Some tests for the constancy of regressions under heteroscedasticity

For testing the equality of coefficients of a linear regression model under heteroscedasticity, we suggest an F criterion conditioned on the posterior mean of the ratio of standard deviations of error terms in two subsamples. For pairable subsamples, and exact F test is derived. Sampling experiments show that the Chow test differs substantially from the nominal significance level when the two subsample sizes are unequal, and that the F test conditioned on the posterior mean is superior to other tests when sample sizes are small.     

Tests of non-nested regression models: Small sample adjustments and Monte Carlo evidence

It is known that the small sample significance levels of Cox-type tests of non-nested regression models can be much greater than the nominal level. Adjustments designed to overcome this problem are discussed and two tests are proposed. Monte Carlo evidence on the performance of the tests derived in this paper, the Davidson-MacKinnon J-test and the Fisher-McAleer test is presented. The F-test applied to the comprehensive model is also included in the simulation experiments.     

Joint one-sided tests of linear regression coefficients

The main example of the class of problems considered below is that of testing whether a subset of regression coefficients are jointly zero assuming knowledge of the coefficients' signs. If this knowledge is ignored, the likelihood ratio, Wald, and Lagrange multiplier tests are each equivalent to the F-test. We propose a new test which can be applied as a one-sided t-test and which is UMPI in a subspace of the parameter space. Empirical power comparisons with the power envelope, the F-test, and the exact one-sided likelihood ratio test show that the new test can have exceptionally good power over a wide range of the parameter space.     

Simple and powerful GMM over-identification tests with accurate size

Based on the series long run variance estimator, we propose a new class of over-identification tests that are robust to heteroscedasticity and autocorrelation of unknown forms. We show that when the number of terms used in the series long run variance estimator is fixed, the conventional J statistic, after a simple correction, is asymptotically F-distributed. We apply the idea of the F-approximation to the conventional kernel-based J tests. Simulations show that the J^∗ tests based on the finite sample corrected J statistic and the F-approximation have virtually no size distortion, and yet are as powerful as the standard J tests.     

Tests for two separate regressions

Several recently proposed tests for separate regressions in econometrics are re-examined in the light of recommendations by Cox (1961). This re-examination points to simplified criteria and emphasizes the unity underlying the tests. The exact distributions of some of the tests are developed under the tested hypothesis. These are given a geometrical characterization which is helpful in exploring relations with the classical F-test. An orthogonal decomposition is proposed which provides a direct link between the F-test and tests based upon artificial nesting.     

A lagrange multiplier interpretation of disturbance estimators with an application to testing for nonlinearity

This paper describes a Lagrange multiplier interpretation of LUF disturbance estimators and an associated means of constructing residuals and procedures for testing for certain kinds of misspecification. Together with a recent technique for partially ordering multivariate data, the methods are used to devise an F test for nonlinearity in some or all of the explanatory variables in multiple regression. Unlike several alternatives, the proposed test does not require all misspecified variables to be positively correlated. Power computations, and comparisons with other procedures, yield encouraging results.     

Hypothesis testing based on goodness-of-fit in the moving average time series model

Small sample properties of t-tests are compared with those of tests based on relative goodness- of-fit in the context of the first order moving average time series model. Monte Carlo experiments reported in the paper suggest that the actual size of these t-tests greatly exceeds theoretical large sample significance levels, while conformity of goodness-of-fit statistics to the appropriate chi-square or F-distributions is much closer. The evidence presented suggests that practitioners are well advised to employ goodness-of-fit tests as a check on results of t-tests particularly when the latter indicate ‘significance’.     

M-tests in multivariate models

Summary For testing homogeneity in multivariatek sample model, robust tests based onM-estimators are proposed and their asymptoticx ²-distributions are investigated. FurthermoreM-tests in multivariate regression models are discussed.     

