold for effective sample size.In practice, we set an i-specific number of particles equal to S i = 50T i , where T i is the number of observations of household i.We set the threshold for effective sample size to S i /2.
Practical issues: specification.In practice we set the following polynomial degrees K for our baseline specification, chosen after some experimentation: Practical issues: starting values.In practice, we start the algorithm from different parameter values.For example, for the initial values of the quantile parameters in η it , we run quantile regressions of log-earnings on lagged log-earnings and age.
We proceed similarly to set other starting parameter values, including those for the proposal distributions.In addition, we use latent draws from the income model as initial draws when estimating the consumption model.We experimented with a number of other choices.
Practical issues: numerical performance.Our aim is to ensure that the stochastic EM parameter Markov chains mix well.Among the factors that influence mixing (as measured by the decay rate of auto-correlations along the parameter Markov chains), we found three key ones to be the number of particles, the length of the Metropolis chains, and the number of iterations in the overall EM algorithm.Given our experiments, we found that setting moderate numbers for the first two (we set S i = 50T i particles, as indicated above, and we run each Metropolis chain for 50 iterations), and relatively large numbers for the third (we run the stochastic EM for 2000 iterations), gave best performance given computation constraints in our short panel data setting.
25 When ℓ > 1 we simply take ξ 0 i to be the ξ i draw from the previous (ℓ − 1) step.When ℓ = 1 we always accept the first proposal.In both cases, we run an SMC algorithm (see line 6 in the pseudo-code) based on ξ 0 i to generate a draw (η 0 i,t i , ..., η 0 i,t i +T i −1 ).income persistence measure proposed by ABB is, in the case of the persistent income component η it , ρ(η, age, τ ) = ∂Q η (η, age, τ ) ∂η , τ ∈ (0, 1), (B.4) where Q η is the quantile function appearing in (4). 27he income persistence results reported in ABB are based on comparing various estimation runs, and selecting the one that provides the highest value of the likelihood.However, compared to Metropolis Hastings used in ABB, we found the SMC approach to be more effective at reducing the numerical instability across estimation runs.To illustrate this, in Lastly, although reported estimates in ABB appear reliable in the shorter balanced sample, in our experience increasing the number of households and the length of the panel makes it more challenging to rely on Metropolis Hastings for sampling.In contrast, we found our SMC implementation to remain numerically stable in such cases.

Appendix C. Which features of the consumption policy rule can be identified?
Consider a structural policy rule of the form where ν, of unrestricted dimension, is independent of X .To simplify the presentation we assume that X is scalar.In this paper, C denotes consumption, and X contains all state variables, including the income components.Denote the conditional quantile function of C given X as Q (X , τ ).Hence, for U uniform independent of X , we can write We are interested in moments of the marginal effects The key challenge is that, while Q is identified from data on (C , X ), g is generally not.
Average responses.We have, under standard conditions, or, equivalently, that is, ] .
Hence, average marginal effects are identified, irrespective of the dimensionality of ν and the monotonicity properties of g.
Variance of responses.By Theorem 2.1 in Hoderlein and Mammen (2007) we have for all τ and x.We thus can write ]  has mean zero, and variance ) , where R 2 corresponds to the nonparametric regression of ∆ x C on C and X .Hence, the variance of ∂ ∂x Q (x, U) underestimates the variance of ∆ x C , by an amount that depends on how well C and X explain ∆ x C .
For example, if ν is scalar and has a monotone effect on g, then R 2 = 1 and the variances are equal.In that case, Q = g, and g is identified.More generally, even though g is may not be identified, the mean of         Notes: Average non-residualized log-consumption, for different ages and percentiles of ξ i (10%, Median, 90%).The dashed lines show the age-specific and ξ i -specific 10th and 90th percentiles of log-consumption.Notes: Average non-residualized log-assets and persistent latent component of log-income, for different ages and percentiles of ξ i (10%, Median, 90%).The dashed lines show the age-specific and ξ i -specific 10th and 90th percentiles for each outcome measure.Here we report the results by percentiles of heterogeneity ξ i in consumption.

Fig. 1 .
Fig. 1.Average derivative of log-consumption with respect to log-income.Notes: The graph shows averages of the derivative of log-consumption with respect to log-income, conditional on log-income, age and log-assets.Estimates are based on a linear regression of log-consumption on a second-order polynomial in log-income, age, and log-assets.The two horizontal axes show age and assets percentiles.µ and σ denote the mean and standard deviation of the average derivatives, respectively.

Fig. 2 .
Fig. 2. Quantile derivatives of log-consumption with respect to log-income.Notes: The graphs show averages of the derivatives of quantile functions of log-consumption with respect to log-income, conditional on income, age and assets.In the left graph we report results for the bottom tercile (averaged over a fine grid of percentiles), in the right graph we report results for the top tercile.Estimates are based on quantile regressions of log-consumption on a second-order polynomial in log-income, age, and log-assets.The two horizontal axes show age and assets percentiles.

