Information Matrix Test Research Articles

Testing distributional assumptions in CUB models for the analysis of rating data

In this paper, we propose a portmanteau test for misspecification in combination of uniform and binomial (CUB) models for the analysis of ordered rating data. Specifically, the test we build belongs to the class of information matrix (IM) tests that are based on the information matrix equality. Monte Carlo evidence indicates that the test has excellent properties in finite samples in terms of actual size and power versus several alternatives. Differently from other tests of the IM family, finite-sample adjustments based on the bootstrap seem to be unnecessary. An empirical application is also provided to illustrate how the IM test can be used to supplement model validation and selection.

Multivariate Hermite polynomials and information matrix tests

The information matrix test for a normal random vector is shown to coincide with the sum of the moment tests for all third- and fourth-order multivariate Hermite polynomials. The statistic is decomposed as the sum of the marginal information matrix test for a subvector, the conditional information matrix test for the complementary subvector, and a third leftover component. It is also shown that exact finite sample distributions can be obtained by drawing spherical Gaussian vectors and orthogonalising them using sample moments. These tests are applied to assess the implications of Gibrat’s law for US city sizes using the three most recent censuses.

Bootstrap specification tests for dynamic conditional distribution models

This paper proposes bootstrap based tests for the specification of a given parametric conditional distribution in autoregressive time series with GARCH-type disturbances. The tests are based on an estimated residual empirical process and are implemented by parametric bootstrap. We show that the proposed tests are asymptotically valid, consistent, and have nontrivial asymptotic power against a large proportion of local alternatives. Our approach relies on non-primitive regularity conditions and certain properties of exponential almost sure convergence. The regularity conditions are shown to be satisfied by GARCH(p,q); this technique of verification is applicable to other models as well. In our Monte Carlo study, the proposed tests performed well and better than several competing tests, including the information matrix test. A real data example illustrates the testing procedure.

Beta distribution misspecification tests with application to Covid-19 mortality rates in the United States.

The beta distribution is routinely used to model variables that assume values in the standard unit interval, (0, 1). Several alternative laws have, nonetheless, been proposed in the literature, such as the Kumaraswamy and simplex distributions. A natural and empirically motivated question is: does the beta law provide an adequate representation for a given dataset? We test the null hypothesis that the beta model is correctly specified against the alternative hypothesis that it does not provide an adequate data fit. Our tests are based on the information matrix equality, which only holds when the model is correctly specified. They are thus sensitive to model misspecification. Simulation evidence shows that the tests perform well, especially when coupled with bootstrap resampling. We model state and county Covid-19 mortality rates in the United States. The misspecification tests indicate that the beta law successfully represents Covid-19 death rates when they are computed using either data from prior to the start of the vaccination campaign or data collected when such a campaign was under way. In the latter case, the beta law is only accepted when the negative impact of vaccination reach on death rates is moderate. The beta model is rejected under data heterogeneity, i.e., when mortality rates are computed using information gathered during both time periods.

Application of the Information Matrix Test on the National Productivity Analysis as the Cause of International Trade

Purpose In order to analyze the national competitiveness of the Korean economy, it is necessary to estimate the total factor productivity correctly, because incorrect assumptions in the model can lead to serious estimation errors. Therefore, in order to estimate the suitability of the model, an information matrix test was proposed as an applicable method to the actual stochastic frontier production function. Design/Methodology/Approach We first examine the large sample distribution properties of maximum likelihood estimation (MLE) and quasi-maximum likelihood estimation (QMLE), and introduce several versions of the stochastic frontier product function models proposed by different scholars. Finally, we derive the Information Matrix Test (Battese and Coelli, 1992) for the stochastic frontier production function model. Also, a Monte Carlo simulation was applied to verify the correctness of the model. Findings From the results, we derive an analytical form of the Information Matrix Test (Battese and Coelli, 1992) on the stochastic frontier production function model using the simplified form proposed by Lancaster (1984). and as a result of empirical analysis by applying this process to each of the seven countries (Korea, China, Japan, the United States, United Kingdom, Germany, France), each  value was below the 95% significance level in all countries, indicating that the translog production function model was specified correctly. Research Implications The Information Matrix Test by White (1982) is an important solution for misspecified regression models. To solve this issue, Lancaster (1984) suggested the simpler form of the Information Matrix Test of White (1982). However, this method is also complex to apply in a real model. Therefore, until now, the Information Matrix Test has not been properly implemented in a regression model. This study presents an analytical form of the Information Matrix Test for the stochastic frontier production function, which is very important for analyzing national, industrial, and corporate competitiveness.

