Simultaneous optimal estimation in linear mixed models

Simultaneous optimal estimation in linear mixed models is considered. A necessary and sufficient condition is presented for the least squares estimator of the fixed effects and the analysis of variance estimator of the variance components to be of uniformly minimum variance simultaneously in a general variance components model. That is, the matrix obtained by orthogonally projecting the covariance matrix onto the orthogonal complement space of the column space of the design matrix is symmetric, each eigenvalue of the matrix is a linear combinations of the variance components and the number of all distinct eigenvalues of the matrix is equal to the the number of the variance components. Under this condition, uniformly optimal unbiased tests and uniformly most accurate unbiased confidence intervals are constructed for the parameters of interest. A necessary and sufficient condition is also given for the equivalence of several common estimators of variance components. Two examples of their application are given.     

Multivariate ‘analysis of dispersion’ in the presence of intra-class correlation

Summary Subsequent to a review of the effects of familial or intra-class correlation (=ϱ) on the univariateF, or analysis of variance tests, and of methods for obtaining confidence limits for ϱ, results are presented on the effects of familial correlations in tests in multivariate ‘analysis of dispersion’. Methods for obtaining confidence limits are given in the case where a common variance-covariance matrix may be assumed for the successive multivariate samples.     

The determination of expected mean squares for anova in balanced designs

Summary This paper deals with the construction of expected mean squares in the analysis of variance. Rather simple rules, which cover the experimental case with unit-treatment-interactions and an arbitrary number of nested blockfactors are given. The rules are applied to an example from steel industry. From the rules it can be seen that as soon as the factor of experimental plots (units) or at least one of certain blockfactors is random with infinitely many levels in the population one can use the simple expressions, given in textbooks, for the case with no unit-treatment-interactions.     

On estimation of variance components*

In this survey paper the estimation of variance components is given. The least squares approach in variance component estimation is a unifying principle which includes the analysis of variance estimators and the MINQUE. When normality is assumed the maximum likelihood estimators can be used. Many variance component estimators are not permissible because they are not non-negative. The development of non-negative variance component estimators is indicated.     

Functional Principal Component Regression and Functional Partial Least‐squares Regression: An Overview and a Comparative Study

Functional data analysis is a field of growing importance in Statistics. In particular, the functional linear model with scalar response is surely the model that has attracted more attention in both theoretical and applied research. Two of the most important methodologies used to estimate the parameters of the functional linear model with scalar response are functional principal component regression and functional partial least‐squares regression. We provide an overview of estimation methods based on these methodologies and discuss their advantages and disadvantages. We emphasise that the role played by the functional principal components and by the functional partial least‐squares components that are used in estimation appears to be very important to estimate the functional slope of the model. A functional version of the best subset selection strategy usual in multiple linear regression is also analysed. Finally, we present an extensive comparative simulation study to compare the performance of all the considered methodologies that may help practitioners in the use of the functional linear model with scalar response.     

Optimal nested row-column designs for blocks of size 2×2 under dependence

This paper examines nested 2 ×2 row-column designs when within-block observations are assumed to be dependent. The model considered has fixed block effects, which may also include row and/or column effects. Optimal binary and non-binary designs, constructed from semi-balanced arrays, are given under both generalised and ordinary least squares estimation. It is shown that binary designs are optimal when dependence is low. In general, however, the optimal designs are highly specific to the correlation values. Received: October 1999     

The impact of major global events on volatility shifts: Evidence from the Asian crisis and 9/11

We analyze whether the Asian crisis and the terrorist attacks of September 11 caused permanent volatility shifts in the world stock markets. In doing so, we test for the presence of structural breaks in volatility during 1997–2002 by resorting to the iterative cumulative sum of squares (ICSS) algorithm and wavelet-based variance analysis.We find that the number of shifts detected by the two methods decreases substantially when both correlated volatility and inertia are taken into account. Specifically, the ICSS algorithm fails to find any breakpoints, while a wavelet-based variance test detects breakpoints at the high-frequency components of the filtered data.     

Quasi-likelihood inference in a generalized linear mixed model for balanced data

For modelling the effect of crossed, fixed factors on the response variable in balanced designs with nested stratifications, a generalized linear mixed model is proposed. This model is based on a set of quasi-likelihood assumptions which imply quadratic variance functions. From these variance functions, deviances are obtained to quantify the variation per stratification. The effects of the fixed factors will be tested, an dispersion components will be estimated. The practical use of the model is illustrated by reanalysing a soldering failures problem.     

