THE SCORE OF CONDITIONALLY HETEROSKEDASTIC DYNAMIC REGRESSION MODELS WITH STUDENT T INNOVATIONS, AN LM TEST FOR MULTIVARIATE NORMALITY

We provide numerically reliable analytical expressions for the score of conditionally heteroskedastic dynamic regression models when the conditional distribution is multivariate $t$. We also derive one-sided and 2-sided LM tests for multivariate normality versus multivariate $t$ based on the first two moments of the (squared) norm of the standardised innovations evaluated at the Gaussian quasi-ML estimators of the conditional mean and variance parameters. We reinterpret them as specification tests for multivariate excess kurtosis, and show that they have power against leptokurtic alternatives. Finally, we analyse UK stock returns, and confirm that their conditional distribution has fat tails.Kurtosis, Inequality Constraints, ARCH, Financial Returns.

Regression models under competing covariance matrices: A Bayesian perspective

Economics

Testing the Capital Asset Pricing Model Efficiently Under Elliptical Symmetry: A Semiparametric Approach

We develop new tests of the capital asset pricing model (CAPM) that take account of and are valid under the assumption that the distribution generating returns is elliptically symmetric; this assumption is necessary and sufficient for the validity of the CAPM. Our test is based on semiparametric efficient estimation procedures for a seemingly unrelated regression model where the multivariate error density is elliptically symmetric, but otherwise unrestricted. The elliptical symmetry assumption allows us to avert the curse of dimensionality problem that typically arises in multivariate semiparametric estimation procedures, because the multivariate elliptically symmetric density function can be written as a function of a scalar transformation of the observed multivariate data. The elliptically symmetric family includes a number of thick-tailed distributions and so is potentially relevant in financial applications. Our estimated betas are lower than the OLS estimates, and our parameter estimates are much less consistent with the CAPM restrictions than the corresponding OLS estimates. Nous développons de nouveaux tests du modèle d'évaluation des actifs financiers (" CAPM ") qui tiennent compte de, et sont valides sous, l'hypothèse que les retours des actifs découlent d'un loi de probabilité elliptiquement symétrique. Cette hypothèse est nécessaire et suffisante pour la validité du CAPM. Notre test utilise un estimateur des paramètres du modèle qui a l'efficacité semiparamétrique quand on a un modèle de régression apparemment sans relation et qui a des erreurs qui suivent une loi elliptiquement symétrique. L'hypothèse de la symétrie elliptique nous permet d'éviter le problème d'estimer non-paramétriquement une fonction de haute dimension parce qu'on peut écrire la densité d'une loi elliptique comme une fonction d'une transformation unidimensionnelle de la variable aléatoire multidimensionnelle. La famille des lois elliptiquement symétriques inclue plusieurs lois leptokurtiques, donc elle est pertinente à des applications financières. Les bêtas obtenus avec notre estimateur sont plus bas que ceux qui sont obtenus en utilisant des moindres carrés, et sont moins compatibles avec le CAPM.Adaptive estimation, capital asset pricing model, elliptical symmetry, semiparametric efficiency

Semi-conjugate prior densities in multivariate t regression models

Regression Analysis

Inequality restricted and pre-test estimation in a mis-specified econometric model

