# Jushan Bai's research while affiliated with Columbia University and other places

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## Publications (88)

This article provides methods for flexibly capturing unobservable heterogeneity from longitudinal data in the context of an exponential family of distributions. The group memberships of individual units are left unspecified, and their heterogeneity is influenced by group-specific unobservable factor structures. The model includes, as special cases,...

A factor model with a break in its factor loadings is observationally equivalent to a model without changes in the loadings but a change in the variance of its factors. This effectively transforms a structural change problem of high dimension into a problem of low dimension. This paper considers the likelihood ratio (LR) test for a variance change...

Economists are blessed with a wealth of data for analysis, but more often than not, values in some entries of the data matrix are missing. Various methods have been proposed to handle missing observations in a few variables. We exploit the factor structure in panel data of large dimensions. Our tall-project algorithm first estimates the factors fro...

This paper proposes a quasi-maximum likelihood (QML) estimator of the break point for large-dimensional factor models with a single structural break in the factor loading matrix. We show that the QML estimator is consistent for the true break point when the covariance matrix of the pre- or post-break factor loading (or both) is singular. Consistenc...

Pervasive cross-section dependence is increasingly recognized as an appropriate characteristic of economic data and the approximate factor model provides a useful framework for analysis. Assuming a strong factor structure, early work established convergence of the principal component estimates of the factors and loadings to a rotation matrix. This...

This paper proposes an imputation procedure that uses the factors estimated from a tall block along with the re-rotated loadings estimated from a wide block to impute missing values in a panel of data. Assuming that a strong factor structure holds for the full panel of data and its sub-blocks, it is shown that the common component can be consistent...

This paper considers the estimation and inference procedures for the case of a logistic panel regression model with interactive fixed effects, where multiple individual effects are allowed and the model is capable of capturing high-dimensional cross-section dependence. The proposed model also allows for heterogeneous regression coefficients. New Ba...

Economists are blessed with a wealth of data for analysis, but more often than not, values in some entries of the data matrix are missing. Various methods have been proposed to impute missing observations with iterative estimates of their unconditional or conditional means. We exploit the factor structure in panel data of large dimensions. We first...

This paper estimates the break point for large-dimensional factor models with a single structural break in factor loadings at a common unknown date. First, we propose a quasi-maximum likelihood (QML) estimator of the change point based on the second moments of factors, which are estimated by principal component analysis. We show that the QML estima...

This paper studies dynamic spatial panel data models with common shocks to deal with both weak and strong cross-sectional correlations. Weak correlations are captured by a spatial structure and strong correlations are captured by a factor structure. The proposed quasi-maximum likelihood estimator (QMLE) is capable of handling both types of cross se...

This paper considers generalized least squares (GLS) estimation for linear panel data models. By estimating the large error covariance matrix consistently, the proposed feasible GLS estimator is more efficient than the ordinary least squares in the presence of heteroskedasticity, serial and cross-sectional correlations. The covariance matrix used f...

This paper develops a new standard-error estimator for linear panel data models. The proposed estimator is robust to heteroskedasticity, serial correlation, and cross-sectional correlation of unknown forms. The serial correlation is controlled by the Newey–West method. To control for cross-sectional correlations, we propose to use the thresholding...

Estimates of the approximate factor model are increasingly used in empirical work. Their theoretical properties, studied some twenty years ago, also laid the ground work for analysis on large dimensional panel data models with cross-section dependence. This paper presents simplified proofs for the estimates by using alternative rotation matrices, e...

In this paper, we consider the estimation of break points in high-dimensional factor models where the unobserved factors are estimated by principal component analysis (PCA). The factor loading matrix is assumed to have a structural break at an unknown time. We establish the conditions under which the least squares (LS) estimator is consistent for t...

This paper considers generalized least squares (GLS) estimation for linear panel data models. By estimating the large error covariance matrix consistently, the proposed feasible GLS (FGLS) estimator is robust to heteroskedasticity, serial correlation, and cross-sectional correlation. It is more efficient than the OLS estimator. To control serial co...

