A note on the covariance matrix of the maximum likelihood estimator in constrained multivariate linear regression

A useful result concerning variances and covariances of a linear function of a random matrix is applied to find the variance–covariance matrix of the maximum likelihood estimator in multivariate linear regression subject to zero constraints.     

On the strong uniform consistency of a new kernel density estimator

Summary A new multivariate kernel probability density estimator is introduced and its strong uniform consistency is proved under certain regularity conditions. This result is then applied particularly to a kernel estimator whose mean vector and covariance matrix areμ _n andV _n, respectively, whereμ _n is an unspecified estimator of the mean vector andV _n, up to a multiplicative constant, the sample covariance matrix of the probability density to be estimated, respectively. Work supported by the Natural Sciences and Engineering Research Council of Canada and by the Fonds F.C.A.R. of the Province of Quebec.     

Missing in Asynchronicity: A Kalman‐em Approach for Multivariate Realized Covariance Estimation

Motivated by the need for a positive‐semidefinite estimator of multivariate realized covariance matrices, we model noisy and asynchronous ultra‐high‐frequency asset prices in a state‐space framework with missing data. We then estimate the covariance matrix of the latent states through a Kalman smoother and expectation maximization (KEM) algorithm. Iterating between the two EM steps, we obtain a covariance matrix estimate which is robust to both asynchronicity and microstructure noise, and positive‐semidefinite by construction. We show the performance of the KEM estimator using extensive Monte Carlo simulations that mimic the liquidity and market microstructure characteristics of the S&P 500 universe as well as in a high‐dimensional application on US stocks. KEM provides very accurate covariance matrix estimates and significantly outperforms alternative approaches recently introduced in the literature. Copyright © 2014 John Wiley & Sons, Ltd.     

Estimation of the trace of the scaled covariance matrix of a multivariate t-model using a known information

The trace of the scaled covariance matrix of the multivariate t-distribution is considered for estimation using a power transformation. The proposed estimator always dominates the usual maximum likelihood estimator in the sense of having smaller risk under a quadratic loss function. The dominance behaviour is proved analytically as well as computationally by using Monte-Carlo simulation.     

Statistical properties of the total variation estimator for compositional data

The sample space of compositional data, a simplex, induces a different kind of geometry, known as Aitchison geometry, with the Euclidean space property. For this reason, the standard statistical analysis is not meaningful here, and this is also true for measures of location and covariance. The measure of location, called centre, is the best linear unbiased estimator of the central tendency of the distribution of a random composition with respect to the geometry on the simplex (Pawlowsky-Glahn and Egozcue in Stoch Envir Res Risk Ass, 15:384–398, 2001; Math Geol, 34:259–274, 2002). Its covariance structure is described through a variation matrix, which induces the so called total variation as a measure of dispersion. The aim of the paper is to show that its sample counterpart has theoretical properties, corresponding to the standard multivariate case, like unbiasedness and convergence in probability. Moreover, its distribution in the case of normality on the simplex is developed.     

Pre-averaging estimators of the ex-post covariance matrix in noisy diffusion models with non-synchronous data

We show how pre-averaging can be applied to the problem of measuring the ex-post covariance of financial asset returns under microstructure noise and non-synchronous trading. A pre-averaged realised covariance is proposed, and we present an asymptotic theory for this new estimator, which can be configured to possess an optimal convergence rate or to ensure positive semi-definite covariance matrix estimates. We also derive a noise-robust Hayashi–Yoshida estimator that can be implemented on the original data without prior alignment of prices. We uncover the finite sample properties of our estimators with simulations and illustrate their practical use on high-frequency equity data.     

