Estimating structural VARMA models with uncorrelated but non-independent error terms

The asymptotic properties of the quasi-maximum likelihood estimator (QMLE) of vector autoregressive moving-average (VARMA) models are derived under the assumption that the errors are uncorrelated but not necessarily independent nor martingale differences. Relaxing the martingale difference assumption on the errors considerably extends the range of application of the VARMA models, and allows one to cover linear representations of general nonlinear processes. Conditions are given for the asymptotic normality of the QMLE. Particular attention is given to the estimation of the asymptotic variance matrix, which may be very different from that obtained in the standard framework.     

Covariance estimation under spatial dependence

Correlated multivariate processes have a dependence structure which must be taken into account when estimating the covariance matrix. The natural estimator of the covariance matrix is introduced and is shown that to be biased under the dependence structure. This bias is studied under two different asymptotic models, namely increasing the domain by increasing the number of observations, and increasing the number of observations in the fixed domain. Using the first asymptotic model, we quantify the convergence rate of the bias and of the covariance between the components of the estimated covariance matrix. The second asymptotic model serves to derive a fast and accurate bias correction. As shown, under mild hypotheses, the asymptotic normality of the estimated covariance matrix holds and can be used to test whether the bias is significant, for example, in the sense that the eigenvectors of the estimated and true covariance matrices are significantly different.     

Asymptotic distribution of the weighted least squares estimator

This paper derives the asymptotic distribution of the weighted least squares estimator (WLSE) in a heteroscedastic linear regression model. A consistent estimator of the asymptotic covariance matrix of the WLSE is also obtained. The results are obtained under weak conditions on the design matrix and some moment conditions on the error distributions. It is shown that most of the error distributions encountered in practice satisfy these moment conditions. Some examples of the asymptotic covariance matrices are also given.     

Estimation de la matrice de variance asymptotique des estimateurs du QMV de modèles ARMA faibles multivariés

In this Note, we consider the problems of estimating the asymptotic variance of the quasi-maximum likelihood estimator (QMLE) of vector autoregressive moving-average (VARMA) models under the assumption that the errors are uncorrelated but not necessarily independent (i.e. weak VARMA). We first give expressions for the derivatives of the VARMA residuals in terms of the parameters of the models. Secondly we give an explicit expression of the asymptotic variance of the QMLE, in terms of the VAR and MA polynomials, and of the second- and fourth-order structure of the noise. We deduce a consistent estimator of the asymptotic variance of the QMLE.     

On asymptotic theory for multivariate GARCH models

The paper investigates the asymptotic theory for a multivariate GARCH model in its general vector specification proposed by Bollerslev, Engle and Wooldridge (1988) [4], known as the VEC model. This model includes as important special cases the so-called BEKK model and many versions of factor GARCH models, which are often used in practice. We provide sufficient conditions for strict stationarity and geometric ergodicity. The strong consistency of the quasi-maximum likelihood estimator (QMLE) is proved under mild regularity conditions which allow the process to be integrated. In order to obtain asymptotic normality, the existence of sixth-order moments of the process is assumed.     

Variable selection using penalized empirical likelihood

This paper considers variable selection for moment restriction models.We propose a penalized empirical likelihood(PEL) approach that has desirable asymptotic properties comparable to the penalized likelihood approach,which relies on a correct parametric likelihood specification.In addition to being consistent and having the oracle property,PEL admits inference on parameter without having to estimate its estimator's covariance.An approximate algorithm,along with a consistent BIC-type criterion for selecting ...     

Asymptotic expansions for the pivots using log-likelihood derivatives with an application in item response theory

Asymptotic expansions of the distributions of the pivotal statistics involving log-likelihood derivatives under possible model misspecification are derived using the asymptotic cumulants up to the fourth-order and the higher-order asymptotic variance. The pivots dealt with are the studentized ones by the estimated expected information, the negative Hessian matrix, the sum of products of gradient vectors, and the so-called sandwich estimator. It is shown that the first three asymptotic cumulants are the same over the pivots under correct model specification with a general condition of the equalities. An application is given in item response theory, where the observed information is usually used rather than the estimated expected one.     

