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.     

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.     

A new bounded log-linear regression model

In this paper we introduce a new regression model in which the response variable is bounded by two unknown parameters. A special case is a bounded alternative to the four parameter logistic model. The four parameter model which has unbounded responses is widely used, for instance, in bioassays, nutrition, genetics, calibration and agriculture. In reality, the responses are often bounded although the bounds may be unknown, and in that situation, our model reflects the data-generating mechanism better. Complications arise for the new model, however, because the likelihood function is unbounded, and the global maximizers are not consistent estimators of unknown parameters. Although the two sample extremes, the smallest and the largest observations, are consistent estimators for the two unknown boundaries, they have a slow convergence rate and are asymptotically biased. Improved estimators are developed by correcting for the asymptotic biases of the two sample extremes in the one sample case; but even these consistent estimators do not obtain the optimal convergence rate. To obtain efficient estimation, we suggest using the local maximizers of the likelihood function, i.e., the solution to the likelihood equations. We prove that, with probability approaching one as the sample size goes to infinity, there exists a solution to the likelihood equation that is consistent at the rate of the square root of the sample size and it is asymptotically normally distributed.     

Conditional empirical likelihood estimation and inference for quantile regression models

This paper considers two empirical likelihood-based estimation, inference, and specification testing methods for quantile regression models. First, we apply the method of conditional empirical likelihood (CEL) by Kitamura et al. [2004. Empirical likelihood-based inference in conditional moment restriction models. Econometrica 72, 1667–1714] and Zhang and Gijbels [2003. Sieve empirical likelihood and extensions of the generalized least squares. Scandinavian Journal of Statistics 30, 1–24] to quantile regression models. Second, to avoid practical problems of the CEL method induced by the discontinuity in parameters of CEL, we propose a smoothed counterpart of CEL, called smoothed conditional empirical likelihood (SCEL). We derive asymptotic properties of the CEL and SCEL estimators, parameter hypothesis tests, and model specification tests. Important features are (i) the CEL and SCEL estimators are asymptotically efficient and do not require preliminary weight estimation; (ii) by inverting the CEL and SCEL ratio parameter hypothesis tests, asymptotically valid confidence intervals can be obtained without estimating the asymptotic variances of the estimators; and (iii) in contrast to CEL, the SCEL method can be implemented by some standard Newton-type optimization. Simulation results demonstrate that the SCEL method in particular compares favorably with existing alternatives.     

A regularization approach to the many instruments problem

This paper focuses on the estimation of a finite dimensional parameter in a linear model where the number of instruments is very large or infinite. In order to improve the small sample properties of standard instrumental variable (IV) estimators, we propose three modified IV estimators based on three different ways of inverting the covariance matrix of the instruments. These inverses involve a regularization or smoothing parameter. It should be stressed that no restriction on the number of instruments is needed and that all the instruments are used in the estimation. We show that the three estimators are asymptotically normal and attain the semiparametric efficiency bound. Higher-order analysis of the MSE reveals that the bias of the modified estimators does not depend on the number of instruments. Finally, we suggest a data-driven method for selecting the regularization parameter. Interestingly, our regularization techniques lead to a consistent nonparametric estimation of the optimal instrument.     

Least absolute deviations estimation for the censored regression model

This paper proposes an alternative to maximum likelihood estimation of the parameters of the censored regression (or censored ‘Tobit’) model. The proposed estimator is a generalization of least absolute deviations estimation for the standard linear model, and, unlike estimation methods based on the assumption of normally distributed error terms, the estimator is consistent and asymptotically normal for a wide class of error distributions, and is also robust to heteroscedasticity. The paper gives the regularity conditions and proofs of these large-sample results, and proposes classes of consistent estimators of the asymptotic covariance matrix for both homoscedastic and heteroscedastic disturbances.     

Spatial heteroskedasticity and autocorrelation consistent estimation of covariance matrix

This paper considers spatial heteroskedasticity and autocorrelation consistent (spatial HAC) estimation of covariance matrices of parameter estimators. We generalize the spatial HAC estimator introduced by Kelejian and Prucha (2007) to apply to linear and nonlinear spatial models with moment conditions. We establish its consistency, rate of convergence and asymptotic truncated mean squared error (MSE). Based on the asymptotic truncated MSE criterion, we derive the optimal bandwidth parameter and suggest its data dependent estimation procedure using a parametric plug-in method. The finite sample performances of the spatial HAC estimator are evaluated via Monte Carlo simulation.     

