Analysis of Deviance for Hypothesis Testing in Generalized Partially Linear Models

In this study, we develop nonparametric analysis of deviance tools for generalized partially linear models based on local polynomial fitting. Assuming a canonical link, we propose expressions for both local and global analysis of deviance, which admit an additivity property that reduces to analysis of variance decompositions in the Gaussian case. Chi-square tests based on integrated likelihood functions are proposed to formally test whether the nonparametric term is significant. Simulation results are shown to illustrate the proposed chi-square tests and to compare them with an existing procedure based on penalized splines. The methodology is applied to German Bundesbank Federal Reserve data.     

Variance Estimation in Spatial Regression Using a Non-parametric Semivariogram Based on Residuals

Abstract.  The empirical semivariogram of residuals from a regression model with stationary errors may be used to estimate the covariance structure of the underlying process. For prediction (kriging) the bias of the semivariogram estimate induced by using residuals instead of errors has only a minor effect because the bias is small for small lags. However, for estimating the variance of estimated regression coefficients and of predictions, the bias due to using residuals can be quite substantial. Thus we propose a method for reducing this bias. The adjusted empirical semivariogram is then isotonized and made conditionally negative-definite and used to estimate the variance of estimated regression coefficients in a general estimating equations setup. Simulation results for least squares and robust regression show that the proposed method works well in linear models with stationary correlated errors.     

Weighted composite quantile regression analysis for nonignorable missing data using nonresponse instrument

Efficient statistical inference on nonignorable missing data is a challenging problem. This paper proposes a new estimation procedure based on composite quantile regression (CQR) for linear regression models with nonignorable missing data, that is applicable even with high-dimensional covariates. A parametric model is assumed for modelling response probability, which is estimated by the empirical likelihood approach. Local identifiability of the proposed strategy is guaranteed on the basis of an instrumental variable approach. A set of data-based adaptive weights constructed via an empirical likelihood method is used to weight CQR functions. The proposed method is resistant to heavy-tailed errors or outliers in the response. An adaptive penalisation method for variable selection is proposed to achieve sparsity with high-dimensional covariates. Limiting distributions of the proposed estimators are derived. Simulation studies are conducted to investigate the finite sample performance of the proposed methodologies. An application to the ACTG 175 data is analysed.     

Effects of model misspecification on tests of no randomized treatment effect arising from Cox's proportional hazards model

We examine the asymptotic and small sample properties of model-based and robust tests of the null hypothesis of no randomized treatment effect based on the partial likelihood arising from an arbitrarily misspecified Cox proportional hazards model. When the distribution of the censoring variable is either conditionally independent of the treatment group given covariates or conditionally independent of covariates given the treatment group, the numerators of the partial likelihood treatment score and Wald tests have asymptotic mean equal to 0 under the null hypothesis, regardless of whether or how the Cox model is misspecified. We show that the model-based variance estimators used in the calculation of the model-based tests are not, in general, consistent under model misspecification, yet using analytic considerations and simulations we show that their true sizes can be as close to the nominal value as tests calculated with robust variance estimators. As a special case, we show that the model-based log-rank test is asymptotically valid. When the Cox model is misspecified and the distribution of censoring depends on both treatment group and covariates, the asymptotic distributions of the resulting partial likelihood treatment score statistic and maximum partial likelihood estimator do not, in general, have a zero mean under the null hypothesis. Here neither the fully model-based tests, including the log-rank test, nor the robust tests will be asymptotically valid, and we show through simulations that the distortion to test size can be substantial.     

An Empirical Likelihood Estimate of the Finite Population Correlation Coefficient

In this article, the problem of the estimation of finite population correlation coefficient is considered using the empirical likelihood method. A new estimator that makes the use of both the known mean and variance of an auxiliary variable is proposed. The percent relative bias and percent relative efficiency of the proposed new estimator with respect to the usual estimator of the correlation coefficient is investigated through extensive simulation study for values of the correlation coefficient from ?0.90 to +0.90. The proposed estimator is found to perform better than the simple correlation coefficient from both the bias and relative efficiency points of views, for the population, considered in the investigation. At the end, the proposed estimator has been extended to complex survey designs. Supplementary materials for this article are available online.     

Maximum likelihood estimation of linear continuous time long memory processes with discrete time data

Summary.  We develop a new class of time continuous autoregressive fractionally integrated moving average (CARFIMA) models which are useful for modelling regularly spaced and irregu-larly spaced discrete time long memory data. We derive the autocovariance function of a stationary CARFIMA model and study maximum likelihood estimation of a regression model with CARFIMA errors, based on discrete time data and via the innovations algorithm. It is shown that the maximum likelihood estimator is asymptotically normal, and its finite sample properties are studied through simulation. The efficacy of the approach proposed is demonstrated with a data set from an environmental study.     

