A Lagrange multiplier test for testing the adequacy of constant conditional correlation GARCH model

ABSTRACT

A Lagrange multiplier test for testing the parametric structure of a constant conditional correlation-generalized autoregressive conditional heteroskedasticity (CCC-GARCH) model is proposed. The test is based on decomposing the CCC-GARCH model multiplicatively into two components, one of which represents the null model, whereas the other one describes the misspecification. A simulation study shows that the test has good finite sample properties. We compare the test with other tests for misspecification of multivariate GARCH models. The test has high power against alternatives where the misspecification is in the GARCH parameters and is superior to other tests. The test is not greatly affected by misspecification in the conditional correlations and is therefore well suited for considering misspecification of GARCH equations.     

Testing for a change lit the regression matrix of a multivariate linear model

A Bayesian test procedure Is developed to test; the null hypothesis of no change In the regression matrix of a multivariate lin¬ear model against the alternative hypothesis of exactly one change The resulting test is based on the marginal posterior distribution of the change point; To illustrate the test procedure a numerical example using a bivariate regression model is considered.     

CONDITIONAL INFERENCE ABOUT AN EQUICORRELATION COEFFICIENT

SenGupta (1987) proposed a locally most powerful test which is globally (one sided) unbiased, and an estimator of p, the equicorrelation coefficient of a standard symmetric multivariate normal (SSMN) distribution. Here we use the idea in Williams (1984) to illustrate the construction and use of ancillary statistics to make inference about p. The test and confidence intervals based on this construction are conditionally optimal.     

Bayesian outlier testing using the predictive distribution for a linear model op constant intraclass form

The problem of testing suspected outliers from a linear model with constant intraclass correlation is considered from a Bayesian viewpoint. The main objective of this paper is to develop an outlier test procedure based on the predictive distribution of suspected outlier observations given a set of existing inlier observations. The test procedure is easily performed with the usual F and t distributions.     

Mixture inverse Gaussian for unobserved heterogeneity in the autoregressive conditional duration model

In this paper, we assume that the duration of a process has two different intrinsic components or phases which are independent. The first is the time it takes for a trade to be initiated in the market (for example, the time during which agents obtain knowledge about the market in which they are operating and accumulate information, which is coherent with Brownian motion) and the second is the subsequent time required for the trade to develop into a complete duration. Of course, if the first time is zero then the trade is initiated immediately and no initial knowledge is required. If we assume a specific compound Bernoulli distribution for the first time and an inverse Gaussian distribution for the second, the resulting convolution model has a mixture of an inverse Gaussian distribution with its reciprocal, which allows us to specify and test the unobserved heterogeneity in the autoregressive conditional duration (ACD) model.

Our proposals make it possible not only to capture various density shapes of the durations but also easily to accommodate the behaviour of the tail of the distribution and the non monotonic hazard function. The proposed model is easy to fit and characterizes the behaviour of the conditional durations reasonably well in terms of statistical criteria based on point and density forecasts.     

Approximate conditional inference and the linear functional model

Approximate conditional inference is developed for the slope parameter of the linear functional model with two variables. It is shown that the model can be transformed so that the slope parameter becomes an angle and nuisance parameters are radial distances. If the nuisance parameters are known an exact confidence interval based on a location-type conditional distribution is available for the angle. More gen¬erally, confidence distributions are used to average the conditional distribution over the nuisance parameters yielding an approximate conditional confidence interval that reflects the precision indicated by the data. An example is analyzed.     

Multiple linear regression model with stochastic design variables

In a simple multiple linear regression model, the design variables have traditionally been assumed to be non-stochastic. In numerous real-life situations, however, they are stochastic and non-normal. Estimators of parameters applicable to such situations are developed. It is shown that these estimators are efficient and robust. A real-life example is given.     

A semi-parametric additive model for variance heterogeneity

This paper presents a flexible model for variance heterogeneity in a normal error model. Specifically, both the mean and variance are modelled using semi-parametric additive models. We call this model a Mean And Dispersion Additive Model (MADAM). A successive relaxation algorithm for fitting the model is described and justified as maximizing a penalized likelihood function with penalties for lack of smoothness in the additive non-parametric functions in both mean and variance models. The algorithm is implemented in GLIM4, allowing flexible and interactive modelling of variance heterogeneity. Two data sets are used for demonstration.     

Testing for lack of fit in linear multiresponse models based on exact or near replicates

Three procedures for testing the adequacy of a proposed linear multiresponse regression model against unspecified general alternatives are considered. The model has an error structure with a matrix normal distribution which allows the vector of responses for a particular run to have an unknown covariance matrix while the responses for different runs are uncorrelated. Furthermore, each response variable may be modeled by a separate design matrix. Multivariate statistics corresponding to the classical univariate lack of fit and pure error sums of squares are defined and used to determine the multivariate lack of fit tests. A simulation study was performed to compare the power functions of the test procedures in the case of replication. Generalizations of the tests for the case in which there are no independent replicates on all responses are also presented.     

