Information Matrix Test Research Articles

A note on misspecification in joint modeling of correlated data with informative cluster sizes

Clustered data are commonly generated in biomedical, economic, and social science studies. When cluster sizes are correlated with primary outcomes, data are identified as correlated data with “informative cluster sizes”. For such data, standard data analysis methods do not apply because of induced biases. Joint modeling approach has been proposed as one of the strategies to analyze informative cluster size data. However, cluster size model misspecification may affect the reliability of joint modeling approach in statistical inference. This article focuses on the joint modeling approach that combines generalized linear mixed models and cluster size models. It reveals the prodigious impact of cluster size model misspecification on the statistical inferences in joint models and offers solutions to this misspecification problem. A modified information matrix test and a sandwich estimator test are adopted, and their effectiveness as diagnostic tools in detecting cluster size model misspecification is demonstrated. Numerical studies show that the two diagnostic tests are highly favorable, but they might inflate Type I errors when the sample size is small. We also recommend an alternative approach to draw inferences by using frequentist model averaging if there is strong confidence that the suitable model is included in the pool of candidate cluster size models. This article reviews a set of related theorems as well as complements them with comprehensive numerical studies, which together provide a methodological solution to the cluster size model misspecification in joint modeling.

Assessing Person Fit With the Information Matrix Test

In this manuscript, a new approach to the analysis of person fit is presented that is based on the information matrix test of White (1982) . This test can be interpreted as a test of trait stability during the measurement situation. The test follows approximately a χ2-distribution. In small samples, the approximation can be improved by a higher-order expansion. The performance of the test is explored in a simulation study. This simulation study suggests that the test adheres to the nominal Type-I error rate well, although it tends to be conservative in very short scales. The power of the test is compared to the power of four alternative tests of person fit. This comparison corroborates that the power of the information matrix test is similar to the power of the alternative tests. Advantages and areas of application of the information matrix test are discussed.

How Robust Standard Errors Expose Methodological Problems They Do Not Fix, and What to Do About It

“Robust standard errors” are used in a vast array of scholarship to correct standard errors for model misspecification. However, when misspecification is bad enough to make classical and robust standard errors diverge, assuming that it is nevertheless not so bad as to bias everything else requires considerable optimism. And even if the optimism is warranted, settling for a misspecified model, with or without robust standard errors, will still bias estimators of all but a few quantities of interest. The resulting cavernous gap between theory and practice suggests that considerable gains in applied statistics may be possible. We seek to help researchers realize these gains via a more productive way to understand and use robust standard errors; a new general and easier-to-use “generalized information matrix test” statistic that can formally assess misspecification (based on differences between robust and classical variance estimates); and practical illustrations via simulations and real examples from published research. How robust standard errors are used needs to change, but instead of jettisoning this popular tool we show how to use it to provide effective clues about model misspecification, likely biases, and a guide to considerably more reliable, and defensible, inferences. Accompanying this article is software that implements the methods we describe.

Testing Equality of Covariance Matrices via Pythagorean Means

We provide a new test for equality of covariance matrices that leads to a convenient mechanism for testing specification using the information matrix equality. The test relies on a new characterization of equality between twok dimensional positive-definite matrices A and B : the traces of AB 1 and BA 1 are equal tok if and only if A = B. Using this criterion, we introduce a class of omnibus test statistics for equality of covariance matrices and examine their null, local, and global approximations under some mild regularity conditions. Monte Carlo experiments are conducted to explore the performance characteristics of the test criteria and provide comparisons with existing tests under the null hypothesis and local and global alternatives. The tests are applied to the classic empirical models for voting turnout investigated by Wolfinger and Rosenstone (1980) and Nagler (1991, 1994). Our tests show that all classic models for the 1984 presidential voting turnout are misspecified in the sense that the information matrix equality fails.

