Panel Data Model With Interactive Fixed Effects And Its Application In Causal Inference

In the traditional panel model with fixed effects,individual fixed effects and time fixed effects are usually introduced as dummy variables to control for unknown shocks at different periods and unknown heterogeneity among individuals,respectively.However,time-varying shocks may be multidimensional,and different individuals may have different responsiveness to shocks in different dimensions.Pesaran(2006)and Bai(2009)introduced a panel data model containing the interactive term of multi-dimensional individual fixed effects and multidimensional time fixed effects,which is also known as the panel data model with interactive fixed effects.Compared with the panel data model with fixed effect,this model overcomes the weaknesses of the panel data model with fixed effects,describes the multi-dimensional timevarying shocks and the heterogeneous responses of individuals to the multi-dimensional shocks,and uses a small number of factors to summarize the information contained in a large number of potential variables.Due to its simplicity and scientificity in design,the panel data model with interactive fixed effect has gradually become one of the important research directions in theoretical econometrics,and is widely used in the empirical research of applied econometrics such as parameter estimation and causal inference.This paper reviews the existing literature on the panel data model with interactive fixed effects,and summarizes the research status of the panel data model with interactive fixed effects from three perspectives:coefficient estimation,factor number estimation and causal inference.In view of the issues existing in the current research,this paper conducts an in-depth study on the panel data model with interactive fixed effect,and the main contents include:1.Inspired by the least square principal components approach of Bai(2009)and the transformed approach of Hsiao et al.(2021),this paper proposes the full information transformed approach for the estimation of coefficient based on the idea of matrix elimination,provides the asymptotic properties of the full information transformed estimator,and proves the numerical equivalence between this estimator and the least square principal components estimator.2.On the basis of the full information transformed approach,this paper further extends the idea of matrix elimination,proposes the iterative full information transformed estimator,provides the asymptotic properties of this estimator,and shows the finite sample performance of this estimator by Monte Carlo simulations.In addition,this paper illustrates the empirical application of this estimator by reproducing the study of Eberhardt et al.(2013).3.Based on the quantile control method of Chen et al.(2022)and the counterfactual construction framework of Hsiao et al.(2019),this paper proposes the quantile control method with covariates for constructing the pointwise confidence intervals of the treatment effects,provides the asymptotic properties of the pointwise confidence interval estimator for the treatment effects,and shows the finite sample performance of this estimator for the treatment effects by Monte Carlo simulations.In addition,this paper illustrates the empirical application of the quantile control method with covariates by reproducing the study of Andersson(2019).4.In view of the statistical inference problem of in-time placebo test used in the synthetic control method and the regression control method,this paper proposes the mixed placebo test,and discusses the applications and matters needing attention of the mixed placebo test.In addition,this paper demonstrates the empirical application of mixed placebo test based on synthetic control method and regression control method,by reproducing the cases of Abadie et al.(2010)and Hsiao et al.(2012),respectively.The innovation of this paper includes the following:1.This paper proposes the full information transformed approach to estimate the coefficients,provides the asymptotic properties of the full information transformed estimator,and proves the numerical equivalence between this estimator and the least square principal components estimator.The numerical equivalence explains the profound concept of"control is equivalent to elimination".It not only provides a new way for understanding of the least square principal components estimator,but also allows the further construction of elimination matrix in theory to accelerate the convergence and improve the efficiency of estimation.2.This paper proposes the iterative full information transformed approach for coefficient estimation,and provides the asymptotic properties of the iterative full information transformed estimator.Compared with the full information transformed estimator,the iterative full information transformed estimator can eliminate the interactive fixed effects in the model and the cross-sectional correlation in errors through the repeated iteration of matrix elimination method,so as to improve the efficiency of the estimator in different situation.Monte Carlo simulation results show that the iterative full information transformed estimator has better efficiency than other estimators in the case of heteroscedasticity and/or cross-section correlation of random error in the model,and the model containing lagged explanatory variables and additive effects.3.This paper proposes the quantile control method with covariates that used to construct pointwise confidence intervals for the treatment effects,and provide the asymptotic properties of the pointwise confidence interval estimator for the treatment effects.The quantile control method with covariates extends the linear model framework derived by matrix elimination method to a nonlinear framework,and uses the quantile regression forest of machine learning to construct robust pointwise confidence intervals for treatment effects,allowing covariates to be added to the model to improve estimation efficiency.The Monte Carlo simulation results show that the quantile control method with covariates has better efficiency than the quantile control method in finite samples when the covariates are included in the model,and the noise variables unrelated to the outcome do not affect the consistency of the pointwise confidence intervals for treatment effects.4.To address the problem that the in-time placebo test cannot provide P values,this paper proposes the mixed placebo test.The mixed placebo test uses the fake treatment period to run the in-space placebo test,so that when the significance of the in-time placebo effect is difficult to judge by inspection,mixed placebo test can provide the p-value for the in-time placebo test.In addition,there are still some unsolved problems in this dissertation,which are worthy of in-depth research and systematic exploration in the future,such as the application of the iterative full information transformed estimator in the dynamic panel data model with interactive fixed effects,the coefficient estimation of panel data model with hierarchical multifactor error structure and the application of the quantile control method with covariate for unbalanced panel data model.In view of these problems,the theory related to the iterative full information transformation estimator and the quantile control method with covariate will continue to be explored in the future research,so that it can be applied to more situation of the empirical study.