Testing normalization and overidentification of cointegrating vectors in vector autoregressive processes

This paper develops test procedures for testing the validity of general linear identifying restrictions imposed on cointegrating vectors in the context of a vector autoregressive model. In addition to overidentifying restrictions the considered restrictions may also involve normalizing restrictions. Tests for both types of restrictions are developed and their asymptotic properties are obtained. Under the null hypothesis tests for normalizing restrictions have an asymptotic "multivariate unit root distribution", similar to that obtained for the likelihood ratio test for cointegration, while tests for overidentifying restrictions have a standard chi-square limiting distribution. Since these two types of tests are asymptotically independent they are easy to cotnbine to an overall test for the spccifed identifying restrictions. An overall test of this kind can consistently reveal the failure of the identifying restrictions in a wider class of cases than previous tests which only test for overidentifying restrictions.     

The effect of non-normal disturbances and conditional heteroskedasticity on multiple cointegration tests

!t is well-known that Johansen's multiple cointegration tests' results and those of Johansen and Juselius' tests for restricrions on cointegrating vectors and their weights have far-reaching implications for economic modelling and analysis. Therefore, it is important to ensure that the tests have desirable finite sample properties. Although the statistics are derived under Gaussian distribution,the asympotic results are derived under a much wider class of distributions. Using simulation, this paper investigates the effect of non-normal disturbances on these tests in finite samples. Further, ARCH/GARCH type conditional heteroskedasticity is present in many economic and financial time series. This paper examines the finite properties of the tests when the error term follows ARCH/GARCH type processes. From the evidence, it appears that researchers should not be overly concerned by the possibility of small departures from non-normality when using Johansen's suggested techniques even in finite samples. ARCH and GARCH effects may be more problematic, however. In particular it becomes more important ro test whether the restriction implicit in the integrated (or near-integrated) ARCH-type Drocess actually holds in time series for the application of the cointegraiion rank tests and the test for restrictions on cointegrating weights. The tests for restrictions on cointegrating vectors apper to be robust for non-normal errors and for all ARCH and GARCH type processes considered.     

协整平滑转移回归中的线性检验 ——基于完全修正最小二乘法的扩展

 在解释变量内生条件下,Choi,Saikkonen(2004)使用动态最小二乘法估计协整平滑转移回归模型,并基于动态最小二乘的估计结果构造 统计量检验协整向量的非线性。本文系统解析了 的构造并指出其不足,针对这一不足,本文将动态最小二乘法扩展为完全修正的最小二乘法,并进而基于完全修正的最小二乘法估计结果构造 统计量检验协整向量的非线性。本文的仿真试验表明,在有限样本下, 与 的检验势没有显著差异,但 的水平扭曲小于 。     

A Model Specification Test For GARCH(1,1) Processes

We provide a consistent specification test for generalized autoregressive conditional heteroscedastic (GARCH (1,1)) models based on a test statistic of Cramér‐von Mises type. Because the limit distribution of the test statistic under the null hypothesis depends on unknown quantities in a complicated manner, we propose a model‐based (semiparametric) bootstrap method to approximate critical values of the test and to verify its asymptotic validity. Finally, we illuminate the finite sample behaviour of the test by some simulations.     

Percentage points of the statistics for testing hypotheses on mean vectors of multivariate normal distributions with missing observations

A problem of testing of hypotheses on the mean vector of a multivariate normal distribution with unknown and positive definite covariance matrix is considered when a sample with a special, though not unusual, pattern of missing observations from that population is available. The approximate percentage points of the test statistic are obtained and their accuracy has been checked by comparing them with some exact percentage points which are calculated for complete samples and some special incomplete samples. The approximate percentage points are in good agreement with exact percentage points. The above work is extended to the problem of testing the hypothesis of equality of two mean vectors of two multivariate normal distributions with the same, unknown covariance matrix     

A Wavelet-Based Approach of Testing for Granger Causality in the Presence of GARCH Effects

The size and power of the most commonly used tests and a new wavelet-based approach of testing for Granger causality is evaluated by means of a Monte Carlo study in which the error term follows a generalized autoregressive conditional heteroscedasticity consistent (GARCH) process. In the simulation study it is shown that the commonly used causality tests tend to overreject the true null hypothesis in the presence of GARCH errors and that the new wavelet-based approach improves the size properties of the Granger causality test for all of the different situations evaluated.     

Maximum Likelihood Unit Root Testing in the Presence of GARCH: A New Test with Increased Power

The literature on testing the unit root hypothesis in the presence of GARCH errors is extended. A new test based upon the combination of local-to-unity detrending and joint maximum likelihood estimation of the autoregressive parameter and GARCH process is presented. The finite sample distribution of the test is derived under alternative decisions regarding the deterministic terms employed. Using Monte Carlo simulation, the newly proposed ML t-test is shown to exhibit increased power of relative to rival tests. Finally, the empirical relevance of the simulation results is illustrated via an application to real GDP for the UK.     

