Quasi-likelihood Ratio Statistic Research Articles

Hypotheses Testing from Complex Survey Data Using Bootstrap Weights: A Unified Approach

ABSTRACT Standard statistical methods without taking proper account of the complexity of a survey design can lead to erroneous inferences when applied to survey data due to unequal selection probabilities, clustering, and other design features. In particular, the Type I error rates of hypotheses tests using standard methods can be much larger than the nominal significance level. Methods incorporating design features in testing hypotheses have been proposed, including Wald tests and quasi-score tests that involve estimated covariance matrices of parameter estimates. In this article, we present a unified approach to hypothesis testing without requiring estimated covariance matrices or design effects, by constructing bootstrap approximations to quasi-likelihood ratio statistics and quasi-score statistics and establishing its asymptotic validity. The proposed method can be easily implemented without a specific software designed for complex survey sampling. We also consider hypothesis testing for categorical data and present a bootstrap procedure for testing simple goodness of fit and independence in a two-way table. In simulation studies, the Type I error rates of the proposed approach are much closer to their nominal significance level compared with the naive likelihood ratio test and quasi-score test. An application to an educational survey under a logistic regression model is also presented. Supplementary materials for this article are available online.

Simple Adaptive Size-Exact Testing for Full-Vector and Subvector Inference in Moment Inequality Models

Abstract We propose a simple test for moment inequalities that has exact size in normal models with known variance and has uniformly asymptotically exact size under asymptotic normality. The test compares the quasi-likelihood ratio statistic to a chi-squared critical value, where the degree of freedom is the rank of the inequalities that are active in finite samples. The test requires no simulation and thus is computationally fast and especially suitable for constructing confidence sets for parameters by test inversion. It uses no tuning parameter for moment selection and yet still adapts to the slackness of the moment inequalities. Furthermore, we show how the test can be easily adapted to inference on subvectors in the common empirical setting of conditional moment inequalities with nuisance parameters entering linearly. User-friendly Matlab code to implement the test is provided.

Penalized sieve GEL for weighted average derivatives of nonparametric quantile IV regressions

This paper considers estimation and inference for a weighted average derivative (WAD) of a nonparametric quantile instrumental variables regression (NPQIV). NPQIV is a non-separable and nonlinear ill-posed inverse problem, which might be why there is no published work on the asymptotic properties of any estimator of its WAD. We first characterize the semiparametric efficiency bound for a WAD of a NPQIV, which, unfortunately, depends on an unknown conditional derivative operator and hence an unknown degree of ill-posedness, making it difficult to know if the information bound is singular or not. In either case, we propose a penalized sieve generalized empirical likelihood (GEL) estimation and inference procedure, which is based on the unconditional WAD moment restriction and an increasing number of unconditional moments that are implied by the conditional NPQIV restriction, where the unknown quantile function is approximated by a penalized sieve. Under some regularity conditions, we show that the self-normalized penalized sieve GEL estimator of the WAD of a NPQIV is asymptotically standard normal. We also show that the quasi likelihood ratio statistic based on the penalized sieve GEL criterion is asymptotically chi-square distributed regardless of whether or not the information bound is singular.

Robust inference for threshold regression models

This paper considers robust inference in threshold regression models when the practitioners do not know whether at the threshold point the true specification has a kink or a jump, nesting previous works that assume either continuity or discontinuity at the threshold. We find that the parameter values under the kink restriction are irregular points of the Hessian matrix, destroying the asymptotic normality and inducing the cube-root convergence rate for the threshold estimate. However, we are able to obtain the same asymptotic distribution as Hansen (2000) for the quasi-likelihood ratio statistic for the unknown threshold. We propose to construct confidence intervals for the threshold by bootstrap test inversion. Finite sample performances of the proposed procedures are examined through Monte Carlo simulations and an economic empirical application is given.

