Statistical estimation of multivariate Ornstein–Uhlenbeck processes and applications to co-integration

Ornstein–Uhlenbeck models are continuous-time processes which have broad applications in finance as, e.g., volatility processes in stochastic volatility models or spread models in spread options and pairs trading. The paper presents a least squares estimator for the model parameter in a multivariate Ornstein–Uhlenbeck model driven by a multivariate regularly varying Lévy process with infinite variance. We show that the estimator is consistent. Moreover, we derive its asymptotic behavior and test statistics. The results are compared to the finite variance case. For the proof we require some new results on multivariate regular variation of products of random vectors and central limit theorems. Furthermore, we embed this model in the setup of a co-integrated model in continuous time.     

Asymptotics for panel quantile regression models with individual effects

This paper studies panel quantile regression models with individual fixed effects. We formally establish sufficient conditions for consistency and asymptotic normality of the quantile regression estimator when the number of individuals, n$n$, and the number of time periods, T$T$, jointly go to infinity. The estimator is shown to be consistent under similar conditions to those found in the nonlinear panel data literature. Nevertheless, due to the non-smoothness of the objective function, we had to impose a more restrictive condition on T$T$ to prove asymptotic normality than that usually found in the literature. The finite sample performance of the estimator is evaluated by Monte Carlo simulations.     

GMM estimators with improved finite sample properties using principal components of the weighting matrix,with an application to the dynamic panel data model

GMM estimators have poor finite sample properties in highly overidentified models. With many moment conditions the optimal weighting matrix is poorly estimated. We suggest using principal components of the weighting matrix. This effectively drops some of the moment conditions. Our simulations, done in the context of the dynamic panel data model, show that the resulting GMM estimator has better finite sample properties than the usual two-step GMM estimator, in the sense of smaller bias and more reliable standard errors.     

A zero inefficiency stochastic frontier model

Traditional stochastic frontier models impose inefficient behavior on all firms in the sample of interest. If the data under investigation represent a mixture of both fully efficient and inefficient firms then off-the-shelf frontier models are statistically inadequate. We introduce the zero inefficiency stochastic frontier model which can accommodate the presence of both efficient and inefficient firms in the sample. We derive the corresponding log-likelihood function, conditional mean of inefficiency, to estimate observation-specific inefficiency and discuss testing for the presence of fully efficient firms. We provide both simulated evidence as well as an empirical example which demonstrates the applicability of the proposed method.     

Pseudo conditional maximum likelihood estimation of the dynamic logit model for binary panel data

We show how the dynamic logit model for binary panel data may be approximated by a quadratic exponential model. Under the approximating model, simple sufficient statistics exist for the subject-specific parameters introduced to capture the unobserved heterogeneity between subjects. The latter must be distinguished from the state dependence which is accounted for by including the lagged response variable among the regressors. By conditioning on the sufficient statistics, we derive a pseudo conditional likelihood estimator of the structural parameters of the dynamic logit model, which is simple to compute. Asymptotic properties of this estimator are studied in detail. Simulation results show that the estimator is competitive in terms of efficiency with estimators recently proposed in the econometric literature.     

Panel data models with spatially correlated error components

In this paper we consider a panel data model with error components that are both spatially and time-wise correlated. The model blends specifications typically considered in the spatial literature with those considered in the error components literature. We introduce generalizations of the generalized moments estimators suggested in Kelejian and Prucha (1999. A generalized moments estimator for the autoregressive parameter in a spatial model. International Economic Review 40, 509–533) for estimating the spatial autoregressive parameter and the variance components of the disturbance process. We then use those estimators to define a feasible generalized least squares procedure for the regression parameters. We give formal large sample results for the proposed estimators. We emphasize that our estimators remain computationally feasible even in large samples.     

Maximum likelihood estimation and inference methods for the covariance stationary panel AR(1)/unit root model

This paper considers Maximum Likelihood (ML) based estimation and inference procedures for linear dynamic panel data models with fixed effects.     

Heteroskedasticity and spatiotemporal dependence robust inference for linear panel models with fixed effects

This paper studies robust inference for linear panel models with fixed effects in the presence of heteroskedasticity and spatiotemporal dependence of unknown forms. We propose a bivariate kernel covariance estimator that nests existing estimators as special cases. Our estimator improves upon existing estimators in terms of robustness, efficiency, and adaptiveness. For distributional approximations, we considered two types of asymptotics: the increasing-smoothing asymptotics and the fixed-smoothing asymptotics. Under the former asymptotics, the Wald statistic based on our covariance estimator converges to a chi-square distribution. Under the latter asymptotics, the Wald statistic is asymptotically equivalent to a distribution that can be well approximated by an F distribution. Simulation results show that our proposed testing procedure works well in finite samples.     

