Choice of Parametric Accelerated Life and Proportional Hazards Models for Survival Data: Asymptotic Results

We discuss the impact of misspecifying fully parametric proportional hazards and accelerated life models. For the uncensored case, misspecified accelerated life models give asymptotically unbiased estimates of covariate effect, but the shape and scale parameters depend on the misspecification. The covariate, shape and scale parameters differ in the censored case. Parametric proportional hazards models do not have a sound justification for general use: estimates from misspecified models can be very biased, and misleading results for the shape of the hazard function can arise. Misspecified survival functions are more biased at the extremes than the centre. Asymptotic and first order results are compared. If a model is misspecified, the size of Wald tests will be underestimated. Use of the sandwich estimator of standard error gives tests of the correct size, but misspecification leads to a loss of power. Accelerated life models are more robust to misspecification because of their log-linear form. In preliminary data analysis, practitioners should investigate proportional hazards and accelerated life models; software is readily available for several such models.     

A composite quantile function estimator with applications in bootstrapping

In this note we define a composite quantile function estimator in order to improve the accuracy of the classical bootstrap procedure in small sample setting. The composite quantile function estimator employs a parametric model for modelling the tails of the distribution and uses the simple linear interpolation quantile function estimator to estimate quantiles lying between 1/(n+1) and n/(n+1). The method is easily programmed using standard software packages and has general applicability. It is shown that the composite quantile function estimator improves the bootstrap percentile interval coverage for a variety of statistics and is robust to misspecification of the parametric component. Moreover, it is also shown that the composite quantile function based approach surprisingly outperforms the parametric bootstrap for a variety of small sample situations.     

On measuring sensitivity to parametric model misspecification

In settings where parametric inference is inconsistent under model misspecification, the discrepancy between correct and misspecified inferences is compared with the discrepancy between correct and misspecified models. To make the comparison tractable, large sample and small misspecification approximations are employed. The ratio of the approximate discrepancy between inferences to the approximate discrepancy between models is regarded as a relative measure of sensitivity to model misspecification. The maximum ratio over a family of correct distributions is determined as a measure of worst case sensitivity. As well, the distribution producing this maximum can be examined, to see how a particular combination of a parametric family and estimand is susceptible to model misspecifications.     

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.     

BOOTSTRAP TESTS FOR THE ERROR DISTRIBUTION IN LINEAR AND NONPARAMETRIC REGRESSION MODELS

In this paper we investigate several tests for the hypothesis of a parametric form of the error distribution in the common linear and non‐parametric regression model, which are based on empirical processes of residuals. It is well known that tests in this context are not asymptotically distribution‐free and the parametric bootstrap is applied to deal with this problem. The performance of the resulting bootstrap test is investigated from an asymptotic point of view and by means of a simulation study. The results demonstrate that even for moderate sample sizes the parametric bootstrap provides a reliable and easy accessible solution to the problem of goodness‐of‐fit testing of assumptions regarding the error distribution in linear and non‐parametric regression models.     

On Robustness of Model-Based Bootstrap Schemes in Nonparametric Time Series Analysis

Theory in time series analysis is often developed under the assumption of finite-dimensional models for the data generating process. Whereas corresponding estimators such as those of a conditional mean function are reasonable even if the true dependence mechanism is more complex, it is usually necessary to capture the whole dependence structure asymptotically for the bootstrap to be valid. In contrast, we show that certain simplified bootstrap schemes which imitate only some aspects of the time series are consistent for quantities arising in nonparametric statistics. To this end, we generalize the well-known "whitening by windowing" principle to joint distributions of nonparametric estimators of the autoregression function. Consequently, we obtain that model-based nonparametric bootstrap schemes remain valid for supremum-type functionals as long as they mimic those finite-dimensional joint distributions consistently which determine the quantity of interest. As an application, we show that simple regression-type bootstrap schemes can be applied for the determination of critical values for nonparametric tests of parametric or semiparametric hypotheses on the autoregression function in the context of a general process.     

Estimation and Inference for Linear Panel Data Models Under Misspecification When Both n and T are Large

This article considers fixed effects (FE) estimation for linear panel data models under possible model misspecification when both the number of individuals, n, and the number of time periods, T, are large. We first clarify the probability limit of the FE estimator and argue that this probability limit can be regarded as a pseudo-true parameter. We then establish the asymptotic distributional properties of the FE estimator around the pseudo-true parameter when n and T jointly go to infinity. Notably, we show that the FE estimator suffers from the incidental parameters bias of which the top order is O(T^{? 1}), and even after the incidental parameters bias is completely removed, the rate of convergence of the FE estimator depends on the degree of model misspecification and is either (nT)^{? 1/2} or n^{? 1/2}. Second, we establish asymptotically valid inference on the (pseudo-true) parameter. Specifically, we derive the asymptotic properties of the clustered covariance matrix (CCM) estimator and the cross-section bootstrap, and show that they are robust to model misspecification. This establishes a rigorous theoretical ground for the use of the CCM estimator and the cross-section bootstrap when model misspecification and the incidental parameters bias (in the coefficient estimate) are present. We conduct Monte Carlo simulations to evaluate the finite sample performance of the estimators and inference methods, together with a simple application to the unemployment dynamics in the U.S.     

