A test for the presence of conditional heteroskedasticity within arch-m framework

This paper is concerned with testing the presence of ARCH within the ARCH-M model as the alternative hypothesis. Standard testing procedures are inapplicable since a nuisance parameter is unidentified under the null hypothesis. Nonetheless, the diagnostic tests for the presence of the conditional variance is very important since any misspecification in the conditional variance equation leads to inconsistent estimates of the conditional mean parameters. BTo resolve the problem of unidentified nuisance parameter, 'Ne apply Davies' approach, and investigate its finite sample performance through a Monte Carlo study.     

Maximizing a Family of Optimal Statistics over a Nuisance Parameter with Applications to Genetic Data Analysis

In this article, a simple algorithm is used to maximize a family of optimal statistics for hypothesis testing with a nuisance parameter not defined under the null hypothesis. This arises from genetic linkage and association studies and other hypothesis testing problems. The maximum of optimal statistics over the nuisance parameter space can be used as a robust test in this situation. Here, we use the maximum and minimum statistics to examine the sensitivity of testing results with respect to the unknown nuisance parameter. Examples from genetic linkage analysis using affected sub pairs and a candidate-gene association study in case-parents trio design are studied.     

Detecting parameter shift in garch models

This paper applies recent theories of testing for parameter constancy to the conditional variance in a GARCH model. The supremum Lagrange multiplier test for conditional Gaussian GARCH models and its robustified variants are discussed. The asymptotic null distribution of the test statistics are derived from the weak convergence of the scores, and the critical values from the hitting probability of squared Bessel process.

Monte Carlo studies on the finite sample size and power performance of the supremum LM tests are conducted. Applications of these tests to S&P 500 indicate that the hypothesis of stable conditional variance parameters can be rejected.     

Detecting parameter shift in garch models

This paper applies recent theories of testing for parameter constancy to the conditional variance in a GARCH model. The supremum Lagrange multiplier test for conditional Gaussian GARCH models and its robustified variants are discussed. The asymptotic null distribution of the test statistics are derived from the weak convergence of the scores, and the critical values from the hitting probability of squared Bessel process.

Monte Carlo studies on the finite sample size and power performance of the supremum LM tests are conducted. Applications of these tests to S&P 500 indicate that the hypothesis of stable conditional variance parameters can be rejected.     

Hypothesis testing in the presence of nuisance parameters

How to deal with nuisance parameters is an important problem in econometrics because of the non-experimental nature of economic data. This paper suggests a new approach to dealing with such parameters in the context of hypothesis testing. It involves calculating p-values conditional on values for key nuisance parameters and then taking a weighted average of these values with the weights reflecting the likelihood or posterior probabilities of these values being true. Two specific applications are discussed. These are testing linear regression coefficients in the presence of first-order autoregressive (AR(1)) disturbances and testing for AR(1) disturbances in the dynamic linear regression model. For the former testing problem, a Monte Carlo experiment demonstrates that the new procedure typically provides more accurate inferences than the accepted conventional tests.     

Sequential parameter structure,conditional inference,and likelihood drop

Successive tests of hypotheses, as exemplified with an analysis of variance table, impose a set theoretic structure on the parameter space and yet allow much arbitrariness in the definition of nuisance parameters. Two major types of statistical model, the exponential and transformation, are shown to have by basic theory well defined conditional testing procedures. The two types of testing procedure are then shown to have opposite forms of set theoretic structure on the sample space, and to differ sharply from the commonly used deviance or likelihood drop methods. The two types of model have the normal linear model as the intersection model and the two opposite forms of testing procedure manage to coincide by product space structure and independence. Details of the two types of testing procedure are discussed, related to the arbitrariness in nuisance parameter definition, and organized to provide a general-case pattern for the development of conditional procedures as an alternative to the default likelihood-drop methods.     

Robust Tests Against Smooth Transition Autoregressive Models

In this paper, we consider testing for linearity against a well-known class of regime switching models known as the smooth transition autoregressive (STAR) models. Apart from the model selection issues, one reason for interest in testing for linearity in time-series models is that non-linear models such as the STAR are considerably more difficult to use. This testing problem is non-standard because a nuisance parameter becomes unidentified under the null hypothesis. In this paper, we further explore the class of tests proposed by Luukkonen, Saikonnen and Terasvirta (1988). Luukkonen et al . (1988) proposed LM tests for linearity against STAR models. A potential difficulty here is that the linear approximation introduces high leverage points, and hence outliers are likely to be quite influential. To overcome this difficulty, we use the same approximating linear model of Luukkonen et al . (1988), but we apply Wald and F -tests based on l 1 - and bounded influence estimates. The efficiency gains of this procedure cannot be easily deduced from the existing theoretical results because the test is based on a misspecified model under H 1 . Therefore, we carried out a simulation study, in which we observed that the robust tests have desirable properties compared to the test of Luukkonen et al . (1988) for a range of error distributions in the STAR model, in particular the robust tests have power advantages over the LM test.     

