This survey study discusses main aspects to optimal estimation methodologies for panel data regression models. In particular, we present current methodological developments for modeling stationary panel data as well as robust methods for estimation and inference in nonstationary panel data regression models. Some applications from the network econometrics and high dimensional statistics literature are also discussed within a stationary time series environment.
This paper develops bootstrap methods for practical statistical inference in panel data quantile regression models with fixed effects. We consider random-weighted bootstrap resampling and formally establish its validity for asymptotic inference. The bootstrap algorithm is simple to implement in practice by using a weighted quantile regression estimation for fixed effects panel data. We provide results under conditions that allow for temporal dependence of observations within individuals, thus encompassing a large class of possible empirical applications. Monte Carlo simulations provide numeric…
Canay (2011)'s two-step estimator of quantile panel data models, due to its simple intuition and low computational cost, has been widely used in empirical studies in recent years. In this paper, we revisit the estimator of Canay (2011) and point out that in his asymptotic analysis the bias of his estimator due to the estimation of the fixed effects is mistakenly omitted, and that such omission will lead to invalid inference on the coefficients. To solve this problem, we propose a similar easy-to-implement estimator based on smoothed quantile regressions. The asymptotic distribution of the new …
Threshold Regression for Fixed-T Panel Data with Interactive Fixed Effects
Free University of Bozen-Bolzanno（博尔扎诺自由大学）；Brunel University of London（伦敦布鲁内尔大学）；University of Birmingham（伯明翰大学）；Lund University（隆德大学）；Deakin University（迪肯大学）
This paper considers panel data models where the conditional quantiles of the dependent variables are additively separable as unknown functions of the regressors and the individual effects. We propose two estimators of the quantile partial effects while controlling for the individual heterogeneity. The first estimator is based on local linear quantile regressions, and the second is based on local linear smoothed quantile regressions, both of which are easy to compute in practice. Within the large T framework, we provide sufficient conditions under which the two estimators are shown to be asymp…
We study linear panel regression models in which the unobserved error term is an unknown smooth function of two-way unobserved fixed effects. In standard additive or interactive fixed effect models the individual specific and time specific effects are assumed to enter with a known functional form (additive or multiplicative). In this paper, we allow for this functional form to be more general and unknown. We discuss two different estimation approaches that allow consistent estimation of the regression parameters in this setting as the number of individuals and the number of time periods grow t…
This paper introduces estimation methods for grouped latent heterogeneity in panel data quantile regression. We assume that the observed individuals come from a heterogeneous population with a finite number of types. The number of types and group membership is not assumed to be known in advance and is estimated by means of a convex optimization problem. We provide conditions under which group membership is estimated consistently and establish asymptotic normality of the resulting estimators. Simulations show that the method works well in finite samples when T is reasonably large. To illustrate…
The paper introduces structured machine learning regressions for heavy-tailed dependent panel data potentially sampled at different frequencies. We focus on the sparse-group LASSO regularization. This type of regularization can take advantage of the mixed frequency time series panel data structures and improve the quality of the estimates. We obtain oracle inequalities for the pooled and fixed effects sparse-group LASSO panel data estimators recognizing that financial and economic data can have fat tails. To that end, we leverage on a new Fuk-Nagaev concentration inequality for panel data cons…
This paper provides a method to construct simultaneous confidence bands for quantile functions and quantile effects in nonlinear network and panel models with unobserved two-way effects, strictly exogenous covariates, and possibly discrete outcome variables. The method is based upon projection of simultaneous confidence bands for distribution functions constructed from fixed effects distribution regression estimators. These fixed effects estimators are debiased to deal with the incidental parameter problem. Under asymptotic sequences where both dimensions of the data set grow at the same rate,…
In this study, we develop a novel estimation method for quantile treatment effects (QTE) under rank invariance and rank stationarity assumptions. Ishihara (2020) explores identification of the nonseparable panel data model under these assumptions and proposes a parametric estimation based on the minimum distance method. However, when the dimensionality of the covariates is large, the minimum distance estimation using this process is computationally demanding. To overcome this problem, we propose a two-step estimation method based on the quantile regression and minimum distance methods. We then…
