A small sample study of the moments and distribution of the log likelihood ratio statistic

A study of the first four moments of the log likelihood ratio statistic suggests that a transformed Beta ranks as a serious competitor to the Chi-square as an approximating distribution for certain multinomials. An error analysis of the series approximations of these moments indicates that the equiprobable case is not of sufficient generality for the study of multinomial type statistics. This research was partially supported by the Air Force Office of Scientific Research under grant number AFOSR-85-0161.     

Analysis of Divergence in Loglinear Models When Expected Frequencies are Subject to Linear Constraints

Consider the loglinear model for categorical data under the assumption of multinomial sampling. We are interested in testing between various hypotheses on the parameter space when we have some hypotheses relating to the parameters of the models that can be written in terms of constraints on the frequencies. The usual likelihood ratio test, with maximum likelihood estimator for the unspecified parameters, is generalized to tests based on -divergence statistics, using minimum -divergence estimator. These tests yield the classical likelihood ratio test as a special case. Asymptotic distributions for the new -divergence test statistics are derived under the null hypothesis.     

Testing for equality of variance of correlated normal variables

A standard test exists for whether bivariate normal data of arbitrary correlation have equal variances. An extension of this model is useful to test whether two measuring instruments, with which repeated measurements have been made on each of n units, have equal error variance. It is shown that one or two simple F -distributed statistics yield performance comparable with that of the generalized likelihood ratio statistic.     

Specification testing when score test statistics are identically zero

We investigate the problem of specification testing when the score vector evaluated at the restricted MLE is identically zero. Several econometric examples are provided. A general test procedure which generalizes the geometric principle of the score test is proposed. The Wald and the likelihood ratio tests are also analyzed. Even under such irregularities, the usual asymptotic distribution of the likelihood ratio statistics is still valid. However, the Wald-type statistics need to be modified. The generalized score, the likelihood ratio and the modified Wald tests are shown to be asymptotically equivalent. The asymptotic efficiency of the tests is derived.     

Generalizing weak instrument robust IV statistics towards multiple parameters,unrestricted covariance matrices and identification statistics

We generalize the weak instrument robust score or Lagrange multiplier and likelihood ratio instrumental variables (IV) statistics towards multiple parameters and a general covariance matrix so they can be used in the generalized method of moments (GMM). The GMM extension of Moreira's [2003. A conditional likelihood ratio test for structural models. Econometrica 71, 1027–1048] conditional likelihood ratio statistic towards GMM preserves its expression except that it becomes conditional on a statistic that tests the rank of a matrix. We analyze the spurious power decline of Kleibergen's [2002. Pivotal statistics for testing structural parameters in instrumental variables regression. Econometrica 70, 1781–1803, 2005. Testing parameters in GMM without assuming that they are identified. Econometrica 73, 1103–1124] score statistic and show that an independent misspecification pre-test overcomes it. We construct identification statistics that reflect if the confidence sets of the parameters are bounded. A power study and the possible shapes of confidence sets illustrate the analysis.     

Contemporaneous threshold autoregressive models: Estimation,testing and forecasting

This paper proposes a contemporaneous smooth transition threshold autoregressive model (C-STAR) as a modification of the smooth transition threshold autoregressive model surveyed in Teräsvirta [1998. Modelling economic relationships with smooth transition regressions. In: Ullah, A., Giles, D.E.A. (Eds.), Handbook of Applied Economic Statistics. Marcel Dekker, New York, pp. 507–552.], in which the regime weights depend on the ex ante probability that a latent regime-specific variable will exceed a threshold value. We argue that the contemporaneous model is well suited to rational expectations applications (and pricing exercises), in that it does not require the initial regimes to be predetermined. We investigate the properties of the model and evaluate its finite-sample maximum likelihood performance. We also propose a method to determine the number of regimes based on a modified Hansen [1992. The likelihood ratio test under nonstandard conditions: testing the Markov switching model of GNP. Journal of Applied Econometrics 7, S61–S82.] procedure. Furthermore, we construct multiple-step ahead forecasts and evaluate the forecasting performance of the model. Finally, an empirical application of the short term interest rate yield is presented and discussed.     