MCMC Con dence Sets for Identi ed Sets

In complicated/nonlinear parametric models, it is generally hard to determine whether the model parameters are (globally) point identified. We provide computationally attractive procedures to construct confidence sets (CSs) for identified sets of parameters in econometric models defined through a likelihood or a vector of moments. The CSs for the identified set or for a function of the identified set (such as a subvector) are based on inverting an optimal sample criterion (such as likelihood or continuously updated GMM), where the cutoff values are computed via Monte Carlo simulations directly from a quasi posterior distribution of the criterion. We establish new Bernstein-von Mises type theorems for the posterior distributions of the quasi-likelihood ratio (QLR) and profile QLR statistics in partially identified models, allowing for singularities. These results imply that the Monte Carlo criterion-based CSs have correct frequentist coverage for the identified set as the sample size increases, and that they coincide with Bayesian credible sets based on inverting a LR statistic for point-identified likelihood models. We also show that our Monte Carlo optimal criterion-based CSs are uniformly valid over a class of data generating processes that include both partially- and point-identified models. We demonstrate good finite sample coverage properties of our proposed methods in four non-trivial simulation experiments: missing data, entry game with correlated payoff shocks, Euler equation and finite mixture models. Finally, our proposed procedures are applied in two empirical examples.

Conditional Inference With a Functional Nuisance Parameter

This paper shows that the problem of testing hypotheses in moment condition models without any assumptions about identification may be considered as a problem of testing with an infinite-dimensional nuisance parameter. We introduce a sufficient statistic for this nuisance parameter in a Gaussian problem and propose conditional tests. These conditional tests have uniformly correct asymptotic size for a large class of models and test statistics. We apply our approach to construct tests based on quasi-likelihood ratio statistics, which we show are efficient in strongly identified models and perform well relative to existing alternatives in two examples.

Structural-break models under mis-specification: Implications for forecasting

This paper revisits the least squares estimator of the linear regression with a structural break. We view the model as an approximation to the true data generating process whose exact nature is unknown but perhaps changing over time either continuously or with some jumps. This view is widely held in the forecasting literature and under this view, the time series dependence property of all the observed variables is unstable as well. We establish that the rate of convergence of the estimator to a properly defined limit is at most the cube root of T, where T is the sample size, which is much slower than the standard super consistent rate. We also provide an asymptotic distribution of the estimator and that of the Gaussian quasi likelihood ratio statistic for a certain class of true data generating processes. We relate our finding to current forecast combination methods and propose a new averaging scheme. Our method compares favourably with various contemporary forecasting methods in forecasting a number of macroeconomic series.

Testing linearity using power transforms of regressors

We develop a method of testing linearity using power transforms of regressors, allowing for stationary processes and time trends. The linear model is a simplifying hypothesis that derives from the power transform model in three different ways, each producing its own identification problem. We call this modeling difficulty the trifold identification problem and show that it may be overcome using a test based on the quasi-likelihood ratio (QLR) statistic. More specifically, the QLR statistic may be approximated under each identification problem and the separate null approximations may be combined to produce a composite approximation that embodies the linear model hypothesis. The limit theory for the QLR test statistic depends on a Gaussian stochastic process. In the important special case of a linear time trend regressor and martingale difference errors asymptotic critical values of the test are provided. Test power is analyzed and an empirical application to crop-yield distributions is provided. The paper also considers generalizations of the Box–Cox transformation, which are associated with the QLR test statistic.

Conditional Inference with a Functional Nuisance Parameter

This paper shows that the problem of testing hypotheses in moment condition models without any assumptions about identification may be considered as a problem of testing with an infinite-dimensional nuisance parameter. We introduce a sufficient statistic for this nuisance parameter and propose conditional tests. These conditional tests have uniformly correct asymptotic size for a large class of models and test statistics. We apply our approach to construct tests based on quasi-likelihood ratio statistics, which we show are efficient in strongly identified models and perform well relative to existing alternatives in two examples.

