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Empirical Economics

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01-03-2015

Panel data dynamics with mis-measured variables: modeling and GMM estimation

Author: Erik Biørn

Published in: Empirical Economics | Issue 2/2015

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Abstract

Generalized Method of Moments (GMM) estimation is discussed under the joint occurrence of fixed effects and random measurement errors in an autoregressive panel data model. Finite memory of measurement errors is allowed for. Two GMM specializations are considered: (i) using instruments (IVs) in levels for a differenced version of the equation and (ii) using IVs in differences for the level version. Index sets for lags and leads are convenient in examining how the potential IV-set is affected by changes in the memory pattern. While measurement errors with long memory may give an IV-set too small for identification, problems of “IV proliferation” and “weak IVs” may arise unless the panel is short. An application based on data for (log-transformed) capital stock and output from Norwegian manufacturing firms, supplemented with Monte Carlo simulations, to illustrate finite sample biases, is considered. Overall, with respect to bias and IV strength, GMM specialization (ii) seems superior to inference using GMM specialization (i).

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Consistent IV estimation of static errors in variables models in non-panel-data contexts is discussed in e.g., (Fuller (1987), Sections 1.4 and 2.4), while Grether and Maddala (1973); Pagano (1974), and Staudenmayer and Buonaccorsi (2005) discuss distributed lag models for pure time series combining errors in variables and serially correlated disturbances. Maravall and Aigner (1977); Maravall (1979) and Nowak (1993) discuss identification problems for such models.

“\(\cdots \) to ensure identification ... there should not be “too much structure” on the second order moments of the latent exogenous regressors along the time dimension, and not “too little structure” on the second order moments of the errors and disturbances along the time dimension”; see (Biørn (2000), p. 398).

ARX and ARMAX are acronyms for AR and ARMA models with exogenous variables.

Subscript convention: IVs are placed before \(\bullet \), the composite error/disturbance and the instrumented variable are placed after \(\bullet \) in the \(\varvec{Z}\)-sets and the \(\varvec{S}\)-sets, respectively.

Application of \(\widetilde{\varvec{\gamma }}_L\) is presumes that \(\varvec{\xi }_{it}\) (and hence \(\varvec{q}_{it}\)) is stationary in the mean. See (Blundell and Bond (1998), Sect. 4) and (Biørn (2000), Sections 2.3 and 4.d), for discussions of, and remedies, relating to, respectively, AR models without measurement error, and static models with measurement error.

A computer program in the Gauss software code, version 7.0, cf. Gauss (2006), constructed by Xuehui Han in cooperation with the author, is applied. The reported standard errors are calculated from the GMM formulae as described in (Biørn and Krishnakumar (2008), Section 10.2.5).

The \(\mu _{it}\) process is initialized by using its “long-run expectation” \(\mu _{i0}\!=\!\mathsf{{E}}[\mu _{it}/(1\!-\!\lambda \mathsf{{L}})]\!=\!\mathsf{{E}}(\chi _i)\!= \!\beta /(1\!-\!\lambda )\).

Note that GMM estimation as described in Sect. 3 is infeasible unless \(N_\xi \!\ge \!N_\nu \!+\!3\).

Since \(T\) is as low as 10, we have refrained from doing a formal (panel data) stationarity test.

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Title: Panel data dynamics with mis-measured variables: modeling and GMM estimation
Author: Erik Biørn
Publication date: 01-03-2015
Publisher: Springer Berlin Heidelberg
Published in: Empirical Economics / Issue 2/2015
Print ISSN: 0377-7332
Electronic ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-014-0802-1

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