This paper develops a new standard-error estimator for linear panel data models. The proposed estimator is robust to heteroskedasticity, serial correlation, and cross-sectional correlation of unknown form. Serial correlation is controlled by the Newey-West method. To control cross-sectional correlations, we propose to use the thresholding method, w...

This paper suggests a factor-based imputation procedure that uses the factors estimated from a TALL block along with the re-rotated loadings estimated from a WIDE block to impute missing values in a panel of data. Under a strong factor assumption, it is shown that the common component can be consistently estimated but there will be four different c...

It is known that the common factors in a large panel of data can be consistently estimated by the method of principal components, and principal components can be constructed by iterative least squares regressions. Replacing least squares with ridge regressions turns out to have the effect of removing the contribution of factors associated with smal...

This paper introduces a new procedure for analyzing the quantile co-movement of a large number of financial time series based on a large-scale panel data model with factor structures. The proposed method attempts to capture the unobservable heterogeneity of each of the financial time series based on sensitivity to explanatory variables and to the u...

It is known that the common factors in a large panel of data can be consistently estimated by the method of principal components, and principal components can be constructed by iterative least squares regressions. Replacing least squares with ridge regressions turns out to have the effect of shrinking the singular values of the common component and...

We consider efficient estimation of panel data models with interactive effects, which relies on a high-dimensional inverse covariance matrix estimator. By using a consistent estimator of the error covariance matrix, we can take into account both cross-sectional correlations and heteroskedasticity. In the presence of cross-sectional correlations, th...

Large factor models use a few latent factors to characterize the co-movement of economic variables in a high-dimensional data set. High dimensionality brings challenges as well as new insights into the advancement of econometric theory. Because of their ability to effectively summarize information in large data sets, factor models have been increas...

This paper introduces a new procedure for clustering a large number of financial time series based on high-dimensional panel data with grouped factor structures. The proposed method attempts to capture the level of similarity of each of the time series based on sensitivity to observable factors as well as to the unobservable factor structure. The p...

This paper considers panel data models in the presence of a large number of potential predictors and unobservable common factors. The model is estimated by the regularization method together with the principal components procedure. We propose a panel information criterion for selecting the regularization parameter and the number of common factors u...

We consider the problem of testing for slope homogeneity in high-dimensional panel data models with a factor error structure. We consider the Swamy-type test for slope homogeneity by incorporating interactive fixed effects. We show that the proposed test statistic is asymptotically normal. Our test allows the explanatory variables to be correlated...

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 numb...

This article introduces a new procedure for clustering a large number of financial time series based on high-dimensional panel data with grouped factor structures. The proposed method attempts to capture the level of similarity of each of the time series based on sensitivity to observable factors as well as to the unobservable factor structure. The...

The factor-augmented vector autoregressive (FAVAR) model, first proposed by Bernanke, Bovin, and Eliasz (2005, QJE), is now widely used in macroeconomics and finance. In this model, observable and unobservable factors jointly follow a vector autoregressive process, which further drives the comovement of a large number of observable variables. We st...

The factor-augmented vector autoregressive (FAVAR) model, first proposed by Bernanke, Bovin, and Eliasz (2005, QJE), is now widely used in macroeconomics and finance. In this model, observable and unobservable factors jointly follow a vector autoregressive process, which further drives the comovement of a large number of observable variables. We st...

We consider a set of minimal identification conditions for dynamic factor models. These conditions have economic interpretations and require fewer restrictions than the static factor framework. Under these restrictions, a standard structural vector autoregression (SVAR) with measurement errors can be embedded into a dynamic factor model. More gener...

This article analyzes multifactor models in the presence of a large number of potential observable risk factors and unobservable
common and group-specific factors. We show how relevant observable factors can be found from a large given set and how to
determine the number of common and group-specific unobservable factors. The method allows consisten...

This paper considers the maximum likelihood estimation of panel data models
with interactive effects. Motivated by applications 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 co...

High dimensional factor models can involve thousands of parameters. The Jacobian matrix for identification is of a large dimension. It can be difficult and numerically inaccurate to evaluate the rank of such a Jacobian matrix. We reduce the identification problem to a small rank problem, which is easy to check. The identification conditions allow b...

