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Empirical Economics

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Open Access 08-10-2020

Alternative estimation approaches for the factor augmented panel data model with small T

Authors: Jörg Breitung, Philipp Hansen

Published in: Empirical Economics | Issue 1/2021

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Abstract

In this paper, we compare alternative estimation approaches for factor augmented panel data models. Our focus lies on panel data sets where the number of panel groups (N) is large relative to the number of time periods (T). The principal component (PC) and common correlated effects (CCE) estimators were originally developed for panel data with large N and T, whereas the GMM approaches of Ahn et al. (J Econ 728 174:1–14, 2013) and Robertson and Sarafidis (J Econ 185(2):526–541, 2015) assume that T is small (that is T is fixed in the asymptotic analysis). Our comparison of existing methods addresses three different issues. First, we analyze the possibility of an inappropriate normalization of the factor space (the so-called normalization failure). In particular we propose a variant of the CCE estimator that avoids the normalization failure by adapting a weighting scheme inspired by the analysis of Mundlak (Econometrica 46(1):69–85, 1978). Second, we analyze the effects of estimating versus fixing the number of factors in advance. Third, we demonstrate how the design of the Monte Carlo simulations favors some estimators, which explains the conflicting findings from existing Monte Carlo experiments.

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The model may include further terms such as \(\varvec{\gamma }_i'\varvec{d}_t\), where \(\varvec{d}_t\) is some observed strictly exogenous regressor, cf. Pesaran (2006). As such additional terms are easily accounted for without affecting the main results, these extensions are ignored.

In panels with individual specific parameters and fixed T, including weakly dependent regressors (such as lagged dependent variables) results in a bias of order 1/T (the incidental parameter problem). The GMM-based estimators of Sect. 2.3 are able to cope with this bias by introducing time-dependent vectors of instruments. In this paper we abstract from such complications. The reader interested in dynamic models is referred to Juodis and Sarafidis (2018).

This does not imply, however, that the CCE estimator is always inefficient whenever \(\varvec{\lambda }\ne \varvec{\lambda }_0\). As shown by Westerlund et al. (2019) the CCE estimator is asymptotically efficient if \(r=k+1\) and \(u_{it}\) is i.i.d. across i and t.

The restricted version of the CCE estimator is considered in Everaert and De Groote (2016). In our experience, imposing the nonlinear restriction does not result in an important gain in efficiency. In the model with \(r> 1\) the restriction cannot be imposed anyway.

A practical solution is to reduce the set of instruments (cf. Juodis and Sarafidis 2018) or applying other methods of dimensionality reduction (Breitung 2015, Section 15.2.3).

This estimator can be seen as a special case of the combination-CCE estimator proposed by Karabiyik et al. (2019).

Note that for finite N the matrix \(\varvec{\Xi }\) is almost surely invertible, even if \(\lambda _i\) and \(\varvec{x}_{it}\) are uncorrelated for all i and t. To establish consistency we require that the probability limit of \(\varvec{\Xi }\) is invertible as \(N\rightarrow \infty \).

The alert reader may have noticed that the linear combination does not involve the ordinary cross-section averages \(N^{-1}\sum _i y_{it}\), \(N^{-1}\sum _i x_{1,it}\) and \(N^{-1}\sum _i x_{2,it}\) that are employed in the CCE estimator. These additional averages are not required for identification but often improve the statistical properties of the estimator. They may also help to escape the problems resulting from a (nearly) singular matrix \(\varvec{\Xi }\).

The proof of Moon and Weidner (2015) requires \(T\rightarrow \infty \) and is based on the i.i.d. assumption but they note that it appears that their results extend to a less restrictive setting.

Following Ahn et al. (2013), we use \( \beta = 0 \) as starting value for the iterative ALS\(^*\) procedure.

The performance is similar to the case where \(\beta \) is known (not shown). Therefore, the estimation of \(\beta \) does not seem to have an important effect on the performance of the BN and AH selection criteria. Furthermore, the growth ratio statistic of Ahn and Horenstein (2013) performs similar to the eigenvalue ratio statistic. For reasons of space we do not show the respective results.

To save space, we do not show results for the estimators based on the BN criterion, since the hit rates are either 0% or (close to) 100%.

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Title: Alternative estimation approaches for the factor augmented panel data model with small T
Authors: Jörg Breitung
Philipp Hansen
Publication date: 08-10-2020
Publisher: Springer Berlin Heidelberg
Published in: Empirical Economics / Issue 1/2021
Print ISSN: 0377-7332
Electronic ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-020-01948-7

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