Tests In Finite Samples Research Articles

Factor double autoregressive models with application to simultaneous causality testing

Testing causality-in-mean and causality-in-variance has been largely studied. However, none of the tests can detect causality-in-mean and causality-in-variance simultaneously. In this paper, we introduce a factor double autoregressive (FDAR) model. Based on this model, a score test is proposed to detect causality-in-mean and causality-in-variance simultaneously. Furthermore, strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) for the FDAR model are established. A small simulation study shows good performances of the QMLE and the score test in finite samples. A real data example on the causal relationship between Hong Kong stock market and US stock market is given.

Testing for spherical symmetry via the empirical characteristic function

Kolmogorov–Smirnov-type and Cramér–von Mises-type goodness-of-fit tests are proposed for the null hypothesis that the distribution of a random vector X is spherically symmetric. The test statistics utilize the fact that X has a spherical symmetric distribution if, and only if, the characteristic function of X is constant over surfaces of spheres centred at the origin. Both tests come in convenient forms that are straightforwardly applicable with the computer. The asymptotic null distribution of the test statistics as well as the consistency of the tests is investigated under general conditions. Since both the finite sample and the asymptotic null distribution depend on the unknown distribution of the Euclidean norm of X, a conditional Monte Carlo procedure is used to actually carry out the tests. Results on the behaviour of the test in finite-samples are included along with a real-data example.

On a family of test statistics for discretely observed diffusion processes

We consider parametric hypotheses testing for multidimensional ergodic diffusion processes observed at discrete time. We propose a family of test statistics related to the so called ϕ-divergence measures. It is proved that the test statistics in this family are all asymptotically distribution free and converge to the chi squared distribution. In the case of contiguous alternatives, it is also possible to study in detail the power function of the tests.Although all the tests in this family are asymptotically equivalent, the second order expansion of the test statistics suggest a different behavior for the power function of the test in finite samples. We show by the Monte Carlo analysis that, in the small sample case, the performance of the test depends on the choice of the function ϕ and on the statistical model. The simulations show further that there is no uniformly most powerful test in this class.

Constructing smooth tests without estimating the eigenpairs of the limiting process

Based on the well known Karhunen–Loève expansion, it can be shown that many omnibus tests lack power against “high frequency” alternatives. The smooth tests of Neyman (1937) may be employed to circumvent this power deficiency problem. Yet, such tests may be difficult to compute in many applications. In this paper, we propose a more operational approach to constructing smooth tests. This approach hinges on a Fourier representation of the postulated empirical process with known Fourier coefficients, and the proposed test is based on the normalized principal components associated with the covariance matrix of finitely many Fourier coefficients. The proposed test thus needs only standard principal component analysis that can be carried out using most econometric packages. We establish the asymptotic properties of the proposed test and consider two data-driven methods for determining the number of Fourier coefficients in the test statistic. Our simulations show that the proposed tests compare favorably with the conventional smooth tests in finite samples.

A Generalized Kruskal–Wallis Test Incorporating Group Uncertainty with Application to Genetic Association Studies

Motivated by genetic association studies of SNPs with genotype uncertainty, we propose a generalization of the Kruskal-Wallis test that incorporates group uncertainty when comparing k samples. The extended test statistic is based on probability-weighted rank-sums and follows an asymptotic chi-square distribution with k - 1 degrees of freedom under the null hypothesis. Simulation studies confirm the validity and robustness of the proposed test in finite samples. Application to a genome-wide association study of type 1 diabetic complications further demonstrates the utilities of this generalized Kruskal-Wallis test for studies with group uncertainty. The method has been implemented as an open-resource R program, GKW.

Consistent testing for a constant copula under strong mixing based on the tapered block multiplier technique

Considering multivariate strongly mixing time series, nonparametric tests for a constant copula with specified or unspecified change point (candidate) are derived; the tests are consistent against general alternatives. A tapered block multiplier technique based on serially dependent multiplier random variables is provided to estimate p-values of the test statistics. Size and power of the tests in finite samples are evaluated with Monte Carlo simulations. The block multiplier technique might have several other applications for statistical inference on copulas of serially dependent data.

Estimating critical values for testing the i.i.d. in standardized residuals from GARCH models in finite samples

Taking into account that the BDS test--which is used as a misspecification test applied to standardized residuals from the GARCH(1,1) model--is characterized by size distortion and departure from normality in finite samples, this paper obtains the critical values for the finite sample distribution of the BDS test. We focus on bootstrap simulation to avoid the sampling uncertainty of parameter estimation and make use of estimated response surface regressions (RSR) derived from the experimental results. We consider an extensive grid of models to obtain critical values with the results of the bootstrap experiments. The RSR used to estimate them is an artificial neural network (ANN) model, instead of the traditional linear regression models. Specifically, we estimate critical values by using a bootstrap aggregated neural network (BANN) and by employing functions of the sample size and parameters used in the experiment as the embedding dimension and proximity parameters in the BDS statistic, GARCH parameters and even the q-quantiles of the BDS distributions. The main results confirm that the sample size and BDS parameters play a role in size distortion. Finally, an empirical application to three price indexes is performed, to highlight the differences between decisions made using the asymptotic or our predicted critical values for the BDS test in finite samples.