Exact permutation tests for non-nested non-linear regression models

This paper proposes exact distribution-free permutation tests for the specification of a non-linear regression model against one or more possibly non-nested alternatives. The new tests may be validly applied to a wide class of models, including models with endogenous regressors and lag structures. These tests build on the well-known J test developed by Davidson and MacKinnon [1981. Several tests for model specification in the presence of alternative hypotheses. Econometrica 49, 781–793] and their exactness holds under broader assumptions than those underlying the conventional J test. The J-type test statistics are used with a randomization or Monte Carlo resampling technique which yields an exact and computationally inexpensive inference procedure. A simulation experiment confirms the theoretical results and also shows the performance of the new procedure under violations of the maintained assumptions. The test procedure developed is illustrated by an application to inflation dynamics.     

Estimation of variance after a preliminary test of homogeneity and optimal levels of significance for the pre-test

The question of whether to pool two samples in variance estimation is often decided via a preliminary F test. In this paper we show that the optimal pre-test F value is unity for a one- sided alternative, where the objective function is to minimize average relative risk. The outcome is independent of numbers of degrees of freedom in each sample. Optimal significance levels vary somewhat but are close to $\text{1}\text{2}$ for most d.f. and equal to $\text{1}\text{2}$ when numerator and denominator d.f. are equal. The results also apply to regression variance estimation across two data regimes.     

Testing for functional misspecification in regression analysis

Recursive residuals may be used to detect functional misspecification in a regression equation. A simple t-statistic and a related Sign test may be constructed from the residuals. The powers of these tests compare favourably with the Durbin–Watson and other tests commonly used to detect functional misspecification from residuals. In addition the tests are relatively robust to serial correction in an otherwise correctly specified model, and this is a further point in their favour.     

Multivariate ‘analysis of dispersion’ in the presence of intra-class correlation

Summary Subsequent to a review of the effects of familial or intra-class correlation (=ϱ) on the univariateF, or analysis of variance tests, and of methods for obtaining confidence limits for ϱ, results are presented on the effects of familial correlations in tests in multivariate ‘analysis of dispersion’. Methods for obtaining confidence limits are given in the case where a common variance-covariance matrix may be assumed for the successive multivariate samples.     

Multivariate t and beta distributions associated with the multivariate F distribution

Relationships between F, skew t and beta distributions in the univariate case are in this paper extended in a natural way to the multivariate case. The result is two new distributions: a multivariate t/skew t distribution (on ℜ^m) and a multivariate beta distribution (on (0,1)^m). A special case of the former distribution is a new multivariate symmetric t distribution. The new distributions have a natural relationship to the standard multivariate F distribution (on (ℜ⁺)^m) and many of their properties run in parallel. We look at: joint distributions, mathematically and graphically; marginal and conditional distributions; moments; correlations; local dependence; and some limiting cases. Received: March 2001     

Local rank tests in a multivariate nonparametric relationship

Consider a multivariate nonparametric model where the unknown vector of functions depends on two sets of explanatory variables. For a fixed level of one set of explanatory variables, we provide consistent statistical tests, called local rank tests, to determine whether the multivariate relationship can be explained by a smaller number of functions. We also provide estimators for the smallest number of functions, called local rank, explaining the relationship. The local rank tests and the estimators of local rank are defined in terms of the eigenvalues of a kernel-based estimator of some matrix. The asymptotics of the eigenvalues is established by using the so-called Fujikoshi expansion along with some techniques of the theory of U-statistics. We present a simulation study which examines the small sample properties of local rank tests. We also apply the local rank tests and the local rank estimators to a demand system given by a newly constructed data set. This work can be viewed as a “local” extension of the tests for a number of factors in a nonparametric relationship introduced by Stephen Donald.     