While most of the convergence results in the literature on high-dimensional
covariance matrix are concerned about the accuracy of estimating the covariance
matrix (and precision matrix), relatively less is known about the effect of
estimating large covariances on statistical inferences. We study two important
models: factor analysis and panel data...

Summary The paper proposes statistics to test the null hypothesis of no cointegration in panel data when common factors drive the cross‐sectional dependence. We focus on the case in which regressors and the common factors are correlated, although the uncorrelated case is also discussed. Both endogenous and strictly exogenous regressors are consider...

We consider the estimation of dynamic panel data models in the presence of incidental parameters in both dimensions: individual fixed-effects and time fixed-effects, as well as incidental parameters in the variances. We adopt the factor analytical approach by estimating the sample variance of individual effects rather than the effects themselves. I...

This paper studies panel data models with unobserved group factor structures. The group membership of each unit and the number of groups are left unspecified. We estimate the model by minimizing the sum of least squared errors with a shrinkage penalty. The regressions coefficients can be homogeneous or group specific. The consistency and asymptotic...

This paper analyzes multifactor models in the presence of a large number of potential observable risk factors and unobservable common and group-specific pervasive factors. We show how relevant observable factors can be found from a large given set and how to determine the number of common and group-specific unobservable factors. The method allows c...

While most of the convergence results in the literature on high dimensional covariance matrix are concerned about the accuracy of estimating the covariance matrix (and precision matrix), relatively less is known about the effect of estimating large covariances on statistical inferences. We study two important models: factor analysis and panel data...

We study the estimation of a high dimensional approximate factor model in the
presence of both cross sectional dependence and heteroskedasticity. The
classical method of principal components analysis (PCA) does not efficiently
estimate the factor loadings or common factors because it essentially treats
the idiosyncratic error to be homoskedastic an...

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 set...

A growing body of threshold models has been developed over the past two decades to capture the nonlinear movement of financial time series. Most of these models, however, contain a single threshold variable only. In many empirical applications, models with two or more threshold variables are needed. This article develops a new threshold autoregress...

Motivated by the great moderation in major US macroeconomic time series, we formulate the regime switching problem through a conditional Markov chain. We model the long-run volatility change as a recurrent structure change, while short-run changes in the mean growth rate as regime switches. Both structure and regime are unobserved. The structure is...

It is known that the principal component estimates of the factors and the loadings are rota-tions of the underlying latent factors and loadings. We study conditions under which the latent factors can be estimated asymptotically without rotation. We derive the limiting distributions for the factor estimates when N and T are large and make precise ho...

We consider estimation of parameters in a regression model in which the endogenous re- gressors are just a few of the many other endogenous variables driven by a small number of unobservable exogenous common shocks. We show the method of principal components can be used to estimate factors that can be used as instrumental variables. These are not o...

An effective way to control for cross-section correlation when conducting a panel unit root test is to remove the common factors from the data. However, there remain many ways to use the defactored residuals to construct a test. In this paper, we use the panel analysis of nonstationarity in idiosyncratic and common components (PANIC) residuals to f...

This paper establishes the consistency of the estimated common break point in panel data. Consistency is obtainable even when a regime contains a single observation, making it possible to quickly identify the onset of a new regime. We also propose a new framework for developing the limiting distribution for the estimated break point, and show how t...

We propose a simple method for estimating betas (factor loadings) when factors are measured with error: Ordinary Least-squares Instrumental Variable Estimator (OLIVE). OLIVE is intuitive and easy to implement. OLIVE performs well when the number of instruments becomes large (can be larger than the sample size), while the performance of conventional...

This paper considers large N and large T panel data models with unobservable multiple interactive effects, which are correlated with the regressors. In earnings studies, for example, workers' motivation, persistence, and diligence combined to influence the earnings in addition to the usual argument of innate ability. In macroeconomics, interactive...

In forecasting and regression analysis, it is often necessary to select predictors from a large feasible set. When the predictors have no natural ordering, an exhaustive evaluation of all possible combinations of the predictors can be computationally costly. This paper considers 'boosting' as a methodology of selecting the predictors in factor-augm...