Fitted Copula Statistical Models for Four African and Four Major Stock Markets

The application of copula has become popular in recent years. The use of correlation as a dependence measure has several pitfalls and hence the application of regression prediction model using this correlation may not be an appropriate method. In financial markets, there is often a non-linear dependence between returns. Thus, alternative methods for capturing co-dependency should be considered, such as copula based ones. This paper studies the dependence structure between the four largest African stock markets in terms of market capitalization and other developed stock markets over the period 2003 to 2018 using copula models. The value at risk was used to determine the risk associated with the stock. The ten copula models were fitted to the log returns calculated from the data, two countries at a time of the twenty-eight pairs and examined. The Gumbel copula gives the best fit in terms of log-likelihood values, value of the Akaike information criterion, value of the Bayesian information criterion, value of the consistent Akaike information criterion, value of the corrected Akaike information criterion, value of the Hannan Quinn criterion and p-value of the information matrix equality of White. Estimates of value at risk with probability p for daily returns were computed using the best fitted copula model, from these value at risk, it is seen that SA/FTSE100 have the least risk while EGY/KEN has the highest risk. Prediction is given in terms of correlation and value at risk.

The Behaviour of the Dispersion Matrix of the Information Matrix Test under the Wrong Logistic Regression Model

The Information Matrix Tests (IMT) considers as one of the important global goodness of fit test. The IMT provides a unified framework for specification goodness of fit tests for a wide variety of distribution, multivariate or univariate, discrete or continuous. Many researchers discussed the IMT in cases of the outcome covariate is a continuous variable which reported it has reasonable behaviour. This article considers using IMT as a goodness of fit test for the logistic regression mode, to investigate the behaviour of this statistic under the wrong model. Moreover, we are interested to examine the behaviour of the dispersion matrix under wrong logistic model and compute alternative formula of variance, empirical variance of IMT and examine it by simulation.

Assessment the Distribution of the Information Matrix Test By Different Level of Elements of the Matrix to Avoid the Singularity Problem

The behaviour of the distribution of goodness-of-fit tests is an important statistical problem especially in dichotomous data. Many authors, discussed methods to examine and compare between different goodness-of-fit tests, where they show the Information matrix test (IMT) and the Information Matrix Diagonal test (IMTDIAG) statistics have reasonable power even for a logistic model with very sparse data. However, many issues related to the elements of the log-likelihood functions and covariance matrix which may be quite close to zero, these elements leading to singularity and have an effect on the behaviour of the distribution of statistics. In this paper, we are interested to investigate the behaviour of the distribution of the $IMT$ statistic and IMTDIAG statistic, by using different levels of elements of the matrix. Moreover, the mean and the variance of the distribution of statistics should be examined, by the simulation to compare between different levels of elements of the matrix to find a reasonable new form of $IMT$ to avoid the Singularity Problem forecasting.

Asymptotic Variance of Test Statistics in the ML and QML Frameworks

In this study, we consider the test statistics that can be written as the sample average of data and derive their limiting distributions under the maximum likelihood (ML) and the quasi-maximum likelihood (QML) frameworks. We first generalize the asymptotic variance formula suggested in Pierce (Ann Stat 10(2):475–478, 1982) in the ML framework and illustrate its applications through some well-known test statistics: (1) the skewness statistic, (2) the kurtosis statistic, (3) the Cox statistic, (4) the information matrix test statistic, and (5) the Durbin’s h-statistic. We next provide a similar result in the QML setting and illustrate its applications by providing two examples. Illustrations show the simplicity and the effectiveness of our results for the asymptotic variance of test statistics, and therefore, they are recommended for practical applications.

Left-censored dementia incidences in estimating cohort effects

We estimate the dementia incidence hazard in Germany for the birth cohorts 1900 until 1954 from a simple sample of Germany’s largest health insurance company. Followed from 2004 to 2012, 36,000 uncensored dementia incidences are observed and further 200,000 right-censored insurants included. From a multiplicative hazard model we find a positive and linear trend in the dementia hazard over the cohorts. The main focus of the study is on 11,000 left-censored persons who have already suffered from the disease in 2004. After including the left-censored observations, the slope of the trend declines markedly due to Simpson’s paradox, left-censored persons are imbalanced between the cohorts. When including left-censoring, the dementia hazard increases differently for different ages, we consider omitted covariates to be the reason. For the standard errors from large sample theory, left-censoring requires an adjustment to the conditional information matrix equality.