Robustifying multivariate trend tests to nonstationary volatility

This article studies inference of multivariate trend model when the volatility process is nonstationary. Within a quite general framework we analyze four classes of tests based on least squares estimation, one of which is robust to both weak serial correlation and nonstationary volatility. The existing multivariate trend tests, which either use non-robust standard errors or rely on non-standard distribution theory, are generally non-pivotal involving the unknown time-varying volatility function in the limit. Two-step residual-based i.i.d. bootstrap and wild bootstrap procedures are proposed for the robust tests and are shown to be asymptotically valid. Simulations demonstrate the effects of nonstationary volatility on the trend tests and the good behavior of the robust tests in finite samples.

Robust and Efficient One-Way MANOVA Tests

We propose robust tests as alternatives to the classical Wilks’ Lambda test in one-way MANOVA. The robust tests use highly robust and efficient multisample multivariate S-estimators or MM-estimators instead of the empirical covariances. The properties of several robust test statistics are compared. Under the null hypothesis, the distribution of the test statistics is proportional to a chi-square distribution. As an alternative to the asymptotic distribution, we develop a fast robust bootstrap method to estimate the distribution under the null hypothesis. We show when it is asymptotically correct to estimate the null distribution in this way and we use simulations to verify the performance of the bootstrap based tests in finite samples. We also investigate the power of the new tests, as well as their robustness against outliers. Finally, we illustrate the use of these robust test statistics on a real data example. Some additional results are provided as supplemental material.

A Test for Comparing Two Discrete Stochastic Dynamical Systems Under Heteroskedasticity

In this paper we develop a statistic test in order to compare two discrete stochastic dynamical systems with autoregressive structures under heteroskedasticity, more specifically the variance of the errors follows a general function depending on time. The random disturbances or errors can also be contemporaneously correlated through some structure varying in time. The approach is based on a Wald test and we present three consistent estimates for the asymptotic covariance matrix. The study of the asymptotic properties is based on the Lp-mixingale processes theory since we consider errors generated from martingale difference processes. In order to evaluate the performance of the test in finite samples, we ran some simulations with satisfactory results. A real application is also provided.

Testing Linear Factor Pricing Models With Large Cross Sections: A Distribution-Free Approach

In this article, we develop a finite-sample distribution-free procedure to test the beta-pricing representation of linear factor pricing models. In sharp contrast to extant finite-sample tests, our framework allows for unknown forms of nonnormalities, heteroscedasticity, and time-varying covariances. The power of the proposed test procedure increases as the time series lengthens and/or the cross section becomes larger. So the criticism sometimes heard that nonparametric tests lack power does not apply here, since the number of test assets is chosen by the user. This also stands in contrast to the usual tests that lose power or may not even be computable if the number of test assets is too large. Supplementary materials for this article are available online.

Stationarity testing under nonlinear models. Some asymptotic results

Stationarity testing for nonlinear time series models which include several smooth trend components with (possibly) unknown parameters is considered. A pseudo-Lagrange multiplier stationarity test is proposed and its asymptotic behaviour is derived. The limiting null distribution generally depends on the unknown parameters of the model. A bootstrap approach permits this problem to be circumvented and consistency of the bootstrapped test is obtained. The theoretical analysis is complemented with a simulation study which allows us to check the performance of the test in finite samples. The article ends with an empirical application.

Distributional Tests in Multivariate Dynamic Models with Normal and Student t Innovations

We derive Lagrange Multiplier and Likelihood Ratio specifi cation tests for the null hypotheses of multivariate normal and Student t innovations using the Generalised Hyperbolic distribution as our alternative hypothesis. We decompose the corresponding Lagrange Multiplier-type tests into skewness and kurtosis components, from which we obtain more powerful one-sided Kuhn-Tucker versions that are equivalent to the Likelihood Ratio test, whose asymptotic distribution we provide. We conduct detailed Monte Carlo exercises to study our proposed tests in finite samples. Finally, we present an empirical application to ten US sectoral stock returns, which indicates that their conditional distribution is mildly asymmetric and strongly leptokurtic.