Empirical characteristic function tests for GARCH innovation distribution using multipliers

Goodness-of-fit tests for the innovation distribution in GARCH models based on measuring deviations between the empirical characteristic function of the residuals and the characteristic function under the null hypothesis have been proposed in the literature. The asymptotic distributions of these test statistics depend on unknown quantities, so their null distributions are usually estimated through parametric bootstrap (PB). Although easy to implement, the PB can become very computationally expensive for large sample sizes, which is typically the case in applications of these models. This work proposes to approximate the null distribution through a weighted bootstrap. The procedure is studied both theoretically and numerically. Its asymptotic properties are similar to those of the PB, but, from a computational point of view, it is more efficient.     

Parametric and semiparametric hypotheses in the linear model

The independent additive errors linear model consists of a structure for the mean and a separate structure for the error distribution. The error structure may be parametric or it may be semiparametric. Under alternative values of the mean structure, the best fitting additive errors model has an error distribution which can be represented as the convolution of the actual error distribution and the marginal distribution of a misspecification term. The model misspecification term results from the covariates' distribution. Conditions are developed to distinguish when the semiparametric model yields sharper inference than the parametric model and vice versa. The main conditions concern the actual error distribution and the covariates' distribution. The theoretical results explain a paradoxical finding in semiparametric Bayesian modelling, where the posterior distribution under a semiparametric model is found to be more concentrated than is the posterior distribution under a corresponding parametric model. The paradox is illustrated on a set of allometric data. The Canadian Journal of Statistics 39: 165–180; 2011 ©2011 Statistical Society of Canada     

Second-order least squares estimation of censored regression models

This paper proposes the second-order least squares estimation, which is an extension of the ordinary least squares method, for censored regression models where the error term has a general parametric distribution (not necessarily normal). The strong consistency and asymptotic normality of the estimator are derived under fairly general regularity conditions. We also propose a computationally simpler estimator which is consistent and asymptotically normal under the same regularity conditions. Finite sample behavior of the proposed estimators under both correctly and misspecified models are investigated through Monte Carlo simulations. The simulation results show that the proposed estimator using optimal weighting matrix performs very similar to the maximum likelihood estimator, and the estimator with the identity weight is more robust against the misspecification.     

Parametric simultaneous robust inferences for regression coefficient under generalized linear models

In this article, the parametric robust regression approaches are proposed for making inferences about regression parameters in the setting of generalized linear models (GLMs). The proposed methods are able to test hypotheses on the regression coefficients in the misspecified GLMs. More specifically, it is demonstrated that with large samples, the normal and gamma regression models can be properly adjusted to become asymptotically valid for inferences about regression parameters under model misspecification. These adjusted regression models can provide the correct type I and II error probabilities and the correct coverage probability for continuous data, as long as the true underlying distributions have finite second moments.     

Recent and classical goodness-of-fit tests for the Poisson distribution

We give a critical synopsis of classical and recent tests for Poissonity, our emphasis being on procedures which are consistent against general alternatives. Two classes of weighted Cramér–von Mises type test statistics, based on the empirical probability generating function process, are studied in more detail. Both of them generalize already known test statistics by introducing a weighting parameter, thus providing more flexibility with regard to power against specific alternatives. In both cases, we prove convergence in distribution of the statistics under the null hypothesis in the setting of a triangular array of rowwise independent and identically distributed random variables as well as consistency of the corresponding test against general alternatives. Therefore, a sound theoretical basis is provided for the parametric bootstrap procedure, which is applied to obtain critical values in a large-scale simulation study. Each of the tests considered in this study, when implemented via the parametric bootstrap method, maintains a nominal level of significance very closely, even for small sample sizes. The procedures are applied to four well-known data sets.     

A CONSISTENT MODEL SPECIFICATION TEST BASED ON THE KERNEL SUM OF SQUARES OF RESIDUALS

This paper constructs a consistent model specification test based on the difference between the nonparametric kernel sum of squares of residuals and the sum of squares of residuals from a parametric null model. We establish the asymptotic normality of the proposed test statistic under the null hypothesis of correct parametric specification and show that the wild bootstrap method can be used to approximate the null distribution of the test statistic. Results from a small simulation study are reported to examine the finite sample performance of the proposed tests.     

