Bias From Censored Regressors

We study the bias that arises from using censored regressors in estimation of linear models. We present results on bias in ordinary least aquares (OLS) regression estimators with exogenous censoring and in instrumental variable (IV) estimators when the censored regressor is endogenous. Bound censoring such as top-coding results in expansion bias, or effects that are too large. Independent censoring results in bias that varies with the estimation method—attenuation bias in OLS estimators and expansion bias in IV estimators. Severe biases can result when there are several regressors and when a 0–1 variable is used in place of a continuous regressor.     

Asymptotic Theory for the QMLE in GARCH-X Models With Stationary and Nonstationary Covariates

This article investigates the asymptotic properties of the Gaussian quasi-maximum-likelihood estimators (QMLE’s) of the GARCH model augmented by including an additional explanatory variable—the so-called GARCH-X model. The additional covariate is allowed to exhibit any degree of persistence as captured by its long-memory parameter d_x; in particular, we allow for both stationary and nonstationary covariates. We show that the QMLE’s of the parameters entering the volatility equation are consistent and mixed-normally distributed in large samples. The convergence rates and limiting distributions of the QMLE’s depend on whether the regressor is stationary or not. However, standard inferential tools for the parameters are robust to the level of persistence of the regressor with t-statistics following standard Normal distributions in large sample irrespective of whether the regressor is stationary or not. Supplementary materials for this article are available online.     

Estimation of semi-varying coefficient models with nonstationary regressors

We study a semivarying coefficient model where the regressors are generated by the multivariate unit root I(1) processes. The influence of the explanatory vectors on the response variable satisfies the semiparametric partially linear structure with the nonlinear component being functional coefficients. A semiparametric estimation methodology with the first-stage local polynomial smoothing is applied to estimate both the constant coefficients in the linear component and the functional coefficients in the nonlinear component. The asymptotic distribution theory for the proposed semiparametric estimators is established under some mild conditions, from which both the parametric and nonparametric estimators are shown to enjoy the well-known super-consistency property. Furthermore, a simulation study is conducted to investigate the finite sample performance of the developed methodology and results.     

Inferences for a Partially Varying Coefficient Model With Endogenous Regressors

In this article, we propose a new class of semiparametric instrumental variable models with partially varying coefficients, in which the structural function has a partially linear form and the impact of endogenous structural variables can vary over different levels of some exogenous variables. We propose a three-step estimation procedure to estimate both functional and constant coefficients. The consistency and asymptotic normality of these proposed estimators are established. Moreover, a generalized F-test is developed to test whether the functional coefficients are of particular parametric forms with some underlying economic intuitions, and furthermore, the limiting distribution of the proposed generalized F-test statistic under the null hypothesis is established. Finally, we illustrate the finite sample performance of our approach with simulations and two real data examples in economics.     

Confidence Sets Based on Thresholding Estimators in High-Dimensional Gaussian Regression Models

We study confidence intervals based on hard-thresholding, soft-thresholding, and adaptive soft-thresholding in a linear regression model where the number of regressors k may depend on and diverge with sample size n. In addition to the case of known error variance, we define and study versions of the estimators when the error variance is unknown. In the known-variance case, we provide an exact analysis of the coverage properties of such intervals in finite samples. We show that these intervals are always larger than the standard interval based on the least-squares estimator. Asymptotically, the intervals based on the thresholding estimators are larger even by an order of magnitude when the estimators are tuned to perform consistent variable selection. For the unknown-variance case, we provide nontrivial lower bounds and a small numerical study for the coverage probabilities in finite samples. We also conduct an asymptotic analysis where the results from the known-variance case can be shown to carry over asymptotically if the number of degrees of freedom n ? k tends to infinity fast enough in relation to the thresholding parameter.     

Testing parameter significance in instrumental variables probit estimators: some simulation

Binary choice models that contain endogenous regressors can now be estimated routinely using modern software. Each of the two packages, Stata 11 [Stata Statistical Software: Release 11, StataCorp LP, College Station, TX, 2009] and Limdep 9 [Econometric Software Inc., Plainview, NY, 2008], contains two estimators that can be used to estimate such a model. This paper compares the performance of maximum likelihood, Newey's Amemiya's generalized least-squares (AGLS) estimator, an instrumental variables plug-in estimator and others in samples of sizes 200 and 1000 using simulation. Specifically, this paper focuses on tests of parameter significance under various degrees of instrument strength and severity of endogeneity. Although the maximum-likelihood estimator performs well in large samples, there is some evidence that the more computationally robust AGLS estimator may perform better in smaller samples when instruments are weak. It also appears that instruments in endogenous probit estimation need to be even stronger than when used in linear instrumental variables (IV) estimation.     

