Information-Theoretic Distribution Test with Application to Normality

We derive general distribution tests based on the method of Maximum Entropy density. The proposed tests are derived from maximizing the differential entropy subject to moment constraints. By exploiting the equivalence between the Maximum Entropy and Maximum Likelihood estimates of the general exponential family, we can use the conventional Likelihood Ratio, Wald and Lagrange Multiplier testing principles in the maximum entropy framework. In particular, we use the Lagrange Multiplier method to derive tests for normality and their asymptotic properties. Monte Carlo evidence suggests that the proposed tests have desirable small sample properties and often outperform commonly used tests such as the Jarque-Bera test and the Komogorov-Smirnov-Lillie test for normality. We show that the proposed tests can be extended to tests based on regression residuals and non-iid data in a straightforward manner. We apply the proposed tests to the residuals from a stochastic production frontier model and reject the normality hypothesis.

Analysis of co-explosive processes

A vector autoregressive model allowing for unit roots as well as explosive characteristic roots is developed. The Granger-Johansen representation shows that this results in processes with two common features: a random walk and an explosively growing process. Co-integrating and co-explosive vectors can be found which eliminate these common factors. Likelihood ratio tests for linear restrictions on the co-explosive vectors are derived. As an empirical illustration the method is applied to data from the extreme Yugoslavian hyper-inflation of the 1990s.Asymptotic normality, Co-explosiveness, Cointegration, Explosive processes, Hyper-inflation, Likelihood ratio tests, Vector autoregression

Consistency and asymptotic normality of the maximum likelihood estimator in a zero-inflated generalized Poisson regression

Poisson regression models for count variables have been utilized in many applications. However, in many problems overdispersion and zero-inflation occur. We study in this paper regression models based on the generalized Poisson distribution (Consul (1989)). These regression models which have been used for about 15 years do not belong to the class of generalized linear models considered by (McCullagh and Nelder (1989)) for which an established asymptotic theory is available. Therefore we prove consistency and asymptotic normality of a solution to the maximum likelihood equations for zero-inflated generalized Poisson regression models. Further the accuracy of the asymptotic normality approximation is investigated through a simulation study. This allows to construct asymptotic confidence intervals and likelihood ratio tests

Testing for zero-modification in count regression models

Count data often exhibit overdispersion and/or require an adjustment for zero outcomes with respect to a Poisson model. Zero-modified Poisson (ZMP) and zero-modified generalized Poisson (ZMGP) regression models are useful classes of models for such data. In the literature so far only score tests are used for testing the necessity of this adjustment. For this testing problem we show how poor the performance of the corresponding score test can be in comparison to the performance of Wald and likelihood ratio (LR) tests through a simulation study. In particular, the score test in the ZMP case results in a power loss of 47% compared to the Wald test in the worst case, while in the ZMGP case the worst loss is 87%. Therefore, regardless of the computational advantage of score tests, the loss in power compared to the Wald and LR tests should not be neglected and these much more powerful alternatives should be used instead. We also prove consistency and asymptotic normality of the maximum likelihood estimators in the above mentioned regression models to give a theoretical justification for Wald and likelihood ratio tests

Sinh-arcsinh distributions

We introduce the sinh-arcsinh transformation and hence, by applying it to a generating distribution with no parameters other than location and scale, usually the normal, a new family of sinh-arcsinh distributions. This four-parameter family has symmetric and skewed members and allows for tailweights that are both heavier and lighter than those of the generating distribution. The central place of the normal distribution in this family affords likelihood ratio tests of normality that are superior to the state-of-the-art in normality testing because of the range of alternatives against which they are very powerful. Likelihood ratio tests of symmetry are also available and are very successful. Three-parameter symmetric and asymmetric subfamilies of the full family are also of interest. Heavy-tailed symmetric sinh-arcsinh distributions behave like Johnson SU distributions, while their light-tailed counterparts behave like sinh-normal distributions, the sinh-arcsinh family allowing a seamless transition between the two, via the normal, controlled by a single parameter. The sinh-arcsinh family is very tractable and many properties are explored. Likelihood inference is pursued, including an attractive reparameterization. Illustrative examples are given. A multivariate version is considered. Options and extensions are discussed

Testing mean-variance efficiency in CAPM with possibly non-gaussian errors: an exact simulation-based approach