This paper considers estimating functional-coefficient models in panel quantile regression with individual effects, allowing the cross-sectional and temporal dependence for large panel observations. A latent group structure is imposed on the heterogenous quantile regression models so that the number of nonparametric functional coefficients to be estimated can be reduced considerably. With the preliminary local linear quantile estimates of the subject-specific functional coefficients, a classic agglomerative clustering algorithm is used to estimate the unknown group structure and an easy-to-imp…
In this paper, we propose a class of low-rank panel quantile regression models which allow for unobserved slope heterogeneity over both individuals and time. We estimate the heterogeneous intercept and slope matrices via nuclear norm regularization followed by sample splitting, row- and column-wise quantile regressions and debiasing. We show that the estimators of the factors and factor loadings associated with the intercept and slope matrices are asymptotically normally distributed. In addition, we develop two specification tests: one for the null hypothesis that the slope coefficient is a co…
This paper introduces the problem of coresets for regression problems to panel data settings. We first define coresets for several variants of regression problems with panel data and then present efficient algorithms to construct coresets of size that depend polynomially on 1/$\varepsilon$ (where $\varepsilon$ is the error parameter) and the number of regression parameters - independent of the number of individuals in the panel data or the time units each individual is observed for. Our approach is based on the Feldman-Langberg framework in which a key step is to upper bound the "total sensiti…
Nonlinear panel data models with fixed individual effects provide an important set of tools for describing microeconometric data. In a large class of such models (including probit, proportional hazard and quantile regression to name just a few) it is impossible to difference out individual effects, and inference is usually justified in a `large n large T' asymptotic framework. However, there is a considerable gap in the type of assumptions that are currently imposed in models with smooth score functions (such as probit, and proportional hazard) and quantile regression. In the present paper we …
Department of Economics, University of North Carolina – Chapel Hill（北卡罗来纳大学教堂山分校经济学系）；European Commission, Joint Research Centre (JRC)（欧盟委员会联合研究中心）；Department of Finance, Copenhagen Business School（哥本哈根商学院金融系）
We propose a novel two-regime regression model where regime switching is driven by a vector of possibly unobservable factors. When the factors are latent, we estimate them by the principal component analysis of a panel data set. We show that the optimization problem can be reformulated as mixed integer optimization, and we present two alternative computational algorithms. We derive the asymptotic distribution of the resulting estimator under the scheme that the threshold effect shrinks to zero. In particular, we establish a phase transition that describes the effect of first-stage factor estim…
Institute of Statistics and Data Science, Nanjing Audit University（南京审计大学统计与数据科学学院）
China has made great achievements in electric power industry during the long-term deepening of reform and opening up. However, the complex regional economic, social and natural conditions, electricity resources are not evenly distributed, which accounts for the electricity deficiency in some regions of China. It is desirable to develop a robust electricity forecasting model. Motivated by which, we propose a Panel Semiparametric Quantile Regression Neural Network (PSQRNN) by utilizing the artificial neural network and semiparametric quantile regression. The PSQRNN can explore a potential linear…
This paper studies a panel data setting where the goal is to estimate causal effects of an intervention by predicting the counterfactual values of outcomes for treated units, had they not received the treatment. Several approaches have been proposed for this problem, including regression methods, synthetic control methods and matrix completion methods. This paper considers an ensemble approach, and shows that it performs better than any of the individual methods in several economic datasets. Matrix completion methods are often given the most weight by the ensemble, but this clearly depends on …
This article develops a Bayesian approach for estimating panel quantile regression with binary outcomes in the presence of correlated random effects. We construct a working likelihood using an asymmetric Laplace (AL) error distribution and combine it with suitable prior distributions to obtain the complete joint posterior distribution. For posterior inference, we propose two Markov chain Monte Carlo (MCMC) algorithms but prefer the algorithm that exploits the blocking procedure to produce lower autocorrelation in the MCMC draws. We also explain how to use the MCMC draws to calculate the margin…
