CONDITIONAL LIKELIHOOD RATIO TEST WITH MANY WEAK INSTRUMENTS

This paper investigates the asymptotic size properties of a two-stage test in the linear instrumental variables model when in the first stage a Hausman (1978) specification test is used as a pretest of exogeneity of a regressor. In the second stage, a simple hypothesis about a component of the structural parameter vector is tested, using a t-statistic that is based on either the ordinary least squares (OLS) or the two-stage least squares estimator (2SLS), depending on the outcome of the Hausman pretest. The asymptotic size of the two-stage test is derived in a model where weak instruments are ruled out by imposing a positive lower bound on the strength of the instruments. The asymptotic size equals 1 for empirically relevant choices of the parameter space. The size distortion is caused by a discontinuity of the asymptotic distribution of the test statistic in the correlation parameter between the structural and reduced form error terms. The Hausman pretest does not have sufficient power against correlations that are local to zero while the OLS-based t-statistic takes on large values for such nonzero correlations. Instead of using the two-stage procedure, the recommendation then is to use a t-statistic based on the 2SLS estimator or, if weak instruments are a concern, the conditional likelihood ratio test by Moreira (2003).

The Rasch model allows for a conditional likelihood ratio goodness of fit test. The speed of approximation of the test statistic to the limiting distribution as a function of sample size and test length has not been analyzed so far. Three bootstrap simulation methods are analyzed with respect to their performance in providing a proper distribution of the test statistic under the null- and the alternative hypothesis. We found a stable approximation to the limiting χ 2-distribution for sample sizes of at least 500 and 10 items. The three bootstrap algorithms rendered consistent results for the H 0-cases but not for the H 1-cases. A sequential probability sampling scheme proves sufficiently apt for generating samples under the alternative hypothesis. This superiority can be justified from a theoretical point of view.

Loglinear unidimensional and multidimensional Rasch models are considered for the analysis of repeated observations of polytomous indicators with ordered response categories. Reparameterizations and parameter restrictions are provided which facilitate specification of a variety of hypotheses about latent processes of change. Models of purely quantitative change in latent traits are proposed as well as models including structural change. A conditional likelihood ratio test is presented for the comparison of unidimensional and multiple scales Rasch models. In the context of longitudinal research, this renders possible the statistical test of homogeneity of change against subject-specific change in latent traits. Applications to two empirical data sets illustrate the use of the models.

This paper provides an insight to the time-varying dynamics of the shape of the distribution of financial return series by proposing an exponential weighted moving average (EWMA) model that jointly estimates volatility, skewness and kurtosis over time using a modified form of the Gram–Charlier density in which skewness and kurtosis appear directly in the functional form of this density. In this setting, Value-at-Risk (VaR) can be described as a function of the time-varying higher moments by applying the Cornish-Fisher expansion series of the first four moments. An evaluation of the predictive performance of the proposed model in the estimation of 1-day and 10-day VaR forecasts is performed in comparison with the historical simulation, filtered historical simulation and generalized autoregressive conditional heteroscedasticity (GARCH) model. The adequacy of the VaR forecasts is evaluated under the unconditional, independence and conditional likelihood ratio tests as well as Basel II regulatory tests. The results presented have significant implications for risk management, trading and hedging activities as well as in the pricing of equity derivatives.

We describe a software tool to perform haplotype-based association analysis, for quantitative and qualitative traits, in population and family samples, using single nucleotide polymorphism or multiallelic marker data. A range of tests is offered: omnibus and haplotype-specific tests; prospective and retrospective likelihoods; covariates and moderators; sliding window analyses; permutation P-values. We focus on the ability to flexibly impose constraints on haplotype effects, which allows for a range of conditional haplotype-based likelihood ratio tests: for example, whether an allele has an effect independent of its haplotypic background, or whether a single variant can explain the overall association at a locus. We illustrate using these tests to dissect a multi-locus association. WHAP is a C/C++ program, freely available from the author's website: http://pngu.mgh.harvard.edu/purcell/whap/

This paper studies the applications of extreme value theory on analysis for closing price data of the Dow-Jones industrial index and Danish fire insurance claims data. The generalized extreme value (GEV) distribution is considered in analyzing the real data, and the hypothesis testing problem for the shape parameter of GEV distribution is investigated based on a new test statistic—the $$L_q$$ -likelihood ratio ( $$L_q$$ R) statistic. The $$L_q$$ R statistic can be treated as a generalized form of the classical likelihood ratio (LR) statistic. We show that the asymptotic behavior of proposed statistic is characterized by the degree of distortion $$q$$ . For small and modest sample sizes, the $$L_q$$ R statistic is still available when $$q$$ is properly chosen. By simulation studies, the proposed statistic not only performs the asymptotic properties, but also outperforms the classical LR statistic as the sample sizes are modest or even small. Meanwhile, the test power based on the new statistic is also simulated by Monte Carlo methods. At last, the models are diagnosed by graphical methods.