Variance estimation using auxiliary information: An almost unbiased multivariate ratio estimator

The goal of this paper is to investigate the repeated substitution method (seeSrivastava, 1967) estimating population variance in finite population sample surveys. We propose an almost unbiased multivariate ratio estimator that has a smaller mean squared error than the conventional biased multivariate ratio estimator (established byIsaki (1983)) and with the same precision as the multivariate regression estimator. Furthermore, it is a computationally much more interesting estimator since to compute it we only need to have knowledge of correlation among available variables, which it is common to have in several practical situations. A comparison of the multivariate ratio estimator proposed and the multivariate regression estimator is given.     

Optimum allocation in multivariate stratified random sampling: stochastic matrix mathematical programming

The allocation problem for multivariate stratified random sampling as a problem of stochastic matrix integer mathematical programming is considered, minimizing the estimated covariance matrix of estimated means subject to fixed cost or fixed total sample size. With these aims the asymptotic normality of sample covariance matrices for each strata is established. Some alternative approaches are suggested for its solution. An example is solved by applying the proposed techniques.     

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.     

Two estimators of the long-run variance: Beyond short memory

This paper deals with the estimation of the long-run variance of a stationary sequence. We extend the usual Bartlett-kernel heteroskedasticity and autocorrelation consistent (HAC) estimator to deal with long memory and antipersistence. We then derive asymptotic expansions for this estimator and the memory and autocorrelation consistent (MAC) estimator introduced by Robinson [Robinson, P. M., 2005. Robust covariance matrix estimation: HAC estimates with long memory/antipersistence correction. Econometric Theory 21, 171–180]. We offer a theoretical explanation for the sensitivity of HAC to the bandwidth choice, a feature which has been observed in the special case of short memory. Using these analytical results, we determine the MSE-optimal bandwidth rates for each estimator. We analyze by simulations the finite-sample performance of HAC and MAC estimators, and the coverage probabilities for the studentized sample mean, giving practical recommendations for the choice of bandwidths.     

HAC estimation in a spatial framework

We suggest a non-parametric heteroscedasticity and autocorrelation consistent (HAC) estimator of the variance–covariance (VC) matrix for a vector of sample moments within a spatial context. We demonstrate consistency under a set of assumptions that should be satisfied by a wide class of spatial models. We allow for more than one measure of distance, each of which may be measured with error. Monte Carlo results suggest that our estimator is reasonable in finite samples. We then consider a spatial model containing various complexities and demonstrate that our HAC estimator can be applied in the context of that model.     

High dimensional covariance matrix estimation using a factor model

High dimensionality comparable to sample size is common in many statistical problems. We examine covariance matrix estimation in the asymptotic framework that the dimensionality p$p$ tends to ∞$\infty$ as the sample size n$n$ increases. Motivated by the Arbitrage Pricing Theory in finance, a multi-factor model is employed to reduce dimensionality and to estimate the covariance matrix. The factors are observable and the number of factors K$K$ is allowed to grow with p$p$. We investigate the impact of p$p$ and K$K$ on the performance of the model-based covariance matrix estimator. Under mild assumptions, we have established convergence rates and asymptotic normality of the model-based estimator. Its performance is compared with that of the sample covariance matrix. We identify situations under which the factor approach increases performance substantially or marginally. The impacts of covariance matrix estimation on optimal portfolio allocation and portfolio risk assessment are studied. The asymptotic results are supported by a thorough simulation study.     

Underestimation of mean square error matrix in misspecified linear models

In this paper a necessary and sufficient condition is derived for the mean square error matrix to be overestimated on average by the covariance matrix of the OLS estimator when the linear model is misspecified by omitting at least one independent variable. A sufficient condition for this is also considered and the situation is illustrated by simple examples showing, as previously noted by Gupta and Maasoumi (1979), that underestimation is the prevailing case in practice.     

