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Identi
f
cation and estimation of ‘irregular’ correlated random
coe
ﬃ
cient models
1
Bryan S. Graham
†
and James L. Powell
♦
Initial Draft:
December 2007
This Draft:
July 3, 2008
Preliminary and Incomplete
Abstract
In this paper we study identi
f
cation and estimation of the causal e
f
ect of a small change in
an endogenous regressor on a continuouslyvalued outcome of interest using panel data. We focus
on average e
f
ects, over either the full population distribution of unobserved heterogeneity (the
average partial e
f
ect, APE), or over subpopulations de
f
ned by their regressor values (local average
responses, LARs). In our primary model the outcome of interest varies linearly with a (scalar)
regressor, but with an intercept and slope coe
ﬃ
cient that may vary across units and over time in a
way which depends on the regressor. Our model is a special case of Chamberlain’s (1992a) correlated
random coe
ﬃ
cients (CRC) model, but not does not satisfy the regularity conditions he imposes. We
show how two measures of the outcome and regressor for each unit are su
ﬃ
cient for identi
f
cation
of the APE and LAR, as well as aggregate trends that are meanindependent of the regressors. We
identify aggregate trends using units with a zero
f
rst di
f
erence in the regressor, in the language of
Chamberlain (1980b, 1982) ‘stayers’, and the average partial e
f
ect using units with nonzero
f
rst
di
f
erences or ‘movers’. We discuss extensions of our approach to models with multiple regressors
and more than two time periods, periods. We use our methods to estimate the average elasticity of
calorie consumption with respect to total outlay for a sample of poor Nicaraguan households (cf.,
Strauss and Thomas, 1995, Subramanian and Deaton, 1996).
JEL Classification: C14, C23, C33
Key Words: Panel Data, Correlated Random Coefficients, Control Function,
Average Partial Effects, Local Average Response, Kernel Regression, Calorie De
mand
1
We would like to thank seminar participants at UC  Berkeley, UCLA, USC and Harvard, members of the Berkeley
Econometrics Reading Group and participants in the Conference in Economics and Statistics in honor of Theodore
W. Anderson’s 90th Birthday (Stanford University) and the JAE Conference on Distributional Dynamics (CEMFI,
Madrid) for comments and feedback. Discussions with Manuel Arellano, Stéphane Bonhomme, Gary Chamberlain,
Michael Jansson and Edward Vytlacil were helpful in numerous ways. Max Kasy provided excellent research assistance.
All the usual disclaimers apply.
†
Department of Economics, University of California  Berkeley, 5081 Evans Hall #3880,
Berkeley,
CA 94720 and National Bureau of Economic Research.
Email:
[email protected]
.
Web:
http://www.econ.berkeley.edu/~bgraham/.
♦
Department of Economics, University of California  Berkeley, 5081 Evans Hall #3880, Berkeley, CA 94720.
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This note was uploaded on 12/26/2011 for the course ECON 245a taught by Professor Staff during the Fall '08 term at UCSB.
 Fall '08
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 Normal Distribution, The American, Xt

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