Model Selection for the Trend Vector Model

Model selection is an important component of data analysis. This study focuses on issues of model selection for the trend vector model, a model for the analysis of longitudinal multinomial outcomes. The trend vector model is a so-called marginal model, focusing on population averaged evolutions over time. A quasi-likelihood method is employed to obtain parameter estimates. Such an optimization function in theory invalidates likelihood-based statistics, such as the likelihood ratio statistic. Moreover, standard errors obtained from the Hessian are biased. In this paper, the performances of different model selection methods for the trend vector model are studied in detail. We specifically focused on two aspects of model selection: variable selection and dimensionality determination. Based on the quasi-likelihood function, selection criteria analogous to the likelihood ratio statistics, AIC and BIC, were employed. Additionally, Wald and resampling statistics were included as variable selection criteria. A series of simulations were carried out to evaluate the relative performance of these criteria. The results suggest that model selection can be best performed using either the quasi likelihood ratio statistic or the quasi-BIC. A special study on dimensionality selection found that the quasi-AIC also performs well for cases with degrees of freedom greater than 8. Another important finding is that the sandwich estimator for standard errors used in Wald statistics does not perform well. Even for larger sample sizes, the bias-correction procedure for the sandwich estimator is needed to give satisfactory results.

Structural-Break Models under Mis-Specification: Implications for Forecasting

This paper revisits the least squares estimator of the linear regression with a structural break. We view the model as an approximation to the true data generating process whose exact nature is unknown but perhaps changing over time either continuously or with some jumps. This view is widely held in the forecasting literature and under this view, the time series dependence property of all the observed variables is unstable as well. We establish that the rate of convergence of the estimator to a properly defined limit is much slower than the standard super consistent rate, even slower than the square root of the sample size T and as slow as the cube root of T. We also provide an asymptotic distribution of the estimator and that of the Gaussian quasi likelihood ratio statistic for a certain class of true data generating process. We relate our finding to current forecast combination methods and bagging and propose a new averaging scheme. The performance of various contemporary forecasting methods is compared to ours using a number of macroeconomic data.

Testing Linearity Using Power Transforms of Regressors

We develop a method of testing linearity using power transforms of regressors, allowing for stationary processes and time trends. The linear model is a simplifying hypothesis that derives from the power transform model in three different ways, each producing its own identification problem. We call this modeling difficulty the trifold identification problem and show that it may be overcome using a test based on the quasi-likelihood ratio (QLR) statistic. More specifically, the QLR statistic may be approximated under each identification problem and the separate null approximations may be combined to produce a composite approximation that embodies the linear model hypothesis. The limit theory for the QLR test statistic depends on a Gaussian stochastic process. In the important special case of a linear time trend regressor and martingale difference errors asymptotic critical values of the test are provided. The paper also considers generalizations of the Box-Cox transformation, which are associated with the QLR test statistic.

Testing for the effects of omitted power transformations

We use the quasi-likelihood ratio statistic to test the hypothesis that a power transformation is ineffective. The null is not standard because a parameter is unidentified, and we can also let the power coefficient be zero for the null.

Testing correct model specification using extreme learning machines

Testing the correct model specification hypothesis for artificial neural network (ANN) models of the conditional mean is not standard. The traditional Wald, Lagrange multiplier, and quasi-likelihood ratio statistics weakly converge to functions of Gaussian processes, rather than to convenient chi-squared distributions. Also, their large-sample null distributions are problem dependent, limiting applicability. We overcome this challenge by applying functional regression methods of Cho et al. [8] to extreme learning machines (ELM). The Wald ELM (WELM) test statistic proposed here is easy to compute and has a large-sample standard chi-squared distribution under the null hypothesis of correct specification. We provide associated theory for time-series data and affirm our theory with some Monte Carlo experiments.

Revisiting Tests for Neglected Nonlinearity Using Artificial Neural Networks

Tests for regression neglected nonlinearity based on artificial neural networks (ANNs) have so far been studied by separately analyzing the two ways in which the null of regression linearity can hold. This implies that the asymptotic behavior of general ANN-based tests for neglected nonlinearity is still an open question. Here we analyze a convenient ANN-based quasi-likelihood ratio statistic for testing neglected nonlinearity, paying careful attention to both components of the null. We derive the asymptotic null distribution under each component separately and analyze their interaction. Somewhat remarkably, it turns out that the previously known asymptotic null distribution for the type 1 case still applies, but under somewhat stronger conditions than previously recognized. We present Monte Carlo experiments corroborating our theoretical results and showing that standard methods can yield misleading inference when our new, stronger regularity conditions are violated.