The econometrics of mean‐variance efficiency tests: a survey

Summary This paper provides a comprehensive survey of the econometrics of mean-variance efficiency tests. Starting with the classic F-test of Gibbons et al. (1989) and its generalized method of moments version, I analyse the effects of the number of assets and portfolio composition on test power. I then discuss asymptotically equivalent tests based on portfolio weights, and study the trade-offs between efficiency and robustness of using parametric and semi-parametric likelihood procedures that assume either elliptical innovations or elliptical returns. After reviewing finite sample tests, I conclude with a discussion of mean-variance-skewness efficiency and spanning tests, and other interesting extensions.

Statistical inference procedure for the mean–variance efficient frontier with estimated parameters

We suggest finite sample tests for the location of the efficient frontier with the estimated parameters in mean–variance space. The exact densities of the test statistics are derived. We implement the introduced testing procedure empirically by considering monthly returns of ten developed stock markets. It is shown that ignoring the uncertainty about the estimated parameters leads to a more frequent reconstruction of the efficient frontier.

An exact test on structural changes in the weights of the global minimum variance portfolio

In the paper, a finite sample test is suggested for detecting changes in the composition of the global minimum variance portfolio. The exact density of the test statistic is calculated. It appears that under the null hypothesis of no change, it is independent of the parameters of the asset returns distribution. The testing procedure is implemented in a situation that is practically relevant. We show that ignoring the uncertainty about the estimated weights of the holding portfolio leads to misleading results, i.e. to a more frequent reallocation of the investor's wealth.

Distributional Tests in Multivariate Dynamic Models with Normal and Student T Innovations

We derive Lagrange Multiplier and Likelihood Ratio specifi cation tests for the null hypotheses of multivariate normal and Student t innovations using the Generalised Hyperbolic distribution as our alternative hypothesis. We decompose the corresponding Lagrange Multiplier-type tests into skewness and kurtosis components, from which we obtain more powerful one-sided Kuhn-Tucker versions that are equivalent to the Likelihood Ratio test, whose asymptotic distribution we provide. We conduct detailed Monte Carlo exercises to study our proposed tests in finite samples. Finally, we present an empirical application to ten US sectoral stock returns, which indicates that their conditional distribution is mildly asymmetric and strongly leptokurtic.

Size and power properties of tests of the martingale difference hypothesis: a Monte Carlo study

This paper compares the performance of a wide range of approaches to testing the martingale difference hypothesis in economic and financial time series. An extensive Monte Carlo experiment is conducted to evaluate and compare the alternative tests under a martingale difference null hypothesis, which allows for conditional heteroskedasticity and a fat tailed marginal distribution, against alternative hypotheses which receive empirical support in the financial markets. It is found that the wild bootstrap (WB) test, Chow and Denning's variant of a sign test with zero drift, CD(S1), and the bootstrap adjusted sign tests, BS(S1) and BS(S2), are superior to others in that they show desirable size properties under the martingale difference null and excellent power properties against a variety of non-martingale alternatives. In addition, they are all non-parametric finite sample tests and so do not rely on large sample theories for statistical inference: indeed, the BS(S2) test requires no strong assumptions on the distribution of the time series being tested.

DETECTION OF FUNCTIONAL FORM MISSPECIFICATION IN COINTEGRATING RELATIONS

A simple specification test based on fully modified residuals and the cumulative sum (CUSUM) test for cointegration of Xiao and Phillips (2002, Journal of Econometrics, 108, 43–61) are considered as means of testing for functional form in long-run cointegrating relations. It is shown that both tests are consistent under functional form misspecification and lack of cointegration. A simulation experiment is carried out to assess the properties of the tests in finite samples. The Dickey–Fuller test is also considered. The simulation results reveal that the first two tests perform reasonably well. However, the Dickey–Fuller test performs poorly under functional form misspecification.

Comparing Density Forecasts via Weighted Likelihood Ratio Tests: Asymptotic and Bootstrap Methods

This paper proposes and analyzes tests that can be used to compare the accuracy of alternative conditional density forecasts of a variable. The tests are also valid in the broader context of model selection based on out-of-sample predictive ability. We restrict attention to the case of density forecasts derived from non-nested parametric models, with known or estimated parameters. The evaluation makes use of scoring rules, which are loss functions defined over the density forecast and the realizations of the variable. In particular, we consider the logarithmic scoring rule, which leads to the development of asymptotic and bootstrap 'weighted likelihood ratio' tests. The name comes from the fact that the tests compare weighted averages of the scores over the available sample, as a way to focus attention on different regions of the distribution of the variable. For a uniform weight function, the asymptotic test can be interpreted as an extension of Vuong (1989)' s likelihood ratio test for non-nested hypotheses to time series data and to an out-of-sample testing framework. A Monte Carlo simulation explores the size and power properties of this last test in finite samples. An application using S&P500 daily returns shows how the tests can be used to compare the performance of density forecasts obtained from GARCH models with different distributional assumptions.