Simple Estimators for Invertible Index Models

This article considers estimation of the unknown linear index coefficients of a model in which a number of nonparametrically identified reduced form parameters are assumed to be smooth and invertible function of one or more linear indices. The results extend the previous literature by allowing the number of reduced form parameters to exceed the number of indices (i.e., the indices are “overdetermined” by the reduced form parameters. The estimator of the unknown index coefficients (up to scale) is the eigenvector of a matrix (defined in terms of a first-step nonparametric estimator of the reduced form parameters) corresponding to its smallest (in magnitude) eigenvalue. Under suitable conditions, the proposed estimator is shown to be root-n-consistent and asymptotically normal, and under additional restrictions an efficient choice of a “weight matrix” is derived in the overdetermined case.     

A Semiparametric Estimation Approach for Linear Mixed Models

Maximum likelihood approach is the most frequently employed approach for the inference of linear mixed models. However, it relies on the normal distributional assumption of the random effects and the within-subject errors, and it is lack of robustness against outliers. This article proposes a semiparametric estimation approach for linear mixed models. This approach is based on the first two marginal moments of the response variable, and does not require any parametric distributional assumptions of random effects or error terms. The consistency and asymptotically normality of the estimator are derived under fairly general conditions. In addition, we show that the proposed estimator has a bounded influence function and a redescending property so it is robust to outliers. The methodology is illustrated through an application to the famed Framingham cholesterol data. The finite sample behavior and the robustness properties of the proposed estimator are evaluated through extensive simulation studies.     

Preliminary Data Errors and Their Impact on the Forecast Error of Simultaneous-Equations Models

Econometric models, especially when designed for forecasting purposes, tend to use updated economic series, the last figures(s) of which embed first-published or preliminary data errors. This article identifies and assesses the contribution of errors in preliminary data to the forecast error and to the forecast error variance of linear dynamic simultaneous equation models. The effect of preliminary data errors is shown to be pervasive, although not necessarily weighty. The suggested decomposition of the forecast error is applied to a small macroeconometric model of the Italian economy.     

Conditional Likelihood Estimators for Hidden Markov Models and Stochastic Volatility Models

ABSTRACT.  This paper develops a new contrast process for parametric inference of general hidden Markov models, when the hidden chain has a non-compact state space. This contrast is based on the conditional likelihood approach, often used for ARCH-type models. We prove the strong consistency of the conditional likelihood estimators under appropriate conditions. The method is applied to the Kalman filter (for which this contrast and the exact likelihood lead to asymptotically equivalent estimators) and to the discretely observed stochastic volatility models.     

Bias-Corrected Moment-Based Estimators for Parametric Models Under Endogenous Stratified Sampling

This paper provides an integrated approach for estimating parametric models from endogenous stratified samples. We discuss several alternative ways of removing the bias of the moment indicators usually employed under random sampling for estimating the parameters of the structural model and the proportion of the strata in the population. Those alternatives give rise to a number of moment-based estimators that are appropriate for both cases where the marginal strata probabilities are known and unknown. The derivation of our estimators is very simple and intuitive and incorporates as particular cases most of the likelihood-based estimators previously suggested by other authors.     

A Prediction Interval for M Estimators*

There has been much work in the area of estimating the center of a symmetric population. If one allows for the possibility that the population may be heavy-tailed then robust procedures, and in particular M estimators, have proven quite popular. In this paper we consider the following problem: given a random sample, produce an interval such that the M estimator derived from a future random sample (from the same population) will lie in that interval with some preassigned probability. Clearly such an interval is of use, especially in quality control where prediction is vital. In this paper such an interval is proposed based on asymptotic theory. A simulation study was run for a variety of sample sizes (the sizes of the observed and future samples need not be equal) and distributions. The particular M estimator of choice is that based on the biweight ψ function. The proposed interval performs reasonably well relative to the best that can be achieved asymptotically.     

Goodness-of-Fit Methods for Probabilistic Index Models

A class of semiparametric regression models, called probabilistic index models, has been recently proposed. Because these models are semiparametric, inference is only valid when the proposed model is consistent with the underlying data-generating model. However, no formal goodness-of-fit methods for these probabilistic index models exist yet. We propose a test and a graphical tool for assessing the model adequacy. Simulation results indicate that both methods succeed in detecting lack-of-fit. The methods are also illustrated on a case study.     