The Time-Varying-Parameter Model for Modeling Changing Conditional Variance: The Case of the Lucas Hypothesis

The main econometric issue in testing the Lucas (1973) hypothesis in a time series context is estimation of the forecast-error variance conditional on past information. The conditional variance may vary through time as monetary policy evolves and agents are obliged to infer its present state. Under the assumption that a monetary policy regime is continuously changing, a time-varying-parameter model is proposed for the monetary-growth function. Based on Kalman-filtering estimation of recursive forecast errors and their conditional variances, the Lucas hypothesis is tested for the U.S. economy (1964:1–1985:4) using monetary growth as aggregate demand variable. The Lucas hypothesis is rejected in favor of Friedman's (1977) hypothesis—the conditional variance of monetary growth affects real output directly, not through the coefficients on the forecast-error term in the Lucas-type output equation.     

Robust Bayesian hypothesis testing in the presence of nuisance parameters

Robust Bayesian testing of point null hypotheses is considered for problems involving the presence of nuisance parameters. The robust Bayesian approach seeks answers that hold for a range of prior distributions. Three techniques for handling the nuisance parameter are studied and compared. They are (i) utilize a noninformative prior to integrate out the nuisance parameter; (ii) utilize a test statistic whose distribution does not depend on the nuisance parameter; and (iii) use a class of prior distributions for the nuisance parameter. These approaches are studied in two examples, the univariate normal model with unknown mean and variance, and a multivariate normal example.     

Extending permutation conditional inference to unconditional ones

In this presentation we discuss the extension of permutation conditional inferences to unconditional or population ones. Within the parametric approach this extension is possible when the data set is randomly selected by well-designed sampling procedures on well-defined population distributions, provided that their nuisance parameters have boundely complete statistics in the null hypothesis or are provided with invariant statistics. When these conditions fail, especially if selection-bias procedures are used for data collection processes, in general most of the parametric inferential extensions are wrong or misleading. We will see that, since they are provided with similarity and conditional unbiasedness properties and if correctly applicable, permutation tests may extend, at least in a weak sense, conditional to unconditional inferences.     

Asymptotic properties of a conditional maximum-likelihood estimator

Inference for a scalar interest parameter in the presence of nuisance parameters is considered in terms of the conditional maximum-likelihood estimator developed by Cox and Reid (1987). Parameter orthogonality is assumed throughout. The estimator is analyzed by means of stochastic asymptotic expansions in three cases: a scalar nuisance parameter, m nuisance parameters from m independent samples, and a vector nuisance parameter. In each case, the expansion for the conditional maximum-likelihood estimator is compared with that for the usual maximum-likelihood estimator. The means and variances are also compared. In each of the cases, the bias of the conditional maximum-likelihood estimator is unaffected by the nuisance parameter to first order. This is not so for the maximum-likelihood estimator. The assumption of parameter orthogonality is crucial in attaining this result. Regardless of parametrization, the difference in the two estimators is first-order and is deterministic to this order.     

Variance Shifts,Structural Breaks,and Stationarity Tests

This article considers the problem of testing the null hypothesis of stochastic stationarity in time series characterized by variance shifts at some (known or unknown) point in the sample. It is shown that existing stationarity tests can be severely biased in the presence of such shifts, either oversized or undersized, with associated spurious power gains or losses, depending on the values of the breakpoint parameter and on the ratio of the prebreak to postbreak variance. Under the assumption of a serially independent Gaussian error term with known break date and known variance ratio, a locally best invariant (LBI) test of the null hypothesis of stationarity in the presence of variance shifts is then derived. Both the test statistic and its asymptotic null distribution depend on the breakpoint parameter and also, in general, on the variance ratio. Modifications of the LBI test statistic are proposed for which the limiting distribution is independent of such nuisance parameters and belongs to the family of Cramér–von Mises distributions. One such modification is particularly appealing in that it is simultaneously exact invariant to variance shifts and to structural breaks in the slope and/or level of the series. Monte Carlo simulations demonstrate that the power loss from using our modified statistics in place of the LBI statistic is not large, even in the neighborhood of the null hypothesis, and particularly for series with shifts in the slope and/or level. The tests are extended to cover the cases of weakly dependent error processes and unknown breakpoints. The implementation of the tests are illustrated using output, inflation, and exchange rate data series.     

