A note on nonparametric testing for Gaussian innovations in AR-ARCH models

In this paper we consider autoregressive models with conditional autoregressive variance, including the case of homoscedastic AR-models and the case of ARCH models. Our aim is to test the hypothesis of normality for the innovations in a completely nonparametric way, i. e. without imposing parametric assumptions on the conditional mean and volatility functions. To this end the Cram\'er-von Mises test based on the empirical distribution function of nonparametrically estimated residuals is shown to be asymptotically distribution-free. We demonstrate its good performance for finite sample sizes in a simulation study

A simple and general test for white noise

This article considers testing that a time series is uncorrelated when it possibly exhibits some form of dependence. Contrary to the currently employed tests that require selecting arbitrary user-chosen numbers to compute the associated tests statistics, we consider a test statistic that is very simple to use because it does not require any user chosen number and because its asymptotic null distribution is standard under general weak dependent conditions, and hence, asymptotic critical values are readily available. We consider the case of testing that the raw data is white noise, and also consider the case of applying the test to the residuals of an ARMA model. Finally, we also study finite sample performance

Are benefits from oil - stocks diversification gone? New evidence from a dynamic copula and high frequency data

Oil is perceived as a good diversification tool for stock markets. To fully understand this potential, we propose a new empirical methodology that combines generalized autoregressive score copula functions with high frequency data and allows us to capture and forecast the conditional time-varying joint distribution of the oil -- stocks pair accurately. Our realized GARCH with time-varying copula yields statistically better forecasts of the dependence and quantiles of the distribution relative to competing models. Employing a recently proposed conditional diversification benefits measure that considers higher-order moments and nonlinear dependence from tail events, we document decreasing benefits from diversification over the past ten years. The diversification benefits implied by our empirical model are, moreover, strongly varied over time. These findings have important implications for asset allocation, as the benefits of including oil in stock portfolios may not be as large as perceived

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.

A Simple Asymptotic Analysis of Residual-Based Statistics

What s the asymptotic null distribution of a rank-based serial autocorrelation test applied to residuals of an estimated GARCH model?What s the limiting distribution of estimated ACD parameters applied to the residuals of some first-stage modelling procedure?This paper addresses the often occurring situation in econometrics of applying standard statistics to residuals instead of innovations.The paper provides a simple and unified way of calculating the necessary adjustment in the limiting distribution, be it of tests or estimators. On the technical side, we also provide a novel approach to this problem using Le Cam s theory of convergence of experiments (in this paper restricted to Gaussian shift experiments).The resulting formula is simple and the regularity conditions required fairly minimal.Numerous examples show the strength and wide applicability of our approach.statistics;estimation;ranking

Simultaneous Testing of Mean and Variance Structures in Nonlinear Time Series Models

This paper proposes a nonparametric simultaneous test for parametric specification of the conditional mean and variance functions in a time series regression model. The test is based on an empirical likelihood (EL) statistic that measures the goodness of fit between the parametric estimates and the nonparametric kernel estimates of the mean and variance functions. A unique feature of the test is its ability to distribute natural weights automatically between the mean and the variance components of the goodness of fit. To reduce the dependence of the test on a single pair of smoothing bandwidths, we construct an adaptive test by maximizing a standardized version of the empirical likelihood test statistic over a set of smoothing bandwidths. The test procedure is based on a bootstrap calibration to the distribution of the empirical likelihood test statistic. We demonstrate that the empirical likelihood test is able to distinguish local alternatives which are different from the null hypothesis at an optimal rate.Bootstrap, empirical likelihood, goodness{of{t test, kernel estimation, least squares empirical likelihood, rate-optimal test

Confidence bands in nonparametric time series regression

We consider nonparametric estimation of mean regression and conditional variance (or volatility) functions in nonlinear stochastic regression models. Simultaneous confidence bands are constructed and the coverage probabilities are shown to be asymptotically correct. The imposed dependence structure allows applications in many linear and nonlinear auto-regressive processes. The results are applied to the S&P 500 Index data.Comment: Published in at http://dx.doi.org/10.1214/07-AOS533 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org