Joint and Marginal Diagnostic Tests for Conditional Mean and Variance Specifications

This article proposes a general class of joint and marginal diagnostic tests for parametric conditional mean and variance models of possibly nonlinear non-Markovian time series sequences. The use of joint and marginal tests is motivated from the fact that marginal tests for the conditional variance may lead misleading conclusions when the conditional mean is misspecified. The new tests are based on a generalized spectral approach and, contrary to existing procedures, they do not need to choose a lag order depending on the sample size or to smooth the data. Moreover, the proposed tests are robust to higher order dependence of unknown form, in particular to conditional skewness and kurtosis. It turns out that the asymptotic null distributions of the new tests depend on the data generating process, so a new bootstrap procedure is proposed and theoretically justified. A simulation study compares the finite sample performance of the proposed and competing tests and shows that our tests can play a valuable role in time series modeling. Finally, an application to the S&P 500 highlights the merits of our approach.

Generalized spectral tests for the martingale difference hypothesis

This article proposes a test for the martingale difference hypothesis (MDH) using dependence measures related to the characteristic function. The MDH typically has been tested using the sample autocorrelations or in the spectral domain using the periodogram. Tests based on these statistics are inconsistent against uncorrelated non-martingales processes. Here, we generalize the spectral test of Durlauf (1991) for testing the MDH taking into account linear and nonlinear dependence. Our test considers dependence at all lags and is consistent against general pairwise nonparametric Pitman's local alternatives converging at the parametric rate n-1/2, with n the sample size. Furthermore, with our methodology there is no need to choose a lag order, to smooth the data or to formulate a parametric alternative. Our approach could be extended to specification testing of the conditional mean of possibly nonlinear models. The asymptotic null distribution of our test depends on the data generating process, so a bootstrap procedure is proposed and theoretically justified. Our bootstrap test is robust to higher order dependence, in particular to conditional heteroskedasticity. A Monte Carlo study examines the finite sample performance of our test and shows that it is more powerful than some competing tests. Finally, an application to the S&P 500 stock index and exchange rates highlights the merits of our approach.Publicad

On the Asymptotic Power Properties of Specification Tests for Dynamic Parametric Regressions

Economic theories in dynamic contexts usually impose certain restrictions on the conditional mean of the underlying economic variables. Omnibus specification tests are the primary tools to test such restrictions when there is no information on the possible alternative. In this paper we study in detail the power properties of a large class of omnibus specification tests for parametric conditional means under time series processes. We show that all omnibus specification tests have a preference for a finite-dimensional space of alternatives (usually unknown to the practitioner) and we characterize such space for Cramér-von Mises (CvM) tests. This fact motivates the use of optimal tests against such preferred spaces instead of the omnibus tests. We proposed new asymptotically optimal directional and smooth tests that are optimally designed for cases in which a finite-dimensional space of alternatives is in mind. The new proposed optimal procedures are asymptotically distribution-free and are valid under weak assumptions on the underlying data generating process. In particular, they are valid under possibly time varying higher conditional moments of unknown form, e.g., conditional heteroskedasticity. A Monte Carlo experiment shows that previous asymptotic results provide good approximations in small sample sizes. Finally, an application of our theory to test the martingale difference hypothesis of some exchange rates provides new information on the rejection of omnibus tests and illustrates the relevance of our results for practitioners.

Generalized spectral tests for the martingale difference hypothesis

^aThis article proposes a test for the Martingale Difference Hypothesis (MDH) using dependence measures related to the characteristic function. The MDH typically has been tested using the sample autocorrelations or in the spectral domain using the periodogram. Tests based on these statistics are inconsistent against uncorrelated non-martingales processes. Here, we generalize the spectral test of Durlauf (1991) for testing the MDH taking into account linear and nonlinear dependence. Our test considers dependence at all lags and is consistent against general pairwise nonparametric Pitman's local alternatives converging at the parametric rate n^(-1/2), with n the sample size. Furthermore, with our methodology there is no need to choose a lag order, to smooth the data or to formulate a parametric alternative. Our approach can be easily extended to specification testing of the conditional mean of possibly nonlinear models. The asymptotic null distribution of our test depends on the data generating process, so a bootstrap procedure is proposed and theoretically justified. Our bootstrap test is robust to higher order dependence, in particular to conditional heteroskedasticity. A Monte Carlo study examines the finite sample performance of our test and shows that it is more powerful than some competing tests. Finally, an application to the S and P 500 stock index and exchange rates highlights the merits of our approach

Model Checks Using Residual Marked Empirical Processes

This paper proposes omnibus and directional tests for testing the goodness-of-fit of a parametric regression time series model. We use a general class of residual marked empirical processes as the building-blocks for estimation and testing of such models. First, we establish a weak convergence theorem under mild assumptions, which allows us to study in a unified way the asymptotic null distribution of the test statistics and their asymptotic behavior against Pitman's local alternatives. To approximate the asymptotic null distribution of test statistics we justify theoretically a bootstrap procedure. Also, some asymptotic theory for the estimation of the principal components of the residual marked processes is considered. This asymptotic theory is used to derive optimal directional tests and efficient estimation of regression parameters. Finally, a Monte Carlo study shows that the bootstrap and the asymptotic results provide good approximations for small sample sizes and an empirical application to the Canadian lynx data set is considered.