Stepwise Regression in Mixed Quantitative Linear Models with Autocorrelated Errors

ABSTRACT

In the stepwise procedure of selection of a fixed or a random explanatory variable in a mixed quantitative linear model with errors following a Gaussian stationary autocorrelated process, we have studied the efficiency of five estimators relative to Generalized Least Squares (GLS): Ordinary Least Squares (OLS), Maximum Likelihood (ML), Restricted Maximum Likelihood (REML), First Differences (FD), and First-Difference Ratios (FDR). We have also studied the validity and power of seven derived testing procedures, to assess the significance of the slope of the candidate explanatory variable x ₂ to enter the model in which there is already one regressor x ₁. In addition to five testing procedures of the literature, we considered the FDR t-test with n ? 3 df and the modified t-test with n? ? 3 df for partial correlations, where n? is Dutilleul's effective sample size. Efficiency, validity, and power were analyzed by Monte Carlo simulations, as functions of the nature, fixed vs. random (purely random or autocorrelated), of x ₁ and x ₂, the sample size and the autocorrelation of random terms in the regression model. We report extensive results for the autocorrelation structure of first-order autoregressive [AR(1)] type, and discuss results we obtained for other autocorrelation structures, such as spherical semivariogram, first-order moving average [MA(1)] and ARMA(1,1), but we could not present because of space constraints. Overall, we found that:

1. the efficiency of slope estimators and the validity of testing procedures depend primarily on the nature of x ₂, but not on that of x ₁;

2. FDR is the most inefficient slope estimator, regardless of the nature of x ₁ and x ₂;

3. REML is the most efficient of the slope estimators compared relative to GLS, provided the specified autocorrelation structure is correct and the sample size is large enough to ensure the convergence of its optimization algorithm;

4. the FDR t-test, the modified t-test and the REML t-test are the most valid of the testing procedures compared, despite the inefficiency of the FDR and OLS slope estimators for the former two;

5. the FDR t-test, however, suffers from a lack of power that varies with the nature of x ₁ and x ₂; and

6. the modified t-test for partial correlations, which does not require the specification of an autocorrelation structure, can be recommended when x ₁ is fixed or random and x ₂ is random, whether purely random or autocorrelated. Our results are illustrated by the environmental data that motivated our work.

     

Unconditional Quantile Treatment Effects Under Endogeneity

This article develops estimators for unconditional quantile treatment effects when the treatment selection is endogenous. We use an instrumental variable (IV) to solve for the endogeneity of the binary treatment variable. Identification is based on a monotonicity assumption in the treatment choice equation and is achieved without any functional form restriction. We propose a weighting estimator that is extremely simple to implement. This estimator is root n consistent, asymptotically normally distributed, and its variance attains the semiparametric efficiency bound. We also show that including covariates in the estimation is not only necessary for consistency when the IV is itself confounded but also for efficiency when the instrument is valid unconditionally. An application of the suggested methods to the effects of fertility on the family income distribution illustrates their usefulness. Supplementary materials for this article are available online.     

Kernel Estimators for Distribution Functions on Dependent Random Fields

Consider observations (representing lifelengths) taken on a random field indexed by lattice points. Estimating the distribution function F(x) = P(X _i  ≤ x) is an important problem in survival analysis. We propose to estimate F(x) by kernel estimators, which take into account the smoothness of the distribution function. Under some general mixing conditions, our estimators are shown to be asymptotically unbiased and consistent. In addition, the proposed estimator is shown to be strongly consistent and sharp rates of convergence are obtained.     

PMSE dominance of the positive-part shrinkage estimator in a regression model with proxy variables

Consider a linear regression model with some relevant regressors are unobservable. In such a situation, we estimate the model by using the proxy variables as regressors or by simply omitting the relevant regressors. In this paper, we derive the explicit formula of predictive mean squared error (PMSE) of a general family of shrinkage estimators of regression coefficients. It is shown analytically that the positive-part shrinkage estimator dominates the ordinary shrinkage estimator even when proxy variables are used in place of the unobserved variables. Also, as an example, our result is applied to the double k-class estimator proposed by Ullah and Ullah (Double k-class estimators of coefficients in linear regression. Econometrica. 1978;46:705–722). Our numerical results show that the positive-part double k-class estimator with proxy variables has preferable PMSE performance.     

Optimality of quasi-score in the multivariate mean–variance model with an application to the zero-inflated Poisson model with measurement errors

In a multivariate mean–variance model, the class of linear score (LS) estimators based on an unbiased linear estimating function is introduced. A special member of this class is the (extended) quasi-score (QS) estimator. It is ‘extended’ in the sense that it comprises the parameters describing the distribution of the regressor variables. It is shown that QS is (asymptotically) most efficient within the class of LS estimators. An application is the multivariate measurement error model, where the parameters describing the regressor distribution are nuisance parameters. A special case is the zero-inflated Poisson model with measurement errors, which can be treated within this framework.     

On additive and block decompositions of WLSEs under a multiple partitioned regression model

While considering the mechanism of weighted least-squares estimators (WLSEs) of regression coefficients in a partitioned linear model, Tian and Takane [On sum decompositions of weighted least-squares estimators under the partitioned linear model, Comm. Statist. Theory Methods 37 (2008), pp. 55–69] gave some identifying conditions for the WLSEs to be the sum of WLSEs under its two small models based on orthogonality of regressors with respect to the given weight matrix. The purpose of this paper is to show how to establish additive and block decompositions of WLSEs under a multiple partitioned linear model and its k small models based on orthogonality of regressors with respect to a given weight matrix.     

