A Semi-Parametric Bayesian Generalized Least Squares Estimator

In this paper we propose a semi-parametric Bayesian Generalized Least Squares estimator. In a generic GLS setting where each error is a vector, parametric GLS maintains the assumption that each error vector has the same covariance matrix. In reality however, the observations are likely to be heterogeneous regarding their distributions. To cope with such heterogeneity, a Dirichlet process prior is introduced for the covariance matrices of the errors, leading to the error distribution being a mixture of a variable number of normal distributions. Our methods let the number of normal components be data driven. Two specific cases are then presented: the semi-parametric Bayesian Seemingly Unrelated Regression (SUR) for equation systems; as well as the Random Effects Model (REM) and Correlated Random Effects Model (CREM) for panel data. A series of simulation experiments is designed to explore the performance of our methods. The results demonstrate that our methods obtain smaller posterior standard deviations than the parametric Bayesian GLS. We then apply our semi-parametric Bayesian SUR and REM/CREM methods to empirical examples.Comment: 32 pages, 2 figures, 18 table

Estimating the intercept in an orthogonally blocked experiment when the block effects are random.

Abstract: For an orthogonally blocked experiment, Khuri (1992) has shown that the ordinary least squares estimator and the generalized least squares estimator of the factor effects in a response surface model with random block effects coincide. However, the equivalence does not hold for the estimation of the intercept when the block sizes are heterogeneous. When the block sizes are homogeneous, ordinary and generalized least squares provide an identical estimate for the intercept.Effects;

Asymptotic normality and consistency of a two-stage generalized least squares estimator in the growth curve model

Let \mathbf{Y}=\mathbf{X}\bolds{\Theta}\mathbf{Z}'+\bolds{\mathcal {E}} be the growth curve model with \bolds{\mathcal{E}} distributed with mean $\mathbf{0}$ and covariance \mathbf{I}_n\otimes\bolds{\Sigma}, where \bolds{\Theta}, \bolds{\Sigma} are unknown matrices of parameters and $\mathbf{X}$, $\mathbf{Z}$ are known matrices. For the estimable parametric transformation of the form \bolds {\gamma}=\mathbf{C}\bolds{\Theta}\mathbf{D}' with given $\mathbf{C}$ and $\mathbf{D}$, the two-stage generalized least-squares estimator \hat{\bolds \gamma}(\mathbf{Y}) defined in (7) converges in probability to \bolds\gamma as the sample size $n$ tends to infinity and, further, \sqrt{n}[\hat{\bolds{\gamma}}(\mathbf{Y})-\bolds {\gamma}] converges in distribution to the multivariate normal distribution \ma thcal{N}(\mathbf{0},(\mathbf{C}\mathbf{R}^{-1}\mathbf{C}')\otimes(\mat hbf{D}(\mathbf{Z}'\bolds{\Sigma}^{-1}\mathbf{Z})^{-1}\mathbf{D}')) under the condition that $\lim_{n\to\infty}\mathbf{X}'\mathbf{X}/n=\mathbf{R}$ for some positive definite matrix $\mathbf{R}$. Moreover, the unbiased and invariant quadratic estimator \hat{\bolds{\Sigma}}(\mathbf{Y}) defined in (6) is also proved to be consistent with the second-order parameter matrix \bolds{\Sigma}.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ128 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

The effect of a Durbin-Watson pretest on confidence intervals in regression

Consider a linear regression model and suppose that our aim is to find a confidence interval for a specified linear combination of the regression parameters. In practice, it is common to perform a Durbin-Watson pretest of the null hypothesis of zero first-order autocorrelation of the random errors against the alternative hypothesis of positive first-order autocorrelation. If this null hypothesis is accepted then the confidence interval centred on the Ordinary Least Squares estimator is used; otherwise the confidence interval centred on the Feasible Generalized Least Squares estimator is used. We provide new tools for the computation, for any given design matrix and parameter of interest, of graphs of the coverage probability functions of the confidence interval resulting from this two-stage procedure and the confidence interval that is always centred on the Feasible Generalized Least Squares estimator. These graphs are used to choose the better confidence interval, prior to any examination of the observed response vector

A STUDY OF INDONESIA’S STOCK MARKET: HOW PREDICTABLE IS IT?

Using monthly data from January 1995 to December 2017, this paper tests whetherIndonesian stock index returns are predictable. In particular, we use eight macrovariables to predict the Indonesian composite and six sectoral index returns using thefeasible generalized least squares estimator. Our results suggest that the Indonesianstock index returns are predictable. However, the predictability depends not only onthe macro predictor used but also on the indexes examined. Second, we find that themost popular predictor is the exchange rate, followed by the interest rate. Finally, ourmain findings hold for a number of robustness tests.Using monthly data from January 1995 to December 2017, this paper tests whetherIndonesian stock index returns are predictable. In particular, we use eight macrovariables to predict the Indonesian composite and six sectoral index returns using thefeasible generalized least squares estimator. Our results suggest that the Indonesianstock index returns are predictable. However, the predictability depends not only onthe macro predictor used but also on the indexes examined. Second, we find that themost popular predictor is the exchange rate, followed by the interest rate. Finally, ourmain findings hold for a number of robustness tests

Estimated generalized least squares estimation for the heterogeneous measurement error model

The measurement error model of interest is (UNFORMATTED TABLE OR EQUATION FOLLOWS)\eqalign y[subscript]t &= [beta][subscript]0 + x[subscript]t[beta][subscript]1 + q[subscript]t, Y[subscript]t &= y[subscript]t + w[subscript]t, X[subscript]t &= x[subscript]t + u[subscript]t, (TABLE/EQUATION ENDS)where Z[subscript]t = ( Y[subscript]t, X[subscript]t) is the observed p-dimensional vector, z[subscript]t = (y[subscript]t, x[subscript]t) is the true unknown random vector, q[subscript]t is the equation error, and the measurement errors a[subscript]t = (w[subscript]t, u[subscript]t) are distributed with mean zero and known variance [sigma][subscript]aatt. An estimated generalized least squares estimator of the mean and variance of z[subscript]t, denoted by [mu] and [sigma][subscript]zz respectively, is shown to have a limiting normal distribution under mild regularity conditions. An estimator of [beta][superscript]\u27 = ([beta][subscript]0, [beta][subscript]sp1\u27) based upon the proposed estimator of [mu] and [sigma][subscript]zz is constructed and shown to have a limiting normal distribution. The variances of the limiting distribution are less than or equal to the corresponding variances for other estimators that have been suggested for the heterogeneous error model. The estimated generalized least squares estimator also displayed smaller mean square error than other estimators in a Monte Carlo study. A program to implement the proposed estimators is developed. The iterated estimated generalized least squares estimators of the measurement error model are investigated. The limit of a modified iteration procedure is shown to be the maximum likelihood estimator for the normal distribution;Estimated generalized least squares estimation is considered for the general linear model, Y = X[beta] + u, where the variance of u is denoted by V[subscript]uu and the elements of X[superscript]\u27 V[subscript]spuu-1 X may increase at different rates. Sufficient conditions are given for the estimated generalized least squares estimator to be consistent and asymptotically equivalent to the generalized least squares estimator constructed with known V[subscript]uu. Consistent estimators of the normalizing matrix are developed, and the asymptotic distribution of a linear combination of the elements of the estimator is considered. The model where V[subscript]uu is a function of a fixed, finite number of parameters and the use of ordinary least squares residuals to estimate the parameters of V[subscript]uu are examined. Applications of the results to the trend model with first order autoregressive errors and to the measurement error model are given