## Instrumental variable with interactive fixed effects

##### Resumo

The biggest challenge of the instrumental variable model is finding a valid instrument, one that satisfies the assumptions of relevance and, especially, exogeneity. To impose a less restrictive assumption, some estimators combine unobservable effects with instrumental variables, as example IV-FE. In this paper, we study a large-panel instrumental variable model where the instrument may be correlated with unobservable variables, even when they vary across both dimensions. This variation implies that standard approaches in the literature, such as IV, IV-FE, and linear factor models (Pesaran, 2006; Bai, 2009), are inconsistent. We construct two and show their √ NT convergence under both large N and large T. Our exogeneity assumption is less restrictive than the standard instrumental variable model assumption since the instrument shall be exogenous given the factor structure. We show their asymptotic normality distribution for some rate of N/T even when the errors have autocorrelation and heteroskedasticity in both dimensions. We also study the trade-off between our estimators and a standard IV estimator when the error has an interactive fixed-effect structure, and the instrument is valid. In this case, we show that our estimator is more efficient than IV if the variance of the factor structure is sufficiently large than the error variance and less efficient if the variance of factor structure is sufficiently smaller. The biggest challenge of the instrumental variable model is finding a valid instrument, one that satisfies the assumptions of relevance and, especially, exogeneity. To impose a less restrictive assumption, some estimators combine unobservable effects with instrumental variables, as example IV-FE. In this paper, we study a large-panel instrumental variable model where the instrument may be correlated with unobservable variables, even when they vary across both dimensions. This variation implies that standard approaches in the literature, such as IV, IV-FE, and linear factor models (Pesaran, 2006; Bai, 2009), are inconsistent. We construct two and show their √ NT convergence under both large N and large T. Our exogeneity assumption is less restrictive than the standard instrumental variable model assumption since the instrument shall be exogenous given the factor structure. We show their asymptotic normality distribution for some rate of N/T even when the errors have autocorrelation and heteroskedasticity in both dimensions. We also study the trade-off between our estimators and a standard IV estimator when the error has an interactive fixed-effect structure, and the instrument is valid. In this case, we show that our estimator is more efficient than IV if the variance of the factor structure is sufficiently large than the error variance and less efficient if the variance of factor structure is sufficiently smaller.