Fourier–type tests involving martingale difference processes

We develop testing procedures which detect if the observed time series is a martingale difference sequence. Furthermore, tests are developed that detect change–points in the conditional expectation of the series given its past. The test statistics are formulated following the approach of Fourier–type conditional expectations first proposed by Bierens (1982 Bierens, H. J. (1982). Consistent model speci?cation tests. J. Econometr. 20:105–134.[Crossref], [Web of Science ®] , [Google Scholar]) and have the advantage of computational simplicity. The limit behavior of the test statistics is investigated under the null hypothesis as well as under alternatives. Since the asymptotic null distribution contains unknown parameters, a bootstrap procedure is proposed in order to actually perform the test. The performance of the bootstrap version of the test is compared in finite samples with other methods for the same problem. A real–data application is also included.     

Book Review: New Introduction to Multiple Time Series Analysis

ABSTRACT

This paper proposes a test for the null hypothesis of periodic stationarity against the alternative hypothesis of periodic integration. We derive the limiting distribution of the test statistic and its characteristic function, which are the same as those of the test developed in Kwiatkowski, Phillips, Schmidt and Shin.[15] Kwiatkowski, D., Phillips, P.C. B., Schmidt, P. and Shin, Y. 1992. Testing the Null Hypothesis of Stationarity against the Alternative of a Unit Root.. Journal of Econometrics, 54: 159–178. [Crossref], [Web of Science ®] , [Google Scholar] We find that some parameters, which we must assume under the alternative, have an important effect on the limiting power, so we should choose such parameters carefully. A Monte Carlo simulation reveals that the test has reasonable power but may be affected by the lag truncation parameter that is used for the correction of nuisance parameters.     

Data-Driven Tests for Trend

This article proposes a new nonparametric test for the ordered alternatives problem in the k-sample setting for null hypothesis of lack of trend. This article further elaborates upon and extends the results of Ledwina and Wy?upek (2012a Ledwina , T. , Wy?upek , G. ( 2012a ). Two-sample test against one-sided alternatives . Scand. J. Statist. 39 : 358 – 381 .[Crossref], [Web of Science ®] , [Google Scholar]) obtained for k = 2. Simulations show that the new test has high and stable power and is able to control the Type I error to satisfactory extent, thus solving the problem posed in Terpstra and Magel (2003 Terpstra , J. T. , Magel , R. C. ( 2003 ). A new nonparametric test for the ordered alternative problem . J. Nonparametr. Statist. 15 : 289 – 301 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Our theoretical results say that asymptotic errors of both kinds do not exceed significance level, thus implying that the test is asymptotically unbiased.     

Bootstrap for generalized linear models

We consider the distribution of the (standardized) ML-estimator of the unknown parameter vector in a Generalized Linear Model with canonical link function. It will be shown that its (parametric) Bootstrap estimator is consistent under the same assumptions needed by Fahrmeir & Kaufmann (1985, 1986) to show its asymptotic normality.     

Minimax sequential tests of composite hypotheses on the drift of a Wiener process

A Wiener process with unknown drift parameter μ is, beginning at O, observed continuously and one has to decide between the hypotheses μ≤0 and μ>0. For loss functions of the form sμ^r and linear cost functions one wants to determine a minimax sequential test. Generalizing the results of DeGroot (1960) a minimax test in the class of all symmetrical SPRT’s is given in an explicit form. On the other hand it is shown that this SPRT is, in general, no longer minimax in the class of all sequential tests.     

Shrinkage of Variance for Minimum Distance Based Tests

This paper promotes information theoretic inference in the context of minimum distance estimation. Various score test statistics differ only through the embedded estimator of the variance of estimating functions. We resort to implied probabilities provided by the constrained maximization of generalized entropy to get a more accurate variance estimator under the null. We document, both by theoretical higher order expansions and by Monte-Carlo evidence, that our improved score tests have better finite-sample size properties. The competitiveness of our non-simulation based method with respect to bootstrap is confirmed in the example of inference on covariance structures previously studied by Horowitz (1998 Horowitz , J. ( 1998 ). Bootstrap methods for covariance structures . The Journal of Human Resources 33 : 39 – 61 .[Crossref], [Web of Science ®] , [Google Scholar]).     