Prequential omnibus goodness-of-fit tests for stochastic processes: A numerical study

This article is a contribution to the study of an omnibus goodness-of-fit (Gof) test based on Rosenblatt Probability Integral Transform (RPIT) within Dawid's prequential framework. This Gof test is easy to use since it has a common test statistic (with apparently the same asymptotic distribution) for a wide range of stochastic models. Intensive Monte-Carlo simulations are presented to investigate the behavior of this test for several stochastic models: renewal, autoregressive (AR, ARMA, ARCH, GARCH) and Poisson processes, generalized linear models... These simulations suggest that the RPIT test could be used to test the fit of a wide range of stochastic models but it may be not powerful when compared to Gof tests specifically designed for the tested processes. It is also conjectured that this test is still appropriate for testing the Gof of any discrete-time stochastic process provided that efficient estimators are used.     

Frequency Domain Tests of Semiparametric Hypotheses for Locally Stationary Processes

Abstract.  Many time series in applied sciences obey a time-varying spectral structure. In this article, we focus on locally stationary processes and develop tests of the hypothesis that the time-varying spectral density has a semiparametric structure, including the interesting case of a time-varying autoregressive moving-average (tvARMA) model. The test introduced is based on a L ₂ -distance measure of a kernel smoothed version of the local periodogram rescaled by the time-varying spectral density of the estimated semiparametric model. The asymptotic distribution of the test statistic under the null hypothesis is derived. As an interesting special case, we focus on the problem of testing for the presence of a tvAR model. A semiparametric bootstrap procedure to approximate more accurately the distribution of the test statistic under the null hypothesis is proposed. Some simulations illustrate the behaviour of our testing methodology in finite sample situations.     

Comment on “recent developments in bootstrapping time series”

The paper develops a general framework for identification, estimation, and hypothesis testing in cointegrated systems when the cointegrating coefficients are subject to (possibly) non-linear and cross-equation restrictions, obtained from economic theory or other relevant a priori information. It provides a proof of the consistency of the quasi maximum likelihood estimators (QMLE), establishes the relative rates of convergence of the QMLE of the short-run and the long-run parameters, and derives their asymptotic distributions; thus generalizing the results already available in the literature for the linear case. The paper also develops tests of the over-identifying (possibly) non-linear restrictions on the cointegrating vectors. The estimation and hypothesis testing procedures are applied to an Almost Ideal Demand System estimated on U.K. quarterly observations. Unlike many other studies of consumer demand this application does not treat relative prices and real per capita expenditures as exogenously given.     

LONG-RUN STRUCTURAL MODELLING

The paper develops a general framework for identification, estimation, and hypothesis testing in cointegrated systems when the cointegrating coefficients are subject to (possibly) non-linear and cross-equation restrictions, obtained from economic theory or other relevant a priori information. It provides a proof of the consistency of the quasi maximum likelihood estimators (QMLE), establishes the relative rates of convergence of the QMLE of the short-run and the long-run parameters, and derives their asymptotic distributions; thus generalizing the results already available in the literature for the linear case. The paper also develops tests of the over-identifying (possibly) non-linear restrictions on the cointegrating vectors. The estimation and hypothesis testing procedures are applied to an Almost Ideal Demand System estimated on U.K. quarterly observations. Unlike many other studies of consumer demand this application does not treat relative prices and real per capita expenditures as exogenously given.     

A note on the Jarque–Bera normality test for GARCH innovations

In this paper, we consider the validity of the Jarque–Bera normality test whose construction is based on the residuals, for the innovations of GARCH (generalized autoregressive conditional heteroscedastic) models. It is shown that the asymptotic behavior of the original form of the JB test adopted in this paper is identical to that of the test statistic based on true errors. The simulation study also confirms the validity of the original form since it outperforms other available normality tests.     

A corrected likelihood ratio statistic for the multivariate regression model with antedependent errors

This paper addresses the problem of inference for the antedependence model (Gabriel, 1961, 1962). Antedependence can be formulated as an autoregressive process of general order which is non-stationary in time. Its primary application is in the analysis of repeated measurements data, that is, data consisting of independent replicates of relatively short time series. Our focus is on testing a general linear hypothesis in the context of a multivariate regression model with multivariate normal antedependent errors. Although the relevant likelihood ratio statistic was first presented by Gabriel (1961), the distribution of this statistics has not yet been derived. We present this derivation and show how this result leads to a simple correction factor to improve the x² approximation of the likelihood ratio statistic.     

