Covariance matrix estimation in a seemingly unrelated regression model under Stein’s loss

Journal Article Nonparametric estimation of large covariance matrices of longitudinal data Get access Wei Biao Wu, Wei Biao Wu Search for other works by this author on: Oxford Academic Google Scholar Mohsen Pourahmadi Mohsen Pourahmadi Search for other works by this author on: Oxford Academic Google Scholar Biometrika, Volume 90, Issue 4, December 2003, Pages 831–844, https://doi.org/10.1093/biomet/90.4.831 Published: 01 December 2003

In a general normal regression model, this paper first derives the least upper bound (LUB) for the covariance matrix of a generalized least squares estimator (GLSE) relative to the covariance matrix of the Gauss-Markov estimator. Second the result is applied to the (unrestricted) Zellner estimator in an N-equation seemingly unrelated regression (SUR) model and to the GLSE in a heteroscedastic model.

Estimating seemingly unrelated regression models with vector autoregressive disturbances

Covariance matrix estimation is one of the most important problems in statistics. To accommodate the complexity of modern datasets, it is desired to have estimation procedures that not only can incorporate the structural assumptions of covariance matrices, but are also robust to outliers from arbitrary sources. In this paper, we define a new concept called matrix depth and then propose a robust covariance matrix estimator by maximizing the empirical depth function. The proposed estimator is shown to achieve minimax optimal rate under Huber’s $\\varepsilon$-contamination model for estimating covariance/scatter matrices with various structures including bandedness and sparsity.

In transportation studies, variables of interest are often influenced by similar factors and have correlated latent terms (errors). In such cases, a seemingly unrelated regression (SUR) model is normally used. However, most studies ignore the potential temporal and spatial autocorrelations across observations, which may lead to inaccurate conclusions. In contrast, the SUR model proposed in this study also considers these correlations, making the model more behaviorally convincing and applicable to circumstances where a three-dimensional correlation exists, across time, space, and equations. An example of crash rates in Chinese cities is used. The results show that incorporation of spatial and temporal effects significantly improves the model. Moreover, investment in transportation infrastructure is estimated to have statistically significant effects on reducing severe crash rates, but with an elasticity of only −0.078. It is also observed that, while vehicle ownership is associated with higher per capita crash rates, elasticities for severe and non-severe crashes are just 0.13 and 0.18, respectively; much lower than one. The techniques illustrated in this study should contribute to future studies requiring multiple equations in the presence of temporal and spatial effects.

The omission of important variables is often encountered in econometrics, which often results in the misspecification of the model. This study considered a misspecified joint seemingly unrelated regression (SUR) model within the Bayesian context. The explanatory variables were generated from a uniform distribution for sample sizes 5, 10, 30, 50, 100 and 1000. The coefficients of were generated from the multivariate normal distribution. Different levels of correlation (0.2, 0.4, 0.5, 0.7 and 0.9) among the independent variables were considered. The choice of hyperprior distribution was explored. Due to the incorrectly defined model, the intended attribute of unbiasedness in the estimated coefficients, which depends on the fitted model's correspondence with the actual underlying data-generating process, was not attained. Irregular patterns were seen in the posterior means and standard deviations except for a few large sample sizes. Keywords: Misspecification, Multicollinearity, Posterior mean, Posterior standard deviation, Seemingly unrelated regression.

Predicting the distribution of households and employment: a seemingly unrelated regression model with two spatial processes

On the Efficiencies of Several Generalized Least Squares Estimators in a Seemingly Unrelated Regression Model and a Heteroscedastic Model

This paper addresses the experimental wireless MIMO channel modeling and validation based on channel sounding data using the approximation of the full channel correlation matrix. Measurement were carried out in indoor laboratory environment at central frequency 2.3 GHz. An analytical MIMO channel model is presented based on optimal approximation of channel covariance matrix. Approximation of a full channel covariance matrix is based on the optimal Kronecker product series expansion of the sample covariance matrix. The channel correlation matrices calculated from the measured channel coefficients were decomposed using Van Loan and Pitsanis approximation algorithm. Experimental validation of such model is presented. The accuracy of the MIMO channel modeling was evaluated by the correlation matrix distance and by calculating of CDF of channel capacity. The results show that these models have good agreement with the MIMO channel measured data. Also two popular analytical MIMO channel models – Kronecker and Weichselberger models are evaluated and compared with the presented channel model.