This paper studies the problem of unit root testing in the presence of multiple structural changes and common dynamic factors.
Structural breaks represent infrequent regime shifts, while dynamic factors capture common shocks underlying the comovement
of economic time series. We examine the modified Sargan-Bhargava (MSB) test in the panel data setti...

This paper studies estimation of panel cointegration models with cross-sectional dependence generated by unobserved global stochastic trends. The standard least squares estimator is, in general, inconsistent owing to the spuriousness induced by the unobservable I(1) trends. We propose two iterative procedures that jointly estimate the slope paramet...

Practitioners often have at their disposal a large number of instruments that are weakly exogenous for the parameter of interest. However, not every instrument has the same predictive power for the endogenous variable, and using too many instruments can induce bias. We consider two ways of handling these problems. The first is to form principal com...

Much is written about the use of factors estimated by the method of principal components from large panels in linear regression models. In this paper, we provide an analysis for non-linear estimation and establish the conditions under which the estimated factors can be treated as though they were observable. The results can be used to estimate prob...

This paper studies two refinements to the method of factor forecasting. First, we consider the method of quadratic principal components that allows the link function between the predictors and the factors to be non-linear. Second, the factors used in the forecasting equation are estimated in a way to take into account that the goal is to forecast a...

In this paper, we consider testing distributional assumptions in multivariate GARCH models based on empirical processes. Using the fact that joint distribution carries the same amount of information as the marginal together with conditional distributions, we first transform the multivariate data into univariate independent data based on the margina...

This paper considers the estimation of multiple-structural-break models under specification errors. A common example in economics is that the true model is measured in level, but a linear-log model is estimated. We show that, under specification errors, if there are more than one break points and if a single-break model is estimated, the estimated...

Econometric analysis of large dimensional factor models has been a heavily researched topic in recent years. This review surveys the main theoretical results that relate to static factor models or dynamic fac- tor models that can be cast in a static framework. Among the topics covered are how to determine the number of factors, how to conduct infer...

A widely held but untested assumption underlying macroeconomic analysis is that the number of shocks driving economic ∞uctuations, q, is small. In this paper, we associate q with the number of dynamic factors in a large panel of data. We pro- pose a methodology to determine q without having to estimate the dynamic factors. We flrst estimate a VAR i...

Most of the existing literature on panel data cointegration assumes cross-sectional independence, an assumption that is difficult to satisfy. This paper studies panel cointegration under cross-sectional dependence, which is characterized by a factor structure. We derive the limiting distribution of a fully modified estimator for the panel cointegra...

In a recent paper, Bai and Perron ( 1998 ) considered theoretical issues related to the limiting distribution of estimators and test statistics in the linear model with multiple structural changes. In this companion paper, we consider practical issues for the empirical applications of the procedures. We first address the problem of estimation of th...

We consider the situation when there is a large number of series, N, each with T observations, and each series has some predictive ability for some variable of interest. A methodology of growing interest is first to estimate common factors from the panel of data by the method of principal components and then to augment an otherwise standard regress...

INTRODUCTION
Both the statistics and econometrics literature contain a vast amount of work on issues related to structural change, most of it specifically designed for the case of a single change. The problem of multiple structural changes, however, has received considerably less attention. Recently, Bai and Perron (1998, 2003a) provided a comprehe...

We present the sampling distributions for the coefficient of skewness, kurtosis, and a joint test of normal- ity for time series observations. We show that when the data are serially correlated, consistent estimates of three-dimensional long-run covariance matrices are needed for testing symmetry or kurtosis. These tests can be used to make inferen...

Common factors play an important role in many disciplines of social science. In economics, the factors are the common shocks that underlie the co-movements of the large number of economic time series. The question of interest is whether some observable economic variables are in fact the underlying unobserved factors. We consider statistics to deter...

We consider the situation when there is a large number of series, $N$, each with $T$ observations, and each series has some predictive ability for the variable of interest, $y$. A methodology of growing interest is to first estimate common factors from the panel of data by the method of principal components, and then augment an otherwise standard r...

