Alan T.K. Wan

Optimal Weight Choice for Frequentist Model Average Estimators

There has been increasing interest recently in model averaging within the frequentist paradigm. The main benefit of model averaging over model selection is that it incorporates rather than ignores the uncertainty inherent in the model selection process. One of the most important, yet challenging, aspects of model averaging is how to optimally combine estimates from different models. In this work, we suggest a procedure of weight choice for frequentist model average estimators that exhibits optimality properties with respect to the estimator’s mean squared error (MSE). As a basis for demonstrating our idea, we consider averaging over a sequence of linear regression models. Building on this base, we develop a model weighting mechanism that involves minimizing the trace of an unbiased estimator of the model average estimator’s MSE. We further obtain results that reflect the finite sample as well as asymptotic optimality of the proposed mechanism. A Monte Carlo study based on simulated and real data evaluates and compares the finite sample properties of this mechanism with those of existing methods. The extension of the proposed weight selection scheme to general likelihood models is also considered. This article has supplementary material online.

Focused Information Criteria, Model Selection, and Model Averaging in a Tobit Model With a Nonzero Threshold

Claeskens and Hjort (2003) have developed a focused information criterion (FIC) for model selection that selects different models based on different focused functions with those functions tailored to the parameters singled out for interest. Hjort and Claeskens (2003) also have presented model averaging as an alternative to model selection, and suggested a local misspecification framework for studying the limiting distributions and asymptotic risk properties of post-model selection and model average estimators in parametric models. Despite the burgeoning literature on Tobit models, little work has been done on model selection explicitly in the Tobit context. In this article we propose FICs for variable selection allowing for such measures as mean absolute deviation, mean squared error, and expected expected linear exponential errors in a type I Tobit model with an unknown threshold. We also develop a model average Tobit estimator using values of a smoothed version of the FIC as weights. We study the finite-sample performance of model selection and model average estimators resulting from various FICs via a Monte Carlo experiment, and demonstrate the possibility of using a model screening procedure before combining the models. Finally, we present an example from a well-known study on married womens working hours to illustrate the estimation methods discussed. This article has supplementary material online.

Estimating Equations Inference With Missing Data

There is a large and growing body of literature on estimating equation (EE) as an estimation approach. One basic property of EE that has been universally adopted in practice is that of unbiasedness, and there are deep conceptual reasons why unbiasedness is a desirable EE characteristic. This article deals with inference from EEs when data are missing at random. The investigation is motivated by the observation that direct imputation of missing data in EEs generally leads to EEs that are biased and, thus, violates a basic assumption of the EE approach. The main contribution of this article is that it goes beyond existing imputation methods and proposes a procedure whereby one mitigates the effects of missing data through a reformulation of EEs imputed through a kernel regression method. These (modified) EEs then constitute a basis for inference by the generalized method of moments (GMM) and empirical likelihood (EL). Asymptotic properties of the GMM and EL estimators of the unknown parameters are derived and analyzed. Unlike most of the literature, which deals with missingness in either covariate values or response data, our method allows for missingness in both sets of variables. Another important strength of our approach is that it allows auxiliary information to be handled successfully. We illustrate the method using a well-known wormy-fruits dataset and data from a study on Duchenne muscular dystrophy detection and compare our results with several existing methods via a simulation study.

Frequentist Model Averaging with missing observations

Model averaging or combining is often considered as an alternative to model selection. Frequentist Model Averaging (FMA) is considered extensively and strategies for the application of FMA methods in the presence of missing data based on two distinct approaches are presented. The first approach combines estimates from a set of appropriate models which are weighted by scores of a missing data adjusted criterion developed in the recent literature of model selection. The second approach averages over the estimates of a set of models with weights based on conventional model selection criteria but with the missing data replaced by imputed values prior to estimating the models. For this purpose three easy-to-use imputation methods that have been programmed in currently available statistical software are considered, and a simple recursive algorithm is further adapted to implement a generalized regression imputation in a way such that the missing values are predicted successively. The latter algorithm is found to be quite useful when one is confronted with two or more missing values simultaneously in a given row of observations. Focusing on a binary logistic regression model, the properties of the FMA estimators resulting from these strategies are explored by means of a Monte Carlo study. The results show that in many situations, averaging after imputation is preferred to averaging using weights that adjust for the missing data, and model average estimators often provide better estimates than those resulting from any single model. As an illustration, the proposed methods are applied to a dataset from a study of Duchenne muscular dystrophy detection.