A simple approach for the analysis of generalizea linear mixed models

A broad class of generalized linear mixed models, e.g. variance components models for binary data, percentages or count data, will be introduced by incorporating additional random effects into the linear predictor of a generalized linear model structure. Parameters are estimated by a combination of quasi-likelihood and iterated MINQUE (minimum norm quadratic unbiased estimation), the latter being numerically equivalent to REML (restricted, or residual, maximum likelihood). First, conditional upon the additional random effects, observations on a working variable and weights are derived by quasi-likelihood, using iteratively re-weighted least squares. Second, a linear mixed model is fitted to the working variable, employing the weights for the residual error terms, by iterated MINQUE. The latter may be regarded as a least squares procedure applied to squared and product terms of error contrasts derived from the working variable. No full distributional assumptions are needed for estimation. The model may be fitted with standardly available software for weighted regression and REML.     

Efficiency of iterated WLS in the linear model with completely unknown heteroskedasticity

Iterated weighted least squares (IWLS) is investigated for estimating the regression coefficients in a linear model with symmetrically distributed errors. The variances of the errors are not specified; it is not assumed that they are unknown functions of the explanatory variables nor that they are given in some parametric way.
IWLS is carried out in a random number of steps, of which the first one is OLS. In each step the error variance at time t is estimated with a weighted sum of m squared residuals in the neighbourhood of t and the coefficients are estimated using WLS. Furthermore an estimate of the co-variance matrix is obtained. If this estimate is minimal in some way the iteration process is stopped.
Asymptotic properties of IWLS are derived for increasing sample size n . Some particular cases show that the asymptotic efficiency can be increased by allowing more than two steps. Even asymptotic efficiency with respect to WLS with the true error variances can be obtained if m is not fixed but tends to infinity with n and if the heteroskedasticity is smooth.     

Robust Non‐nested Testing for Ordinary Least Squares Regression when Some of the Regressors are Lagged Dependent Variables*

The problem of testing non‐nested regression models that include lagged values of the dependent variable as regressors is discussed. It is argued that it is essential to test for error autocorrelation if ordinary least squares and the associated J and F tests are to be used. A heteroskedasticity–robust joint test against a combination of the artificial alternatives used for autocorrelation and non‐nested hypothesis tests is proposed. Monte Carlo results indicate that implementing this joint test using a wild bootstrap method leads to a well‐behaved procedure and gives better control of finite sample significance levels than asymptotic critical values.     

The Use of Hierarchical Linear Modeling to Address Lack-of-Independence in Empirical Auditing Research

Prior empirical auditing research has typically used linear regression analysis to analyze auditor relationships. However, because audit firms, audit partners, and audit clients are nested and clustered, data on them lacks independence, and violates the assumptions necessary for valid tests using simple linear regressions. This deficiency can be overcome by employing the hierarchical linear modeling (HLM) technique to conduct empirical tests. We illustrate this by employing HLM to explain the relationship between audit quality and audit firm, and audit partner tenure. We show that employing HLM yields different results than those found using ordinary least squares.     

The analysis of ranked data in blocked factorial experiments

A non-parametric method for the analysis of blocked factorial experiments, based on ranking within blocks, is proposed and shown to be equivalent to partitioning Friedman's test statistic into a set of contrasts reflecting polynomial components of the main effects and interaction. A slightly modified version of the procedure is suggested to partially overcome the problem of loss of power to detect one component when the model includes other components. This alternative procedure is shown to be equivalent to applying a standard normal theory analysis of variance to the ranks. The null distributions and power comparisons are investigated using simulation methods, and it is shown that the non-parametric methods are almost as powerful as the analysis of variance. Received: February 1999     

Designs for Variance Components Estimation: Past and Present

This article provides (1) a comprehensive coverage of the literature on designs for estimating variance components, and (2) a review of recent applications of such designs in genetics, statistical process control, and quality improvement. In addition, recent methods of estimation of variance components and model forms, other than the linear, are discussed. The latter developments have a direct effect on the choice of design. Some suggested ideas for future research directions are also given.     