A simple iterative procedure is found for the exact null and alternative distributions of likelihood ratio, cumulative sum and related statistics for testing for a change in probability of a sequence of independent binomial random variables. It is concluded that the likelihood ratio test is slightly less powerful than the cumulative sum test near the middle of the sequence, but that the likelihood ratio test is much more powerful near the ends. An example from epidemiology is used to illustrate the results. Under certain conditions the cumulative sum test is identical to a two-sample Kolmogorov-Smirnov test and so the iterative procedure can be used to find the null distribution of Kolmogorov-Smirnov and related statistics.

In this research paper various new advanced inferential tools namely modified likelihood ratio (LR), Ward and Lagrange Multiplier test statistics have been proposed for testing general linear hypothesis in stochastic linear regression model. In this process internally studentized residuals have been used. This research study has brought out some new advance tools for analysing inferential aspects of stochastic linear regression models by using internally studentized residuals. Miguel Fonseca et.al [1] developed statistical inference in linear models dealing with the theory of maximum likelihood estimates and likelihood ratio tests under some linear inequality restrictions on the regression coefficients. Tim Coelli [2] used Monte carlo experimentation to investigate the finite sample properties of maximum likelihood (ML) and correct ordinary least squares (COLS) estimators of the half –normal stochastic frontier production function. In 2011, p. Bala siddamuni et.al [3] have developed advanced tools for mathematical and stochastic modelling.

This article evaluates the accuracy of approximate confidence intervals for parameters and quantiles of the smallest extreme value and normal distributions. The findings also apply to the Weibull and the lognormal distributions. The interval estimates are based on (a) the asymptotic normality of the maximum likelihood estimator, (b) the asymptotic x 2 distribution of the likelihood ratio (LR) statistic, (c) a mean and variance correction to the signed square roots of the LR statistic, and (d) the Bartlett correction to the LR statistic. The extensive Monte Carlo results about true error probabilities and average lengths under various degrees of censoring show advantages of the LR-based intervals. For complete or moderately censored samples, the mean and variance correction to the LR statistic gives nearly exact and symmetric error probabilities. In small samples with heavy censoring, the Bartlett correction tends to give conservative error probabilities, whereas the uncorrected LR interval is often anticonservative. The results also indicate that LR-based methods have longer interval lengths than intervals based on the asymptotic normality of the maximum likelihood estimator.

From first principles, we derive a quadratic nonlinear, first-order dynamical system capable of performing exact Bayes-Markov inferences for a wide class of biologically plausible stimulus-dependent patterns of activity while simultaneously providing an online estimate of model performance. This is accomplished by constructing a dynamical system that has solutions proportional to the probability distribution over the stimulus space, but with a constant of proportionality adjusted to provide a local estimate of the probability of the recent observations of stimulus-dependent activity-given model parameters. Next, we transform this exact equation to generate nonlinear equations for the exact evolution of log likelihood and log-likelihood ratios and show that when the input has low amplitude, linear rate models for both the likelihood and the log-likelihood functions follow naturally from these equations. We use these four explicit representations of the probability distribution to argue that, in contrast to the arguments of previous work, the dynamical system for the exact evolution of the likelihood (as opposed to the log likelihood or log-likelihood ratios) not only can be mapped onto a biologically plausible network but is also more consistent with physiological observations.

The limit distributions of likelihood ratio and cumulative sum tests for a change in a binomial probability

Two step-down tests for equality of covariance matrices

Tests for a family of random-effects covariance structures in a multivariate growth curve model

In this article, we focus on the integer valued autoregressive model, INAR (1), with Poisson innovations. We test the null of serial independence, where the INAR parameter is zero, versus the alternative of a positive INAR parameter. To this end, we propose different explicit approximations of the likelihood ratio (LR) statistic. We derive the limiting distributions of our statistics under the null. In a simulation study, we compare size and power of our tests with the score test, proposed by Sun and McCabe [2013. Score statistics for testing serial dependence in count data. Journal of Time Series Analysis 34 (3):315–29]. The size is either asymptotic or derived via response surface regressions of critical values. We find that our statistics are superior to score in terms of power and work just as well in terms of size. Another finding is that the powers of our approximate LR statistics compare well with the power of the numerical LR statistic. Power simulations are also performed under an INAR(2) framework, with similar outcome.

In this paper, we examine by Monte Carlo experiments the small sample properties of the W (Wald), LM (Lagrange Multiplier) and LR (Likelihood Ratio) tests for equality between sets of coefficients in two linear regressions under heteroscedasticity. The small sample properties of the size-corrected W, LM and LR tests proposed by Rothenberg (1984) are also examined and it is shown that the performances of the size-corrected W and LM tests are very good. Further, we examine the two-stage test which consists of a test for homoscedasticity followed by the Chow (1960) test if homoscedasticity is indicated or one of the W, LM or LR tests if heteroscedasticity should be assumed. It is shown that the pretest does not reduce much the bias in the size when the sizecorrected citical values are used in the W, LM and LR tests.