A Goodness-of-fit Test for Copulas

We propose a new rank-based goodness-of-fit test for copulas. It uses the information matrix equality and so relates to the White (1982) specification test. The test avoids parametric specification of marginal distributions, it does not involve kernel weighting, bandwidth selection, or any other strategic choices, it is asymptotically pivotal with a standard distribution, and it is simple to compute compared to available alternatives. The finite-sample size of this type of tests is known to deviate from their nominal size based on asymptotic critical values, and bootstrapping critical values could be a preferred alternative. A power study shows that, in a bivariate setting, the test has reasonable properties compared to its competitors. We conclude with an application in which we apply the test to two stock indices.

A test for bivariate normality with applications in microeconometric models

In this paper, we propose a test for bivariate normality in imperfectly observed models, based on the information matrix test for censored models with bootstrap critical values. In order to evaluate its properties, we run a comprehensive Monte Carlo experiment, in which we use the bivariate probit model and Heckman sample selection model as examples. We find that, while asymptotic critical values can be seriously misleading, the use of bootstrap critical values results in a test that has excellent size and power properties even in small samples. Since this procedure is relatively inexpensive from a computational viewpoint and is easy to generalise to models with arbitrary censoring schemes, we recommend it as an important and valuable testing tool.

Variance Corrected Proportional Hazard Model for the Analysis of Recurrent Multiple Failure Modes

Automobiles do fail repetitively owing to different types of failures. Modeling of such products should concern dissimilar recurrent failure types. Failure types are habitually correlated to each other. This study has utilized variance corrected proportional hazard models (VCPH) for modeling multiple failure occurrences of different failure types of automobiles taking into account the correlated nature of the failures. Though original Cox proportional hazard (PH) models require failure events to be independent, VCPH can handle non independency. In this study, this model is utilized for the analysis of multi-type, multiple occurrences of failures in automobiles, where the assumption of independence among failure times is violated. The VCPH model obtains parameter estimates by first fitting a Cox PH model that ignores the dependence structure and then replaces the naive standard errors with estimates from empirical sandwich variance estimation in order to incorporate the non-independence of times between failures. This study applies the Prentice, Williams and Petersen (PWP) models to model multiple occurrences of different failures in automobiles as proportionality among failure events were well demonstrated when risk interval is taken as ‘gap time’ and since PWP models are specified with a gap time risk interval. The Information Matrix (IM) test of White is applied for the checking of the PH model specification with multivariate failure time data. White’s paper on inference from missspecified models presented the IM test as a test for correct model specification. The objective of the study was to find out how automobile type and type of failure affect the time to failure. Applying the best suitable VCPH model to the data, it was revealed that both automobile and failure type have an impact on timing of failure however this effect doesn’t change over multiple failure occurrences of the automobile.

Assessing Fit of Item Response Models Using the Information Matrix Test

The information matrix can equivalently be determined via the expectation of the Hessian matrix or the expectation of the outer product of the score vector. The identity of these two matrices, however, is only valid in case of a correctly specified model. Therefore, differences between the two versions of the observed information matrix indicate model misfit. The equality of both matrices can be tested with the so‐called information matrix test as a general test of misspecification. This test can be adapted to item response models in order to evaluate the fit of single items and the fit of the whole scale. The performance of different versions of the test is compared in a simulation study with existing tests of model fit, among them the test of Orlando and Thissen, the score test of local independence due to Glas and Suarez‐Falcon, and the limited information approach of Maydeu‐Olivares and Joe. In general, the different versions of the information matrix test adhere to the nominal Type I error rate and have high power for detecting misspecified item characteristic curves. Additionally, some versions of the test can be used in order to detect violations of the local independence assumption.

Test of misspecification with application to negative binomial distribution

A misspecification test based directly on Bartlett’s First Identity is examined. This test is exemplified by the negative binomial distribution. A Monte Carlo simulation study has been conducted, in the context of testing distributional misspecification, and the performance of the proposed test has been benchmarked with some goodness-of-fit tests based on the empirical distribution function. The results suggest that the proposed test is viable in terms of computational speed and statistical power, and has the advantage that complications arising from the use of the covariance matrix in White’s information matrix test are avoided.