In this paper we propose exact likelihood-based mean-variance efficiency tests of the market portfolio in the context of Capital Asset Pricing Model (CAPM), allowing for a wide class of error distributions which include normality as a special case. These tests are developed in the framework of multivariate linear regressions (MLR). It is well known however that despite their simple statistical structure, standard asymptotically justified MLR-based tests are unreliable. In financial econometrics, exact tests have been proposed for a few specific hypotheses [Jobson and Korkie (Journal of Financial Economics, 1982), MacKinlay (Journal of Financial Economics, 1987), Gibbons, Ross and Shanken (Econometrica, 1989), Zhou (Journal of Finance 1993)] most of which depend on normality. For the gaussian model, our tests correspond to Gibbons, Ross and Shanken's mean-variance efficiency tests. In non-gaussian contexts, we reconsider mean-variance efficiency tests allowing for multivariate Student-t and gaussian mixture errors. Our framework allows to cast more evidence on whether the normality assumption is too restrictive when testing the CAPM. We also propose exact multivariate diagnostic checks (including tests for multivariate GARCH and multivariate generalization of the well known variance ratio tests) and goodness of fit tests as well as a set estimate for the intervening nuisance parameters. Our results [over five-year subperiods] show the following: (i) multivariate normality is rejected in most subperiods, (ii) residual checks reveal no significant departures from the multivariate i.i.d. assumption, and (iii) mean-variance efficiency tests of the market portfolio is not rejected as frequently once it is allowed for the possibility of non-normal errors. -- In diesem Papier schlagen wir exakte likelihood-basierte Tests auf Mittelwert-Varianz- Effizienz im Rahmen des CAPM vor. Dabei wird eine breite Klasse von Verteilungen für den stochastischen Term zugelassen. Normalverteilung ist ein Spezialfall. Die Tests werden im Rahmen von multivariablen linearen Regressionen (MLR) entwickelt. Bekanntlich sind Standardtests, die auf MLR basieren und asymptotisch gerechtfertigt werden, nicht zuverlässig. In der Finanzökonometrie sind exakte Tests für einige wenige Hypothesen vorgeschlagen worden. Die meisten hängen von der Annahme der Normalverteilung ab (Jobson und Korkie (1982), Mac Kinley (1987), Gibbons, Ross und Shanken (1989), Zhou (1993)). Für das gaussianische Modell entsprechen unsere Tests denen von Gibbons, Ross und Shanken. Im nichtgaussianischen Modell betrachten wir Mittelwert-Varianz-Effizienz-Tests, wobei multivariate-Student-t und ?gemischte? Normalverteilungen zugelassen werden. Unser Ansatz gibt mehr Aufschluß darüber, ob die Annahme der Normalverteilung zu restriktiv ist, wenn das CAPM gestestet wird. Wir schlagen auch exakte multivariate Diagnosen (einschließlich Tests für multivariate GARCH-Modelle und multivariate Verallgemeinerungen der bekannten Varianz- Relationen-Tests) sowie Tests auf die Anpassungsgüte und eine Schätzung für die störenden Verschmutzungsparameter vor. Unsere Ergebnisse (für 5-Jahres-Perioden) zeigen das Folgende: (i) multivariate Normalität wird für die meisten Perioden verworfen (ii) die Überprüfung der Residuen zeigt keine signifikante Abweichung von der Annahme einer multivariaten i.i.d. Verteilung (iii), wenn man nichtnormalverteilte Fehler zulässt, werden Mittelwert-Varianz-Effizienz Tests des Marktportfolios seltener verworfen.capital assed pricing model,CAPM,mean-variance efficiency,nonnormality,multivariate linear regression,uniform linear hypothesis,exact test

Robust MM-Estimation and Inference in Mixed Linear Models

Mixed linear models are used to analyse data in many settings. These models generally rely on the normality assumption and are often fitted by means of the maximum likelihood estimator (MLE) or the restricted maximum likelihood estimator (REML). However, the sensitivity of these estimation techniques and related tests to this underlying assumption has been identified as a weakness that can even lead to wrong interpretations. Recently Copt and Victoria-Feser(2005) proposed a high breakdown estimator, namely an S-estimator, for general mixed linear models. It has the advantage of being easy to compute - even for highly structured variance matrices - and allow the computation of a robust score test. However this proposal cannot be used to define a likelihood ratio type test which is certainly the most direct route to robustify an F-test. As the latter is usually a key tool to test hypothesis in mixed linear models, we propose two new robust estimators that allow the desired extension. They also lead to resistant Wald-type tests useful for testing contrasts and covariate efects. We study their properties theoretically and by means of simulations. An analysis of a real data set illustrates the advantage of the new approach in the presence of outlying observations.