On Comparing Cumulative Incidence Functions Using an Empirical Likelihood Ratio Type Test

We consider the competing risks problem with two risks and develop empirical likelihood ratio type tests for testing the null hypothesis that the cumulative incidence functions corresponding to these two risks are equal against the alternatives: (a) they are not equal and (b) they are linearly ordered. The asymptotic null distributions of the proposed test statistics are shown to have simple distribution-free representations in terms of a standard Brownian motion process. The results of a simulation study indicate that the proposed test for testing for the presence of the linear order is more powerful than a test designed for the same situation in Aly et al. (1994 Aly, E.A.A., Kochar, S.C., McKeague, I.W. (1994). Some tests for comparing cumulative incidence functions and cause-specific hazard rates. J. Am. Stat. Assoc. 89:994–999.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). To illustrate the theoretical results, we discuss an example involving survival times of mice exposed to radiation.     

Relatioinships between economic and statistical price indices

In the following, the economic counterparts of Eichhorn's and Voeller's tests for statistical price indices will be studied. We will see that replacing the statistical Commensurability Axiom in the economic price index theory by a property which is only concerned with price changes leads to similar relationships between this one and several other tests as in the statistical price index theory.     

Statistical tests for the comparison of two samples: The general alternative

It is common to test the null hypothesis that two samples were drawn from identical distributions; and the Smirnov (sometimes called Kolmogorov–Smirnov) test is conventionally applied. We present simulation results to compare the performance of this test with three recently introduced alternatives. We consider both continuous and discrete data. We show that the alternative methods preserve type I error at the nominal level as well as the Smirnov test but offer superior power. We argue for the routine replacement of the Smirnov test with the modified Baumgartner test according to Murakami (2006) Murakami, H. (2006). A k-sample rank test based on a modified Baumgartner statistic and its power comparison. Journal of the Japanese Society of Computational Statistics 19:1–13.[Crossref] , [Google Scholar], or with the test proposed by Zhang (2006) Zhang, J. (2006). Powerful two-sample tests based on the likelihood ratio. Technometrics 48:95–103.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar].     

Testing homogeneity in semiparametric mixture case–control models

Parametric and semiparametric mixture models have been widely used in applications from many areas, and it is often of interest to test the homogeneity in these models. However, hypothesis testing is non standard due to the fact that several regularity conditions do not hold under the null hypothesis. We consider a semiparametric mixture case–control model, in the sense that the density ratio of two distributions is assumed to be of an exponential form, while the baseline density is unspecified. This model was first considered by Qin and Liang (2011 Qin, J., Liang, K.Y. (2011). Hypothesis testing in a mixture case–control model. Biometrics 67(1):182–198.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar], biometrics), and they proposed a modified score statistic for testing homogeneity. In this article, we consider alternative testing procedures based on supremum statistics, which could improve power against certain types of alternatives. We demonstrate the connection and comparison among the proposed and existing approaches. In addition, we provide a unified theoretical justification of the supremum test and other existing test statistics from an empirical likelihood perspective. The finite-sample performance of the supremum test statistics was evaluated in simulation studies.     

Detecting structural change with heteroskedasticity

The hypothesis of structural stability that the regression coefficients do not change over time is central to all applications of linear regression models. It is rather surprising that existing theory as well as practice focus on testing for structural change under homoskedasticity – that is, regression coefficients may change, but the variances remain the same. Since structural change can, and often does, involve changes in variances, this is a puzzling gap in the literature. Our main focus in this paper is to utilize a newly developed test (MZ) by Maasoumi et al. (2010 Maasoumi, E., Zaman, A., Ahmed, M. (2010). Tests for structural change, aggregation, and homogeneity. Econ Model. 27(6):1382–1391.[Crossref], [Web of Science ®] , [Google Scholar]) that tests simultaneously for break in regression coefficients as well as in variance. Currently, the sup F test is most widely used for structural change. This has certain optimality properties shown by Andrews (1993 Andrews, D.W.K. (1993). Test for parameter instability and structural change with unknown change point. Econometrica. 61(4):821–856.[Crossref], [Web of Science ®] , [Google Scholar]). However, this test assumes homoskedasticity across the structural change. We introduce the sup MZ test which caters to unknown breakpoints, and also compare it to the sup F. Our Monte Carlo results show that sup MZ test incurs only a low cost in case of homoskedasticity while having hugely better performance in case of heteroskedasticity. The simulation results are further supported by providing a real-world application. In real-world datasets, we find that structural change often involves heteroskedasticity. In such cases, the sup F test can fail to detect structural breaks and give misleading results, while the sup MZ test works well. We conclude that the sup MZ test is superior to current methodology for detecting structural change.     