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     

On kernel nonparametric regression designed for complex survey data

In this article, we consider nonparametric regression analysis between two variables when data are sampled through a complex survey. While nonparametric regression analysis has been widely used with data that may be assumed to be generated from independently and identically distributed (iid) random variables, the methods and asymptotic analyses established for iid data need to be extended in the framework of complex survey designs. Local polynomial regression estimators are studied, which include as particular cases design-based versions of the Nadaraya–Watson estimator and of the local linear regression estimator. In this paper, special emphasis is given to the local linear regression estimator. Our estimators incorporate both the sampling weights and the kernel weights. We derive the asymptotic mean squared error (MSE) of the kernel estimators using a combined inference framework, and as a corollary consistency of the estimators is deduced. Selection of a bandwidth is necessary for the resulting estimators; an optimal bandwidth can be determined, according to the MSE criterion in the combined mode of inference. Simulation experiments are conducted to illustrate the proposed methodology and an application with the Canadian survey of labour and income dynamics is presented.     

Semiparametric econometric estimators for a truncated regression model: a review with an extension

Econometric estimators for a truncated regression model are reviewed. For each estimator, the motivations, the key assumptions, the asymptotic distribution and estimates for the asymptotic variance matrix are presented; also a new estimator is suggested. We select five practical estimators among those, and compare them through a Monte Carlo study where the response variable is simulated but the covariates are drawn from a real data set. Some practical and computational issues are addressed as well.     

Instrumental variable estimation based on grouped data

The paper considers the estimation of the coefficients of a single equation in the presence of dummy intruments. We derive pseudo ML and GMM estimators based on moment restrictions induced either by the structural form or by the reduced form of the model. The performance of the estimators is evaluated for the non-Gaussian case. We allow for heteroscedasticity. The asymptotic distributions are based on parameter sequences where the number of instruments increases at the same rate as the sample size. Relaxing the usual Gaussian assumption is shown to affect the normal asymptotic distributions. As a result also recently suggested new specification tests for the validity of instruments depend on Gaussianity. Monte Carlo simulations confirm the accuracy of the asymptotic approach.     

Shrinkage estimation of the exponentiated Weibull regression model for time-to-event data

The exponentiated Weibull distribution is a convenient alternative to the generalized gamma distribution to model time-to-event data. It accommodates both monotone and nonmonotone hazard shapes, and flexible enough to describe data with wide ranging characteristics. It can also be used for regression analysis of time-to-event data. The maximum likelihood method is thus far the most widely used technique for inference, though there is a considerable body of research of improving the maximum likelihood estimators in terms of asymptotic efficiency. For example, there has recently been considerable attention on applying James–Stein shrinkage ideas to parameter estimation in regression models. We propose nonpenalty shrinkage estimation for the exponentiated Weibull regression model for time-to-event data. Comparative studies suggest that the shrinkage estimators outperform the maximum likelihood estimators in terms of statistical efficiency. Overall, the shrinkage method leads to more accurate statistical inference, a fundamental and desirable component of statistical theory.     

Partially varying coefficient instrumental variables models

In this article, we study a new class of semiparametric instrumental variables models, in which the structural function has a partially varying coefficient functional form. Under this specification, the model is linear in the endogenous/exogenous components with unknown constant or functional coefficients. As a result, the ill‐posed inverse problem in a general non‐parametric model with continuous endogenous variables can be avoided. We propose a three‐step estimation procedure for estimating both constant and functional coefficients and establish their asymptotic properties such as consistency and asymptotic normality. We develop consistent estimators for their error variances. We demonstrate that the constant coefficient estimators achieve the optimal ‐convergence rate, and the functional coefficient estimators are oracle. In addition, efficiency issue of the parameter estimation is discussed and a simple efficient estimator is proposed. The proposed procedure is illustrated via a Monte Carlo simulation and an application to returns to education.     