Nonparametric trending regression with cross-sectional dependence

Panel data, whose series length T$T$ is large but whose cross-section size N$N$ need not be, are assumed to have common time trend, of unknown form. The model includes additive, unknown, individual-specific components and allows for spatial or other cross-sectional dependence and/or heteroscedasticity. A simple smoothed nonparametric trend estimate is shown to be dominated by an estimate which exploits availability of cross-sectional data. Asymptotically optimal bandwidth choices are justified for both estimates. Feasible optimal bandwidths, and feasible optimal trend estimates, are asymptotically justified, finite sample performance of the latter being examined in a Monte Carlo study. Potential extensions are discussed.     

Micro versus macro cointegration in heterogeneous panels

We consider the issue of cross-sectional aggregation in nonstationary and heterogeneous panels where each unit cointegrates. We derive asymptotic properties of the aggregate estimate, and necessary and sufficient conditions for cointegration to hold in the aggregate relationship. We then analyze the case when cointegration does not carry through the aggregation process, and we investigate whether the violation of the formal conditions for perfect aggregation can still lead to an aggregate equation that is observationally equivalent to a cointegrated relationship. We derive a measure of the degree of noncointegration of the aggregate relationship and we explore its asymptotic properties. We propose a valid bootstrap approximation of the test. A Monte Carlo exercise evaluates size and power properties of the bootstrap test.     

Efficient estimation and inference in linear pseudo-panel data models

We consider pseudo-panel data models constructed from repeated cross sections in which the number of individuals per group is large relative to the number of groups and time periods. First, we show that, when time-invariant group fixed effects are neglected, the OLS estimator does not converge in probability to a constant but rather to a random variable. Second, we show that, while the fixed-effects (FE) estimator is consistent, the usual t statistic is not asymptotically normally distributed, and we propose a new robust t statistic whose asymptotic distribution is standard normal. Third, we propose efficient GMM estimators using the orthogonality conditions implied by grouping and we provide t tests that are valid even in the presence of time-invariant group effects. Our Monte Carlo results show that the proposed GMM estimator is more precise than the FE estimator and that our new t test has good size and is powerful.     

Quasi ML estimation of the panel AR(1) model with arbitrary initial conditions

In this paper we show that the Quasi ML estimation method yields consistent Random and Fixed Effects estimators for the autoregression parameter ρ$\rho$ in the panel AR(1) model with arbitrary initial conditions and possibly time-series heteroskedasticity even when the error components are drawn from heterogeneous distributions. We investigate both analytically and by means of Monte Carlo simulations the properties of the QML estimators for ρ$\rho$. The RE(Q)MLE for ρ$\rho$ is asymptotically at least as robust to individual heterogeneity and, when the data are i.i.d. and normal, at least as efficient as the FE(Q)MLE for ρ$\rho$. Furthermore, the QML estimators for ρ$\rho$ only suffer from a ‘weak moment conditions’ problem when ρ$\rho$ is close to one if the cross-sectional average of the variances of the errors is (almost) constant over time, e.g. under time-series homoskedasticity. However, in this case the QML estimators for ρ$\rho$ are still consistent when ρ$\rho$ is local to or equal to one although they converge to a non-normal possibly asymmetric distribution at a rate that is lower than N^1/2$N^{1/2}$ but at least N^1/4$N^{1/4}$. Finally, we study the finite sample properties of two types of estimators for the standard errors of the QML estimators for ρ$\rho$, and the bounds of QML based confidence intervals for ρ$\rho$.     

Estimation of a nonlinear panel data model with semiparametric individual effects

This paper investigates identification and estimation of a class of nonlinear panel data, single-index models. The model allows for unknown time-specific link functions, and semiparametric specification of the individual-specific effects. We develop an estimator for the parameters of interest, and propose a powerful new kernel-based modified backfitting algorithm to compute the estimator. We derive uniform rates of convergence results for the estimators of the link functions, and show the estimators of the finite-dimensional parameters are root-N$N$ consistent with a Gaussian limiting distribution. We study the small sample properties of the estimator via Monte Carlo techniques.     

The common and specific components of dynamic volatility

This paper develops a dynamic approximate factor model in which returns are time-series heteroskedastic. The heteroskedasticity has three components: a factor-related component, a common asset-specific component, and a purely asset-specific component. We develop a new multivariate GARCH model for the factor-related component. We develop a univariate stochastic volatility model linked to a cross-sectional series of individual GARCH models for the common asset-specific component and the purely asset-specific component. We apply the analysis to monthly US equity returns for the period January 1926 to December 2000. We find that all three components contribute to the heteroskedasticity of individual equity returns. Factor volatility and the common component in asset-specific volatility have long-term secular trends as well as short-term autocorrelation. Factor volatility has correlation with interest rates and the business cycle.