Hostname: page-component-7dd5485656-wjfd9 Total loading time: 0 Render date: 2025-10-31T22:16:01.108Z Has data issue: false hasContentIssue false
  • English
  • Français

ROBUST TESTS FOR WHITE NOISE AND CROSS-CORRELATION

Published online by Cambridge University Press:  21 September 2020

Violetta Dalla ,
Liudas Giraitis  and
Peter C. B. Phillips
Violetta Dalla
Affiliation:
National and Kapodistrian University of Athens
Liudas Giraitis*
Affiliation:
Queen Mary University of London
Peter C. B. Phillips
Affiliation:
Yale University, University of Auckland, University of Southampton, Singapore Management University
*
Address correspondence to Liudas Giraitis, Queen Mary University of London, London, UK; e-mail: l.giraitis@qmul.ac.uk.
Get access Rights & Permissions [Opens in a new window]

Abstract

Commonly used tests to assess evidence for the absence of autocorrelation in a univariate time series or serial cross-correlation between time series rely on procedures whose validity holds for i.i.d. data. When the series are not i.i.d., the size of correlogram and cumulative Ljung–Box tests can be significantly distorted. This paper adapts standard correlogram and portmanteau tests to accommodate hidden dependence and nonstationarities involving heteroskedasticity, thereby uncoupling these tests from limiting assumptions that reduce their applicability in empirical work. To enhance the Ljung–Box test for non-i.i.d. data, a new cumulative test is introduced. Asymptotic size of these tests is unaffected by hidden dependence and heteroskedasticity in the series. Related extensions are provided for testing cross-correlation at various lags in bivariate time series. Tests for the i.i.d. property of a time series are also developed. An extensive Monte Carlo study confirms good performance in both size and power for the new tests. Applications to real data reveal that standard tests frequently produce spurious evidence of serial correlation.

Information

Type
ARTICLES
Information
Econometric Theory , Volume 38 , Issue 5: YALE 2018 CONFERENCE IN HONOR OF PETER C. B. PHILLIPS: PART I , October 2022 , pp. 913 - 941
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

Dalla acknowledges financial support from ELKE-EKPA. Phillips acknowledges support from the Kelly Fund at the University of Auckland, a KLC Fellowship at Singapore Management University, and the NSF under Grant No. SES 18-50860.