Asymptotic inference of unstable periodic ARCH processes

This paper studies asymptotic properties of the quasi maximum likelihood and weighted least squares estimates (QMLE and WLSE) of the conditional variance slope parameters of a strictly unstable ARCH model with periodically time varying coefficients (PARCH in short). The model is strictly unstable in the sense that its parameters lie outside the strict periodic stationarity domain and its boundary. Obtained from the regression form of the PARCH, the WLSE is a variant of the least squares method weighted by the square of the conditional variance evaluated at any fixed value in the parameter space. In calculating the QMLE and WLSE, the conditional variance intercepts are set to any arbitrary values not necessarily the true ones. The theoretical finding is that the QMLE and WLSE are consistent and asymptotically Gaussian with the same asymptotic variance irrespective of the fixed conditional variance intercepts and the weighting parameters. So because of its numerical complexity, the QMLE may be dropped in favor of the WLSE which enjoys closed form.     

Multistage weighted least squares estimation of ARCH processes in the stable and unstable cases

For an ARCH model, we propose a multistage weighted least squares (WLS) estimate which consists of repeated WLS procedures until the corresponding asymptotic variance equals that of the quasi-maximum likelihood estimate (QMLE). At every stage, the current estimate is of a WLS type weighted by the squared conditional variance evaluated at the estimate of the previous stage. Initially, the weighting parameter is any fixed and known value in the parameter space. The procedure provides, without any moment requirement, an asymptotically Gaussian estimate having the same asymptotic distribution as the QMLE even in the unstable case. Apart from the initialization stage, two additional stages are required in the stable case to obtain the same asymptotic distribution as the QMLE, while in the unstable case only one stage is enough. So in all, the proposed procedure involves three stages WLS in the stable case and two stages WLS in the unstable case.     

Truncated Estimator of Asymptotic Covariance Matrix in Partially Linear Models with Heteroscedastic Errors

A partially linear regression model with heteroscedastic and/or serially correlated errors is studied here. It is well known that in order to apply the semiparametric least squares estimation （SLSE） to make statistical inference a consistent estimator of the asymptotic covariance matrix is needed. The traditional residual-based estimator of the asymptotic covariance matrix is not consistent when the errors are heteroscedastic and/or serially correlated. In this paper we propose a new estimator by truncating, which is an extension of the procedure in White. This estimator is shown to be consistent when the truncating parameter converges to infinity with some rate.     

Normal distribution based pseudo ML for missing data: With applications to mean and covariance structure analysis

When missing data are either missing completely at random (MCAR) or missing at random (MAR), the maximum likelihood (ML) estimation procedure preserves many of its properties. However, in any statistical modeling, the distribution specification for the likelihood function is at best only an approximation to the real world. In particular, since the normal-distribution-based ML is typically applied to data with heterogeneous marginal skewness and kurtosis, it is necessary to know whether such a practice still generates consistent parameter estimates. When the manifest variables are linear combinations of independent random components and missing data are MAR, this paper shows that the normal-distribution-based MLE is consistent regardless of the distribution of the sample. Examples also show that the consistency of the MLE is not guaranteed for all nonnormally distributed samples. When the population follows a confirmatory factor model, and data are missing due to the magnitude of the factors, the MLE may not be consistent even when data are normally distributed. When data are missing due to the magnitude of measurement errors/uniqueness, MLEs for many of the covariance parameters related to the missing variables are still consistent. This paper also identifies and discusses the factors that affect the asymptotic biases of the MLE when data are not missing at random. In addition, the paper also shows that, under certain data models and MAR mechanism, the MLE is asymptotically normally distributed and the asymptotic covariance matrix is consistently estimated by the commonly used sandwich-type covariance matrix. The results indicate that certain formulas and/or conclusions in the existing literature may not be entirely correct.     

Adaptive quasi-likelihood estimate in generalized linear models

This paper gives a thorough theoretical treatment on the adaptive quasi-likelihood estimate of the parameters in the generalized linear models. The unknown covariance matrix of the response variable is estimated by the sample. It is shown that the adaptive estimator defined in this paper is asymptotically most efficient in the sense that it is asymptotic normal, and the covariance matrix of the limit distribution coincides with the one for the quasi-likelihood estimator for the case that the covariance matrix of the response variable is completely known.     