Testing for unit roots in time series models with non-stationary volatility

Many of the key macro-economic and financial variables in developed economies are characterized by permanent volatility shifts. It is known that conventional unit root tests are potentially unreliable in the presence of such behaviour, depending on a particular function (the variance profile) of the underlying volatility process. Somewhat surprisingly then, very little work has been undertaken to develop unit root tests which are robust to the presence of permanent volatility shifts. In this paper we fill this gap in the literature by proposing tests which are valid in the presence of a quite general class of permanent variance changes which includes single and multiple (abrupt and smooth-transition) volatility change processes as special cases. Our solution uses numerical methods to simulate the asymptotic null distribution of the statistics based on a consistent estimate of the variance profile which we also develop. The practitioner is not required to specify a parametric model for volatility. An empirical illustration using producer price inflation series from the Stock–Watson database is reported.     

Exact Skewness–Kurtosis Tests for Multivariate Normality and Goodness‐of‐Fit in Multivariate Regressions with Application to Asset Pricing Models*

We study the problem of testing the error distribution in a multivariate linear regression (MLR) model. The tests are functions of appropriately standardized multivariate least squares residuals whose distribution is invariant to the unknown cross‐equation error covariance matrix. Empirical multivariate skewness and kurtosis criteria are then compared with a simulation‐based estimate of their expected value under the hypothesized distribution. Special cases considered include testing multivariate normal and stable error distributions. In the Gaussian case, finite‐sample versions of the standard multivariate skewness and kurtosis tests are derived. To do this, we exploit simple, double and multi‐stage Monte Carlo test methods. For non‐Gaussian distribution families involving nuisance parameters, confidence sets are derived for the nuisance parameters and the error distribution. The tests are applied to an asset pricing model with observable risk‐free rates, using monthly returns on New York Stock Exchange (NYSE) portfolios over 5‐year subperiods from 1926 to 1995.     

Expected Precision of Estimation and Probability of Ruling Out a Hypothesis Based on a Confidence Interval

Interval estimation is an important objective of most experimental and observational studies. Knowing at the design stage of the study how wide the confidence interval (CI) is expected to be and where its limits are expected to fall can be very informative. Asymptotic distribution of the confidence limits can also be used to answer complex questions of power analysis by computing power as probability that a CI will exclude a given parameter value. The CI‐based approach to power and methods of calculating the expected size and location of asymptotic CIs as a measure of expected precision of estimation are reviewed in the present paper. The theory is illustrated with commonly used estimators, including unadjusted risk differences, odds ratios and rate ratios, as well as more complex estimators based on multivariable linear, logistic and Cox regression models. It is noted that in applications with the non‐linear models, some care must be exercised when selecting the appropriate variance expression. In particular, the well‐known ‘short‐cut’ variance formula for the Cox model can be very inaccurate under unequal allocation of subjects to comparison groups. A more accurate expression is derived analytically and validated in simulations. Applications with ‘exact’ CIs are also considered.     

A note on the equivalence of specification tests in the two-factor multivariate variance components model

This note offers a generalization of Hausman and Taylor's equivalence of specification tests in the single-equation variance (error) components model to the two-factor multivariate variance components case. The relationship between the specification tests and the hypothesis test in the model proposed by Mundlak is also discussed.     

Nonparametric estimation of structural change points in volatility models for time series

We propose a hybrid estimation procedure that combines the least squares and nonparametric methods to estimate change points of volatility in time series models. Its main advantage is that it does not require any specific form of marginal or transitional densities of the process. We also establish the asymptotic properties of the estimators when the regression and conditional volatility functions are not known. The proposed tests for change points of volatility are shown to be consistent and more powerful than the nonparametric ones in the literature. Finally, we provide simulations and empirical results using the Hong Kong stock market index (HSI) series.     

Heteroskedasticity and non-normality robust LM tests for spatial dependence

The standard LM tests for spatial dependence in linear and panel regressions are derived under the normality and homoskedasticity assumptions of the regression disturbances. Hence, they may not be robust against non-normality or heteroskedasticity of the disturbances. Following Born and Breitung (2011), we introduce general methods to modify the standard LM tests so that they become robust against heteroskedasticity and non-normality. The idea behind the robustification is to decompose the concentrated score function into a sum of uncorrelated terms so that the outer product of gradient (OPG) can be used to estimate its variance. We also provide methods for improving the finite sample performance of the proposed tests. These methods are then applied to several popular spatial models. Monte Carlo results show that they work well in finite sample.