Testing for unobserved heterogeneity in exponential and Weibull duration models

We examine the use of the likelihood ratio (LR) statistic to test for unobserved heterogeneity in duration models, based on mixtures of exponential or Weibull distributions. We consider both the uncensored and censored duration cases. The asymptotic null distribution of the LR test statistic is not the standard chi-square, as the standard regularity conditions do not hold. Instead, there is a nuisance parameter identified only under the alternative, and a null parameter value on the boundary of the parameter space, as in Cho and White (2007a). We accommodate these and provide methods delivering consistent asymptotic critical values. We conduct a number of Monte Carlo simulations, comparing the level and power of the LR test statistic to an information matrix (IM) test due to Chesher (1984) and Lagrange multiplier (LM) tests of Kiefer (1985) and Sharma (1987). Our simulations show that the LR test statistic generally outperforms the IM and LM tests. We also revisit the work of van den Berg and Ridder (1998) on unemployment durations and of Ghysels et al. (2004) on interarrival times between stock trades, and, as it turns out, affirm their original informal inferences.

A semiparametric Wald statistic for testing logistic regression models based on case-control data

We propose a semiparametric Wald statistic to test the validity of logistic regression models based on case-control data. The test statistic is constructed using a semiparametric ROC curve estimator and a nonparametric ROC curve estimator. The statistic has an asymptotic chisquared distribution and is an alternative to the Kolmogorov-Smirnov-type statistic proposed by Qin and Zhang in 1997, the chi-squared-type statistic proposed by Zhang in 1999 and the information matrix test statistic proposed by Zhang in 2001. The statistic is easy to compute in the sense that it requires none of the following methods: using a bootstrap method to find its critical values, partitioning the sample data or inverting a high-dimensional matrix. We present some results on simulation and on analysis of two real examples. Moreover, we discuss how to extend our statistic to a family of statistics and how to construct its Kolmogorov-Smirnov counterpart.

An Analysis of the Conditional Volatility Dynamics of the Australian Business Cycle

In this paper, we analyse the conditional variance of the Australian real gross domestic product (GDP) and the expenditure components by a variety of generalised autoregressive conditional heteroskedasticity (GARCH) models. First, we test the plausibility of the constant-correlation assumption by employing Tsei¯s (2000) Lagrange Multiplier (LM) test and the Bera and Kimi¯s (2002) Information Matrix (IM) test. Our results indicate that the correlations among the shocks to real GDP and its various expenditure components are invariant over time. In addition, these shocks are not highly correlated with one another. Second, we examine if volatility asymmetry exists in the Australian business cycle by proposing four bivariate asymmetric GARCH specifications. Except for the case of gross fixed capital formation, the evidence of asymmetric conditional volatility in the growth rates of the Australian real GDP and the other components is weak. Despite the weak evidence of asymmetric volatility, higher volatility is generally associated with the contractionary phase of the Australian business cycle. This finding has important implications for macroeconomic policy and forecasting for business cycle.

Misspecification tests for binomial and beta‐binomial models

The IOS test of Presnell and Boss (J. Am. Stat. Assoc. 2004; 99(465):216-227) is a general-purpose goodness-of-fit test based on the ratio of in-sample and out-of-sample likelihoods. For large samples, the IOS statistic can be approximated by a multiplicative contrast between two estimates of the information matrix, and in this way the IOS test is connected to White's (Econometrica 1982; 50:1-26) information matrix test, or IM test, which is based directly on the difference of two estimates of the information matrix. In this paper, we compare the performance of IOS to that of the IM test and of other goodness-of-fit tests for binomial and beta-binomial models, in both examples and simulations. Our findings suggest that IOS is strongly competitive, not only against the IM test but also against tests designed for specific binomial and beta-binomial models.