On Test Statistics in Profile Analysis with High-dimensional Data

We consider profile analysis with unequal covariance matrices under multivariate normality. In particular, we discuss this problem for high-dimensional data where the dimension is larger than the sample size. We propose three test statistics based on Bennett’s (1951) transformation and the Dempster trace criterion proposed by Dempster (1958 Dempster, A.P. (1958). A high dimensional two samples significance test. Annals of Mathematical Statistics 29:995–1010.[Crossref] , [Google Scholar]). We derive the null distributions as well as the nonnull distributions of the test statistics. Finally, in order to investigate the accuracy of the proposed statistics, we perform Monte Carlo simulations for some selected values of parameters.     

Model selection using union-intersection principle for non nested models

In this article, we have extended the Vuong’s (1989 Vuong, Q.H. (1989). Likelihood ratio tests for model selection and non-nested hypothesis. Econometrica. 57:307–333.[Crossref], [Web of Science ®] , [Google Scholar]) model selection test to three models in accordance to union-intersection principle. Using the Kullback–Leibler criterion to measure the closeness of a model to the truth, we propose a simple likelihood ratio-based statistics for testing the null hypothesis that the competing models are equally close to the true data-generating process against the alternative hypothesis that at least one model is closer. We show that the distribution of the test statistic is asymptotically equal to the distribution of the maximum of dependent random variables with bivariate folded standard normal distribution. The density function of the maximum of dependent random variables with elliptically contoured distributions has been obtained by other researchers, but, not for distributions which do not belong to the elliptically contoured distributions family. In this article, the exact distribution of the maximum of dependent random variables with bivariate folded standard normal distribution is calculated as an asymptotic distribution of the proposed test statistic. The test is directional and is derived successively for the cases where the competing models are non nested and whether three, two, one, or none of them are misspecified.     

Power Loss for Inhomogeneous Poisson Processes

In this work, based on a realization of an inhomogeneous Poisson process whose intensity function depends on a real parameter, we consider a simple null hypothesis against the composite one sided alternative. Under certain regularity conditions we will obtain the power loss of the score test which measures its performance with respect to the Neyman-Pearson test. We present the second-order approximation of the power of the score test under the close alternatives by specifying the explicit form of the next term after the Gaussian term.     

Testing Parameter Constancy in Unit Root Autoregressive Models Against Multiple Continuous Structural Changes

This article considers tests for logistic smooth transition autoregressive (LSTAR) models accommodating multiple time dependent transitions between regimes when the data generating process is a random walk. The asymptotic null distributions of the tests, in contrast to the standard results in Lin and Teräsvirta (1994 Lin , C. F. J. , Teräsvirta , T. ( 1994 ). Testing the constancy of regression parameters against continuous structural change . Journal of Econometrics 62 : 211 – 228 .[Crossref], [Web of Science ®] , [Google Scholar]), are nonstandard. Monte Carlo experiments reveal that the tests have modest size distortions and satisfactory power against LSTAR models with multiple smooth breaks. The tests are applied to Swedish unemployment rates and the hysteresis hypothesis is over-turned in favour of an LSTAR model with two transitions between extreme regimes.     