Natural estimation of variances in a general finite discrete spectrum linear regression model

The method of “natural” estimation of variances in a general (orthogonal or nonorthogonal) finite discrete spectrum linear regression model of time series is suggested. Using geometrical language of the theory of projectors a form and properties of the estimators are investigated. Obtained results show that in describing the first and second moment properties of the new estimators the central role plays a matrix known in linear algebra as the Schur complement. Illustrative examples with particular regressors demonstrate direct applications of the results.     

Shrinkage estimation of the proportion in randomized response

This article considers the asymptotic estimation theory for the proportion in randomized response survey usinguncertain prior information (UPI) about the true proportion parameter which is assumed to be available on the basis of some sort of realistic conjecture. Three estimators, namely, the unrestricted estimator, the shrinkage restricted estimator and an estimator based on a preliminary test, are proposed. Their asymptotic mean squared errors are derived and compared. The relative dominance picture of the estimators is presented.     

Variance estimation in the change analysis of a linear regression model

We study a “direct test” of Chu and White (1992) proposed for detecting changes in the trend of a linear regression model. The power of this test strongly depends on a suitable estimation of the variance of the error variables involved. We discuss various types of variance estimators and derive their asymptotic properties under the null-hypothesis of “no change” as well as under the alternative of “a change in linear trend”. A small simulation study illustrates the estimators' finite sample behaviour.     

Statistical inference on restricted partial linear regression models with partial distortion measurement errors

We consider the estimation and hypothesis testing problems for the partial linear regression models when some variables are distorted with errors by some unknown functions of commonly observable confounding variable. The proposed estimation procedure is designed to accommodate undistorted as well as distorted variables. To test a hypothesis on the parametric components, a restricted least squares estimator is proposed under the null hypothesis. Asymptotic properties for the estimators are established. A test statistic based on the difference between the residual sums of squares under the null and alternative hypotheses is proposed, and we also obtain the asymptotic properties of the test statistic. A wild bootstrap procedure is proposed to calculate critical values. Simulation studies are conducted to demonstrate the performance of the proposed procedure, and a real example is analyzed for an illustration.     

Approximate inference in multivariate nonlinear functional relationships

An estimation procedure based on estimating equations is presented for the parameters in a multivariate functional relationship model, where all observations are subject to error. The covariance matrix of the observational errors may be parametrized and is allowed to be different for different sets of observations. Estimators are defined for the unknown relation parameters and error parameters.
For linear models (i.e. where the model function is linear in the incidental parameters) the estimators are consistent and asymptotically normal. A consistent expression for the covariance matrix of the estimators is derived. The results are valid for general error distributions.
For nonlinear models the estimators are based on locally linear approximations to the model function. The afore mentioned properties of the estimators are now only approximately valid. The adequacy of the approximate inference, based on asymptotic theory for the linearized model, needs at least informal check. Some examples are given to illustrate the estimation procedure.     

Model reference adaptive systems applied to regression analyses

The adaptive estimation procedure of model reference adaptive systems is modified and applied to linear models. In general the principle can be used for almost any time series model. Because of the recursive nature of the resulting estimator, it is computationally appealing, especially when a time series is considered as a flow of data. In addition, the estimator turns out to have certain statistical optimality properties.
In the linear regression setting, Ridge estimators turn out to constitute a subclass of the adaptive estimators considered, whereas for unknown measurement variance, the resulting estimators are related to J ames -S tkin type estimators, and have better properties than the latter. The estimator is shown to be strongly consistent and to converge in law to a normal variate under the standard assumptions of linear models. Further it is shown to be admissible and minimax in restricted parameter spaces. The connection between K alman filters and the classical least-squares estimator is also pointed out.     

Large-sample estimation strategies for eigenvalues of a Wishart matrix

The problem of simultaneous asymptotic estimation of eigenvalues of covariance matrix of Wishart matrix is considered under a weighted quadratic loss function. James-Stein type of estimators are obtained which dominate the sample eigenvalues. The relative merits of the proposed estimators are compared to the sample eigenvalues using asymptotic quadratic distributional risk under loal alternatives. It is shown that the proposed estimators are asymptotically superior to the sample eigenvalues. Further, it is demonstrated that the James-Stein type estimator is dominated by its truncated part.