Fig. 3 .
Fig. 3. Nonlinear income persistence.Notes: PSID, 2005-2017 sample, disposable income, dual earners.The left graph shows quantile derivatives of log-income with respect to lagged log-income, Fig. D.3 and the nonparametric bootstrap 95% confidence bands in Appendix Fig. D.4.In addition, comparing Fig. 3 to Appendix Fig. B

Fig. 4 .
Fig. 4. Average consumption responses.Notes: PSID, 2005-2017 sample, dual earners.The graphs show the average derivative of log-consumption with respect to log-income (in the top panel) and the persistent latent component η it (in the bottom panel).The left graphs correspond to a model without unobserved heterogeneity ξ i in consumption, whereas the right graphs correspond to a model with unobserved heterogeneity ξ i .The two horizontal axes show age and assets percentiles, respectively.
Fig. D.5 we report confidence bands based on the nonparametric bootstrap clustered at the household level for the average responses depicted in the lower panel of Fig. 4. We find a 95% confidence interval for the mean across these responses of [0.50, 0.59] in the model without heterogeneity, and of [0.21, 0.44] in the model with heterogeneity.The fact that the two confidence intervals do not overlap represents a formal rejection, at the 5% level, of the null hypothesis of homogeneity.The same conclusion holds when we use the parametric bootstrap to produce confidence intervals, see Appendix Fig. D.6.

Fig. 5 .
Fig. 5. Heterogeneity in consumption responses.Notes: See the notes to Fig. 4.Here we report the results by percentiles of heterogeneity ξ i in consumption.
Notes: See the notes to Fig.4.Here we report average consumption responses for young and low assets households compared to old and high assets households, for different percentiles of heterogeneity ξ i in consumption.Values are calculated by evaluating the average consumption response for households at a fixed percentile of ξ i when assets and age are fixed at the τ th percentile.Reported values for young and low assets households are then shown by averaging over τ ∈ (0, 0.5).Reported values for old and high assets households are then shown by averaging over τ ∈ (0.5, 1).Parametric bootstrap 95% confidence intervals based on 200 replications are shown in brackets.

Fig. 6 .
Fig. 6.Life-cycle profile.Notes: Average non-residualized log-consumption in graph (a), log-assets in graph (b), and persistent latent component of log-income in graph (c), for

Fig. 7 .
Fig. 7. Heterogeneity in impulse responses.Notes: Impulse responses shown for shocks at the 10th (top subpanels) and 90th (bottom subpanels) percentiles, relative to median.See the text for a description.The different lines correspond to different percentiles of ξ i .
In Appendix Fig. D.20 we report the results for income persistence, and in Appendix Fig. D.21 we reproduce our main results on heterogeneity in consumption responses to income shocks.Our findings are qualitatively unchanged relative to our baseline sample of dual earners.
Fig. B.2 we report numerical stability bands that indicate the variability of income persistence estimates obtained from 200 runs of our estimation algorithm using different seeds, based on the two different sampling methods.In the left graph of the figure we report results based on Metropolis Hastings.In the right graph we report results based on the SMC algorithm we rely on in this paper.The SMC results show substantially less numerical variability.

Fig. D. 1 .
Fig. D.1.Consumption responses at various quantiles, confidence bands.Notes: See the notes to Figs. 1 and 2. Bootstrapped pointwise 95% confidence bands clustered at the household level.
the notes to Fig.4.Here we report average consumption responses for young and low assets households compared to old and high assets households, for different percentiles of heterogeneity ξ i in consumption.Values are calculated by evaluating the average consumption response for households at a fixed percentile of ξ i when assets and age are fixed at the τ th percentile.Reported values for young and low assets households are then shown by averaging over τ ∈ (0, 0.5).Reported values for old and high assets households are then shown by averaging over τ ∈ (0.5, 1).Nonparametric bootstrap 95% confidence intervals clustered at the household level based on 200 replications are shown in brackets.

Fig. D. 5 .
Fig. D.5.Average insurance in model with and without heterogeneity 95% pointwise confidence bands (nonparametric bootstrap).Notes: Pointwise 95% confidence bands based on nonparametric bootstrap.200 replications.Bootstrap is clustered at the household level.

Fig. D. 9 .
Fig. D.9.Heterogeneity in residual variation of consumption responses.Notes: See the notes to Fig. 5.The figure shows an upper bound on the proportion of the variation in consumption responses to η it explained by the average consumption response, conditional on age and assets, see Appendix C. The various graphs corresponds to different percentiles of ξ i .

Fig. D. 11 .
Fig. D.11.Life-cycle profiles of log-assets and log-income, for different percentiles of unobserved types.

Fig. D. 12 .
Fig. D.12.Heterogeneity in consumption responses, model with heterogeneity in assets.Notes: See the notes to Fig.5.The results are based on a model with latent heterogeneity ξ i consumption and assets.Here we report the results by percentiles of heterogeneity ξ i .