This paper develops a new methodology that makes use of the factor structure of large dimensional panels to understand the nature of nonstationarity in the data. We refer to it as PANIC-Panel Analysis of Nonstationarity in Idiosyncratic and Common components. PANIC can detect whether the nonstationarity in a series is pervasive, or variable-specifi...

This paper studies large-dimension factor models with nonstationary dynamic factors, also referred to as cross-section common stochastic trends. We consider the problem of estimating the dimension of the common stochastic trends and the stochastic trends themselves. We derive the rates of convergence and the limiting distributions for the estimated...

Bai and Perron (1998) considered theoretical issues related to the limiting distribution of estimators and test statistics in the linear model with multiple structural changes. The asymptotic distributions of the tests depend on a trimming parameter ε and critical values were tabulated for ε= 0.05. As discussed in Bai and Perron (2000), larger valu...

This paper proposes a nonparametric test for parametric conditional distributions of dynamic models. The test is of the Kolmogorov type coupled with Khmaladze's martingale transformation. It is asymptotically distribution-free and has nontrivial power against root-n local alternatives. The method is applicable for various dynamic models, including...

This paper develops an inferential theory for factor models of large dimensions. The principal components estimator is considered because it is easy to compute and is asymptotically equivalent to the maximum likelihood estimator (if normality is assumed). We derive the rate of convergence and the limiting distributions of the estimated factors, fac...

In a recent paper, Bai and Perron (1998) considered theoretical issues related to the limiting distribution of estimators and test statistics in the linear model with multiple structural changes. In this companion paper, we consider practical issues for the empirical applications of the procedures. We first address the problem of estimation of the...

This paper uses a decomposition of the data into common and idiosyncratic components to develop procedures that test if these components satisfy the null hypothesis of stationarity. The decomposition also allows us to construct pooled tests that satisfy the cross-section independence assumption. In simulations, tests on the components separately ge...

In this paper we develop some econometric theory for factor models of large dimensions. The focus is the determination of the number of factors (r), which is an unresolved issue in the rapidly growing literature on multifactor models. We first establish the convergence rate for the factor estimates that will allow for consistent estimation of r. We...

The assumption of conditional symmetry is often invoked to validate adaptive estimation and consistent estimation of ARCH/GARCH models by quasi maximum likelihood. Imposing conditional symmetry can increase the efficiency of bootstraps if the symmetry assumption is valid. This paper proposes a procedure for testing conditional symmetry. The propose...

This paper develops the statistical theory for testing and estimating multiple change points in regression models. The rate of convergence and limiting distribution for the estimated parameters are obtained. Several test statistics are proposed to determine the existence as well as the number of change points. A partial structural change model is c...

In this paper, we derive a general Hajek-Renyi type inequality for vector-valued martingales. Several well known inequalities are shown to be special cases of this general inequality. We also derive a similar inequality for dependent sequences. We then apply the inequality to the problem of strong consistency of least squares estimators for multipl...

The simulation result of Nunes, Kuan, and Newbold suggests that it is possible to estimate a spurious break for a regression model with I(1) disturbances. In this note, we provide a rigorous proof for this phenomenon.

We are interested in obtaining the h-period ahead forecast of a series yt. The information available includes the panel of data on xit (i = 1; 2; :::; N; t = 1; 2; :::; T ) and a smaller set of other variables Wt. For example, Wt might be lags of yt. If N was small, we could formulate a forecasting model with all the xit and Wt as predictors. But t...

Motivated by the great moderation in major U.S. macroeconomic time series, we propose a new type of restrictions, called conditional Markov chain, on the Markov switching model to study the nonstationarity of time series data. We take the long-run volatility change as a recurrent structure change, while the short-run/medium-run change in mean growt...

## Citations

... In the context of functional data, low rank matrix completion is studied by Descary and Panaretos (2019). In this work we employ the framework of approximative factor models (Bai (2003); Fan et al. (2013); Bai and Ng (2023)), which is rather popular in macroeconometrics, and amend the related imputation methods in Cahan et al. (2023) for functional data. While in the classical factor models literature the number of factors (denoted by r) is fixed and small (more than two or three factors can hardly be interpreted) a much larger number of factors is generally required to describe functional data sufficiently well. ...