Structural changes in autoregressive models for binary time series

We study autoregressive models for binary time series with possible changes in their parameters. A procedure for detection and testing of a single change is suggested. The limiting behavior of the test statistic is derived. The performance of the test is analyzed under the null hypothesis as well as under different alternatives via a simulation study. Application of the method to a real data set on US recession is provided as an illustration.     

The power of the circular cone test: A noncentral chi-bar-squared distribution

Pincus (1975) derived the null distribution of the likelihood-ratio test statistic for testing that the mean vector of a multivariate normal distribution is zero against the alternative that the mean vector lies in a circular cone. Under the null hypothesis, the likelihood-ratio test statistic has a chi-bar-squared distribution. We extend the results of Pincus by deriving the distribution of the likelihood-ratio test statistic under the alternative hypothesis. In a special case, the distribution is a “noncentral chi-bar-squared” distribution. To our knowledge, this is the first order-restricted testing problem for which the relationship between the null and alternative distributions of the test statistic is similar to the relationship in the linear-model setting. That is, the distribution of the likelihood-ratio test has a central form of a distribution under the null hypothesis and a noncentral form of the same distribution under the alternative.     

On Testing Changes in Autoregressive Parameters of a VAR Model

The article deals with the problem of testing a change in autoregressive matrices of the p-th order vector autoregressive process, VAR(p). The proposed test statistics are based on the likelihood ratio concept and are studied under the null hypothesis of no change in parameters. Their asymptotic behavior is derived under minimal moment assumptions in both cases where the time point of possible change is known a priori and is undefined. The Gumbel-type approximation of the test statistic is also developed, which previous papers on VAR(p) models do not cover.     

A Lagrange multiplier test for testing the adequacy of constant conditional correlation GARCH model

ABSTRACT

A Lagrange multiplier test for testing the parametric structure of a constant conditional correlation-generalized autoregressive conditional heteroskedasticity (CCC-GARCH) model is proposed. The test is based on decomposing the CCC-GARCH model multiplicatively into two components, one of which represents the null model, whereas the other one describes the misspecification. A simulation study shows that the test has good finite sample properties. We compare the test with other tests for misspecification of multivariate GARCH models. The test has high power against alternatives where the misspecification is in the GARCH parameters and is superior to other tests. The test is not greatly affected by misspecification in the conditional correlations and is therefore well suited for considering misspecification of GARCH equations.     

Testing equality of mean vectors in a one-way MANOVA with monotone missing data

In this study, testing the equality of mean vectors in a one-way multivariate analysis of variance (MANOVA) is considered when each dataset has a monotone pattern of missing observations. The likelihood ratio test (LRT) statistic in a one-way MANOVA with monotone missing data is given. Furthermore, the modified test (MT) statistic based on likelihood ratio (LR) and the modified LRT (MLRT) statistic with monotone missing data are proposed using the decomposition of the LR and an asymptotic expansion for each decomposed LR. The accuracy of the approximation for the Chi-square distribution is investigated using a Monte Carlo simulation. Finally, an example is given to illustrate the methods.     

On the identification of ARMA echelon-form models

An identification procedure for multivariate autoregressive moving average (ARMA) echelon-form models is proposed. It is based on the study of the linear dependence between rows of the Hankel matrix of serial correlations. To that end, we define a statistical test for checking the linear dependence between vectors of serial correlations. It is shown that the test statistic t?_n considered is distributed asymptotically as a finite linear combination of independent chi-square random variables with one degree of freedom under the null hypothesis, whereas under the alternative hypothesis, t?_N/N converges in probability to a positive constant. These results allow us, in particular, to compute the asymptotic probability of making a specification error with the proposed procedure. Links to other methods based on the application of canonical analysis are discussed. A simulation experiment was done in order to study the performance of the procedure. It is seen that the graphical representation of t?_N, as a function of N, can be very useful in identifying the dynamic structure of ARMA models. Furthermore, for the model considered, the proposed identification procedure performs very well for series of 100 observations or more and reasonably well with short series of 50 observations.     

Goodness-of-fit test using residuals in infinite-order autoregressive models

In this paper we study the asymptotic behavior of the Bickel–Rosenblatt test in infinite-order autoregressive models. Under some mild conditions, the test statistic based on residuals is shown to have the same limiting distribution as that based on true iid errors. It is also proved that this result remains the same when the nuisance parameters in the model are to be estimated, which means that the Bickel–Rosenblatt test is easily applicable in composite hypothesis goodness-of-fit testing unlike the Kolmogorov–Smirnov type tests. The result of a simulation study is supplemented to verify the result of this paper.