Engineering robust instruments for GMM estimation of panel data regression models with errors in variables: a note

Abstract

Econometricians have long recognized the need to account in some way for measurement errors, specification errors and endogeneity to ensure that the ordinary least squares estimator is consistent. This article introduces a new generalized method of moments estimator that relies on robust instruments to estimate panel data regression models containing errors in variables. We show how this GMM approach can be generalized for the panel data framework using higher moments and cumulants as instruments. The new instruments, engineered for greater robustness, are proposed to tackle the pervasive problem of weak instruments.

Keywords:

• econometric engineering
• endogeneity
• errors in variables
• GMM
• higher moments
• panel data

JEL Classification:

• C13
• C33

Acknowledgement

I thank William F. Rentz for useful comments.

Notes

¹ See also Dagenais and Dagenais (1994).

² See also Racicot and Rentz (2014) for an application of this method – the Pástor and Stambaugh model.

³ This terminology comes from the field of financial engineering, where the term ‘technology’ is also used in an analogous context (e.g. Neftci, 2008). We define econometric engineering as the process of blending existing methods (i.e. technology) guided by theoretical results (i.e. theorems and lemmas). Podivinsky (1990) uses this terminology in another context.

⁴ See also Stock and Watson (2011).

⁵ Here, we follow Greene (Greene, 2012, 2014). For an introduction to the panel data regression models with applications, see Gujarati and Porter (2009), chap. 16.

⁶ A general framework that permits to vary across equation is the SUR model. This approach is left for further research.

⁷ See Baltagi (2001) and Arellano (2003) for a discussion of the impact of errors in variables on the estimation process of panel data regression models.

⁸ Two period panels involve, for instance, a before and after treatment. The treatment effect – the change in an outcome variable – may be studied in that context (Greene, 2012).

⁹ See Racicot (2013) or Racicot and Théoret (2014) for the case of a single time series estimation.

¹⁰ The group mean deviations constitutes the LSDV approach.

¹¹ That is, if we wrongly choose the random or the fixed-effect model, LSDV estimator remains consistent.

¹² A alternative approach is provided by computing .

¹³ Not to be confused with the well-known bias due to errors in variables called attenuation which results in an estimator that tends to 0.

¹⁴ The OLS estimator is biased and inconsistent in that context.

¹⁵ See also Arellano (2003) for the presentation of the GMM in a panel data context. Racicot and Théoret (2001, 2008) present the basics of the GMM approach with applications to finance.

¹⁶ Note that we use W as a weighting matrix in the GLS estimator (equation 24). As well known, this matrix can be replaced by the White (1980) or the Newey-West (1987) HAC asymptotically consistent variance–covariance matrix. Racicot (1993, 2013) discusses the properties of the estimator in the context of the White (1980) matrix. This estimator is named β_E. In this article, we rely on the HAC matrix. For the problem of cross-sectional correlation (or spatial correlation), see Driscoll and Kraay (1998).

¹⁷ For an application of Dagenais and Dagenais’ higher moments estimator to the Fama and French model, see Racicot and Coën (2007).

¹⁸ SUR stands for seemingly unrelated regression (Zellner, 1962).

¹⁹ See Fomby et al. (1984), Theorem 21.2.1.

²⁰ In a simultaneous equations context, Da Graça and Masson (2014) use a three-stage least-squares estimator for investigating a structural event study of M&As effects. Our instruments may be useful in that context or a similar one to improve to the robustness of the estimation process.