Likelihood Ratio Test for and Against Nonlinear Inequality Constraints

In applied statistics a finite dimensional parameter involved in the distribution function of the observed random variable is very often constrained by a number of nonlinear inequalities. This paper is devoted to studying the likelihood ratio test for and against the hypothesis that the parameter is restricted by some nonlinear inequalities. The asymptotic null distributions of the likelihood ratio statistics are derived by using the limits of the related optimization problems. The author also shows how to compute critical values for the tests.     

Tests based on φ-divergences for bivariate symmetry

In this paper we introduce a family of test statistics for testing symmetry based on φ-divergence families. These test statistics yield the likelihood ratio test and the Pearson test statistic as special cases. Asymptotic distribution for the new test statistics are derived under both the null and the alternative hypotheses. A simulation study is presented to see that some new test statistics offer an attractive alternative to the classical Pearson test statistic for the problem of symmetry. Received: May 2000     

Applications of the Likelihood Theory in Finance: Modelling and Pricing

This paper discusses the connection between mathematical finance and statistical modelling which turns out to be more than a formal mathematical correspondence. We like to figure out how common results and notions in statistics and their meaning can be translated to the world of mathematical finance and vice versa. A lot of similarities can be expressed in terms of LeCam’s theory for statistical experiments which is the theory of the behaviour of likelihood processes. For positive prices the arbitrage free financial assets fit into statistical experiments. It is shown that they are given by filtered likelihood ratio processes. From the statistical point of view, martingale measures, completeness, and pricing formulas are revisited. The pricing formulas for various options are connected with the power functions of tests. For instance the Black–Scholes price of a European option is related to Neyman–Pearson tests and it has an interpretation as Bayes risk. Under contiguity the convergence of financial experiments and option prices are obtained. In particular, the approximation of Itô type price processes by discrete models and the convergence of associated option prices is studied. The result relies on the central limit theorem for statistical experiments, which is well known in statistics in connection with local asymptotic normal (LAN) families. As application certain continuous time option prices can be approximated by related discrete time pricing formulas.     

Testing conditional symmetry against one-sided alternatives in square contingency tables

For a (k×k) square contingency table with ordered categories, letX(Y) denote the row (column) number. The conditional symmetry model is given byP(X=i, Y=j|X<Y)=P(X=j, Y=i |X>Y), ∀i<j. In this paper, we study the likelihood ratio tests of conditional symmetry in a square contingency table against two particular classes of one-sided alternatives. We obtain the maximum likelihood estimators under each alternative. The asymptotic null distributions of the likelihood ratio statistics are shown to have chi-bar square type distributions. A simulation study is performed by comparing the powers of different tests. The theory developed is illustrated by using the famous eye vision data from Stuart (1953).     

A local maximum likelihood estimator for Poisson regression

A local maximum likelihood estimator based on Poisson regression is presented as well as its bias, variance and asymptotic distribution. This semiparametric estimator is intended to be an alternative to the Poisson, negative binomial and zero-inflated Poisson regression models that does not depend on regularity conditions and model specification accuracy. Some simulation results are presented. The use of the local maximum likelihood procedure is illustrated on one example from the literature. This procedure is found to perform well. This research was partially supported by Calouste Gulbenkian Foundation and PRODEP III.     

Discrete choice modeling with nonstationary panels applied to exchange rate regime choice

This paper develops a regression limit theory for discrete choice nonstationary panels with large cross section (N) and time series (T) dimensions. Some results emerging from this theory are directly applicable in the wider context of M-estimation. This includes an extension of work by Wooldridge [Wooldridge, J.M., 1994. Estimation and Inference for Dependent Processes. In: Engle, R.F., McFadden, D.L. (Eds.). Handbook of Econometrics, vol. 4, North-Holland, Amsterdam] on the limit theory of local extremum estimators to multi-indexed processes in nonlinear nonstationary panel data models.It is shown that the maximum likelihood (ML) estimator is consistent without an incidental parameters problem and has a limit theory with a fast rate of convergence N^1/2T^3/4 (in the stationary case, the rate is N^1/2T^1/2) for the regression coefficients and thresholds, and a normal limit distribution. In contrast, the limit distribution is known to be mixed normal in time series modeling, as shown in [Park, J.Y., Phillips, P.C.B., 2000, Nonstationary binary choice. Econometrica, 68, 1249–1280] (hereafter PP), and [Phillips, P.C.B., Jin, S., Hu, L., 2007. Nonstationary discrete choice: A corrigendum and addendum. Journal of Econometrics 141(2), 1115–1130] (hereafter, PJH).The approach is applied to exchange rate regime choice by monetary authorities, and we provide an analysis of the empirical phenomenon known as “fear of floating”.     