... (2021), Duan et al. (2022)). Methods utilizing the estimated pseudo factors from the whole sample will necessarily have power against heteroscedasticity in the factors, even if the loadings are actually time invariant. ...

... Traditionally, missing data problems in factor-based macroeconomic analysis have been addressed using expectation maximisation principal components analysis (EMPCA); see Stock and Watson (2002). More recently, new approaches have been suggested, including the tall wide (TW) algorithm of Bai and Ng (2021) and the tall project (TP) algorithm of Cahan et al. (2023). These are all generic-factor estimation algorithms to handle missing data. ...

... Traditionally, missing data problems in factor-based macroeconomic analysis have been addressed using expectation maximisation principal components analysis (EMPCA); see Stock and Watson (2002). More recently, new approaches have been suggested, including the tall wide (TW) algorithm of Bai and Ng (2021) and the tall project (TP) algorithm of Cahan et al. (2023). These are all generic-factor estimation algorithms to handle missing data. ...

... For linear models with interactive fixed effects, fundamental contributions have been made by Pesaran (2006), Bai (2009 and Moon and Weidner (2015). In recent years, Chen (2016), Boneva and Linton (2017), Chen, Fernández-Val, and Weidner (2021), Ando, Bai, and Li (2022) and Gao, Liu, Peng, and Yan (2023) have investigated the estimation of nonlinear models with interactive fixed effects. 1 Panel data models with interactive fixed effects involve three sets of parameters: a finitedimensional vector of coefficients (denoted by β) for the observed regressors, a T × r matrix of latent factors (denoted by F ) representing the global shocks to all individuals, and an N × r matrix of factors loadings (denoted by Λ) measuring the individual-specific responses to the factors. 2 While the main object of interest is β, the latter two sets of parameters are introduced to account for individual heterogeneity and cross-sectional dependence. To estimate these parameters in nonlinear models, the papers mentioned above usually take two different approaches. ...

... The data was collected from Quantec Easy, with a total of 207 observations. According to Bai and Li (2021), panel data is widely used to estimate dynamic econometric models, which include both time series and cross-sectional dimensions. Panel data analysis is appropriate for this study since it uses South African that have been observed over a period of time as its analytical framework. ...

... More often than not, one has to assume independence along at least one dimension of the dataset (e.g., Assumption 2 of Pesaran 2006 The studies sharing a similar concern with ours are Gonçalves (2011) and Bai et al. (2020). Specifically, Gonçalves (2011) studies a fixed effect panel data model, and proposes using the moving blocks bootstrap (MBB) technique, which allows for the error terms to have correlation over both dimensions. ...

... To calculate the long-run estimates, we used a cross-sectional time-series Feasible Generalized Least Squares (FGLS) and robust dynamic ordinary least squares (DOLS) as well as a panel generalized method of moments (GMM). The error term in FGLS regressions with heteroskedastic crosssectional correlations and panel-specific AR 1 ð Þspeculations reflects a large number of factors, heteroskedasticity, serial correlations and cross-sectional correlations (Bai et al., 2021). Moreover, FGLS method eliminates cross-sectional as well as serial correlation biases by using a high-dimensional error covariance matrix estimator. ...

... We require the number of factors r to be identical before and after the break, because our method relies on subsample estimates after splitting the sample, a common regularity condition found in many other methods utilizing subsample estimates (e.g. Su (2018) andBai et al. (2020)). Throughout the paper, we treat both the number of factors r and the break fraction π as known, as both of these can be consistently estimated (see Remarks 1 and 2) without affecting any of our asymptotic results. ...

... In view of the low-rank structures, we can resort to NNR estimation which has attracted increasing attention recently in panel data analyses. NNR has been used in recent econometric research -see Bai and Ng (2019), Moon andWeidner (2018), Chernozhukov et al. (2020), Belloni et al. (2023), Miao et al. (2023), Feng (2023), and Hong et al. (2023, among others. But none of these papers imposes any latent group structures on the slope coefficients. ...