The impact of integrated measurement errors on modeling long-run macroeconomic time series

ABSTRACT

Data spanning long time periods, such as that over 1860–2012 for the UK, seem likely to have substantial errors of measurement that may even be integrated of order one, but which are probably cointegrated for cognate variables. We analyze and simulate the impacts of such measurement errors on parameter estimates and tests in a bivariate cointegrated system with trends and location shifts which reflect the many major turbulent events that have occurred historically. When trends or shifts therein are large, cointegration analysis is not much affected by such measurement errors, leading to conventional stationary attenuation biases dependent on the measurement error variance, unlike the outcome when there are no offsetting shifts or trends.     

Linear regression analysis with inequality constraints on the regression parameters via empirical likelihood

An empirical likelihood ratio test is developed for testing for or against inequality constraints on regression parameters in linear regression analysis. The proposed approach imposes no parametric model nor identically distributing assumption on the random errors. The asymptotic distribution of the proposed test statistic under null hypothesis is shown to be of chi-bar-squared type. The asymptotic power under contiguous alternatives is also briefly discussed. Moreover, an adjusted empirical likelihood method is adopted to improve the small sample size behaviour of the proposed test. Several simulation studies are carried out to assess the finite sample performance of the proposed tests. The results reveal that the proposed tests could be valuable for improving inference efficiency. A real-life example is discussed to illustrate the theoretical results.     

Conditional and unconditional tests with historical controls

Score statistics utilizing historical control data have been proposed to test for increasing trend in tumour occurrence rates in laboratory carcinogenicity studies. Novel invariance arguments are used to confirm, under slightly weaker conditions, previously established asymptotic distributions (mixtures of normal distributions) of tests unconditional on the tumor response rate in the concurrent control group. Conditioning on the control response rate, an ancillary statistic, leads to a new conditional limit theorem in which the test statistic converges to an unknown random variable. Because of this, a subasymptotic approximation to the conditional limiting distribution is also considered. The adequacy of these large-sample approximations in finite samples is evaluated using computer simulation. Bootstrap methods for use in finite samples are also proposed. The application of the conditional and unconditional tests is illustrated using bioassay data taken from the literature. The results presented in this paper are used to formulate recommendations for the use of tests for trend with historical controls in practice.     

A class of finite mixture of quantile regressions with its applications

Mixture of linear regression models provide a popular treatment for modeling nonlinear regression relationship. The traditional estimation of mixture of regression models is based on Gaussian error assumption. It is well known that such assumption is sensitive to outliers and extreme values. To overcome this issue, a new class of finite mixture of quantile regressions (FMQR) is proposed in this article. Compared with the existing Gaussian mixture regression models, the proposed FMQR model can provide a complete specification on the conditional distribution of response variable for each component. From the likelihood point of view, the FMQR model is equivalent to the finite mixture of regression models based on errors following asymmetric Laplace distribution (ALD), which can be regarded as an extension to the traditional mixture of regression models with normal error terms. An EM algorithm is proposed to obtain the parameter estimates of the FMQR model by combining a hierarchical representation of the ALD. Finally, the iterated weighted least square estimation for each mixture component of the FMQR model is derived. Simulation studies are conducted to illustrate the finite sample performance of the estimation procedure. Analysis of an aphid data set is used to illustrate our methodologies.     

New Robust Variable Selection Methods for Linear Regression Models

Motivated by an entropy inequality, we propose for the first time a penalized profile likelihood method for simultaneously selecting significant variables and estimating unknown coefficients in multiple linear regression models in this article. The new method is robust to outliers or errors with heavy tails and works well even for error with infinite variance. Our proposed approach outperforms the adaptive lasso in both theory and practice. It is observed from the simulation studies that (i) the new approach possesses higher probability of correctly selecting the exact model than the least absolute deviation lasso and the adaptively penalized composite quantile regression approach and (ii) exact model selection via our proposed approach is robust regardless of the error distribution. An application to a real dataset is also provided.     

On Estimation and Tests of Time-Varying Effects in the Proportional Hazards Model

Abstract.  Cox's proportional hazards model is routinely used in many applied fields, some times, however, with too little emphasis on the fit of the model. In this paper, we suggest some new tests for investigating whether or not covariate effects vary with time. These tests are a natural and integrated part of an extended version of the Cox model. An important new feature of the suggested test is that time constancy for a specific covariate is examined in a model, where some effects of other covariates are allowed to vary with time and some are constant; thus making successive testing of time-dependency possible. The proposed techniques are illustrated with the well-known Mayo liver disease data, and a small simulation study investigates the finite sample properties of the tests.     

Maximum Likelihood Unit Root Testing in the Presence of GARCH: A New Test with Increased Power

The literature on testing the unit root hypothesis in the presence of GARCH errors is extended. A new test based upon the combination of local-to-unity detrending and joint maximum likelihood estimation of the autoregressive parameter and GARCH process is presented. The finite sample distribution of the test is derived under alternative decisions regarding the deterministic terms employed. Using Monte Carlo simulation, the newly proposed ML t-test is shown to exhibit increased power of relative to rival tests. Finally, the empirical relevance of the simulation results is illustrated via an application to real GDP for the UK.     

