Multivariate Signed-Rank Tests in Vector Autoregressive Order Identification (original) (raw)

Affine-invariant aligned rank tests for the multivariate general linear model with VARMA errors

Journal of Multivariate Analysis, 2005

We develop optimal rank-based procedures for testing affine-invariant linear hypotheses on the parameters of a multivariate general linear model with elliptical VARMA errors. We propose a class of optimal procedures that are based either on residual (pseudo-)Mahalanobis signs and ranks, or on absolute interdirections and lift-interdirection ranks, i.e., on hyperplanebased signs and ranks. The Mahalanobis versions of these procedures are strictly affineinvariant, while the hyperplane-based ones are asymptotically affine-invariant. Both versions generalize the univariate signed rank procedures proposed by Hallin and Puri (J. Multivar. Anal. 50 (1994) 175), and are locally asymptotically most stringent under correctly specified radial densities. Their AREs with respect to Gaussian procedures are shown to be convex linear combinations of the AREs obtained in Hallin and Paindaveine (Ann. Statist. 30 (2002) 1103; Bernoulli 8 (2002) 787) for the pure location and purely serial models, respectively. The resulting test statistics are provided under closed form for several important particular cases, including multivariate Durbin-Watson tests, VARMA order identification tests, etc. The key technical result is a multivariate asymptotic linearity result proved in Hallin and Paindaveine (Asymptotic linearity of serial and nonserial multivariate signed rank statistics, submitted).

Rank-based testing for semiparametric VAR models: A measure transportation approach

Bernoulli

We develop a class of tests for semiparametric vector autoregressive (VAR) models with unspecified innovation densities based on the recent measure-transportation-based concepts of multivariate center-outward ranks and signs. We show that these concepts, combined with Le Cam's asymptotic theory of statistical experiments, yield novel testing procedures, which (a) are valid under a broad class of innovation densities (possibly non-elliptical, skewed, and/or with infinite moments), (b) are optimal (locally asymptotically maximin or most stringent) at selected ones, and (c) are robust against additive outliers. In order to do so, we establish, for a general class of center-outward rankbased serial statistics, a Hájek asymptotic representation result, of independent interest, which allows for a rank-based reconstruction of central sequences. As an illustration, we consider the problems of testing the absence of serial correlation in multiple-output and possibly non-linear regression (an extension of the classical Durbin-Watson problem) and the sequential identification of the order p of a VAR(p) model. A Monte Carlo comparative study of our tests and their routinely-applied Gaussian competitors demonstrates the benefits (in terms of size, power, and robustness) of our methodology; these benefits are particularly significant in the presence of asymmetric and leptokurtic innovation densities. A real-data application concludes the paper.

Rank Tests for Serial Dependence

Journal of Time Series Analysis, 1981

A family of linear rank statistics is proposed in order to test the independence of a time series, under the assumption that the random variables involved have symmetric distributions with zero medians, without the standard assumptions of normality or identical distributions. The family considered includes analogues of the sign, Wilcoxon signed-rank and van der Waerden tests for symmetry about zero and tables constructed for these tests remain applicable in the present context. The tests proposed are exact and may be applied to assess serial dependence at lag one or greater. The procedures developed are illustrated by a test of the efficiency of forward exhange rates as predictors of future spot rates during the German hyperinflation.

Rank-based Inference for Multivariate Nonlinear and Long-memory Time Series Models

JOURNAL OF THE JAPAN STATISTICAL SOCIETY, 2010

The portfolio of the Japanese Government Pension Investment Fund (GPIF) consists of a linear combination of five benchmarks of financial assets. Some of these exhibit long-memory and nonlinear behavior. Their analysis therefore requires multivariate nonlinear and long-memory time series models. Moreover, the assumption that the innovation densities underlying those models are known seems quite unrealistic. If those densities remain unspecified, the model becomes a semiparametric one, and rank-based inference methods naturally come into the picture. Rank-based inference methods under very general conditions are known to achieve the semiparametric efficiency bounds. Defining ranks in the context of multivariate time series models, however, is not obvious. We propose two distinct definitions. The first one relies on the assumption that the innovation density is some unspecified elliptical density. The second one relies on the assumption that the innovation process is described by some unspecified independent component analysis model. Applications to portfolio management problems are discussed.