Aalen’s Additive Risk Model for Left-Truncated and Right-Censored Data

We analyze left-truncated and right-censored (LTRC) data using Aalen’s linear models. The integrated square error (ISE) is used to select an optimal bandwidth of the weighted least-squared estimator. We also consider a semiparametric approach for the case when the distribution of the left-truncated variable is parameterized. A simulation study is conducted to investigate the performance of the proposed estimators. The approaches are illustrated with the data of Stanford heart transplant.     

A Note on Nonlinear Cointegration,Misspecification, and Bimodality

We derive the asymptotic distribution of the ordinary least squares estimator in a regression with cointegrated variables under misspecification and/or nonlinearity in the regressors. We show that, under some circumstances, the order of convergence of the estimator changes and the asymptotic distribution is non-standard. The t-statistic might also diverge. A simple case arises when the intercept is erroneously omitted from the estimated model or in nonlinear-in-variables models with endogenous regressors. In the latter case, a solution is to use an instrumental variable estimator. The core results in this paper also generalise to more complicated nonlinear models involving integrated time series.     

SHRINKAGE, PRETEST AND ABSOLUTE PENALTY ESTIMATORS IN PARTIALLY LINEAR MODELS

We consider a partially linear model in which the vector of coefficients β in the linear part can be partitioned as ( β ₁, β ₂) , where β ₁ is the coefficient vector for main effects (e.g. treatment effect, genetic effects) and β ₂ is a vector for ‘nuisance’ effects (e.g. age, laboratory). In this situation, inference about β ₁ may benefit from moving the least squares estimate for the full model in the direction of the least squares estimate without the nuisance variables (Steinian shrinkage), or from dropping the nuisance variables if there is evidence that they do not provide useful information (pretesting). We investigate the asymptotic properties of Stein‐type and pretest semiparametric estimators under quadratic loss and show that, under general conditions, a Stein‐type semiparametric estimator improves on the full model conventional semiparametric least squares estimator. The relative performance of the estimators is examined using asymptotic analysis of quadratic risk functions and it is found that the Stein‐type estimator outperforms the full model estimator uniformly. By contrast, the pretest estimator dominates the least squares estimator only in a small part of the parameter space, which is consistent with the theory. We also consider an absolute penalty‐type estimator for partially linear models and give a Monte Carlo simulation comparison of shrinkage, pretest and the absolute penalty‐type estimators. The comparison shows that the shrinkage method performs better than the absolute penalty‐type estimation method when the dimension of the β ₂ parameter space is large.     

Efficient Estimation of the Seemingly Unrelated Regression Cointegration Model and Testing for Purchasing Power Parity

Abstract

This paper studies the efficient estimation of seemingly unrelated linear models with integrated regressors and stationary errors. We consider two cases. The first one has no common regressor among the equations. In this case, we show that by adding leads and lags of the first differences of the regressors and estimating this augmented dynamic regression model by generalized least squares using the long-run covariance matrix, we obtain an efficient estimator of the cointegrating vector that has a limiting mixed normal distribution. In the second case we consider, there is a common regressor to all equations, and we discuss efficient minimum distance estimation in this context. Simulation results suggests that our new estimator compares favorably with others already proposed in the literature. We apply these new estimators to the testing of the proportionality and symmetry conditions implied by purchasing power parity (PPP) among the G-7 countries. The tests based on the efficient estimates easily reject the joint hypotheses of proportionality and symmetry for all countries with either the United States or Germany as numeraire. Based on individual tests, our results suggest that Canada and Germany are the most likely countries for which the proportionality condition holds, and that Italy and Japan for the symmetry condition relative to the United States.     

On the estimation of extropy

Recently, Lad, Sanfilippo, and Agro [(2015), ‘Extropy: Complementary Dual of Entropy’, Statistical Science, 30, 40–58.] showed the measure of entropy has a complementary dual, which is termed extropy. The present article introduces some estimators of the extropy of a continuous random variable. Properties of the proposed estimators are stated, and comparisons are made with Qiu and Jia’s estimators [(2018a), ‘Extropy Estimators with Applications in Testing uniformity’, Journal of Nonparametric Statistics, 30, 182–196]. The results indicate that the proposed estimators have a smaller mean squared error than competing estimators. A real example is presented and analysed.     

Small-Sample Quantile Estimators in a Large Nonparametric Model

The large nonparametric model in this note is a statistical model with the family ? of all continuous and strictly increasing distribution functions. In the abundant literature of the subject, there are many proposals for nonparametric estimators that are applicable in the model. Typically the kth order statistic X _k:n is taken as a simplest estimator, with k = [nq], or k = [(n + 1)q], or k = [nq] + 1, etc. Often a linear combination of two consecutive order statistics is considered. In more sophisticated constructions, different L-statistics (e.g., Harrel–Davis, Kaigh–Lachenbruch, Bernstein, kernel estimators) are proposed. Asymptotically the estimators do not differ substantially, but if the sample size n is fixed, which is the case of our concern, differences may be serious. A unified treatment of quantile estimators in the large, nonparametric statistical model is developed.