Two-Stage Estimation in Copula Models Used in Family Studies

In this paper register based family studies provide the motivation for studying a two-stage estimation procedure in copula models for multivariate failure time data. The asymptotic properties of the estimators in both parametric and semi-parametric models are derived, generalising the approach by Shih and Louis (Biometrics vol. 51, pp. 1384–1399, 1995b) and Glidden (Lifetime Data Analysis vol. 6, pp. 141–156, 2000). Because register based family studies often involve very large cohorts a method for analysing a sampled cohort is also derived together with the asymptotic properties of the estimators. The proposed methods are studied in simulations and the estimators are found to be highly efficient. Finally, the methods are applied to a study of mortality in twins.     

Choice between Semi-parametric Estimators of Markov and Non-Markov Multi-state Models from Coarsened Observations

Abstract.  We consider models based on multivariate counting processes, including multi-state models. These models are specified semi-parametrically by a set of functions and real parameters. We consider inference for these models based on coarsened observations, focusing on families of smooth estimators such as produced by penalized likelihood. An important issue is the choice of model structure, for instance, the choice between a Markov and some non-Markov models. We define in a general context the expected Kullback–Leibler criterion and we show that the likelihood-based cross-validation (LCV) is a nearly unbiased estimator of it. We give a general form of an approximate of the leave-one-out LCV. The approach is studied by simulations, and it is illustrated by estimating a Markov and two semi-Markov illness–death models with application on dementia using data of a large cohort study.     

Bias Reduction of Likelihood Estimators in Semiparametric Frailty Models

Abstract. Frailty models with a non‐parametric baseline hazard are widely used for the analysis of survival data. However, their maximum likelihood estimators can be substantially biased in finite samples, because the number of nuisance parameters associated with the baseline hazard increases with the sample size. The penalized partial likelihood based on a first‐order Laplace approximation still has non‐negligible bias. However, the second‐order Laplace approximation to a modified marginal likelihood for a bias reduction is infeasible because of the presence of too many complicated terms. In this article, we find adequate modifications of these likelihood‐based methods by using the hierarchical likelihood.     

Asymptotic Properties of QML Estimators for VARMA Models with Time‐dependent Coefficients

This paper is about vector autoregressive‐moving average models with time‐dependent coefficients to represent non‐stationary time series. Contrary to other papers in the univariate case, the coefficients depend on time but not on the series' length n. Under appropriate assumptions, it is shown that a Gaussian quasi‐maximum likelihood estimator is almost surely consistent and asymptotically normal. The theoretical results are illustrated by means of two examples of bivariate processes. It is shown that the assumptions underlying the theoretical results apply. In the second example, the innovations are marginally heteroscedastic with a correlation ranging from ?0.8 to 0.8. In the two examples, the asymptotic information matrix is obtained in the Gaussian case. Finally, the finite‐sample behaviour is checked via a Monte Carlo simulation study for n from 25 to 400. The results confirm the validity of the asymptotic properties even for short series and the asymptotic information matrix deduced from the theory.     

Estimating Moments in Linear Mixed Models

In this article, we investigate estimating moments, up to fourth order, in linear mixed models. For this estimation, we only assume the existence of moments. The obtained estimators of the model parameters and the third and fourth moments of the errors and random effects are proved to be consistent or asymptotically normal. The estimation provides a base for further statistical inference such as confidence region construction and hypothesis testing for the parameters of interest. Moreover, the method is readily extended to estimate higher moments. A simulation is carried out to examine the performance of this estimating method.     

On Necessary and Sufficient Conditions for Ordinary Least Squares Estimators to Be Best Linear Unbiased Estimators

Two often-quoted necessary and sufficient conditions for ordinary least squares estimators to be best linear unbiased estimators are described. Another necessary and sufficient condition is described, providing an additional tool for checking to see whether the covariance matrix of a given linear model is such that the ordinary least squares estimator is also the best linear unbiased estimator. The new condition is used to show that one of the two published conditions is only a sufficient condition.     

Robust Variable Selection in Linear Mixed Models

In this article, we develop a robust variable selection procedure jointly for fixed and random effects in linear mixed models for longitudinal data. We propose a penalized robust estimator for both the regression coefficients and the variance of random effects based on a re-parametrization of the linear mixed models. Under some regularity conditions, we show the oracle properties of the proposed robust variable selection method. Simulation study shows the robustness of the proposed method against outliers. In the end, the proposed methods is illustrated in the analysis of a real data set.     