Efficient Quantile Regression Analysis With Missing Observations

This article examines the problem of estimation in a quantile regression model when observations are missing at random under independent and nonidentically distributed errors. We consider three approaches of handling this problem based on nonparametric inverse probability weighting, estimating equations projection, and a combination of both. An important distinguishing feature of our methods is their ability to handle missing response and/or partially missing covariates, whereas existing techniques can handle only one or the other, but not both. We prove that our methods yield asymptotically equivalent estimators that achieve the desirable asymptotic properties of unbiasedness, normality, and -consistency. Because we do not assume that the errors are identically distributed, our theoretical results are valid under heteroscedasticity, a particularly strong feature of our methods. Under the special case of identical error distributions, all of our proposed estimators achieve the semiparametric efficiency bound. To facilitate the practical implementation of these methods, we develop an iterative method based on the majorize/minimize algorithm for computing the quantile regression estimates, and a bootstrap method for computing their variances. Our simulation findings suggest that all three methods have good finite sample properties. We further illustrate these methods by a real data example. Supplementary materials for this article are available online.

Improved Estimators of Hedonic Housing Price Models

In hedonic housing price modeling, real estate researchers and practitioners are often not completely ignorant about the parameters to be estimated. Experience and expertise usually provide them with tacit understanding of the likely values of the true parameters. Under this scenario, the subjective knowledge about the parameter value can be incorporated as non-sample information in the hedonic price model. This paper considers a class of Generalized Stein Variance Double k-class (GSVKK) estimators, which allows real estate practitioners to introduce potentially useful information about the parameter values into the estimation of hedonic pricing models. Data from the Hong Kong real estate market are used to investigate the estimators?performance empirically. Compared with the traditional Ordinary Lease Squares approach, the GSVKK estimators have smaller predictive mean squared errors and lead to more precise parameter estimates. Some results on the theoretical properties of the GSVKK estimators are also presented.

A note on almost unbiased generalized ridge regression estimator under asymmetric loss

Using the asymmetric LINEX loss function, we derive and numerically evaluate the exact risk function of the almost unbiased feasible generalized ridge regression estimator. Contrary to the properties of the (biased) feasible generalized ridge estimator, it is found that regardless of the loss asymmetry,the almost unbiased feasible generalized ridge estimator does not strictly dominate the traditional least squares estimator. Our numerical results show that over a wide range of parameter values, the almost unbiased feasible generalized ridge estimator is inferior to either the least squares or the feasible generalized ridge estimators.

Risk comparison of the inequality constrained least squares and other related estimators under balanced loss

Abstract This paper considers the risks of the inequality constrained least squares and pre-test estimators using the balanced loss function proposed by Zellner. It is found that some of the well-known results which hold under squared error loss do not necessarily hold under a balanced loss function.

A Varying-Coefficient Expectile Model for Estimating Value at Risk

This article develops a nonparametric varying-coefficient approach for modeling the expectile-based value at risk (EVaR). EVaR has an advantage over the conventional quantile-based VaR (QVaR) of being more sensitive to the magnitude of extreme losses. EVaR can also be used for calculating QVaR and expected shortfall (ES) by exploiting the one-to-one mapping from expectiles to quantiles, and the relationship between VaR and ES. Previous studies on conditional EVaR estimation only considered parametric autoregressive model set-ups, which account for the stochastic dynamics of asset returns but ignore other exogenous economic and investment related factors. Our approach overcomes this drawback and allows expectiles to be modeled directly using covariates that may be exogenous or lagged dependent in a flexible way. Risk factors associated with profits and losses can then be identified via the expectile regression at different levels of prudentiality. We develop a local linear smoothing technique for estimating the coefficient functions within an asymmetric least squares minimization set-up, and establish the consistency and asymptotic normality of the resultant estimator. To save computing time, we propose to use a one-step weighted local least squares procedure to compute the estimates. Our simulation results show that the computing advantage afforded by this one-step procedure over full iteration is not compromised by a deterioration in estimation accuracy. Real data examples are used to illustrate our method. Supplementary materials for this article are available online.

Minimum mean-squared error estimation in linear regression with an inequality constraint

Abstract This paper considers adaptive versions of the minimum mean-squared error estimators in models with an inequality constraint. We derive a sufficient condition under which the proposed class of estimators dominates the traditional inequality constrained least-squares estimator in terms of risk under quadratic loss. Numerical calculations of the risks show that over much of the parameter space, the proposed estimators are superior to the inequality constrained estimator, even if the sufficient condition is not satisfied, and some members of this class have risk advantage over the inequality constrained Stein-rule estimator proposed by Judge et al. (1984, J. Econometrics 25, 165–177) over a wide range of parameter values.