The properties of the two stage least squares (TSLS) and limited information maximum likelihood (LIML) estimators in panel data models where the observables are affected by common shocks, modelled through unobservable factors, are studied for the case where the time series dimension is fixed. We show that the key assumption in determining the consistency of the panel TSLS and LIML estimators, as the cross section dimension tends to infinity, is the lack of correlation between the factor

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... in the errors and in the exogenous variables-including the instruments-conditional on the common shocks. If this condition fails, both estimators have degenerate distributions. When the panel TSLS and LIML estimators are consistent, they have covariance-matrix mixed-normal distributions asymptotically. Tests on the coefficients can be constructed in the usual way and have standard distributions under the null hypothesis. Econometrics 2018, 6, 19 2 of 12 very strong assumptions. In panel data with both large time series and cross-sectional dimensions, the assumption that the factor loadings in the disturbances and the regressors are conditionally uncorrelated can be relaxed (e.g., Pesaran 2006; Bai 2009). However, this situation is certainly not the norm in microeconometric applications, where the time dimension tends to be limited. Since the common shocks affect both the errors and the regressors, they induce correlation between some of the regressors and the disturbance term (we will refer to this as factors endogeneity). Econometric models often contain explanatory variables that are endogenous due to simultaneity so that the dependent variable and some of the explanatory variables are co-determined (we will refer to this as classical endogeneity). In the presence of instrumental variables, two standard approaches to endogeneity in panel data are the panel two-stage least squares (TSLS) (e.g., among others Wooldridge 2005; Arellano 2016) and the panel limited information maximum likelihood (LIML) estimators (e.g., Wansbeek and Meijer 2000; Alonso-Borrego and Arellano 1999; and Wansbeek and Prak 2017). This paper investigates how the panel TSLS and LIML estimators are affected by common shocks for which, surprisingly, no results seem available in the literature. The literature on the effects of common shocks in models affected by classical endogeneity is very small. Ahn et al. (2001, 2013) generalize a fixed effects model in which the unobserved individual effects vary over time, and they propose a generalized method of moments (GMM) estimator that generalizes the fixed effects estimator through quasi-differencing. Robertson and Sarafidis (2015) consider linear panel data models with classical endogeneity in which the common factors affect the errors and the factor loading may be correlated with the exogenous variables. Following Ahn et al. (2001, 2013), Robertson and Sarafidis (2015) regard the common factors as unknown parameters, investigate the identification conditions and suggest a GMM estimator. Notice that this literature on the effects of common shocks differs from the one initiated by Andrews (2005) in two fundamental ways: (1) common factors are regarded as parameters not as random variables; (2) the explanatory variables are correlated to the error factor loadings while Andrews (2005) assumes that both the explanatory variables and the error term depend on the same factors. Harding and Lamarche (2011, 2014) extend the model of Pesaran (2006) to allow for classical endogeneity. Precisely, Harding and Lamarche (2011) show that the estimators suggested by Pesaran (2006) also account for classical endogeneity when both the time series and the cross sectional dimensions are large (see also Harding and Lamarche (2014) for an approach based on quantiles). Thus, they investigate how the estimators of Pesaran (2006) are affected by classical endogeneity but are uninformative about how classical estimators are affected by factor endogeneity. Notice also that, by assuming that (T, N) tends to infinity, one allows the information about the shocks to accumulate over time. This is not usually a reasonable assumption in micro-econometric studies where the time dimension tends to be small. This paper investigates the effects of factor endogeneity on standard estimators used in the presence of classical endogeneity by studying the asymptotic properties of the panel TSLS and LIML estimators. Our results, which are in line with those of Andrews (2005) , show that as the cross-sectional dimension tends to infinity (for a fixed time dimension):