A Class of Estimators Using Auxiliary Information in Two-Stage Sampling

This paper presents a general class of estimators for the finite population total when the emphasis is laid on the use of two auxiliary variables in a two-stage sampling.     

PORT Hill and Moment Estimators for Heavy-Tailed Models

In this article, we use the peaks over random threshold (PORT)-methodology, and consider Hill and moment PORT-classes of extreme value index estimators. These classes of estimators are invariant not only to changes in scale, like the classical Hill and moment estimators, but also to changes in location. They are based on the sample of excesses over a random threshold, the order statistic X _[np]+1:n , 0 ≤ p < 1, being p a tuning parameter, which makes them highly flexible. Under convenient restrictions on the underlying model, these classes of estimators are consistent and asymptotically normal for adequate values of k, the number of top order statistics used in the semi-parametric estimation of the extreme value index γ. In practice, there may however appear a stability around a value distant from the target γ when the minimum is chosen for the random threshold, and attention is drawn for the danger of transforming the original data through the subtraction of the minimum. A new bias-corrected moment estimator is also introduced. The exact performance of the new extreme value index PORT-estimators is compared, through a large-scale Monte-Carlo simulation study, with the original Hill and moment estimators, the bias-corrected moment estimator, and one of the minimum-variance reduced-bias (MVRB) extreme value index estimators recently introduced in the literature. As an empirical example we estimate the tail index associated to a set of real data from the field of finance.     

Outlier Resistant Estimation in Difference-Based Semiparametric Partially Linear Models

Partially linear models are extensions of linear models that include a nonparametric function of some covariate allowing an adequate and more flexible handling of explanatory variables than in linear models. The difference-based estimation in partially linear models is an approach designed to estimate parametric component by using the ordinary least squares estimator after removing the nonparametric component from the model by differencing. However, it is known that least squares estimates do not provide useful information for the majority of data when the error distribution is not normal, particularly when the errors are heavy-tailed and when outliers are present in the dataset. This paper aims to find an outlier-resistant fit that represents the information in the majority of the data by robustly estimating the parametric and the nonparametric components of the partially linear model. Simulations and a real data example are used to illustrate the feasibility of the proposed methodology and to compare it with the classical difference-based estimator when outliers exist.     

Estimation,Testing, and Finite Sample Properties of Quasi-Maximum Likelihood Estimators in GARCH-M Models

We provide three new results concerning quasi-maximum likelihood (QML) estimators in generalized autoregressive conditional heteroskedastic in mean (GARCH-M) models. We first show that, depending on the functional form that we impose in the mean equation, the properties of the model may change and the conditional variance parameter space may be restricted, in contrast to the theory of traditional GARCH processes. Second, we also present a new test for GARCH effects in the GARCH-M context which is simpler to implement than alternative procedures such as in Beg et al. (2001 Beg , R. , Silvapulle , M. , Silvapulle , P. ( 2001 ). Tests against inequality constraints when some nuisance parameters are present only under the alternative: test of ARCH in ARCH-M models . Journal of Business and Economic Statistics 19 : 245 – 485 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). We propose a new way of dealing with parameters that are not identified by creating composites of parameters that are identified. Third, the finite sample properties of QML estimators are explored in a restricted ARCH-M model and bias and variance approximations are found which show that the larger the volatility of the process the better the variance parameters are estimated. The invariance properties that Lumsdaine (1995 Lumsdaine , R. L. ( 1995 ). Finite sample properties of the maximum likelihood estimator in GARCH(1,1) and IGARCH(1,1) models: a Monte Carlo investigation . Journal of Business and Economic Statistics 13 ( 1 ): 1 – 10 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) proved for the traditional GARCH are shown not to hold in the GARCH-M. For those researchers who choose not to rely on the first order asymptotic approximation of our proposed test statistic, we also show how our bias expressions can be used to bias correct the QML estimates with a view to improving the finite sample performance of the test. Finally, we show how our new proposed test works in practice in an empirical economic application.     

An Introduction to Modern Matrix Methods and Statistics

In this article, using the representation that the Kalman filter recursions in state-space models can be expressed as a matrix-weighted average of prior and sample estimates, we supplement the usual filtering algorithm by an extreme bounds analysis. Specifically, as the covariance matrix of the state error is varied in the class of symmetric and positive-definite matrices, the filtering estimates are shown to be in an ellipsoid.