TESTING FOR WHITE NOISE UNDER UNKNOWN DEPENDENCE AND ITS APPLICATIONS TO DIAGNOSTIC CHECKING FOR TIME SERIES MODELS

Testing for white noise has been well studied in the literature of econometrics and statistics. For most of the proposed test statistics, such as the well-known Box–Pierce test statistic with fixed lag truncation number, the asymptotic null distributions are obtained under independent and identically distributed assumptions and may not be valid for dependent white noise. Because of recent popularity of conditional heteroskedastic models (e.g., generalized autoregressive conditional heteroskedastic [GARCH] models), which imply nonlinear dependence with zero autocorrelation, there is a need to understand the asymptotic properties of the existing test statistics under unknown dependence. In this paper, we show that the asymptotic null distribution of the Box–Pierce test statistic with general weights still holds under unknown weak dependence as long as the lag truncation number grows at an appropriate rate with increasing sample size. Further applications to diagnostic checking of the autoregressive moving average (ARMA) and fractional autoregressive integrated moving average (FARIMA) models with dependent white noise errors are also addressed. Our results go beyond earlier ones by allowing non-Gaussian and conditional heteroskedastic errors in the ARMA and FARIMA models and provide theoretical support for some empirical findings reported in the literature.

[1]  Jens-Peter Kreiss,et al.  Bootstrap specification tests for linear covariance stationary processes , 2006 .

[2]  Christian Francq,et al.  Covariance matrix estimation for estimators of mixing weak ARMA models , 2000 .

[3]  Siem Jan Koopman,et al.  Periodic Seasonal Reg-ARFIMA–GARCH Models for Daily Electricity Spot Prices , 2007 .

[4]  Emmanuel Guerre,et al.  Data-Driven Rate-Optimal Specification Testing in Regression Models , 2004 .

[5]  M. Bartlett,et al.  An Introduction to Stochastic Processes with Special Reference to Methods and Applications. , 1955 .

[6]  Rohit S. Deo,et al.  Power transformations to induce normality and their applications , 2004 .

[7]  Richard A. Davis,et al.  Least absolute deviation estimation for all-pass time series models , 2001 .

[8]  D. B. Preston.  Spectral Analysis and Time Series , 1983 .

[9]  L. Márkus,et al.  A long range dependent model with nonlinear innovations for simulating daily river flows , 2004 .

[10]  Joseph P. Romano,et al.  Inference for Autocorrelations under Weak Assumptions , 1996 .

[11]  Jan Beran,et al.  Maximum Likelihood Estimation of the Differencing Parameter for Invertible Short and Long Memory Autoregressive Integrated Moving Average Models , 1995 .

[12]  T. Hsing,et al.  On weighted U-statistics for stationary processes , 2004, math/0410157.

[13]  C. Granger,et al.  An introduction to bilinear time series models , 1979 .

[14]  W. Wu,et al.  Nonlinear system theory: another look at dependence. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[15]  Willa W. Chen,et al.  A GENERALIZED PORTMANTEAU GOODNESS-OF-FIT TEST FOR TIME SERIES MODELS , 2004, Econometric Theory.

[16]  Yuzo Hosoya,et al.  A limit theory for long-range dependence and statistical inference on related models , 1997 .

[17]  David R. Brillinger,et al.  Time Series: Data Analysis and Theory. , 1982 .

[18]  Wei Biao Wu,et al.  STRONG INVARIANCE PRINCIPLES FOR DEPENDENT RANDOM VARIABLES , 2007, 0711.3674.

[19]  Mark Bartlett,et al.  An Introduction to Stochastic Processes with Special Reference to Methods and Applications. , 1955 .

[20]  A LIMIT THEOREM FOR QUADRATIC FORMS AND ITS APPLICATIONS , 2007, Econometric Theory.

[21]  Efficiency improvements in inference on stationary and nonstationary fractional time series , 2004, math/0508284.

[22]  Xiaofeng Shao,et al.  NONSTATIONARITY-EXTENDED WHITTLE ESTIMATION , 2009, Econometric Theory.

[23]  J. Zakoian,et al.  Recent Results for Linear Time Series Models with Non Independent Innovations , 2005 .

[24]  Efstathios Paparoditis,et al.  Spectral Density Based Goodness-of-Fit Tests for Time Series Models , 2000 .

[25]  D. Brillinger.  Time series - data analysis and theory , 1981, Classics in applied mathematics.

[26]  Ignacio N. Lobato,et al.  Bootstrapping the Box–Pierce Q test: A robust test of uncorrelatedness , 2006 .

[27]  R. Deo,et al.  THE VARIANCE RATIO STATISTIC AT LARGE HORIZONS , 2006, Econometric Theory.

[28]  Felix Schlenk,et al.  Proof of Theorem 3 , 2005 .

[29]  Joel L. Horowitz,et al.  An Adaptive, Rate-Optimal Test of a Parametric Mean-Regression Model Against a Nonparametric Alternative , 2001 .

[30]  Pierre Duchesne,et al.  Statistical Modeling and Analysis for Complex Data Problems , 2010 .

[31]  Guodong Li,et al.  Least absolute deviation estimation for fractionally integrated autoregressive moving average time series models with conditional heteroscedasticity , 2008 .

[32]  Yongmiao Hong,et al.  Consistent Testing for Serial Correlation of Unknown Form , 1996 .

[33]  Rohit S. Deo,et al.  Spectral Tests of the Martingale Hypothesis Under Conditional Heteroscedasticity , 2000 .

[34]  U. Grenander,et al.  Statistical analysis of stationary time series , 1958 .

[35]  Wei Biao Wu,et al.  Limit theorems for iterated random functions , 2004, Journal of Applied Probability.

[36]  Ke Zhu,et al.  A mixed portmanteau test for ARMA‐GARCH models by the quasi‐maximum exponential likelihood estimation approach , 2013 .

[37]  Joseph P. Romano,et al.  Bootstrap‐assisted Goodness‐of‐fit Tests in the Frequency Domain , 1999 .

[38]  S. Durlauf.  Spectral Based Testing of the Martingale Hypothesis , 1991 .

[39]  P. Hall,et al.  Martingale Limit Theory and Its Application , 1980 .

[40]  J. Beran.  A Goodness‐Of‐Fit Test for Time Series with Long Range Dependence , 1992 .

[41]  G. Box,et al.  Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models , 1970 .

[42]  N. E. Savin,et al.  TESTING FOR ZERO AUTOCORRELATION IN THE PRESENCE OF STATISTICAL DEPENDENCE , 2002, Econometric Theory.

[43]  Christian Francq,et al.  Diagnostic Checking in ARMA Models With Uncorrelated Errors , 2005 .

[44]  W. Wu,et al.  On linear processes with dependent innovations , 2005 .

[45]  Serena Ng,et al.  Testing for ARCH in the presence of a possibly misspecified conditional mean , 1999 .

[46]  Miguel A. Delgado,et al.  Distribution Free Goodness-of-Fit Tests for Linear Processes , 2005, math/0603043.

[47]  X. Shao,et al.  Asymptotic spectral theory for nonlinear time series , 2006, math/0611029.

[48]  Martingale approximations for sums of stationary processes , 2004, math/0410160.

[49]  C. Granger,et al.  Modelling Nonlinear Economic Relationships , 1995 .

[50]  Y. Tse,et al.  Forecasting the Nikkei spot index with fractional cointegration , 1999 .

[51]  Wai Keung Li,et al.  On Fractionally Integrated Autoregressive Moving-Average Time Series Models with Conditional Heteroscedasticity , 1997 .