Statistical Tests and Estimators of the Rank of a Matrix and Their Applications in Econometric Modelling

Econometric Reviews, 2009

Non-technical summary 1 Introduction 2 Rank of a general matrix 2.1 A minimum discrepancy function test 2.2 Cragg and Donald (1996) 2.3 Robin and Smith (2000) 2.4 Bartlett (1947) 3 Rank of a hermitian positive semidefi nite matrix 4 Methods to identify the true rank 4.1 Sequential testing methods 4.2 Information criteria methods 5 Applications of tests of rank 5.1 Identifi cation and specifi cation of IV models 5.2 Demand systems 5.3 Reduced rank VAR models 5.4 State space models 5.5 Cointegration 5.6 Other potential applications 6 Conclusion References Appendix A.1 Local power results A.2 Distribution of Λ for a hermitian positive semidefi nite matrix.

Rank-Based Extensions of the Brock, Dechert, and Scheinkman Test

Journal of The American Statistical Association, 2007

This article proposes new tests of randomness for innovations in a large class of time series models. These tests are based on functionals of empirical processes constructed from either the model residuals or their associated ranks. The asymptotic behavior of these processes is determined under the null hypothesis of randomness. The limiting distributions are seen to be independent of estimation errors under appropriate regularity conditions. Several test statistics are derived from these processes; the classical Brock, Dechert, and Scheinkman statistic and a rank-based analog are included as special cases. Because the limiting distributions of the rank-based test statistics are marginfree, their finite-sample p values can be easily calculated by simulation. Monte Carlo experiments show that these statistics are quite powerful against several classes of alternatives.

Rank-Based Extensions of the BDS Test for Serial Dependence

2006

This paper proposes new tests of randomness for innovations of a large class of time series models. These tests are based on functionals of empirical processes constructed either from the model residuals or from their associated ranks. The asymptotic behavior of these empirical processes is determined under the null hypothesis of randomness. The limiting distributions are seen to be independent of estimation errors when appropriate regularity conditions hold. Several test statistics are derived from these processes; the classical BDS statistic and a rank-based analogue thereof are included as special cases. Since the limiting distributions of the rank-based test statistics are margin-free, their finite-sample P -values can easily be calculated by simulation. Monte Carlo experiments show that these statistics are quite powerful against several alternatives.

Rank-Based Extensions of the BDS Test for Serial Dependence C. Genest, K. Ghoudi

2006

This paper proposes new tests of randomness for innovations of a large class of time series models. These tests are based on functionals of empirical processes constructed either from the model residuals or from their associated ranks. The asymptotic behavior of these empirical processes is determined under the null hypothesis of randomness. The limiting distributions are seen to be independent of estimation errors when appropriate regularity conditions hold. Several test statistics are derived from these processes; the classical BDS statistic and a rank-based analogue thereof are included as special cases. Since the limiting distributions of the rank-based test statistics are margin-free, their finite-sample P-values can easily be calculated by simulation. Monte Carlo experiments show that these statistics are quite powerful against several alternatives.

Testing for changes in the rank correlation of time series

For a bivariate time series ((X i , Y i )) i=1,...,n we want to detect whether the correlation between X i and Y i stays constant for all i = 1, . . . , n. We propose a nonparametric change-point test statistic based on Kendall's tau. The asymptotic distribution under the null hypothesis of no change follows from a new U -statistic invariance principle for dependent processes. Assuming a single change-point, we show that the location of the change-point is consistently estimated. Kendall's tau possesses a high efficiency at the normal distribution, as compared to the normal maximum likelihood estimator, Pearson's moment correlation. Contrary to Pearson's correlation coefficient, it shows no loss in efficiency at heavy-tailed distributions, and is therefore particularly suited for financial data, where heavy tails are common. We assume the data ((X i , Y i )) i=1,...,n to be stationary and P -near epoch dependent on an absolutely regular process. The P -near epoch dependence condition constitutes a generalization of the usually considered L p -near epoch dependence allowing for arbitrarily heavy-tailed data. We investigate the test numerically, compare it to previous proposals, and illustrate its application with two real-life data examples.