Consequences of Model Misspecification for Maximum Likelihood Estimation with Missing Data

Researchers are often faced with the challenge of developing statistical models with incomplete data. Exacerbating this situation is the possibility that either the researcher’s complete-data model or the model of the missing-data mechanism is misspecified. In this article, we create a formal theoretical framework for developing statistical models and detecting model misspecification in the presence of incomplete data where maximum likelihood estimates are obtained by maximizing the observable-data likelihood function when the missing-data mechanism is assumed ignorable. First, we provide sufficient regularity conditions on the researcher’s complete-data model to characterize the asymptotic behavior of maximum likelihood estimates in the simultaneous presence of both missing data and model misspecification. These results are then used to derive robust hypothesis testing methods for possibly misspecified models in the presence of Missing at Random (MAR) or Missing Not at Random (MNAR) missing data. Second, we introduce a method for the detection of model misspecification in missing data problems using recently developed Generalized Information Matrix Tests (GIMT). Third, we identify regularity conditions for the Missing Information Principle (MIP) to hold in the presence of model misspecification so as to provide useful computational covariance matrix estimation formulas. Fourth, we provide regularity conditions that ensure the observable-data expected negative log-likelihood function is convex in the presence of partially observable data when the amount of missingness is sufficiently small and the complete-data likelihood is convex. Fifth, we show that when the researcher has correctly specified a complete-data model with a convex negative likelihood function and an ignorable missing-data mechanism, then its strict local minimizer is the true parameter value for the complete-data model when the amount of missingness is sufficiently small. Our results thus provide new robust estimation, inference, and specification analysis methods for developing statistical models with incomplete data.

Asymptotic Variance of Test Statistics in ML and QML Frameworks

In this study, we consider the test statistics that can be written as the sample average of data and derive their limiting distribution under the maximum likelihood (ML) and the quasi-maximum likelihood (QML) frameworks. We first generalize the asymptotic variance formula suggested in Pierce (1982) in the ML framework and illustrate its applications through some well-known test statistics: (i) the skewness statistic, (ii) the kurtosis statistic, (iii) the Cox statistic, (iv) the information matrix test statistic, and (v) the Durbin's h-statistic. We next provide a similar result in the QML setting and illustrate its applications by providing two examples. Illustrations show the simplicity and the effectiveness of our results for the asymptotic variance of test statistics, and therefore, they are recommended for practical applications.

Generalized information matrix tests for copulas

We propose a family of goodness-of-fit tests for copulas. The tests use generalizations of the information matrix (IM) equality of White and so relate to the copula test proposed by Huang and Prokhorov. The idea is that eigenspectrum-based statements of the IM equality reduce the degrees of freedom of the test’s asymptotic distribution and lead to better size-power properties, even in high dimensions. The gains are especially pronounced for vine copulas, where additional benefits come from simplifications of score functions and the Hessian. We derive the asymptotic distribution of the generalized tests, accounting for the nonparametric estimation of the marginals and apply a parametric bootstrap procedure, valid when asymptotic critical values are inaccurate. In Monte Carlo simulations, we study the behavior of the new tests, compare them with several Cramer–von Mises type tests and confirm the desired properties of the new tests in high dimensions.

An Information Matrix Test for the Collapsing of Categories Under the Partial Credit Model

Collapsing categories is a commonly used data reduction technique; however, to date there do not exist principled methods to determine whether collapsing categories is appropriate in practice. With ordinal responses under the partial credit model, when collapsing categories, the true model for the collapsed data is no longer a partial credit model, and therefore refitting a partial credit model may result in model misspecification. This article details the implementation and performance of an information matrix test (IMT) to assess the implications of collapsing categories for a given data set under the partial credit model and compares its performance to the application of a nominal response model (NRM) and the S − X2 goodness-of-fit statistic. The IMT and NRM-based test are able to correctly determine the true number of categories for an item, given reasonable power through this goodness-of-fit test. We conclude by applying the test to a well-studied data set from the literature.