A model of fractional cointegration, and tests for cointegration using the bootstrap

The paper proposes a framework for modelling cointegration in fractionally integrated processes, and considers methods for testing the existence of cointegrating relationships using the parametric bootstrap. In these procedures, ARFIMA models are fitted to the data, and the estimates used to simulate the null hypothesis of non-cointegration in a vector autoregressive modelling framework. The simulations are used to estimate p-values for alternative regression-based test statistics, including the F goodness-of-fit statistic, the Durbin–Watson statistic and estimates of the residual d. The bootstrap distributions are economical to compute, being conditioned on the actual sample values of all but the dependent variable in the regression. The procedures are easily adapted to test stronger null hypotheses, such as statistical independence. The tests are not in general asymptotically pivotal, but implemented by the bootstrap, are shown to be consistent against alternatives with both stationary and nonstationary cointegrating residuals. As an example, the tests are applied to the series for UK consumption and disposable income. The power properties of the tests are studied by simulations of artificial cointegrating relationships based on the sample data. The F test performs better in these experiments than the residual-based tests, although the Durbin–Watson in turn dominates the test based on the residual d.     

Model selection criteria in multivariate models with multiple structural changes

This paper considers the issue of selecting the number of regressors and the number of structural breaks in multivariate regression models in the possible presence of multiple structural changes. We develop a modified Akaike information criterion (AIC), a modified Mallows’ C_p criterion and a modified Bayesian information criterion (BIC). The penalty terms in these criteria are shown to be different from the usual terms. We prove that the modified BIC consistently selects the regressors and the number of breaks whereas the modified AIC and the modified C_p criterion tend to overfit with positive probability. The finite sample performance of these criteria is investigated through Monte Carlo simulations and it turns out that our modification is successful in comparison to the classical model selection criteria and the sequential testing procedure robust to heteroskedasticity and autocorrelation.     

Developing a multivariate approach to monitor fuzzy quality profiles

Referring to several applications in which the response quality characteristic is fuzzy, this paper studies how the profile functional relationship between a fuzzy response variable and a predictor variable can be monitored by using a fuzzy regression model which is referred to as profile. The purpose of this paper is to develop a multivariate approach for monitoring process/product fuzzy quality profiles in phase I for applications where the quality characteristic is linguistic, imprecise, vague or deficient. The multivariate approach includes three fuzzy multivariate control charts which are developed by using fuzzy set theory to monitor fuzzy profiles in order to achieve an in-control process. The performance of developed approach is investigated on the basis of signal probability in various out-of-control scenarios through a simulation study. Compared with univariate approach, the results indicate a good performance of our multivariate approach in detecting all sized shifts in process profiles. A real case in tourism industry is utilized to show the applicability of the proposed approach.     

Modelling security market events in continuous time: Intensity based,multivariate point process models

A continuous time econometric modelling framework for multivariate financial market event (or ‘transactions’) data is developed in which the model is specified via the vector conditional intensity. Generalised Hawkes models are introduced that incorporate inhibitory events and dependence between trading days. Novel omnibus specification tests based on a multivariate random time change theorem are proposed. A bivariate point process model of the timing of trades and mid-quote changes is then presented for a New York Stock Exchange stock and related to the market microstructure literature. The two-way interaction of trades and quote changes in continuous time is found to be important empirically.     

The power of tests of predictive ability in the presence of structural breaks

This paper presents analytical, Monte Carlo, and empirical evidence on the effects of structural breaks on tests for equal forecast accuracy and encompassing. We show that out-of-sample predictive content can be hard to find because out-of-sample tests are highly dependent on the timing of the predictive ability. Moreover, predictive content is harder to find with some tests than others: in power, F-type tests of equal forecast accuracy and encompassing often dominate t-type alternatives. Based on these results and evidence from an empirical application, we conclude that structural breaks under the alternative may explain why researchers often find evidence of in-sample, but not out-of-sample, predictive content.     

Consistent likelihood-based estimation of a star-shaped distribution

We consider the classical problem of nonparametrically estimating a star-shaped distribution, i.e., a distribution function F on [0,∞) with the property that F(u)/u is nondecreasing on the set {u : F(u) < 1}. This problem is intriguing because of the fact that a well defined maximum likelihood estimator (MLE) exists, but this MLE is inconsistent. In this paper, we argue that the likelihood that is commonly used in this context is somewhat unnatural and propose another, so called ‘smoothed likelihood’. However, also the resulting MLE turns out to be inconsistent. We show that more serious smoothing of the likelihood yields consistent estimators in this model.