Unconditional tests for association in 2 × 2 contingency tables in the total sum fixed design

The asymptotic approach and Fisher's exact approach have often been used for testing the association between two dichotomous variables. The asymptotic approach may be appropriate to use in large samples but is often criticized for being associated with unacceptable high actual type I error rates for small to medium sample sizes. Fisher's exact approach suffers from conservative type I error rates and low power. For these reasons, a number of exact unconditional approaches have been proposed, which have been seen to be generally more powerful than exact conditional counterparts. We consider the traditional unconditional approach based on maximization and compare it to our presented approach, which is based on estimation and maximization. We extend the unconditional approach based on estimation and maximization to designs with the total sum fixed. The procedures based on the Pearson chi‐square, Yates's corrected, and likelihood ratio test statistics are evaluated with regard to actual type I error rates and powers. A real example is used to illustrate the various testing procedures. The unconditional approach based on estimation and maximization performs well, having an actual level much closer to the nominal level. The Pearson chi‐square and likelihood ratio test statistics work well with this efficient unconditional approach. This approach is generally more powerful than the other p‐value calculation methods in the scenarios considered.     

A comparison of the Box-Cox maximum likelihood estimator and the non-linear two-stage least squares estimator

This paper investigates the limiting behaviour of the ‘maximum likelihood estimator’(MLE) based on normality, as well as the nonlinear two-stage least squares estimator (NL2S), for the i.i.d. and regression models in which the Box-Cox transformation is applied to the dependent variable. Since the transformed variable cannot in general be normally distributed, the untransformed variable is assumed to have a two-parameter gamma distribution. Tables of probability limits and asymptotic variance demonstrate that, in this case, the inconsistency of the ‘normal MLE’ is often quite pronounced, while the NL2S is consistent and typically well behaved.     

Identifying, estimating and testing restricted cointegrated systems: An overview

The notion of cointegration has led to a renewed interest in the identification and estimation of structural relations among economic time series. This paper reviews the different approaches that have been put forward in the literature for identifying cointegrating relationships and imposing (possibly over-identifying) restrictions on them. Next, various algorithms to obtain (approximate) maximum likelihood estimates and likelihood ratio statistics are reviewed, with an emphasis on so-called switching algorithms. The implementation of these algorithms is discussed and illustrated using an empirical example.     

Serial correlation in latent discrete variable models

We consider the problems of estimation and testing in models with serially correlated discrete latent variables. A particular case of this is the time series regression model in which a discrete explanatory variable is measured with error. Test statistics are derived for detecting serial correlation in such a model. We then show that the likelihood function can be evaluated by a recurrence relation, and thus maximum likelihood estimation is computationally feasible. An illustrative example of these methods is given, followed by a brief discussion of their applicability to a Markov model of switching regressions.     

Boekbespreking

Book reviewed in this article: Statistiek voor S.P.D., H. J. A. v. D. POST The advanced theory of statistics (Vol. 3), M. G. KENDALL and A. STUART Handbook of tables for probability and statistics, W. H. BEYER The Theory of Probability, B. W. GNEDENKO Elements of Nonparametric Statistics, GOTTFRIED E. NOETHER Elementary Probability, EDWARD O. THORP, Wiley The Design and Analysis of Scientific Experiments, K. C. PENG Scientific inventory management, JOSEPH BUCHAN and ERNEST KOENIGSBERG Time-dependent Results in Storage Theory, N. V. PRABHU Introduction to Operations Research, F. S. HILLIER and G. J. LIEBERMAN Proceedings IBM Scientific computing symposium; Simulation Models and Gaming, IBM Data Processing Division Spatial dispersion of economic activity, H. C. Bos Monopolistic Competition, technical progress and income distribution, J. G. M. HILHORST Statistical Techniques in Business and Economics, R. D. MASON, R. D. Irwin De Genormaliseerde Methode van Werkclassiflcatie als Meetinstrument, A. HAZEWINKEL, J. B. Wolters     