A Note on Tests for Zero-Inflation in Correlated Count Data

Zero-inflated Poisson mixed regression models are popular approaches to analyze clustered count data with excess zeros. Prior to application of these models, it is essential to examine the necessity of the adjustment for zero outcomes. The existing literature, however, has focused only on score tests for testing the suitability of zero-inflated models for correlated count data. In view of the observed bias and non-optimal size of score tests, it deserves further investigation of other alternative ways for the test. This article aims to explore the use of the null Wald and likelihood ratio tests for zero-inflation in correlated count data. Our simulation study shows that both the null Wald and likelihood ratio tests outperform the score test of Xiang et al. (2006 Xiang , L. , Lee , A. H. , Yau , K. K. W. , McLachlan , G. J. ( 2006 ). A score test for zero-inflation in correlated count data . Statistics in Medicine 25 : 1660 – 1671 . [Google Scholar]) in terms of statistical power, regardless of the computational convenience of the score test. A bootstrap null Wald statistic is also proposed, which results in improved performance in terms of the size and power of the test.     

Robust cointegration testing in the presence of weak trends,with an application to the human origin of global warming

Standard tests for the rank of cointegration of a vector autoregressive process present distributions that are affected by the presence of deterministic trends. We consider the recent approach of Demetrescu et al. (2009 Demetrescu, M., Lütkepohl, H., Saikkonen, P. (2009). Testing for the cointegrating rank of a vector autoregressive process with an uncertain deterministic trend term. Econometrics Journal 12:414–35.[Crossref], [Web of Science ®] , [Google Scholar]) who recommend testing a composite null. We assess this methodology in the presence of trends (linear or broken) whose magnitude is small enough not to be always detectable at conventional significance levels. We model them using local asymptotics and derive the properties of the test statistics. We show that whether the trend is orthogonal to the cointegrating vector has a major impact on the distributions but that the test combination approach remains valid. We apply of the methodology to the study of cointegration properties between global temperatures and the radiative forcing of human gas emissions. We find new evidence of Granger Causality.     

Finite-sample distributions of the Wald,likelihood ratio,and Lagrange multiplier test statistics in the classical linear model

In this note, it is shown that the finite-sample distributions of the Wald, likelihood ratio, and Lagrange multiplier statistics in the classical linear regression model are members of the generalized beta model introduced by McDonald and Xu (1995a McDonald, J.B., Xu, Y.J. (1995a). A generalization of the beta distribution with applications. J. Econom. 66:133–152.[Crossref], [Web of Science ®] , [Google Scholar]). This is useful for examining the properties of these test statistics. For example, this characterization makes it easy to find distribution, quantile, and density functions for each test statistic, makes it clear why Wald tests may overreject the null hypothesis using asymptotic critical values, and formalizes the fact that the Lagrange multiplier statistic follows a distribution with bounded support.     

A Nonparametric Test for the Ordered Alternative Based on Kendall's Correlation Coefficient

Several methods have been developed for testing the ordered alternative. These include the Jonckheere–Terpstra (JT) test (Jonckheere, 1954 Jonckheere , A. R. ( 1954 ). A distribution free k-sample test against ordered alternatives . Biometrika 41 : 133 – 145 .[Crossref], [Web of Science ®] , [Google Scholar]; Terpstra, 1952 Terpstra , T. ( 1952 ). The asymptotic normality and consistency of Kendall's test against trend when ties are present in one ranking . Indigationes Mathematicae 14 : 327 – 333 . [Google Scholar]), a modified JT test (MJT) (Tryon and Hettmansperger, 1987 Tryon , V. P. , Hettmansperger , T. P. ( 1987 ). A class of nonparametric tests for homogeneity against ordered alternatives . Annals of Statistics 1 : 1061 – 1070 . [Google Scholar]), and a test proposed by Terpstra and Magel (TM) (Terpstra and Magel, 2003 Terpstra , J. T. , Magel , R. C. ( 2003 ). A new nonparametric test for the ordered alternative problem . Journal of Nonparametric Statistics 15 : 289 – 301 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), among others. This article proposes a new method for testing the ordered alternative. The proposed test is based on Kendall's tau statistic. The asymptotic distribution of the test statistic is given. A Monte Carlo simulation study is conducted comparing the estimated powers of the proposed test with existing tests under a variety of sample sizes and distributions.