Testing for Regime Switching

We analyze use of a quasi-likelihood ratio statistic for a mixture model to test the null hypothesis of one regime versus the alternative of two regimes in a Markov regime-switching context. This test exploits mixture properties implied by the regime-switching process, but ignores certain implied serial correlation properties. When formulated in the natural way, the setting is nonstandard, involving nuisance parameters on the boundary of the parameter space, nuisance parameters identified only under the alternative, or approximations using derivatives higher than second order. We exploit recent advances by Andrews (2001) and contribute to the literature by extending the scope of mixture models, obtaining asymptotic null distributions different from those in the literature. We further provide critical values for popular models or bounds for tail probabilities that are useful in constructing conservative critical values for regime-switching tests. We compare the size and power of our statistics to other useful tests for regime switching via Monte Carlo methods and find relatively good performance. We apply our methods to reexamine the classic cartel study of Porter (1983) and reaffirm Porter's findings.

Threshold Models for Combination Data from Reproductive and Developmental Experiments

Abstract In risk assessment, thresholds are generally believed to exist for toxic reproductive and developmental agents. Therefore, dose combinations exist below the threshold where the response is not distinguishable from background and above the threshold where a dose-response trend results. Because data from combination experiments are more reflective of human exposure to environmental agents, threshold models that incorporate the effect of interactions among these agents are developed. For s agents acting in combination, the threshold is an s-dimensional surface. More specifically, for a combination of two agents, the threshold is a two-dimensional contour of dose combinations, and for a single agent, the threshold is represented by a single dose value, not necessarily a dose level used in the experiment. In toxicological experiments designed to investigate the reproductive or developmental effects of a chemical on laboratory animals, litter effects typically are present, yielding overdispersion. Instead of assuming a particular distributional form for the binary responses, quasi-likelihood methods often are used where parameters are introduced to measure the overdispersion. This article describes threshold models for combination reproductive and developmental experiments and develops parameter estimation techniques using quasi-likelihood methods. Once parameters are estimated, a hypothesis test for an overall dose-response relationship is performed using a quasi-likelihood ratio statistic. Next, a confidence interval about the threshold parameter is constructed using a quasi-likelihood ratio statistic, and finally, a confidence region is constructed about the threshold surface. Illustrative examples are presented that use single-agent reproductive data from the National Toxicology Program and three-agent combination developmental data from the Environmental Protection Agency and ManTech Environmental Technology, Inc.

Threshold Models for Combination Data from Reproductive and Developmental Experiments

Abstract In risk assessment, thresholds are generally believed to exist for toxic reproductive and developmental agents. Therefore, dose combinations exist below the threshold where the response is not distinguishable from background and above the threshold where a dose-response trend results. Because data from combination experiments are more reflective of human exposure to environmental agents, threshold models that incorporate the effect of interactions among these agents are developed. For s agents acting in combination, the threshold is an s-dimensional surface. More specifically, for a combination of two agents, the threshold is a two-dimensional contour of dose combinations, and for a single agent, the threshold is represented by a single dose value, not necessarily a dose level used in the experiment. In toxicological experiments designed to investigate the reproductive or developmental effects of a chemical on laboratory animals, litter effects typically are present, yielding overdispersion. Instead of assuming a particular distributional form for the binary responses, quasi-likelihood methods often are used where parameters are introduced to measure the overdispersion. This article describes threshold models for combination reproductive and developmental experiments and develops parameter estimation techniques using quasi-likelihood methods. Once parameters are estimated, a hypothesis test for an overall dose-response relationship is performed using a quasi-likelihood ratio statistic. Next, a confidence interval about the threshold parameter is constructed using a quasi-likelihood ratio statistic, and finally, a confidence region is constructed about the threshold surface. Illustrative examples are presented that use single-agent reproductive data from the National Toxicology Program and three-agent combination developmental data from the Environmental Protection Agency and ManTech Environmental Technology, Inc.

The Bracketing Condition for Limit Theorems on Stationary Linear Processes

This paper aims at improvement in regularity conditions on spectral densities for the limit theorems of the quasi maximum-likelihood estimator and the quasi likelihood-ratio statistic. The approach parallels the Daniels-Huber-Pollard proofs of central limit theorems under nonstandard conditions for i.i.d. situations. The results of the paper enable one to dispense with excessive regularity conditions on the spectral density.