Elsevier

Journal of Econometrics

Volume 188, Issue 2, October 2015, Pages 378-392
Journal of Econometrics

Nonparametric identification in panels using quantiles

https://doi.org/10.1016/j.jeconom.2015.03.006Get rights and content

Abstract

This paper considers identification and estimation of ceteris paribus effects of continuous regressors in nonseparable panel models with time homogeneity. The effects of interest are derivatives of the average and quantile structural functions of the model. We find that these derivatives are identified with two time periods for “stayers”, i.e. for individuals with the same regressor values in two time periods. We show that the identification results carry over to models that allow location and scale time effects. We propose nonparametric series methods and a weighted bootstrap scheme to estimate and make inference on the identified effects. The bootstrap proposed allows inference for function-valued parameters such as quantile effects uniformly over a region of quantile indices and/or regressor values. An empirical application to Engel curve estimation with panel data illustrates the results.

Section snippets

Identification for panel regression

A frequent object of interest is the ceteris paribus effect of x on y, when observed x is an individual choice variable partly determined by preferences or technology. Panel data holds out the hope of controlling for individual preferences or technology by using multiple observations for a single economic agent. This hope is particularly difficult to realize with discrete or other nonseparable models and/or multidimensional individual effects. These models are, by nature, not additively

The model and conditional mean effects

The data consist of n observations on Yi=(Yi1,,YiT) and Xi=[Xi1,,XiT], for a dependent variable Yit and a vector of regressors Xit. Throughout we assume that the observations (Yi,Xi), (i=1,,n), are independent and identically distributed. The nonparametric models we consider satisfy

Assumption 1

There is a function ϕ and vectors of random variables Ai and Vit such that Yit=ϕ(Xit,Ai,Vit),i=1,,n,t=1,2,,T.

We focus in this paper on the two time period case, T=2, though it is straightforward to extend

Conditional quantile effects

Turning now to the identification of the quantile effects given above, let Qt(τx) denote the τth quantile of Yt conditional on X=x=(x1,x2). It will be the solution to 1(ϕ(xt,u)Qt(τx))f(u|x)du=τ. The pair [Q1(τ|x),Q2(τ|x)] is a quantile analog of Chamberlain’s (1982) multivariate regression for panel data. We can identify a quantile analog of the Hoderlein and White (2012) average derivative effect. We first describe how these multivariate panel quantiles can be used to identify an

Quantiles of transformations of the dependent variable

In this section we answer the question whether we can relate quantiles of the first difference of the dependent variable to causal effects. In fact, the same arguments and assumptions that are used for first differences can also be employed for arbitrary functions of the dependent variables which map the T-vector of dependent variables Y (in our case for simplicity T=2) into a scalar “index”. However, as it turns out, if we restrict ourselves to using only two time periods of the covariates Xt,

Time effects

The time homogeneity assumption is a strong one that often seems not to hold in applications. In this section we consider one way to weaken it, by allowing for additive location effects and multiplicative scale effects. Allowing for such time effects leads to effects of interest being exactly identified, unlike the overidentification we found in Sections  2 The model and conditional mean effects, 3 Conditional quantile effects.

We allow for time effects by replacing Assumption 1 with the

Estimation and inference

The conditional mean and quantile effects of interest are identified by special cases of the functionals: θm(x)=hm({Mt(x,x),Vt(x,x):t=1,2}),xX, and θq(w)=hq({Qt(τx,x):t=1,2}),w=(x,τ)W, respectively, where hm and hq are known smooth functions, X is a region of regressor values of interest, and W is a region of regressor values and quantiles of interest. We consider the estimators of θm and θq based on the plug-in rule: θ̂m(x)=hm({M̂t(x,x),V̂t(x,x):t=1,2}),xX, and θ̂q(w)=hq({Q̂t(τx,x):t=1,2})

Engel curves in panel data

In this section, we illustrate the results with an empirical application on estimation of Engel curves with panel data. The Engel curve relationship describes how a household’s demand for a commodity changes as the household’s expenditure increases. Lewbel (2006) provides a recent survey of the extensive literature on Engel curve estimation. We use data from the 2007 and 2009 waves of the Panel Study of Income Dynamics (PSID). Since 2005, the PSID gathers detailed information on household

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    We thank the editor Qi Li, two anonymous referees, participants at Demand Estimation and Modelling Conference, 2014 Econometric Society European Summer Meeting, 2015 Econometric Society North American Winter Meeting, and Arthur Lewbel for comments. We gratefully acknowledge research support from the NSF.

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