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$\rho$.     

Nonparametric dynamic panel data models: Kernel estimation and specification testing

Motivated by the first-differencing method for linear panel data models, we propose a class of iterative local polynomial estimators for nonparametric dynamic panel data models with or without exogenous regressors. The estimators utilize the additive structure of the first-differenced model—the fact that the two additive components have the same functional form, and the unknown function of interest is implicitly defined as a solution of a Fredholm integral equation of the second kind. We establish the uniform consistency and asymptotic normality of the estimators. We also propose a consistent test for the correct specification of linearity in typical dynamic panel data models based on the _L2$L_2$ distance of our nonparametric estimates and the parametric estimates under the linear restriction. We derive the asymptotic distributions of the test statistic under the null hypothesis and a sequence of Pitman local alternatives, and prove its consistency against global alternatives. Simulations suggest that the proposed estimators and tests perform well for finite samples. We apply our new method to study the relationships among economic growth, the initial economic condition and capital accumulation, and find a significant nonlinear relation between economic growth and the initial economic condition.     

Estimation of possibly misspecified semiparametric conditional moment restriction models with different conditioning variables

Newey and Powell [2003. Instrumental variable estimation of nonparametric models. Econometrica 71, 1565–1578] and Ai and Chen [2003. Efficient estimation of conditional moment restrictions models containing unknown functions. Econometrica 71, 1795–1843] propose sieve minimum distance (SMD) estimation of both finite dimensional parameter (θ)$(\theta )$ and infinite dimensional parameter (h) that are identified through a conditional moment restriction model, in which h could depend on endogenous variables. This paper modifies their SMD procedure to allow for different conditioning variables to be used in different equations, and derives the asymptotic properties when the model may be misspecified  . Under low-level sufficient conditions, we show that: (i) the modified SMD estimators of both θ$\theta$ and h   converge to some pseudo-true values in probability; (ii) the SMD estimators of smooth functionals, including the θ$\theta$ estimator and the average derivative estimator, are asymptotically normally distributed; and (iii) the estimators for the asymptotic covariances of the SMD estimators of smooth functionals are consistent and easy to compute. These results allow for asymptotically valid tests of various hypotheses on the smooth functionals regardless of whether the semiparametric model is correctly specified or not.     

Alternative approximations of the bias and MSE of the IV estimator under weak identification with an application to bias correction

We provide analytical formulae for the asymptotic bias (ABIAS) and mean-squared error (AMSE) of the IV estimator, and obtain approximations thereof based on an asymptotic scheme which essentially requires the expectation of the first stage F-statistic to converge to a finite (possibly small) positive limit as the number of instruments approaches infinity. Our analytical formulae can be viewed as generalizing the bias and MSE results of [Richardson and Wu 1971. A note on the comparison of ordinary and two-stage least squares estimators. Econometrica 39, 973–982] to the case with nonnormal errors and stochastic instruments. Our approximations are shown to compare favorably with approximations due to [Morimune 1983. Approximate distributions of k$k$-class estimators when the degree of overidentifiability is large compared with the sample size. Econometrica 51, 821–841] and [Donald and Newey 2001. Choosing the number of instruments. Econometrica 69, 1161–1191], particularly when the instruments are weak. We also construct consistent estimators for the ABIAS and AMSE, and we use these to further construct a number of bias corrected OLS and IV estimators, the properties of which are examined both analytically and via a series of Monte Carlo experiments.     

Asymptotics of the principal components estimator of large factor models with weakly influential factors

This paper introduces a drifting-parameter asymptotic framework to derive accurate approximations to the finite sample distribution of the principal components (PC) estimator in situations when the factors’ explanatory power does not strongly dominate the explanatory power of the cross-sectionally and temporally correlated idiosyncratic terms. Under our asymptotics, the PC estimator is inconsistent. We find explicit formulae for the amount of the inconsistency, and propose an estimator of the number of factors for which the PC estimator works reasonably well. For the special case when the idiosyncratic terms are cross-sectionally but not temporally correlated (or vice versa), we show that the coefficients in the OLS regressions of the PC estimates of factors (loadings) on the true factors (true loadings) are asymptotically normal, and find explicit formulae for the corresponding asymptotic covariance matrix. We explain how to estimate the parameters of the derived asymptotic distributions. Our Monte Carlo analysis suggests that our asymptotic formulae and estimators work well even for relatively small n$n$ and T$T$. We apply our theoretical results to test a hypothesis about the factor content of the US stock return data.     

