Statistical analysis of factor models of high dimension

This paper considers the maximum likelihood estimation of factor models of high dimension, where the number of variables (N) is comparable with or even greater than the number of observations (T). An inferential theory is developed. We establish not only consistency but also the rate of convergence and the limiting distributions. Five different sets of identification conditions are considered. We show that the distributions of the MLE estimators depend on the identification restrictions. Unlike the principal components approach, the maximum likelihood estimator explicitly allows heteroskedasticities, which are jointly estimated with other parameters. Efficiency of MLE relative to the principal components method is also considered.Comment: Published in at http://dx.doi.org/10.1214/11-AOS966 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Theory and methods of panel data models with interactive effects

This paper considers the maximum likelihood estimation of the panel data models with interactive effects. Motivated in economics and other social sciences, a notable feature of the model is that the explanatory variables are correlated with the unobserved effects. The usual within-group estimator is inconsistent. Existing methods for consistent estimation are either designed for panel data with short time periods or are less efficient. The maximum likelihood estimator has desirable properties and is easy to implement, as illustrated by the Monte Carlo simulations. This paper develops the inferential theory for the maximum likelihood estimator, including consistency, rate of convergence and the limiting distributions. We further extend the model to include time-invariant regressors and common regressors (cross-section invariant). The regression coefficients for the time-invariant regressors are time-varying, and the coefficients for the common regressors are cross-sectionally varying

Maximum likelihood estimation and inference for approximate factor models of high dimension

An approximate factor model of high dimension has two key features. First, the idiosyncratic errors are correlated and heteroskedastic over both the cross-section and time dimensions; the correlations and heteroskedasticities are of unknown forms. Second, the number of variables is comparable or even greater than the sample size. Thus a large number of parameters exist under a high dimensional approximate factor model. Most widely used approaches to estimation are principal component based. This paper considers the maximum likelihood-based estimation of the model. Consistency, rate of convergence, and limiting distributions are obtained under various identification restrictions. Comparison with the principal component method is made. The likelihood-based estimators are more efficient than those of principal component based. Monte Carlo simulations show the method is easy to implement and an application to the U.S. yield curves is considere

Practical notes on panel data models with interactive effects

This note is intended for researchers who want to use the interactive effects model for empirical modeling. We consider how to estimate interactive effects models when some of the factors and factor loading are observable. Observable factors are common regressors which do not vary across individuals such as macroeconomic variables, but their regression coefficients are individual-dependent. Observable factor loadings correspond to time-invariant regressors such that race, gender and education, but their regression coefficients are time dependent. This note elaborates the estimation procedures in Bai (2009) in the presence of such regressors

Practical notes on panel data models with interactive effects

This note is intended for researchers who want to use the interactive effects model for empirical modeling. We consider how to estimate interactive effects models when some of the factors and factor loading are observable. Observable factors are common regressors which do not vary across individuals such as macroeconomic variables, but their regression coefficients are individual-dependent. Observable factor loadings correspond to time-invariant regressors such that race, gender and education, but their regression coefficients are time dependent. This note elaborates the estimation procedures in Bai (2009) in the presence of such regressors

Spatial panel data models with common shocks

Spatial effects and common-shocks effects are of increasing empirical importance. Each type of effect has been analyzed separately in a growing literature. This paper considers a joint modeling of both types. Joint modeling allows one to determine whether one or both of these effects are present. A large number of incidental parameters exist under the joint modeling. The quasi maximum likelihood method (MLE) is proposed to estimate the model. Heteroskedasticity is explicitly estimated. This paper demonstrates that the quasi-MLE is effective in dealing with the incidental parameters problem. An inferential theory including consistency, rate of convergence and limiting distributions is developed. The quasi-MLE can be easily implemented via the EM algorithm, as confirmed by the Monte Carlo simulations. The simulations further reveal the excellent finite sample properties of the quasi-MLE. Some extensions are discussed

Maximum likelihood estimation and inference for approximate factor models of high dimension

An approximate factor model of high dimension has two key features. First, the idiosyncratic errors are correlated and heteroskedastic over both the cross-section and time dimensions; the correlations and heteroskedasticities are of unknown forms. Second, the number of variables is comparable or even greater than the sample size. Thus a large number of parameters exist under a high dimensional approximate factor model. Most widely used approaches to estimation are principal component based. This paper considers the maximum likelihood-based estimation of the model. Consistency, rate of convergence, and limiting distributions are obtained under various identification restrictions. Comparison with the principal component method is made. The likelihood-based estimators are more efficient than those of principal component based. Monte Carlo simulations show the method is easy to implement and an application to the U.S. yield curves is considere

Maximum likelihood estimation and inference for approximate factor models of high dimension

An approximate factor model of high dimension has two key features. First, the idiosyncratic errors are correlated and heteroskedastic over both the cross-section and time dimensions; the correlations and heteroskedasticities are of unknown forms. Second, the number of variables is comparable or even greater than the sample size. Thus a large number of parameters exist under a high dimensional approximate factor model. Most widely used approaches to estimation are principal component based. This paper considers the maximum likelihood-based estimation of the model. Consistency, rate of convergence, and limiting distributions are obtained under various identification restrictions. Comparison with the principal component method is made. The likelihood-based estimators are more efficient than those of principal component based. Monte Carlo simulations show the method is easy to implement and an application to the U.S. yield curves is considere