An application of approximate finite sample results to parameter estimation in a linear errors-in-variables model

In the simple errors-in-variables model the least squares estimator of the slope coefficient is known to be biased towards zero for finite sample size as well as asymptotically. In this paper we suggest a new corrected least squares estimator, where the bias correction is based on approximating the finite sample bias by a lower bound. This estimator is computationally very simple. It is compared with previously proposed corrected least squares estimators, where the correction aims at removing the asymptotic bias or the exact finite sample bias. For each type of corrected least squares estimators we consider the theoretical form, which depends on an unknown parameter, as well as various feasible forms. An analytical comparison of the theoretical estimators is complemented by a Monte Carlo study evaluating the performance of the feasible estimators. The new estimator proposed in this paper proves to be superior with respect to the mean squared error.     

Conditions for invariance of the multivariate versions of grubbs’ test and bartlett’s test under a general dependency structure

Summary This paper generalizes a result by Stadje (1984) by deriving conditions for which a general dependency structure for multivariate observations, given in Pavur (1987), yields a positive definite covariance structure. This general dependency structure allows the sample covariance matrix to be distributed as a constant times a Wishart random matrix. It is then demonstrated that the maximum squared-radii test and a test for equal population covariance matrices have null distributions which remain unchanged when the new general dependency structure, rather than the usual independence structure, for the vector observations, is assumed. Moreover, under a general dependency structure for which the population covariance matrices are unequal, it is shown that the distribution of the test statistic for testing equal covariance matrices is identical to the distribution of the same test statistic when the population covariance matrices are equal and the observations are independent.     

Testing block‐diagonal covariance structure for high‐dimensional data

A test statistic is developed for making inference about a block‐diagonal structure of the covariance matrix when the dimensionality p exceeds n, where n = N ? 1 and N denotes the sample size. The suggested procedure extends the complete independence results. Because the classical hypothesis testing methods based on the likelihood ratio degenerate when p > n, the main idea is to turn instead to a distance function between the null and alternative hypotheses. The test statistic is then constructed using a consistent estimator of this function, where consistency is considered in an asymptotic framework that allows p to grow together with n. The suggested statistic is also shown to have an asymptotic normality under the null hypothesis. Some auxiliary results on the moments of products of multivariate normal random vectors and higher‐order moments of the Wishart matrices, which are important for our evaluation of the test statistic, are derived. We perform empirical power analysis for a number of alternative covariance structures.     

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.     

Misspecified models with dependent observations

General results are given in this paper which allow the development of a theory of estimation and inference for situations in which the model of a data-generating process has been misspecified. Observations may come from time-series, cross-section, panel, or experimental data. The nonlinear regression model is examined in some detail. Conditions are provided which ensure the consistency and asymptotic normality of the least-squares estimator with respect to the parameter vector of a weighted least-squares approximation to the underlying data-generating process. A specification-robust estimator of the asymptotic covariance matrix is given, allowing a proper treatment of inference in potentially misspecified models. The properties of the approximation and the covariance estimator are exploited to yield new tests for model specification.     

Weaker MSE criteria and tests for linear restrictions in regression models with non-spherical disturbances

This paper extend, in an asymptotic sense, the strong and the weaker mean square error criteria and corresponding tests to linear models with non-spherical disturbances where the error covariance matrix is unknown but a consistent estimator for it is available. The mean square error tests of Toro-Vizcorrondo and Wallace (1968) and Wallace (1972) test for the superiority of restricted over unrestricted linear estimators in a least squares context. This generalization of these tests makes them available for use with GLS, Zellner's SUR, 2SLS, 3SLS, tests of over identification, and so forth.     

Matrix exponential GARCH

I propose a new multivariate GARCH specification that maintains positive definiteness of the conditional covariance matrix. The idea is to specify the dynamics in the matrix logarithm of the conditional covariance. Because the matrix exponential transformation ensures positive definiteness, the dynamics can be specified without the positive definiteness constraint. This affords a variety of specifications and, in particular, we can specify element-by-element the dynamics of the matrix logarithm. I discuss specifications with leverage effects, estimation with multivariate Gaussian and t-distributions, and diagnostics that evaluate the appropriateness of the matrix exponential specification.