Semiparametric quantile regression estimation in dynamic models with partially varying coefficients

We study quantile regression estimation for dynamic models with partially varying coefficients so that the values of some coefficients may be functions of informative covariates. Estimation of both parametric and nonparametric functional coefficients are proposed. In particular, we propose a three stage semiparametric procedure. Both consistency and asymptotic normality of the proposed estimators are derived. We demonstrate that the parametric estimators are root-n$n$ consistent and the estimation of the functional coefficients is oracle. In addition, efficiency of parameter estimation is discussed and a simple efficient estimator is proposed. A simple and easily implemented test for the hypothesis of a varying-coefficient is proposed. A Monte Carlo experiment is conducted to evaluate the performance of the proposed estimators.     

Instrumental variable estimation of nonseparable models

There are many environments where knowledge of a structural relationship is required to answer questions of interest. Also, nonseparability of a structural disturbance is a key feature of many models. Here, we consider nonparametric identification and estimation of a model that is monotonic in a nonseparable scalar disturbance, which disturbance is independent of instruments. This model leads to conditional quantile restrictions. We give local identification conditions for the structural equations from those quantile restrictions. We find that a modified completeness condition is sufficient for local identification. We also consider estimation via a nonparametric minimum distance estimator. The estimator minimizes the sum of squares of predicted values from a nonparametric regression of the quantile residual on the instruments. We show consistency of this estimator.     

Distribution free estimation of heteroskedastic binary response models using Probit/Logit criterion functions

In this paper estimators for distribution free heteroskedastic binary response models are proposed. The estimation procedures are based on relationships between distribution free models with a conditional median restriction and parametric models (such as Probit/Logit) exhibiting (multiplicative) heteroskedasticity. The first proposed estimator is based on the observational equivalence between the two models, and is a semiparametric sieve estimator (see, e.g. Gallant and Nychka (1987), Ai and Chen (2003) and Chen et al. (2005)) for the regression coefficients, based on maximizing standard Logit/Probit criterion functions, such as NLLS and MLE. This procedure has the advantage that choice probabilities and regression coefficients are estimated simultaneously. The second proposed procedure is based on the equivalence between existing semiparametric estimators for the conditional median model (,  and ) and the standard parametric (Probit/Logit) NLLS estimator. This estimator has the advantage of being implementable with standard software packages such as Stata. Distribution theory is developed for both estimators and a Monte Carlo study indicates they both perform well in finite samples.     

Nonparametric estimation of time varying parameters under shape restrictions

In recent years, a lot of econometric literature has been devoted to estimating time varying coefficients in regression models. Here, a new method based on smoothers is proposed, which is able to introduce shape restrictions over the coefficients. The statistical properties of the estimator are obtained for very general situations, including locally stationary regressors. In particular, the procedure provides consistent results for time varying autoregressive models. The practical problem of implementation is also addressed. A data-driven method for selecting the control parameters is provided, together with an algorithm that reduces the computational cost. A simulation study and an application to real data supports the theoretical results.     

Stochastic cointegration: estimation and inference

This paper considers the estimation of a stochastically cointegrating regression within the stochastic cointegration modelling framework introduced in McCabe et al. (Stochastic cointegration: testing, 2001). A stochastic cointegrating regression allows some or all of the variables to be conventionally or heteroscedastically integrated. This generalizes Hansen's (J. Econom. 54 (1992) 139) heteroscedastic cointegrating regression model, where the dependent variable is heteroscedastically integrated, but all the regressor variables are restricted to being conventionally integrated. In contrast to conventional and heteroscedastic cointegrating regression, ordinary least-squares (OLS) estimation is shown to be inconsistent, in general, in a stochastically cointegrating regression. As a solution, a new instrumental variables (IVs) estimator is proposed and is shown to be consistent. Under a suitable exogeneity assumption, standard asymptotic inference on the stochastic cointegrating vector can be carried out based on the IV estimator. The finite sample properties of the test statistics, including their robustness to the exogeneity assumption, are examined by simulation.     

Quantile regression under random censoring

Censored regression models have received a great deal of attention in both the theoretical and applied econometric literature. Most of the existing estimation procedures for either cross-sectional or panel data models are designed only for models with fixed censoring. In this paper, a new procedure for adapting these estimators designed for fixed censoring to models with random censoring is proposed. This procedure is then applied to the CLAD and quantile estimators of Powell (J. Econom. 25 (1984) 303, 32 (1986a) 143) to obtain an estimator of the coefficients under a mild conditional quantile restriction on the error term that is applicable to samples exhibiting fixed or random censoring. The resulting estimator is shown to have desirable asymptotic properties, and performs well in a small-scale simulation study.     