Estimation of a Normal Covariance Matrix with Incomplete Data under Stein′s Loss

This paper first develops a valid method for approximations to the pdf's and cdf's of GLSE in linear models and, applying this method to the Zellner estimator with an unrestricted sample covariance in the seemingly unrelated regression model, obtains an approximate pdf with a bound of ordern −2 and an approximate covariance matrix with a bound of ordern −3

For multivariate analysis with p variables the problem that often arises is the ambiguous nature of the correlation or covariance matrix. When p is moderately or very large it is generally difficult to identify the true nature of relationship among the variables as well as observations from the covariance or correlation matrix. Under such situations a very common way to simplify the matter is to reduce the dimension by considering only those variables (actual or derived) which are truly responsible for the overall variation. Important and useful dimension reduction techniques are Principal Component Analysis (PCA), Factor Analysis, Multidimensional Scaling, Independent Component Analysis (ICA), etc. Among them PCA is the most popular one. One may look at this method in three different ways. It may be considered as a method of transforming correlated variables into uncorrelated one or a method of finding linear combinations with relatively small or large variability or a tool for data reduction. The third criterion is more data oriented. In PCA primarily it is not necessary to make any assumption regarding the underlying multivariate distribution but if we are interested in some inference problems related to PCA then assumption of multivariate normality is necessary. The eigen values and eigen vectors of the covariance or correlation matrix are the main contributors of a PCA. The eigen vectors determine the directions of maximum variability whereas the eigen values specify the variances. In practice, decisions regarding the quality of the principal component approximation should be made on the basis of eigen value–eigen vector pairs.

In this article, a seemingly unrelated Poisson regression model is presented as an alternative to using Zellner's seemingly unrelated regression model for estimating a system of recreation demand functions. The seemingly unrelated Poisson regression model provides estimates that are asymptotically more efficient than equation-by-equation Poisson estimates and circumvents the bias and inconsistency problems that result when using A. Zellner's seemingly unrelated regression model. Additionally, the seemingly unrelated Poisson regression model is applied to an empirical problem dealing with the value of recreational boating and the findings indicate that the seemingly unrelated regression model consumer surplus estimates are substantially different from those of the seemingly unrelated Poisson regression model. Copyright 1994 by MIT Press.

A statistically homogeneous pixel selection approach for adaptive estimation of multitemporal InSAR covariance matrix

We introduce nonparametric regularization of the eigenvalues of a sample covariance matrix through splitting of the data (NERCOME), and prove that NERCOME enjoys asymptotic optimal nonlinear shrinkage of eigenvalues with respect to the Frobenius norm. One advantage of NERCOME is its computational speed when the dimension is not too large. We prove that NERCOME is positive definite almost surely, as long as the true covariance matrix is so, even when the dimension is larger than the sample size. With respect to the Stein’s loss function, the inverse of our estimator is asymptotically the optimal precision matrix estimator. Asymptotic efficiency loss is defined through comparison with an ideal estimator, which assumed the knowledge of the true covariance matrix. We show that the asymptotic efficiency loss of NERCOME is almost surely 0 with a suitable split location of the data. We also show that all the aforementioned optimality holds for data with a factor structure. Our method avoids the need to first estimate any unknowns from a factor model, and directly gives the covariance or precision matrix estimator, which can be useful when factor analysis is not the ultimate goal. We compare the performance of our estimators with other methods through extensive simulations and real data analysis.