A class of rectangle-screened multivariate normal distributions and its applications

A screening problem is tackled by proposing a parametric class of distributions designed to match the behavior of the partially observed screened data. This class is obtained from the nontruncated marginal of the rectangle-truncated multivariate normal distributions. Motivations for the screened distribution as well as some of the basic properties, such as its characteristic function, are presented. These allow us a detailed exploration of other important properties that include closure property in linear transformation, in marginal and conditional operations, and in a mixture operation as well as the first two moments and some sampling distributions. Various applications of these results to the statistical modelling and data analysis are also provided.     

A test for the parametric form of the variance function in a partial linear regression model

We consider the problem of testing for a parametric form of the variance function in a partial linear regression model. A new test is derived, which can detect local alternatives converging to the null hypothesis at a rate n^-1/2$n^{-1/2}$ and is based on a stochastic process of the integrated variance function. We establish weak convergence to a Gaussian process under the null hypothesis, fixed and local alternatives. In the special case of testing for homoscedasticity the limiting process is a scaled Brownian bridge. We also compare the finite sample properties with a test based on an L²$L^2$-distance, which was recently proposed by You and Chen [2005. Testing heteroscedasticity in partially linear regression models. Statist. Probab. Lett. 73, 61–70].     

A comparison of parametric conditional error-rate estimators for the two-group linear discriminant function

Using mean absolute deviation, we compare the efficay of two new parametric conditional error rate estimators with six others, four of which are well known.The performance of both new estimators is found to be superior to the six competing estimators examined in this paper, especially when the ratio of the training sample size to the feature dimensionality is small.     

A comparison of selection procedures for selecting the best normal population under heterogeneity of variance

This study examines the comparative probabilities of making a correct selection when using the means procedure (M), the medians procedure (D) and the rank-sum procedure (S) to correctly select the normal population with the largest mean under heterogeneity of variance. The comparison is conducted by using Monte-Carlo simulation techniques for 3, 4, and 5 normal populations under the condition that equal sample sizes are taken from each population. The population means and standard deviations are assumed to be equally-spaced. Two types of heterogeneity of variance are considered: (1) associating larger means with larger variances, and (2) associating larger means with smaller variances.     

A test for variance-covarianch parameters in normal linear models

This paper derives a test statistic for the variance-covariance parameters which is a quadratic function of their MINQUE (Minimum Norm Quadratic Unbiased Estimation) estimates. The test is a Wald-type test, and its development closely parallels the theory used to derive a similar test for the coefficients in linear models. In fact, the derivation proceeds by first setting up the estimation problem in a derived linear model in which the dispersion parameters are the coefficients. The test statistic is shown to be the sum of the squares of independent standardized x² variables.     

A bayesian analysis of the mixed linear model

The mixed model is defined. The exact posterior distribution for the fixed effect vector is obtained. The exact posterior distribution for the error variance is obtained. The exact posterior mean and variance of a Bayesian estimator for the variances of random effects is also derived. All computations are non-iterative and avoid numerical integrations.     

Testing linear inequality constraints in the standard linear model

In this paper we are concerned with the problem of testing whether the â-parameters of the standard linear model satisfy the linear equality constraints R = r when they are known to satisfy the corresponding linear inequality constraints Râ ? r. In particular we will show that the exact finite sample null distributions of the Likelihood Ratio, Wald and Kuhn-Tucker

statistics are known when R is of full row rank but not known when R has less than full row rank. The less than full row rank problem has not been discussed previously but it is of considerable potential importance.

This paper contains several simple numerical examples which illustrate the computational details of the tests     

Correlated linear models for the analysis of familial correlations

The problems of estimation and hypotheses testing on the parameters of two correlated linear models are discussed. Such models are known to have direct applications in epidemiologic research, particularly in the field of family studies. When the data are unbalanced, the maximum-likelihood estimation of the parameters is achieved by adopting a fairly simple numerical algorithm. The asymptotic variances and covariances of the estimators are derived, and the procedures are illustrated on arterial-blood-pressure data from the literature.     

A Unified Approach to the Approximation of Multivariate Densities

Approximation of a density by another density is considered in the case of different dimensionalities of the distributions. The results have been derived by inverting expansions of characteristic functions with the help of matrix techniques. The approximations obtained are all functions of cumulant differences and derivatives of the approximating density. The multivariate Edgeworth expansion follows from the results as a special case. Furthermore, the density functions of the trace and eigenvalues of the sample covariance matrix are approximated by the multivariate normal density and a numerical example is given     

Multivariate tail conditional expectation for scale mixtures of skew-normal distribution

In practice, a financial or actuarial data set may be a skewed or heavy-tailed and this motivates us to study a class of distribution functions in risk management theory that provide more information about these characteristics resulting in a more accurate risk analysis. In this paper, we consider a multivariate tail conditional expectation (MTCE) for multivariate scale mixtures of skew-normal (SMSN) distributions. This class of distributions contains skewed distributions and some members of this class can be used to analyse heavy-tailed data sets. We also provide a closed form for TCE in a univariate skew-normal distribution framework. Numerical examples are also provided for illustration.     

Bartlett-corrected tests for normal linear models when the error covariance matrix is nonscaiar

This paper provides Bartlett corrections to improve likelihood ratio tests for heteroskedastic normal linear models when the error covariance matrix is nonscaiar and depends on a set of unknown parameters. The Bartlett corrections are simple enough to be used algebraically to obtain several closed-form expressions in special cases. The corrections have also advantages for numerical purposes because they involve only simple operations on matrices and vectors.