Fig. D. 13 .
Fig. D.13.Heterogeneity in assets dynamics, model with heterogeneity in assets.Notes: The figure shows the average total derivative of log-assets with respect to lagged log-assets, conditional on lags of log-assets, income components, log-consumption, age, and the latent type.Here we report the results by percentiles of heterogeneity ξ i .

Fig. D. 14 .
Fig. D.14.Heterogeneity in assets responses, model with heterogeneity in assets.Notes: The figure shows the average total derivative of log-assets with respect to lagged η, conditional on lags of log-assets, income components, age and the latent type.Here we report the results by percentiles of heterogeneity ξ i .

Fig. D. 16 .
Fig. D.16.Heterogeneity in impulse responses, model with heterogeneity in assets.Notes: Impulse responses shown for shocks at the 10th (top subpanels) and 90th (bottom subpanels) percentiles, relative to median.

Fig. D. 17 .
Fig. D.17.Heterogeneity in consumption responses based on 19 knots.Notes: See the notes to Fig. 5.In this figure we use 19 knots in estimation.For our baseline results in Fig. 5 we used 11 knots.

Fig. D. 18 .
Fig. D.18.Nonlinear income persistence, labor income.Notes: PSID, 2005-2017 sample, household labor income.The left graph shows quantile derivatives of log-income with respect to lagged log-income.The right graph shows quantile derivatives of the persistent latent component η it with respect to η it−1 , model estimated using sequential Monte Carlo with a stochastic EM algorithm.The two horizontal axes show percentiles of η it−1 (''initial income'') and conditional percentiles of η it given η it−1 (''income shock''), respectively.

Fig. D. 22 .
Fig. D.22.Average consumption responses to labor income.Notes: PSID, 2005-2017 sample, dual earners, labor income.The graph shows the average derivative of log-consumption with respect to the persistent latent component η it in a model without unobserved heterogeneity ξ i in consumption.The two horizontal axes show age and assets percentiles, respectively.

Fig. D. 23 .
Fig. D.23.Quantile-quantile plots for ξ i by observables.Notes: Quantile-quantile plots shown for (a) graduates and non-graduates (b) born before 1969 and born after 1969.The graphs show the quantiles of ξ i indicated on the x-axis against the quantiles of ξ i indicated on the y-axis.

Fig. D. 24 .
Fig. D.24.Heterogeneity in consumption responses controlling for education.Notes: See the notes to Fig.4.We report average derivatives in a regression that includes a full set of interactions with a binary higher education indicator.
where V is standard uniform independent of B, is referred to as the Skorohod representation in the literature, see, e.g., Chernozhukov i denote sequences of lags of η and age.For conciseness we leave the full conditioning implicit in the notation.

Table 3
Heterogeneity and parental outcomes.PSID, 2005-2017 sample, household heads aged 25-60 (top panel) and 25-45 (bottom panel).Regressions of posterior ξ i draws on parental outcomes.Parental links are obtained for approximately 33% of panel.Parental outcomes are obtained as average residuals net of cohort and year effects.Results are based on 10 posterior draws per household.Controls include an education dummy for the household head and a quadratic specification for first period age.Standard errors clustered at the household level do not account for the uncertainty in the posterior parameter estimates.
The target for a given household is the posterior distributionf (ξ i , η i,t i , ..., η i,t i +T i −1 | x i,t i , ..., x i,t i +T i −1 ),where x it = (y it , c it , a it , age it ) is a vector of household i's observed income, consumption, assets and age at time t.Algorithm Algorithm A1 provides a pseudo-code for the implementation.The SMC sampling steps (used to generate efficient proposals within a Metropolis Hastings algorithm) are identical to those outlined above with the exception that re-sampling weights at times t = t i and t > t i are now given by ).The parameters β ξ k and ω 2 are estimated by running OLS counterparts to the corresponding M-step quantile regressions.At subsequent iterations of the stochastic EM we initialize the Metropolis Hastings chains using draws from the previous iteration.After initialization we use a Gaussian random walk proposal with variance 3.5ω 2 .Whilst running the SMC samplers we obtain unbiased estimates of the marginal likelihood which can be calculated recursively as p(x i,t i +r , . . .|ξ * i ) = ∑ S s=1 p(x i,t i +r−1 , . . .|ξ * Run an SMC algorithm targeting p(η i,t i , ..., η i,t i +T i −1 |ξ * i , w i,t i , ..., w i,t i +T i −1 ). ) = p(w i,t i , ..., w i,t i +T i −1 |ξ * i ), and the resulting particles η * i,t i , ..., η * i,t i +T i −1 , both of which are available as output of the SMC algorithm in line 6.Let f denote the density of ξ i , whose expression is given in Appendix A. With probability i

Table D .2
Additional descriptive statistics about the main sample, including negative assets.PSID, 2005-2017.Means of variables, standard deviations in parentheses.