Pythagorean generalization of testing the equality of two symmetric positive definite matrices

We provide a new test for equality of two symmetric positive-definite matrices that leads to a convenient mechanism for testing specification using the information matrix equality or the sandwich asymptotic covariance matrix of the GMM estimator. The test relies on a new characterization of equality between two k dimensional symmetric positive-definite matrices A and B: the traces of AB−1 and BA−1 are equal to k if and only if A=B. Using this simple criterion, we introduce a class of omnibus test statistics for equality and examine their null and local alternative approximations under some mild regularity conditions. A preferred test in the class with good omni-directional power is recommended for practical work. Monte Carlo experiments are conducted to explore performance characteristics under the null and local as well as fixed alternatives. The test is applicable in many settings, including GMM estimation, SVAR models and high dimensional variance matrix settings.

Maximum Likelihood Estimation of Structural Equation Models for Continuous Data: Standard Errors and Goodness of Fit

Classical accounts of maximum likelihood (ML) estimation of structural equation models for continuous outcomes involve normality assumptions: standard errors (SEs) are obtained using the expected information matrix and the goodness of fit of the model is tested using the likelihood ratio (LR) statistic. Satorra and Bentler (1994) introduced SEs and mean adjustments or mean and variance adjustments to the LR statistic (involving also the expected information matrix) that are robust to nonnormality. However, in recent years, SEs obtained using the observed information matrix and alternative test statistics have become available. We investigate what choice of SE and test statistic yields better results using an extensive simulation study. We found that robust SEs computed using the expected information matrix coupled with a mean- and variance-adjusted LR test statistic (i.e., MLMV) is the optimal choice, even with normally distributed data, as it yielded the best combination of accurate SEs and Type I errors.

Generalized Information Matrix Tests for Detecting Model Misspecification

Generalized Information Matrix Tests (GIMTs) have recently been used for detecting the presence of misspecification in regression models in both randomized controlled trials and observational studies. In this paper, a unified GIMT framework is developed for the purpose of identifying, classifying, and deriving novel model misspecification tests for finite-dimensional smooth probability models. These GIMTs include previously published as well as newly developed information matrix tests. To illustrate the application of the GIMT framework, we derived and assessed the performance of new GIMTs for binary logistic regression. Although all GIMTs exhibited good level and power performance for the larger sample sizes, GIMT statistics with fewer degrees of freedom and derived using log-likelihood third derivatives exhibited improved level and power performance.

A goodness-of-fit test for regular vine copula models

ABSTRACTWe introduce a new goodness-of-fit test for regular vine (R-vine) copula models, a very flexible class of multivariate copulas based on a pair-copula construction (PCC). The test arises from White’s information matrix test and extends an existing goodness-of-fit test for copulas. The corresponding critical value can be approximated by asymptotic theory or simulation. The simulation based test shows excellent performance with regard to observed size and power in an extensive simulation study, while the asymptotic theory based test is inadequate for n≤10,000 for a 5-dimensional model (in d = 8 even 20,000 are not enough). The simulation based test is applied to select among different R-vine specifications modeling the dependency among exchange rates.

Directed Tests of No Cross‐Sectional Correlation in Large‐N Panel Data Models

SummaryThe paper proposes tests for no cross‐correlation based on the information matrix equality. The tests rely on suitably weighted residual cross‐covariances or cross‐correlations, and are in this sense a generalization of Pesaran's (2004, CESifo working paper 1229) test for no cross‐sectional dependence. They follow chi‐squared distributions under joint N,T asymptotics without restrictive sphericity or distributional assumptions. When using the outcome of the directed tests to decide whether to use panel‐robust standard errors or not for testing slope parameters, the latter tests are apparently not affected; they can be severely affected, though, if using generic cross‐correlation tests as pretests. Copyright © 2015 John Wiley & Sons, Ltd.

A misspecification test for logit based route choice models

The multinomial logit (MNL) model is often used for analyzing route choices in real networks in spite of the fact that path utilities are believed to be correlated. Yet, statistical tests for model misspecification are rarely used. This paper shows how the information matrix test for model misspecification proposed by White (1982) can be applied to test path-based and link-based MNL route choice models.We present a Monte Carlo experiment using simulated data to assess the size and the power of the test and to compare its performance with the IIA (Hausman and McFadden, 1984) and McFadden–Train Lagrange multiplier (McFadden and Train, 2000) tests. Moreover, we test models estimated on real data and they are all rejected. Attributes correcting for correlation (path size and link size) improve model fit but do not affect the outcome of the test.