A Note on the Relationship between the Information Matrix Test and a Score Test for Parameter Constancy

Information matrix (IM) test (White, 1982) has been used for detecting general model misspecification in the applied econometrics literature. Two of the most commonly used asymptotic covariance matrix estimators (ACMEs) for the IM test are the one that White (1982) proposed in his original paper and Chesher (1983)'s ACME. Chesher (1984) showed that the IM test is in effect a score test for parameter constancy. In this note, I show that the IM test with White's is not only the score test but also a specification robust form of the score test or a score test for quasi-maximum likelihood estimators. Based on this result, it is argued that we should be careful in selecting the ACME for properly interpreting the consequence of the IM test.

Robust M Tests Without Consistent Estimation of the Asymptotic Covariance Matrix

We extend the KVB approach of Kiefer, Vogelsang, and Bunzel (2000) to constructing robust M tests without consistent estimation of the asymptotic covariance matrix. We demonstrate that, when model parameters have to be estimated, the normalizing matrix computed using a full-sample estimator is able to eliminate the nuisance parameters when there is no estimation effect but not otherwise. To circumvent the problem of estimation effect, we propose using recursive estimators to compute the normalizing matrix and show that the resulting M test is asymptotically pivotal. This M test is, thus, robust not only to heteroscedasticity and serial correlations of unknown form but also to the presence of an estimation effect. As examples, we consider robust tests for serial correlations and robust information matrix tests. The former tests extend that of Lobato (2001) and are applicable to model residuals. For testing higher-order moments, we find that the latter tests are also robust when a lower-order moment is misspecified. Our simulations confirm that the proposed M tests are properly sized and have power advantage when other tests are computed based on inappropriate user-chosen parameters.

Testing the information matrix equality with robust estimators

The information matrix (IM) equality can be used to test for misspecification of a parametric model. We study the behavior of the IM test when the maximum-likelihood (ML) estimators used in the construction of this test are replaced with robust estimators. The latter do not suffer from the masking effect in the presence of outliers and can improve the power of the IM test. At the normal location-scale model, the IM test using the ML estimators is known as the Jarque–Bera test, and uses skewness and kurtosis to detect deviations from normality. When robust estimators are employed to test the IM equality, a robust version of the Jarque–Bera test emerges. We investigate in detail the local asymptotic power of the IM test, for various estimators and under a variety of local alternatives. For the normal regression model, it is shown by simulations under fixed alternatives that in many cases the use of robust estimators substantially increases the power of the IM test.

Analysis of low count time series data by poisson autoregression

Abstract. This study provides new methods of assessing the adequacy of the Poisson autoregressive time‐series model for count data. New expressions are given for the score function and the information matrix and these lead to the construction of new types of residuals for this model. However, these residuals often need to be supplemented by formal statistical procedures and an overall test of the model adequacy is given via the information matrix equality that holds for correctly specified models. The techniques are applied to a monthly count data set of claimants for wage loss benefit, in order to estimate the the expected duration of claimants in the system.

The information matrix test with bootstrap-based covariance matrix estimation

We propose an information matrix (IM) test in which the covariance matrix of the vector of indicators is estimated using the parametric bootstrap. Monte Carlo results and theoretical arguments show that its small sample performance is comparable with that of the efficient score form.

A score-type and a Hausman-type statistic based on nonsingular generalized inverses

For a model where a type of information matrix equality holds, the literature provides explicit forms for a score-type and a Hausman-type statistic based on particular nonsingular generalized inverses. This paper provides similar explicit forms for a more general model where a type of information matrix equality does not necessarily (but may) hold.

The equality of comparable extended families of classical-type and Hausman-type statistics

An extended family of statistics is defined as a set of asymptotically-equivalent quadratic forms in a given vector where each component of a weighting matrix is any consistent estimator of an appropriate matrix. Given a type of information matrix (IM) equality, two statistics are said to be comparable if both statistics are constructed with the IM-type equality imposed or both constructed with the IM-type equality not imposed. Similarly, two families are said to be comparable if all members of both families are comparable statistics. It is well known that comparable classical-type and Hausman-type statistics are asymptotically equivalent under appropriate conditions. In contrast, given the convergence in probability of appropriate quantities, this paper shows the equality of comparable extended families of classical-type and Hausman-type statistics, a result that implies (but is not implied by) the well-known asymptotic equivalences.