Hypothesis tests for high-dimensional covariance structures

We consider hypothesis testing for high-dimensional covariance structures in which the covariance matrix is a (i) scaled identity matrix, (ii) diagonal matrix, or (iii) intraclass covariance matrix. Our purpose is to systematically establish a nonparametric approach for testing the high-dimensional covariance structures (i)–(iii). We produce a new common test statistic for each covariance structure and show that the test statistic is an unbiased estimator of its corresponding test parameter. We prove that the test statistic establishes the asymptotic normality. We propose a new test procedure for (i)–(iii) and evaluate its asymptotic size and power theoretically when both the dimension and sample size increase. We investigate the performance of the proposed test procedure in simulations. As an application of testing the covariance structures, we give a test procedure to identify an eigenvector. Finally, we demonstrate the proposed test procedure by using a microarray data set.

     

Consistent estimation and testing in heteroscedastic polynomial errors-in-variables models

The paper concentrates on consistent estimation and testing in functional polynomial measurement errors models with known heterogeneous variances. We rest on the corrected score methodology which allows the derivation of consistent and asymptotically normal estimators for line parameters and also consistent estimators for the asymptotic covariance matrix. Hence, Wald and score type statistics can be proposed for testing the hypothesis of a reduced linear relationship, for example, with asymptotic chi-square distribution which guarantees correct asymptotic significance levels. Results of small scale simulation studies are reported to illustrate the agreement between theoretical and empirical distributions of the test statistics studied. An application to a real data set is also presented.     

Bayesian estimation of a covariance matrix with flexible prior specification

Bayesian analysis for a covariance structure has been in use for decades. The commonly adopted Bayesian setup involves the conjugate inverse Wishart prior specification for the covariance matrix. Here we depart from this approach and adopt a novel prior specification by considering a multivariate normal prior for the elements of the matrix logarithm of the covariance structure. This specification allows for a richer class of prior distributions for the covariance structure with respect to strength of beliefs in prior location hyperparameters and the added ability to model potential correlation amongst the covariance structure. We provide three computational methods for calculating the posterior moment of the covariance matrix. The moments of interest are calculated based upon computational results via Importance sampling, Laplacian approximation and Markov Chain Monte Carlo/Metropolis–Hastings techniques. As a particular application of the proposed technique we investigate educational test score data from the project talent data set.     

纵向数据边际模型非参数光滑方法的比较

对于纵向数据边际模型的均值函数, 有很多非参数估计方法, 其中回归样条, 光滑样条, 似乎不相关(SUR)核估计等方法在工作协方差阵正确指定时具有最小的渐近方差. 回归样条的渐近偏差与工作协方差阵无关, 而SUR核估计和光滑样条估计的渐近偏差却依赖于工作协方差阵. 本文主要研究了回归样条, 光滑样条和SUR核估计的效率问题. 通过模拟比较发现回归样条估计的表现比较稳定, 在大多数情况下比光滑样条估计和SUR核估计的效率高.     

Quasi-maximum likelihood estimation for multiple volatility shifts

We propose the Gaussian quasi-maximum likelihood estimator (QMLE) to detect and locate multiple volatility shifts. Our Gaussian QMLE is shown to be consistent under suitable conditions and the rate of convergence is provided. It is also shown that the binary segmentation procedure provides a consistent estimation for the number of volatility shifts.     

Asymptotic theory in generalized ilinear models with nuisance scale parameters

Summary This paper establishes the asymptotic normality and the consistencyrobustness of the weighted least squares estimator (WLSE) in the generalized linear models with multiple nuisance scale parameters. In addition, noting that the asymptotic robust statistical inference in presence of nuisance scale parameters requires a consistency-robust estimator of the asymptotic covariance matrix of the WLSE, this paper derives a class of covariance estimators and proves their consistency-robustness.This research was supported by an operating grant from the Natural Science and Engineering Research Council of Canada and the United States NSF-AFOSR grant ISSA-860068     

多元极值分布参数的最大似然估计与分步估计

本文考虑多元极值分布的参数估计，给出了分步估计渐近协差阵的近似表示，并对维数P=2，5及相关参数α=0．001，0．01；0．1（0．2），0．9；0．99，0．999的各种组合，计算了分步估计关于最大似然估计的渐近效率，分析了各种参数及维数对渐近效率的影响．分步估计是一种合理、简单而且有较强实用意义的估计方法．     

Asymptotic normality of the sample mean and covariances of evanescent fields in noise

We consider the asymptotic properties of the sample mean and the sample covariance sequence of a field composed of the sum of a purely indeterministic and evanescent components. The asymptotic normality of the sample mean and sample covariances is established. A Bartlett-type formula for the asymptotic covariance matrix of the sample covariances of this field, is derived.