We propose a generalization of the linear panel quantile regression model to accommodate both \textit{sparse} and \textit{dense} parts: sparse means while the number of covariates available is large, potentially only a much smaller number of them have a nonzero impact on each conditional quantile of the response variable; while the dense part is represent by a low-rank matrix that can be approximated by latent factors and their loadings. Such a structure poses problems for traditional sparse estimators, such as the $\ell_1$-penalised Quantile Regression, and for traditional latent factor estim…
This paper develops a new model and estimation procedure for panel data that allows us to identify heterogeneous structural breaks. We model individual heterogeneity using a grouped pattern. For each group, we allow common structural breaks in the coefficients. However, the number, timing, and size of these breaks can differ across groups. We develop a hybrid estimation procedure of the grouped fixed effects approach and adaptive group fused Lasso. We show that our method can consistently identify the latent group structure, detect structural breaks, and estimate the regression parameters. Mon…
Three-dimensional panel models are widely used in empirical analysis. Researchers use various combinations of fixed effects for three-dimensional panels. When one imposes a parsimonious model and the true model is rich, then it incurs mis-specification biases. When one employs a rich model and the true model is parsimonious, then it incurs larger standard errors than necessary. It is therefore useful for researchers to know correct models. In this light, Lu, Miao, and Su (2018) propose methods of model selection. We advance this literature by proposing a method of post-selection inference for …
Heterogeneous panel data models that allow the coefficients to vary across individuals and/or change over time have received increasingly more attention in statistics and econometrics. This paper proposes a two-dimensional heterogeneous panel regression model that incorporate a group structure of individual heterogeneous effects with cohort formation for their time-variations, which allows common coefficients between nonadjacent time points. A bi-integrative procedure that detects the information regarding group and cohort patterns simultaneously via a doubly penalized least square with concav…
This paper studies estimation of linear panel regression models with heterogeneous coefficients, when both the regressors and the residual contain a possibly common, latent, factor structure. Our theory is (nearly) efficient, because based on the GLS principle, and also robust to the specification of such factor structure because it does not require any information on the number of factors nor estimation of the factor structure itself. We first show how the unfeasible GLS estimator not only affords an efficiency improvement but, more importantly, provides a bias-adjusted estimator with the con…
This paper studies the estimation of linear panel data models with interactive fixed effects, where one dimension of the panel, typically time, may be fixed. To this end, a novel transformation is introduced that reduces the model to a lower dimension, and, in doing so, relieves the model of incidental parameters in the cross-section. The central result of this paper demonstrates that transforming the model and then applying the principal component (PC) estimator of \cite{bai_panel_2009} delivers $\sqrt{n}$ consistent estimates of regression slope coefficients with $T$ fixed. Moreover, these e…
Empirical growth analysis has three major problems --- variable selection, parameter heterogeneity and cross-sectional dependence --- which are addressed independently from each other in most studies. The purpose of this study is to propose an integrated framework that extends the conventional linear growth regression model to allow for parameter heterogeneity and cross-sectional error dependence, while simultaneously performing variable selection. We also derive the asymptotic properties of the estimator under both low and high dimensions, and further investigate the finite sample performance…
A central goal in social science is to evaluate the causal effect of a policy. One dominant approach is through panel data analysis in which the behaviors of multiple units are observed over time. The information across time and space motivates two general approaches: (i) horizontal regression (i.e., unconfoundedness), which exploits time series patterns, and (ii) vertical regression (e.g., synthetic controls), which exploits cross-sectional patterns. Conventional wisdom states that the two approaches are fundamentally different. We establish this position to be partly false for estimation but…
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 \textsc{tall-project} algorithm first estimates the factors from a \textsc{tall} block in which data for all rows are observed, and projections of variable specific length are then used to estimate the factor loadings. A missing value is imputed as the estimated common component which we show is consist…