Tests of independence in a bivariate exponential distribution

The problem of testing independence in a two component series system is considered. The joint distribution of component lifetimes is modeled by the Pickands bivariate exponential distribution, which includes the widely used Marshall and Olkins distribution and the Gumbels type II distribution. The case of identical components is first addressed. Uniformly most powerful unbiased test (UMPU) and likelihood ratio test are obtained. It is shown that inspite of a nuisance parameter, the UMPU test is unconditional and this test turns out to be the same as the likelihood ratio test. The case of nonidentical components is also addressed and both UMPU and likelihood ratio tests are obtained. A UMPU test is obtained to test the identical nature of the components and extensions to the type II censoring scheme and multiple component systems are also discussed. Some modifications to account for the difference in parameters under test and use conditions are also discussed.     

Boekbespreking

Book reviewed in this article: Probability and Statistics, Models for Research, D. E. Bailey ; John Wiley & Sons, Inc., New York - Londen, 1971, 686 blz. Kansrekening, E. A. W. Bolle , F. Göbel , J. M. H. Lenoir ; Kluwer-Deventer, 1971, 111 blz. Receuil de problèmes de Calcul des Probabílitès, D. Dacunha -Castelle , D. Revuz et M. Schreiber , préface de A. Tortrat , 2me éd. revue et augmentée, Masson et Cie, Paris, 1970, xii + 234 p. The coordinate free approach to Gauss-Markov estimation, H. Drygas . (Lecture notes in Operations Research and Mathematical Systems, Economics, Computer Science, Information and Control, vol. 40). Springer Verlag, Berlin-Heidelberg-New York, 1970, vii + 113 p., prijs DM 12,-. Beschrijvende Statistiek, E. A. W. Bolle , F. Gobel , J. M. H. Lenoir , Kluwer, Deventer, 1971. 164 blz. Beschrijvende Statistiek voor gedragswetenscbappen, w. H. van den Ende , Agon Elsevier, Amsterdam-Brussel, 1971, XVI + 343 blz. Applied Statistics in Decision Making, George K. Chacko , American Elsevier, New-York-London-Amsterdam, 1971, 491 pag. Introduction to Modern Factor Analysis, Wilson H. Guertin and John P. Bailey Jr ., Edwards Brothers Inc., Ann Arbor (Mich.) 1970, 472 pag., 14. Mathematical Model Building in Economics and Industry, Second Series, Editor M. G. Kendall Charles Griffin and Company Ltd., London 1970, 277 blz. Measuring uncertainty, an elementary introduction to Bayesian statistics. S. A. Schmitt : Addison-Wesley Series in behavioral science: quatitative methods, 1969, 400 p., prijs f 61,-. Practical nonparametric statistics, W. J. Conover ; John Wiley & Sons, Inc. New Tork - London, 1971, 462 blz. The Teaching of Probability and Statistics, onder redactie van L. Råde . Wiley Interscience Division, John Wiley and Sons, Inc. New York-London-Sydney, 1970. 373 blz. Marking decisions, D. Lindley , Wiley - Interscience, London etc. 1971, 195 blz. Odhady parametru v linearnich ekonomickych modelech, door Jan Hurt . Uitg. Ceskoslovenska Akademie ved Ekonomicko-matematicka Laborator pri ekoniomcken ustavu. Nonparametric Methods in Multivariate Analysis. M. L. Puri and P. K. Sen , John Wiley & Sons, Inc., New York - London, 1971. 440 p. Angewandte Statistik, Zweiter Teil, Mehrdimensionale Probleme, Kurt Stange , 1971, Springerverlag, Berlijn-Heidelberg, II + 505 pagina's. Statistische analyse methoden (voortgezette statistiek en waarschijnlijkheidsrekening voor bedrijfsleven en hoger beroepsonderwijs), H. P. Anderson , Universitaire Pers Rotterdam 1971, XII + 187 blz., Fl. 33.50.