Identification and nonparametric estimation of a transformed additively separable model

Let r(x,z)$r(x,z)$ be a function that, along with its derivatives, can be consistently estimated nonparametrically. This paper discusses the identification and consistent estimation of the unknown functions H$H$, M$M$, G$G$ and F$F$, where r(x,z)=H[M(x,z)]$r(x,z)=H\left[M(x,z)\right]$, M(x,z)=G(x)+F(z)$M(x,z)=G\left(x\right)+F\left(z\right)$, and H$H$ is strictly monotonic. An estimation algorithm is proposed for each of the model’s unknown components when r(x,z)$r(x,z)$ represents a conditional mean function. The resulting estimators use marginal integration to separate the components G$G$ and F$F$. Our estimators are shown to have a limiting Normal distribution with a faster rate of convergence than unrestricted nonparametric alternatives. Their small sample performance is studied in a Monte Carlo experiment. We apply our results to estimate generalized homothetic production functions for four industries in the Chinese economy.     

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.     

Asymptotic properties of the efficient estimators for cointegrating regression models with serially dependent errors

In this paper, we analytically investigate three efficient estimators for cointegrating regression models: Phillips and Hansen’s [Phillips, P.C.B., Hansen, B.E., 1990. Statistical inference in instrumental variables regression with I(1) processes. Review of Economic Studies 57, 99–125] fully modified OLS estimator, Park’s [Park, J.Y., 1992. Canonical cointegrating regressions. Econometrica 60, 119–143] canonical cointegrating regression estimator, and Saikkonen’s [Saikkonen, P., 1991. Asymptotically efficient estimation of cointegration regressions. Econometric Theory 7, 1–21] dynamic OLS estimator. We consider the case where the regression errors are moderately serially correlated and the AR coefficient in the regression errors approaches 1 at a rate slower than 1/T$1/T$, where T$T$ represents the sample size. We derive the limiting distributions of the efficient estimators under this system and find that they depend on the approaching rate of the AR coefficient. If the rate is slow enough, efficiency is established for the three estimators; however, if the approaching rate is relatively faster, the estimators will have the same limiting distribution as the OLS estimator. For the intermediate case, the second-order bias of the OLS estimator is partially eliminated by the efficient methods. This result explains why, in finite samples, the effect of the efficient methods diminishes as the serial correlation in the regression errors becomes stronger. We also propose to modify the existing efficient estimators in order to eliminate the second-order bias, which possibly remains in the efficient estimators. Using Monte Carlo simulations, we demonstrate that our modification is effective when the regression errors are moderately serially correlated and the simultaneous correlation is relatively strong.     

Inference on an extended Roy model,with an application to schooling decisions in France

This paper considers the identification and estimation of an extension of Roy’s model (1951) of sectoral choice, which includes a non-pecuniary component in the selection equation and allows for uncertainty on potential earnings. We focus on the identification of the non-pecuniary component, which is key to disentangling the relative importance of monetary incentives versus preferences in the context of sorting across sectors. By making the most of the structure of the selection equation, we show that this component is point identified from the knowledge of the covariate effects on earnings, as soon as one covariate is continuous. Notably, and in contrast to most results on the identification of Roy models, this implies that identification can be achieved without any exclusion restriction nor large support condition on the covariates. As a by-product, bounds are obtained on the distribution of the ex ante   monetary returns. We propose a three-stage semiparametric estimation procedure for this model, which yields root-n$n$ consistent and asymptotically normal estimators. Finally, we apply our results to the educational context, by providing new evidence from French data that non-pecuniary factors are a key determinant of higher education attendance decisions.     