This thesis is concerned with the finite sample properties of some estimators of the unknown parameters in a linear model which is (possibly) mis-specified through the exclusion of relevant regressors. We assume that in addition to sample information, prior information regarding the unknown parameters is available in the form of a linear inequality constraint imposed on the regression coefficients. The combination of this type of prior information and sample information in specifying the corresponding statistical model leads to what has been identified in the literature as the inequality restricted estimator. If the statistical significance of the inequality constraint is tested prior to the estimation process, then the estimator thereby generated is called the inequality pre-test estimator. The properties of these estimators of the coefficient vector in a properly specified model have been examined rather thoroughly in the literature. In this thesis, we extend the results reported in the literature to the case where the underlying regression model is underfitted. We also investigate the sampling performance of the corresponding estimators for the model's disturbance variance, as well as the choice of an optimal size for the pre-test. The general background and motivation for this study are given in Chapter 1. Much of the earlier research on inequality restricted and pre-test estimation are built on results from studies that assume that the prior information is in the form of linear equality restrictions. We survey the relevant literature in this area in Chapter 2. Chapter 3 reviews the literature on inequality restricted and pre-test estimation. We focus on this problem in the context of the standard linear model with a single linear inequality constraint on the coefficient vector, as this is directly related to the theme of this thesis. In Chapter 4, we derive and evaluate the risk, under quadratic loss, of the inequality restricted and pre-test estimators for the regression prediction vector in an underfitted model. This analysis takes the established literature further by allowing for mis-specification in the regressor matrix. We consider the risk of the prediction vector, rather than the coefficient vector itself, so that our results are data independent. The risk functions of the corresponding estimators for the regression disturbance variance in the properly specified and underfitted models are derived in Chapters 6 and 7 respectively. As in the case where the prior information exists as linear equality restrictions, our results show that when the model is underfitted, the use of valid prior information does not necessarily guarantee a reduction in risk. This result holds for the estimation of both the prediction vector and the scale parameter. When one is estimating the regression disturbance variance, with an appropriate choice of test size, the inequality pre-test estimator can uniformly dominate the estimator that uses sample information only. We also find that the risk functions of the estimators of the error variance are affected more by mis-specification than are the corresponding predictive risks. In the case where no strictly dominating estimator exists, the question of the choice of an optimal critical value of the pre-test remains. Chapters 5 and 8 explore this issue when one is estimating the prediction vector and scale parameter respectively. We find that most of our results concur qualitatively with those reported in the literature when the prior information exists as exact equality restrictions. Chapter 9 contains some concluding remarks and a summary of the major results obtained in earlier chapters. We also outline some possible future research topics in this general area

Inequality restricted and pre-test estimation in a mis-specified econometric model

This thesis is concerned with the finite sample properties of some estimators of the unknown parameters in a linear model which is (possibly) mis-specified through the exclusion of relevant regressors. We assume that in addition to sample information, prior information regarding the unknown parameters is available in the form of a linear inequality constraint imposed on the regression coefficients. The combination of this type of prior information and sample information in specifying the corresponding statistical model leads to what has been identified in the literature as the inequality restricted estimator. If the statistical significance of the inequality constraint is tested prior to the estimation process, then the estimator thereby generated is called the inequality pre-test estimator. The properties of these estimators of the coefficient vector in a properly specified model have been examined rather thoroughly in the literature. In this thesis, we extend the results reported in the literature to the case where the underlying regression model is underfitted. We also investigate the sampling performance of the corresponding estimators for the model's disturbance variance, as well as the choice of an optimal size for the pre-test. The general background and motivation for this study are given in Chapter 1. Much of the earlier research on inequality restricted and pre-test estimation are built on results from studies that assume that the prior information is in the form of linear equality restrictions. We survey the relevant literature in this area in Chapter 2. Chapter 3 reviews the literature on inequality restricted and pre-test estimation. We focus on this problem in the context of the standard linear model with a single linear inequality constraint on the coefficient vector, as this is directly related to the theme of this thesis. In Chapter 4, we derive and evaluate the risk, under quadratic loss, of the inequality restricted and pre-test estimators for the regression prediction vector in an underfitted model. This analysis takes the established literature further by allowing for mis-specification in the regressor matrix. We consider the risk of the prediction vector, rather than the coefficient vector itself, so that our results are data independent. The risk functions of the corresponding estimators for the regression disturbance variance in the properly specified and underfitted models are derived in Chapters 6 and 7 respectively. As in the case where the prior information exists as linear equality restrictions, our results show that when the model is underfitted, the use of valid prior information does not necessarily guarantee a reduction in risk. This result holds for the estimation of both the prediction vector and the scale parameter. When one is estimating the regression disturbance variance, with an appropriate choice of test size, the inequality pre-test estimator can uniformly dominate the estimator that uses sample information only. We also find that the risk functions of the estimators of the error variance are affected more by mis-specification than are the corresponding predictive risks. In the case where no strictly dominating estimator exists, the question of the choice of an optimal critical value of the pre-test remains. Chapters 5 and 8 explore this issue when one is estimating the prediction vector and scale parameter respectively. We find that most of our results concur qualitatively with those reported in the literature when the prior information exists as exact equality restrictions. Chapter 9 contains some concluding remarks and a summary of the major results obtained in earlier chapters. We also outline some possible future research topics in this general area

Preliminary-test estimation of a mis-specified linear model with spherically symmetric disturbances