THE ROLE OF CONDITIONAL HETEROSKEDASTICITY IN IDENTIFYING AND ESTIMATING LINEAR TRIANGULAR SYSTEMS,WITH APPLICATIONS TO ASSET PRICING MODELS THAT INCLUDE A MISMEASURED FACTOR

A new estimator is proposed for linear triangular systems, where identification results from the model errors following a bivariate and diagonal GARCH(1,1) process with potentially time‐varying error covariances. This estimator applies when traditional instruments are unavailable. I demonstrate its usefulness on asset pricing models like the capital asset pricing model and Fama–French three‐factor model. In the context of a standard two‐pass cross‐sectional regression approach, this estimator improves the pricing performance of both models. Set identification bounds and an associated estimator are also provided for cases where the conditions supporting point identification fail. Copyright © 2013 John Wiley & Sons, Ltd.     

Local polynomial estimation of nonparametric simultaneous equations models

We define a new procedure for consistent estimation of nonparametric simultaneous equations models under the conditional mean independence restriction of Newey et al. [1999. Nonparametric estimation of triangular simultaneous equation models. Econometrica 67, 565–603]. It is based upon local polynomial regression and marginal integration techniques. We establish the asymptotic distribution of our estimator under weak data dependence conditions. Simulation evidence suggests that our estimator may significantly outperform the estimators of Pinkse [2000. Nonparametric two-step regression estimation when regressors and errors are dependent. Canadian Journal of Statistics 28, 289–300] and Newey and Powell [2003. Instrumental variable estimation of nonparametric models. Econometrica 71, 1565–1578].     

Use of minimum risk approach in the estimation of regression models with missing observations

This article considers a linear regression model with some missing observations on the response variable and presents two estimators of regression coefficients employing the approach of minimum risk estimation. Small disturbance asymptotic properties of these estimators along with the traditional unbiased estimator are analyzed and conditions, that are easy to check in practice, for the superiority of one estimator over the other are derived. Received May 2001     

Functional coefficient regression models with time trend

We consider the problem of estimating a varying coefficient regression model when regressors include a time trend. We show that the commonly used local constant kernel estimation method leads to an inconsistent estimation result, while a local polynomial estimator yields a consistent estimation result. We establish the asymptotic normality result for the proposed estimator. We also provide asymptotic analysis of the data-driven (least squares cross validation) method of selecting the smoothing parameters. In addition, we consider a partially linear time trend model and establish the asymptotic distribution of our proposed estimator. Two test statistics are proposed to test the null hypotheses of a linear and of a partially linear time trend models. Simulations are reported to examine the finite sample performances of the proposed estimators and the test statistics.     

On the asymptotic optimality of the LIML estimator with possibly many instruments

We consider the estimation of the coefficients of a linear structural equation in a simultaneous equation system when there are many instrumental variables. We derive some asymptotic properties of the limited information maximum likelihood (LIML) estimator when the number of instruments is large; some of these results are new as well as old, and we relate them to results in some recent studies. We have found that the variance of the limiting distribution of the LIML estimator and its modifications often attain the asymptotic lower bound when the number of instruments is large and the disturbance terms are not necessarily normally distributed, that is, for the micro-econometric models of some cases recently called many instruments and many weak instruments.     

On finite sample properties of alternative estimators of coefficients in a structural equation with many instruments

We compare four different estimation methods for the coefficients of a linear structural equation with instrumental variables. As the classical methods we consider the limited information maximum likelihood (LIML) estimator and the two-stage least squares (TSLS) estimator, and as the semi-parametric estimation methods we consider the maximum empirical likelihood (MEL) estimator and the generalized method of moments (GMM) (or the estimating equation) estimator. Tables and figures of the distribution functions of four estimators are given for enough values of the parameters to cover most linear models of interest and we include some heteroscedastic cases and nonlinear cases. We have found that the LIML estimator has good performance in terms of the bounded loss functions and probabilities when the number of instruments is large, that is, the micro-econometric models with “many instruments” in the terminology of recent econometric literature.     

An efficient GMM estimator of spatial autoregressive models

In this paper, we consider GMM estimation of the regression and MRSAR models with SAR disturbances. We derive the best GMM estimator within the class of GMM estimators based on linear and quadratic moment conditions. The best GMM estimator has the merit of computational simplicity and asymptotic efficiency. It is asymptotically as efficient as the ML estimator under normality and asymptotically more efficient than the Gaussian QML estimator otherwise. Monte Carlo studies show that, with moderate-sized samples, the best GMM estimator has its biggest advantage when the disturbances are asymmetrically distributed. When the diagonal elements of the spatial weights matrix have enough variation, incorporating kurtosis of the disturbances in the moment functions will also be helpful.     

