Nonparametric identification in panels using quantiles☆
Section snippets
Identification for panel regression
A frequent object of interest is the ceteris paribus effect of on , when observed 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 observations on and , for a dependent variable and a vector of regressors . Throughout we assume that the observations , , are independent and identically distributed. The nonparametric models we consider satisfy
Assumption 1 There is a function and vectors of random variables and such that
We focus in this paper on the two time period case, , though it is straightforward to extend
Conditional quantile effects
Turning now to the identification of the quantile effects given above, let denote the th quantile of conditional on . It will be the solution to The pair 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 -vector of dependent variables (in our case for simplicity ) into a scalar “index”. However, as it turns out, if we restrict ourselves to using only two time periods of the covariates ,
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: and respectively, where and are known smooth functions, is a region of regressor values of interest, and is a region of regressor values and quantiles of interest. We consider the estimators of and based on the plug-in rule: and
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
References (27)
Multivariate regression models for panel data
J. Econometrics
(1982)- et al.
Efficient estimation of semiparametric conditional moment models with possibly nonsmooth residuals
J. Econometrics
(2009) - et al.
Nonparametric identi cation in nonseparable panel data models with generalized fixed effects
J. Econometrics
(2012) - et al.
Robust semiparametric M-estimation and the weighted bootstrap
J. Multivariate Anal.
(2005) - et al.
Cross section and panel data estimators for nonseparable models with endogenous regressors
Econometrica
(2005) - Belloni, A., Chernozhukov, V., Chetverikov, D., Kato, K., 2013. On the asymptotic theory for least squares series:...
- Belloni, A., Chernozhukov, V., Fernandez-Val, I., 2011. Conditional quantile processes based on series or many...
- et al.
Identification of marginal effects in a nonparametric correlated random effects model
J. Bus. Econom. Statist.
(2009) - et al.
Semi-nonparametric IV estimation of shape-invariant engel curves
Econometrica
(2007) Analysis of covariance with qualitative data
Rev. Econom. Stud.
(1980)
Panel data
Nonparametric applications of Bayesian inference
J. Bus. Econom. Statist.
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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.