Goodness-of-fit Tests for Linear and Non-linear Time Series Models

In this article we study a general class of goodness-of-fit tests for the conditional mean of a linear or nonlinear time series model. Among the properties of the proposed tests are that they are suitable when the conditioning set is infinite-dimensional; are consistent against a broad class of alternatives including Pitman's local alternatives converging at the parametric rate; and do not need to choose a lag order depending on the sample size or to smooth the data. It turns out that the asymptotic null distributions of the tests depend on the data generating process, so a new bootstrap procedure is proposed and theoretically justified. The proposed bootstrap tests are robust to higher order dependence, in particular to conditional heteroskedasticity of unknown form. A simulation study compares the finite sample performance of the proposed and competing tests and shows that our tests can play a valuable role in time series modeling. Finally, an application to an economic price series highlights the merits of our approach.

Testing the martingale difference hypothesis using integrated regression functions

An omnibus test for testing a generalized version of the martingale difference hypothesis (MDH) is proposed. This generalized hypothesis includes the usual MDH, testing for conditional moments constancy such as conditional homoscedasticity (ARCH effects) or testing for directional predictability. A unified approach for dealing with all of these testing problems is proposed. These hypotheses are long standing problems in econometric time series analysis, and typically have been tested using the sample autocorrelations or in the spectral domain using the periodogram. Since these hypotheses cover also nonlinear predictability, tests based on those second order statistics are inconsistent against uncorrelated processes in the alternative hypothesis. In order to circumvent this problem pairwise integrated regression functions are introduced as measures of linear and nonlinear dependence. The proposed test does not require to chose a lag order depending on sample size, to smooth the data or to formulate a parametric alternative model. Moreover, the test is robust to higher order dependence, in particular to conditional heteroskedasticity. Under general dependence the asymptotic null distribution depends on the data generating process, so a bootstrap procedure is considered and a Monte Carlo study examines its finite sample performance. Then, the martingale and conditional heteroskedasticity properties of the Pound/Dollar exchange rate are investigated.Publicad

Joint Diagnostic Tests for Conditional Mean and Variance Specifications

This article proposes a general class of joint diagnostic tests for parametric conditional mean and variance models of possibly nonlinear and/or non-Markovian time series sequences. The new tests are based on a generalized spectral approach and, contrary to existing procedures, they do not need to choose a lag order depending on the sample size or to smooth the data. Moreover, they are robust to higher order dependence of unknown form. It turns out that the asymptotic null distributions of the new tests depend on the data generating process, so a bootstrap procedure is proposed and theoretically justified. A simulation study compares the finite sample performance of the proposed and competing tests and shows that our tests can play a valuable role in time series modelling. An application to the S&P500 highlights the merits of our approach.

Testing the martingale difference hypothesis using integrated regression functions.

An omnibus test for testing a generalized version of the martingale difference hypothesis (MDH) is proposed. This generalized hypothesis includes the usual MDH, testing for conditional moments constancy such as conditional homoscedasticity (ARCH effects) or testing for directional predictability. A unified approach for dealing with all of these testing problems is proposed. These hypotheses are long standing problems in econometric time series analysis, and typically have been tested using the sample autocorrelations or in the spectral domain using the periodogram. Since these hypotheses cover also nonlinear predictability, tests based on those second order statistics are inconsistent against uncorrelated processes in the alternative hypothesis. In order to circumvent this problem pairwise integrated regression functions are introduced as measures of linear and nonlinear dependence. The proposed test does not require to chose a lag order depending on sample size, to smooth the data or to formulate a parametric alternative model. Moreover, the test is robust to higher order dependence, in particular to conditional heteroskedasticity. Under general dependence the asymptotic null distribution depends on the data generating process, so a bootstrap procedure is considered and a Monte Carlo study examines its finite sample performance. Then, the martingale and conditional heteroskedasticity properties of the Pound/Dollar exchange rate are investigated.Nonlinear time series; Martingale difference hypothesis; Empirical processes; Exchange rates;

Generalized spectral tests for the martingale difference hypothesis.

This article proposes a test for the martingale difference hypothesis (MDH) using dependence measures related to the characteristic function. The MDH typically has been tested using the sample autocorrelations or in the spectral domain using the periodogram. Tests based on these statistics are inconsistent against uncorrelated non-martingales processes. Here, we generalize the spectral test of Durlauf (1991) for testing the MDH taking into account linear and nonlinear dependence. Our test considers dependence at all lags and is consistent against general pairwise nonparametric Pitman's local alternatives converging at the parametric rate n-1/2, with n the sample size. Furthermore, with our methodology there is no need to choose a lag order, to smooth the data or to formulate a parametric alternative. Our approach could be extended to specification testing of the conditional mean of possibly nonlinear models. The asymptotic null distribution of our test depends on the data generating process, so a bootstrap procedure is proposed and theoretically justified. Our bootstrap test is robust to higher order dependence, in particular to conditional heteroskedasticity. A Monte Carlo study examines the finite sample performance of our test and shows that it is more powerful than some competing tests. Finally, an application to the S&P 500 stock index and exchange rates highlights the merits of our approach.Martingale difference hypothesis; Hilbert spaces; Generalized spectral distribution; Characteristic function; S&P 500 stock index; Exchange rates;