A simple estimator for nonlinear error in variable models

We propose a simple estimator for nonlinear method of moment models with measurement error of the classical type when no additional data, such as validation data or double measurements, are available. We assume that the marginal distributions of the measurement errors are Laplace (double exponential) with zero means and unknown variances and the measurement errors are independent of the latent variables and are independent of each other. Under these assumptions, we derive simple revised moment conditions in terms of the observed variables. They are used to make inference about the model parameters and the variance of the measurement error. The results of this paper show that the distributional assumption on the measurement errors can be used to point identify the parameters of interest. Our estimator is a parametric method of moments estimator that uses the revised moment conditions and hence is simple to compute. Our estimation method is particularly useful in situations where no additional data are available, which is the case in many economic data sets. Simulation study demonstrates good finite sample properties of our proposed estimator. We also examine the performance of the estimator in the case where the error distribution is misspecified.     

Assessing Tail Risk Using Expectile Regressions with Partially Varying Coefficients

To characterize heteroskedasticity, nonlinearity, and asymmetry in tail risk, this study investigates a class of conditional (dynamic) expectile models with partially varying coefficients in which some coefficients are allowed to be constants, but others are allowed to be unknown functions of random variables. A three-stage estimation procedure is proposed to estimate both the parametric constant coefficients and nonparametric functional coefficients. Their asymptotic properties are investigated under a time series context, together with a new simple and easily implemented test for testing the goodness of fit of models and a bandwidth selector based on newly defined cross-validatory estimation for the expected forecasting expectile errors. The proposed methodology is data-analytic and of sufficient flexibility to analyze complex and multivariate nonlinear structures without suffering from the curse of dimensionality. Finally, the proposed model is illustrated by simulated data, and applied to analyzing the daily data of the S&P500 return series.     

A characterization of admissible linear estimators of fixed and random effects in linear models

In the paper the problem of simultaneous linear estimation of fixed and random effects in the mixed linear model is considered. A necessary and sufficient conditions for a linear estimator of a linear function of fixed and random effects in balanced nested and crossed classification models to be admissible are given.     

Robustness checks and robustness tests in applied economics

A common exercise in empirical studies is a “robustness check”, where the researcher examines how certain “core” regression coefficient estimates behave when the regression specification is modified by adding or removing regressors. If the coefficients are plausible and robust, this is commonly interpreted as evidence of structural validity. Here, we study when and how one can infer structural validity from coefficient robustness and plausibility. As we show, there are numerous pitfalls, as commonly implemented robustness checks give neither necessary nor sufficient evidence for structural validity. Indeed, if not conducted properly, robustness checks can be completely uninformative or entirely misleading. We discuss how critical and non-critical core variables can be properly specified and how non-core variables for the comparison regression can be chosen to ensure that robustness checks are indeed structurally informative. We provide a straightforward new Hausman (1978) type test of robustness for the critical core coefficients, additional diagnostics that can help explain why robustness test rejection occurs, and a new estimator, the Feasible Optimally combined GLS (FOGLeSs) estimator, that makes relatively efficient use of the robustness check regressions. A new procedure for Matlab, testrob, embodies these methods.     

The generalized ridge estimator and improved adjustments for regression parameters

Summary The generalized ridge estimator, which considers generalizations of mean square error, is presented, and a mathematical rule of determining the optimalk-value is discussed. The generalized ridge estimator is examined in comparison with the least squares, the pseudoinverse, theJames-Stein-type shrinkage, and the principal component estimators, especially focusing their attention on improved adjustments for regression coefficients. An alternative estimation approach that better integrates a priori information is noted. Finally, combining the generalized ridge and robust regression methods is suggested.     

Efficient estimation of limited dependent variable models with endogenous explanatory variables

This paper discusses asymptotically efficient estimation of the parameters of limited dependent variable models with endogenous explanatory variables. General results on asymptotic efficiency of two-stage and Amemiya GLS estimators are derived and used to obtain a simple, asymptotically efficient estimator of the structural coefficients. This estimator can be calculated by applying GLS to estimates of the reduced form coefficients that are obtained by using reduced form residuals as additional explanatory variables. It is also shown that it is possible to obtain asymptotically efficient estimators of the other coefficients by a modified minimum chi-square method.     

Maximum likelihood and restricted maximum likelihood estimation for a class of Gaussian Markov random fields

This work describes a Gaussian Markov random field model that includes several previously proposed models, and studies properties of its maximum likelihood (ML) and restricted maximum likelihood (REML) estimators in a special case. Specifically, for models where a particular relation holds between the regression and precision matrices of the model, we provide sufficient conditions for existence and uniqueness of ML and REML estimators of the covariance parameters, and provide a straightforward way to compute them. It is found that the ML estimator always exists while the REML estimator may not exist with positive probability. A numerical comparison suggests that for this model ML estimators of covariance parameters have, overall, better frequentist properties than REML estimators.