Testing functional inequalities

This paper develops tests for inequality constraints of nonparametric regression functions. The test statistics involve a one-sided version of _Lp$L_p$-type functionals of kernel estimators (1≤p<∞)$(1\le p<\infty )$. Drawing on the approach of Poissonization, this paper establishes that the tests are asymptotically distribution free, admitting asymptotic normal approximation. In particular, the tests using the standard normal critical values have asymptotically correct size and are consistent against general fixed alternatives. Furthermore, we establish conditions under which the tests have nontrivial local power against Pitman local alternatives. Some results from Monte Carlo simulations are presented.     

Generalized spectral tests for the martingale difference hypothesis

This article proposes a test for the martingale difference hypothesis (MDH) using dependence measures related to the characteristic function. The MDH typically has been tested using the sample autocorrelations or in the spectral domain using the periodogram. Tests based on these statistics are inconsistent against uncorrelated non-martingales processes. Here, we generalize the spectral test of Durlauf (1991) for testing the MDH taking into account linear and nonlinear dependence. Our test considers dependence at all lags and is consistent against general pairwise nonparametric Pitman's local alternatives converging at the parametric rate n^-1/2,$n^{-1/2},$ with n$n$ the sample size. Furthermore, with our methodology there is no need to choose a lag order, to smooth the data or to formulate a parametric alternative. Our approach could be extended to specification testing of the conditional mean of possibly nonlinear models. The asymptotic null distribution of our test depends on the data generating process, so a bootstrap procedure is proposed and theoretically justified. Our bootstrap test is robust to higher order dependence, in particular to conditional heteroskedasticity. A Monte Carlo study examines the finite sample performance of our test and shows that it is more powerful than some competing tests. Finally, an application to the S&P 500 stock index and exchange rates highlights the merits of our approach.     

Dominating estimators for minimum-variance portfolios

In this paper, we derive two shrinkage estimators for minimum-variance portfolios that dominate the traditional estimator with respect to the out-of-sample variance of the portfolio return. The presented results hold for any number of assets d≥4$d\ge 4$ and number of observations n≥d+2$n\ge d+2$. The small-sample properties of the shrinkage estimators as well as their large-sample properties for fixed d$d$ but n→∞$n\to \infty$ and n,d→∞$n,d\to \infty$ but n/d→q≤∞$n/d\to q\le \infty$ are investigated. Furthermore, we present a small-sample test for the question of whether it is better to completely ignore time series information in favor of naive diversification.     

Endogenous selection or treatment model estimation

In a sample selection or treatment effects model, common unobservables may affect both the outcome and the probability of selection in unknown ways. This paper shows that the distribution function of potential outcomes, conditional on covariates, can be identified given an observed variable V$V$ that affects the treatment or selection probability in certain ways and is conditionally independent of the error terms in a model of potential outcomes. Selection model estimators based on this identification are provided, which take the form of simple weighted averages, GMM, or two stage least squares. These estimators permit endogenous and mismeasured regressors. Empirical applications are provided to estimation of a firm investment model and a schooling effects on wages model.     

Trending time-varying coefficient time series models with serially correlated errors

This paper studies a time-varying coefficient time series model with a time trend function and serially correlated errors to characterize the nonlinearity, nonstationarity, and trending phenomenon. A local linear approach is developed to estimate the time trend and coefficient functions. The asymptotic properties of the proposed estimators, coupled with their comparisons with other methods, are established under the α$\alpha$-mixing conditions and without specifying the error distribution. Further, the asymptotic behaviors of the estimators at the boundaries are examined. The practical problem of implementation is also addressed. In particular, a simple nonparametric version of a bootstrap test is adapted for testing misspecification and stationarity, together with a data-driven method for selecting the bandwidth and a consistent estimate of the standard errors. Finally, results of two Monte Carlo experiments are presented to examine the finite sample performances of the proposed procedures and an empirical example is discussed.