This thesis considers some finite sample properties of preliminary test (pre-test) estimators of the unknown parameters of a (possibly) mis-specified linear regression model. We investigate two types of misspecification which may or may not occur simultaneously. The first relates to the distribution of the regression disturbances, which is assumed to be normal, when in fact, the error distribution belongs to a broader family of spherically symmetric distributions. The second mis-specification is that the model's design matrix may exclude relevant regressors. We analyse some finite sample properties of three pre-test estimators. The first is an estimator of the prediction vector after a pre-test for exact linear restrictions on the location vector. Secondly, we consider an estimator of the error variance after the same pre-test. Finally, we analyse an estimator of the error variance after a pre-test for homogeneity of the variances in the two-sample linear regression model. In each case we extend the existing literature by generalising the model's error distribution and allowing for model mis-specification through the omission of regressors. To provide a setting for this research, we survey the relevant pretesting literature in Chapter Two. This discussion assumes that the errors are normally distributed. There is a body of research, however, which proposes that some economic data series may be generated by processes whose underlying distributions have thicker tails than that which would result from a normality assumption. We briefly examine this literature in Chapter Three. One alternative family of distributions, which has received considerable attention, is the spherically symmetric family of distributions. Well known members of this family include the normal and the multivariate Student-t distributions. So, we include in Chapter Three a rationale for investigating spherically symmetric regression disturbances as an alternative to the usual normality assumption. We also discuss several studies which consider the linear regression model under a spherically symmetric disturbance assumption. Having provided a setting and rationale for our research in Chapters Two and Three, Chapters Four, Five and Six present the finite sample properties of the aforementioned pre-test estimators. In each of these chapters we derive the exact bias and the exact risk functions (under quadratic loss) of the estimators under the mis-specified regression model. We also give the non-null distributions of the commonly used test-statistics for the investigated pre-tests, and we generalise many of the results reported in the existing literature. In particular, we derive the critical values of the test which result in a minimum of the bias and of the risk of the pre-test estimators of the error variance. To illustrate the results we assume multivariate Student-t regression disturbances, rather than the general spherically symmetric family, and numerically evaluate the derived expressions for various cases. Our results suggest, when estimating the prediction vector, that the mis-specification of the distribution of the regression disturbances has little impact on the qualitative properties of the predictor pre-test estimator, though there are quantitative effects. However, when estimating the error variance, after either a pre-test for linear restrictions or for homogeneity of the error variances, we find that mis-specifying the error distribution can have a substantial qualitative, and quantitative, impact on the bias and the risk functions of the estimators. Imposing the linear restrictions, even if they are valid, or always pooling the samples, even if the error variances are identical, may often be inappropriate strategies. The final chapter, Chapter Seven, contains some concluding remarks. In particular, we consider some possible future research topics

Robust Bayesian inference in elliptical regression models

Bayesian Statistics

Small Sample Properties of Estimators and Test Statistics in Nonlinear Regression: The Box-Cox Transformation.

The dissertation will address small sample properties of estimators and test statistics in a nonlinear regression model. The Box-Cox transformation is attractive to economists because a family of functional forms can be compared simultaneously within the framework of classical statistical inference. Usually, maximum likelihood (ML) methods are used to estimate the Box-Cox model. In the present study, nonlinear two stage and iterative generalized least squares (IGLS) method are considered. The accuracy of probability statements concerning nonlinear models is often questionable in small samples. Therefore, the finite sample distribution of the asymptotic t-statistic in the Box-Cox model is derived using an Edgeworth expansion. Bootstrapping, the more practical method for obtaining small sample distributions, is also discussed. ML estimation of Box-Cox transformation suffers from a violation of the usual regularity conditions since the likelihood function of the Box-Cox model is not a proper density function. Since it is required that y\sb t $\u3e$ 0 in order for the Box-Cox transformation to be well-defined, the dependent variable is assumed to have a truncated normal distribution. The asymptotically equivalent covariance matrix estimators and test statistics--Lagrange multiplier, likelihood ratio and Wald--are compared in small samples. The risk superiority of the Stein-rule estimator to the ML estimator is known in the context of the linear model. The usefulness of Stein-like estimation in the nonlinear Box-Cox model is investigated by considering the finite sample risk properties of ML and Stein-like estimators. It is expected that this dissertation will make four major contributions to the current econometric literature. First, we introduce IGLS estimation of the Box-Cox model, and thus make liner statistical inference applicable to the nonlinear model. Second, the exact distribution of asymptotic t-ratios is derived and the bootstrap inversion of Edgeworth expansion is used. Third, the small sample distribution and power properties of three asymptotically equivalent test statistics are investigated. Fourth, shrinkage estimation is used in the determination of functional form