Determining the mean-variance relationship in generalized linear models—A parametric robust way

This article introduces a parametric robust way of determining the mean-variance relationship in the setting of generalized linear models. More specifically, the normal likelihood is properly amended to become asymptotically valid even if normality fails. Consequently, legitimate inference for the parametric relationship between mean and variance could be derived under model misspecification. More details are given to the scenario when the variance is proportional to an unknown power of the mean function. The efficacy of the novel technique is demonstrated via simulations and the analysis of two real data sets.     

Linear bootstrap methods for vector autoregressive moving-average models

We provide the theoretical justification of bootstrapping stationary invertible echelon vector autoregressive moving-average (VARMA) models using linear methods. The asymptotic validity of the bootstrap is established with strong white noise under parametric and nonparametric assumptions. Our methods are practical and useful for building reliable simulation-based inference and forecasting without implementing nonlinear estimation techniques such as ML which is usually burdensome, time demanding or impractical, particularly in big or highly persistent systems. The relevance of our procedures is more pronounced in the context of dynamic simulation-based techniques such as maximized Monte Carlo (MMC) tests [see Dufour J-M. Monte Carlo tests with nuisance parameters: a general approach to finite-sample inference and nonstandard asymptotics in econometrics. J Econom. 2006;133(2):443–477 and Dufour J-M, Jouini T. Finite-sample simulation-based tests in VAR models with applications to Granger causality testing. J Econom. 2006;135(1–2):229–254 for the VAR case]. Simulation evidence shows that, compared with conventional asymptotics, our bootstrap methods have good finite-sample properties in approximating the actual distribution of the studentized echelon VARMA parameter estimates, and in providing echelon parameter confidence sets with satisfactory coverage.     

On improving the robustness and reliability of Rao's score test

The results of misspecification tests, based on Rao's score principle, are now routinely reported in applied econometric work. This paper draws together some important recent results which are designed to improve: (a) the robustness of standard score tests; and (b) the reliability of the asymptotic approximations used for inferential purposes. The discussion of robustness includes (i) parametric, (ii) distributional, and (iii) higher-order moment robustness. The issue of finite sample reliability focuses on controlling the size of the score test using (i) different variance estimators in conjunction with standard asymptotic theory, (ii) finite sample corrections obtainable from higher-order asymptotic analysis, and (iii) bootstrap procedures.     

A Goodness-of-Fit Test for a Class of Autoregressive Conditional Duration Models

This article develops a method for testing the goodness-of-fit of a given parametric autoregressive conditional duration model against unspecified nonparametric alternatives. The test statistics are functions of the residuals corresponding to the quasi maximum likelihood estimate of the given parametric model, and are easy to compute. The limiting distributions of the test statistics are not free from nuisance parameters. Hence, critical values cannot be tabulated for general use. A bootstrap procedure is proposed to implement the tests, and its asymptotic validity is established. The finite sample performances of the proposed tests and several other competing ones in the literature, were compared using a simulation study. The tests proposed in this article performed well consistently throughout, and they were either the best or close to the best. None of the tests performed uniformly the best. The tests are illustrated using an empirical example.     

Empirical-distribution-function goodness-of-fit tests for discrete models

We present a simple framework for studying empirical-distribution-function goodness-of-fit tests for discrete models. A key tool is a weak-convergence result for an estimated discrete empirical process, regarded as a random element in some suitable sequence space. Special emphasis is given to the problem of testing for a Poisson model and for the geometric distribution. Simulations show that parametric bootstrap versions of the tests maintain a nominal level of significance very closely even for small samples where reliance upon asymptotic critical values is doubtful.     

Parametric bootstrap and approximate tests for two Poisson variates

The parametric bootstrap tests and the asymptotic or approximate tests for detecting difference of two Poisson means are compared. The test statistics used are the Wald statistics with and without log-transformation, the Cox F statistic and the likelihood ratio statistic. It is found that the type I error rate of an asymptotic/approximate test may deviate too much from the nominal significance level α under some situations. It is recommended that we should use the parametric bootstrap tests, under which the four test statistics are similarly powerful and their type I error rates are all close to α. We apply the tests to breast cancer data and injurious motor vehicle crash data.     

A note reconciling ANOVA tests under unequal error variances

The purpose of this note is to derive simple testing procedures for ANOVA under heteroscedasticity by a single approach that are equivalent to the prior art in the literature obtained by the Parametric Bootstrap and the Generalized Fiducial approach. By similar approach, researchers are encouraged to derive generalized tests in other applications, as alternative to parametric bootstrap tests and fiducial tests, including ANCOVA and MANOVA under heteroscedasticity, especially in Mixed Model applications, where the bootstrap approach fails.