Nonlinear IV panel unit root testing under structural breaks in the error variance

The paper examines the behavior of a generalized version of the nonlinear IV unit root test proposed by Chang (2002) when the series’ errors exhibit nonstationary volatility. The leading case of such nonstationary volatility concerns structural breaks in the error variance. We show that the generalized test is not robust to variance changes in general, and illustrate the extent of the resulting size distortions in finite samples. More importantly, we show that pivotality is recovered when using Eicker-White heteroskedasticity-consistent standard errors. This contrasts with the case of Dickey-Fuller unit root tests, for which Eicker-White standard errors do not produce robustness and thus require computationally costly corrections such as the (wild) bootstrap or estimation of the so-called variance profile. The pivotal versions of the generalized IV tests – with or without the correct standard errors – do however have no power in $1/T$ -neighbourhoods of the null. We also study the validity of panel versions of the tests considered here.     

UNIT ROOT TESTING IN THE PRESENCE OF HEAVY‐TAILED GARCH ERRORS

We derive the asymptotic distributions of the Dickey–Fuller (DF) and augmented DF (ADF) tests for unit root processes with Generalized Autoregressive Conditional Heteroscedastic (GARCH) errors under fairly mild conditions. We show that the asymptotic distributions of the DF tests and ADF t‐type test are the same as those obtained in the independent and identically distributed Gaussian cases, regardless of whether the fourth moment of the underlying GARCH process is finite or not. Our results go beyond earlier ones by showing that the fourth moment condition on the scaled conditional errors is totally unnecessary. Some Monte Carlo simulations are provided to illustrate the finite‐sample‐size properties of the tests.     

A Comparison of Unit-Root Test Criteria

During the past 15 years, the ordinary least squares estimator and the corresponding pivotal statistic have been widely used for testing the unit-root hypothesis in autoregressive processes. Recently, several new criteria, based on maximum likelihood estimators and weighted symmetric estimators, have been proposed. In this article, we describe several different test criteria. Results from a Monte Carlo study that compares the power of the different criteria indicate that the new tests are more powerful against the stationary alternative. Of the procedures studied, the weighted symmetric estimator and the unconditional maximum likelihood estimator provide the most powerful tests against the stationary alternative. As an illustration, the weekly series of one-month treasury-bill rates is analyzed.     

Testing for the Equality of Two Autoregressive Functions Using Quasi-Residuals

This article discusses the problem of testing the equality of two nonparametric autoregressive functions against one-sided alternatives. The heteroscedastic errors and stationary densities of the two independent strong mixing strictly stationary time series can be possibly different. The article adapts the idea of using sum of quasi-residuals to construct the test and derives its asymptotic null distributions. The article also shows that the test is consistent for general alternatives and obtains its limiting distributions under a sequence of local alternatives. Then a Monte Carlo simulation is conducted to study the finite sample level and power behavior of these tests at some alternatives. We also compare the test to an existing lag matched test theoretically and by Monte Carlo experiments.     

Asymptotic Theory for the QMLE in GARCH-X Models With Stationary and Nonstationary Covariates

This article investigates the asymptotic properties of the Gaussian quasi-maximum-likelihood estimators (QMLE’s) of the GARCH model augmented by including an additional explanatory variable—the so-called GARCH-X model. The additional covariate is allowed to exhibit any degree of persistence as captured by its long-memory parameter d_x; in particular, we allow for both stationary and nonstationary covariates. We show that the QMLE’s of the parameters entering the volatility equation are consistent and mixed-normally distributed in large samples. The convergence rates and limiting distributions of the QMLE’s depend on whether the regressor is stationary or not. However, standard inferential tools for the parameters are robust to the level of persistence of the regressor with t-statistics following standard Normal distributions in large sample irrespective of whether the regressor is stationary or not. Supplementary materials for this article are available online.     

A Semiparametric Regression Model for Longitudinal Data with Non‐stationary Errors

Motivated by the need to analyze the National Longitudinal Surveys data, we propose a new semiparametric longitudinal mean‐covariance model in which the effects on dependent variable of some explanatory variables are linear and others are non‐linear, while the within‐subject correlations are modelled by a non‐stationary autoregressive error structure. We develop an estimation machinery based on least squares technique by approximating non‐parametric functions via B‐spline expansions and establish the asymptotic normality of parametric estimators as well as the rate of convergence for the non‐parametric estimators. We further advocate a new model selection strategy in the varying‐coefficient model framework, for distinguishing whether a component is significant and subsequently whether it is linear or non‐linear. Besides, the proposed method can also be employed for identifying the true order of lagged terms consistently. Monte Carlo studies are conducted to examine the finite sample performance of our approach, and an application of real data is also illustrated.     

Restricted likelihood ratio lack-of-fit tests using mixed spline models

Summary.  Penalized regression spline models afford a simple mixed model representation in which variance components control the degree of non-linearity in the smooth function estimates. This motivates the study of lack-of-fit tests based on the restricted maximum likelihood ratio statistic which tests whether variance components are 0 against the alternative of taking on positive values. For this one-sided testing problem a further complication is that the variance component belongs to the boundary of the parameter space under the null hypothesis. Conditions are obtained on the design of the regression spline models under which asymptotic distribution theory applies, and finite sample approximations to the asymptotic distribution are provided. Test statistics are studied for simple as well as multiple-regression models.