Testing and Confidence estimation of the Mean of a multidimensional Gaussian Process

From the sequential observation of a multidimensional continuous time Gaussian process, whose mean vector depends linearly of a multidimensional parameter, we consider the confidential estimation of the parameter value and the testing problem of a simple hypothesis about the parameter, in presence of a nuisance variance parameter. The method is based on a previously obtained [cf. 4] point estimate for the case of a known covariance structure. We first see that this estimate is, in fact, independent of the variance parameter. For the hypotheses testing problem, the invariance under certain groups of transformations and the partial sufficiency allows to construct optimal terminal tests. Furthermore we determine the observation time necessary to control its power function. These testing results may be translated in terms of most accurate confidence sets. If the observation is stopped according to the diameter of the confidence set, under some condition, the confidence level is preserved.     

On the Use of Bayes Factor in Frequentist Testing of a Precise Hypothesis

This article deals with Bayes factors as useful Bayesian tools in frequentist testing of a precise hypothesis. A result and several examples are included to justify the definition of Bayes factor for point null hypotheses, without merging the initial distribution with a degenerate distribution on the null hypothesis. Of special interest is the problem of testing a proportion (joint with a natural criterion to compare different tests), the possible presence of nuisance parameters, or the influence of Bayesian sufficiency on this problem. The problem of testing a precise hypothesis under a Bayesian perspective is also considered and two alternative methods to deal with are given.     

INFERENCE ABOUT THE MEAN OF A POISSON DISTRIBUTION IN THE PRESENCE OF A NUISANCE PARAMETER

In this note we examine the problem of estimating the mean of a Poisson distribution when a nuisance parameter is present. Using a condition of Cox (1958) about ancillarity in the presence of a nuisance parameter, we justify that inference about the parameter should be carried out using the conditional distribution given the appropriate ancillary statistics. A small simulation study has been done to compare the performance of the conditional likelihood approach and the standard likelihood approach.     

一类基于非参数回归的条件异方差检验

现有基于参数模型构造的条件异方差检验往往存在模型设定偏误问题。为了避免模型误设对检验结果的影响,并且同时捕获多种条件异方差现象,本文基于非参数回归构造了不依赖于特定模型形式的条件异方差检验统计量。该统计量可视作条件方差和无条件方差之间差异的加权平均,在原假设成立时渐近服从标准正态分布。数值模拟结果一方面表明本文统计量具有良好的有限样本性质,另一方面也说明条件均值模型误设会导致错误地拒绝条件同方差的原假设,凸显了本文引入非参数方法构造条件异方差检验的必要性。实证分析采用本文统计量探讨了国际主要股指收益率的条件异方差现象,得到了与Engle (1982)不同的检验结果,可能意味着股指收益率呈现出非线性动态特征。     

MINIMUM SIGNIFICANCE AND DISTRIBUTION FREE TESTS1

A general way of testing in the presence of nuisance parameters is to choose from a family of tests the one to maximize evidence against null hypothesis; that is, to minimize the significance level. This method yields exact tests when applied to distribution-free testing in various statistical designs; arbitrary choice of score functions is eliminated. However, the exact null distributions are highly non-normal, and there are problems with both computation and asymptotic theory.     

Approximate conditional inference and the linear functional model

Approximate conditional inference is developed for the slope parameter of the linear functional model with two variables. It is shown that the model can be transformed so that the slope parameter becomes an angle and nuisance parameters are radial distances. If the nuisance parameters are known an exact confidence interval based on a location-type conditional distribution is available for the angle. More gen¬erally, confidence distributions are used to average the conditional distribution over the nuisance parameters yielding an approximate conditional confidence interval that reflects the precision indicated by the data. An example is analyzed.     

Generalizing the Welch Test To Non-Zero Hypotheses on the Variance Component in the One-Way Random Effects Model Under Variance Heterogeneity

In an unbalanced and heteroscedastic one-way random effects model of analysis of variance we consider hypotheses concerning the between group variance where in a classical set-up the hypothesis whether this variance component is zero is tested. We consider the still open problem of testing also extended hypotheses concerning this parameter and generalize the prominent Welch test to deal with these hypotheses, too.     

Irreconcilability of P-value and Bayesian measure in two-sided hypothesis: Pareto distribution with the presence of nuisance parameter

Abstract

This article is concerned with the comparison of Bayesian and classical testing of a point null hypothesis for the Pareto distribution when there is a nuisance parameter. In the first stage, using a fixed prior distribution, the posterior probability is obtained and compared with the P-value. In the second case, lower bounds of the posterior probability of H₀, under a reasonable class of prior distributions, are compared with the P-value. It has been shown that even in the presence of nuisance parameters for the model, these two approaches can lead to different results in statistical inference.