References

REFERENCES

Anderson, T.W. & Walker, A.M. (1964) On the asymptotic distribution of the autocorrelations of a sample from a linear stochastic process. The Annals of Mathematical Statistics 35, 1296–1303.CrossRefGoogle Scholar
Bai, J. & Perron, P. (1998) Estimating and testing linear models with multiple structural changes. Econometrica 66, 47–78.CrossRefGoogle Scholar
Bartlett, M.S. (1946) On the theoretical specification and sampling properties of autocorrelated time-series. Supplement to the Journal of the Royal Statistical Society 9, 27–41.CrossRefGoogle Scholar
Box, G.E.P. & Jenkins, G.M. (1970) Time Series Analysis: Forecasting and Control. Holden-Day.Google Scholar
Box, G.E.P. & Pierce, D.A. (1970) Distribution of residual autocorrelations in autoregressive-integrated moving average time series models. Journal of the American Statistical Association 65, 1509–1526.CrossRefGoogle Scholar
Campbell, J.Y., Lo, A.W., & MacKinlay, A.C. (1997) The Econometrics of Financial Markets. Princeton University Press.CrossRefGoogle Scholar
Cumby, R.E. & Huizinga, J. (1992) Testing the autocorrelation structure of disturbances in ordinary least squares and instrumental variables regressions. Econometrica 60, 185–196.CrossRefGoogle Scholar
Diebold, F.X. (1986) Testing for serial correlation in the presence of ARCH. Proceedings of the American Statistical Association, Business and Economic Statistics Section, pp. 323–328.Google Scholar
Francq, C., Roy, R., & Zakoïan, J.-M. (2005) Diagnostic checking in ARMA models with uncorrelated errors. Journal of the American Statistical Association 100, 532–544.CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.-M. (2009) Bartlett’s formula for a general class of nonlinear processes. Journal of Time Series Analysis 30, 449–465.CrossRefGoogle Scholar
Granger, C.W.J. & Andersen, A.P. (1978) An Introduction to Bilinear Time Series Models. Vandenhoeck and Ruprecht.Google Scholar
Gretton, A. & Györfi, L. (2010) Consistent nonparametric tests of independence. Journal of Machine Learning Research 11, 1391–1423.Google Scholar
Guo, B. & Phillips, P.C.B. (2001) Testing for autocorrelation and unit roots in the presence of conditional heteroskedasticity of unknown form. UC Santa Cruz Economics Working Paper 540, pp. 1–55.Google Scholar
Hannan, E.J. (1970) Multiple Time Series. Wiley.CrossRefGoogle Scholar
Hannan, E.J. & Hyde, C.C. (1972) On limit theorems for quadratic functions of discrete time series. The Annals of Mathematical Statistics 43, 2058–2066.CrossRefGoogle Scholar
Haugh, L.D. (1976) Checking the independence of two covariance-stationary time series: A univariate residual cross-correlation approach. Journal of the American Statistical Association 71, 378–385.CrossRefGoogle Scholar
Haugh, L.D. & Box, G.E.P. (1977) Identification of dynamic regression (distributed lag) models connecting two time series. Journal of the American Statistical Association 72, 121–130.CrossRefGoogle Scholar
Hooker, R.H. (1901) Correlation of the marriage-rate with trade. Journal of the Royal Statistical Society 64, 485–492.Google Scholar
Horowitz, J.L., Lobato, I.N., & Savin, N.I. (2006) Bootstrapping the Box-Pierce Q test: A robust test of uncorrelatedness. Journal of Econometrics 133, 841–862.CrossRefGoogle Scholar
Kokoszka, P.S. & Politis, D.N. (2011) Nonlinearity of ARCH and stochastic volatility models and Bartlett’s formula. Probability and Mathematical Statistics 31, 47–59.Google Scholar
Kyriazidou, E. (1998) Testing for serial correlation in multivariate regression models. Journal of Econometrics 86, 193–220.CrossRefGoogle Scholar
Ljung, G.M. & Box, G.E.P. (1978) On a measure of lack of fit in time series models. Biometrika 65, 297–303.CrossRefGoogle Scholar
Lo, A.W. & MacKinlay, A.G. (1989) The size and power of the variance ratio test in finite samples. A Monte Carlo investigation. Journal of Econometrics 40, 203–238.CrossRefGoogle Scholar
Lobato, I.N., Nankervis, J.C., & Savin, N.E. (2002) Testing for zero autocorrelation in the presence of statistical dependence. Econometric Theory 18, 730–743.CrossRefGoogle Scholar
McLeod, A.I. & Li, W.K. (1983) Diagnostic checking ARMA time series models using squared residual autocorrelations. Journal of Time Series Analysis 4, 269–273.CrossRefGoogle Scholar
Phillips, P.C. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.CrossRefGoogle Scholar
Robinson, P.M. (1991) Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of Econometrics 47, 67–84.CrossRefGoogle Scholar
Romano, J.P. & Thombs, L.R. (1996) Inference for autocorrelations under weak assumptions. Journal of the American Statistical Association 91, 590–600.CrossRefGoogle Scholar
Stout, W. (1974) Almost Sure Convergence. Academic Press.Google Scholar
Taylor, S.J. (1984) Estimating the variances of autocorrelations calculated from financial time series. Journal of the Royal Statistical Society, Series C 33, 300–308.Google Scholar
Tsay, R.S. (2010) Analysis of Financial Time Series, 3rd edition. Wiley.CrossRefGoogle Scholar
Wong, H. & Ling, S. (2005) Mixed portmanteau tests for time-series models. Journal of Time Series Analysis 26, 569—579.CrossRefGoogle Scholar
Yule, G.U. (1921) On the time-correlation problem, with especial reference to the variate-difference correlation method. Journal of the Royal Statistical Society 84, 497–537.CrossRefGoogle Scholar
Yule, G.U. (1926) Why do we sometimes get nonsense-correlations between time-series? A study in sampling and the nature of time-series. Journal of the Royal Statistical Society 89, 1–63.CrossRefGoogle Scholar
Zhu, K. (2013) A mixed portmanteau test for ARMA-GARCH models by the quasi-maximum exponential likelihood estimation approach. Journal of Time Series Analysis 34, 230–237.CrossRefGoogle Scholar
Supplementary material: PDF

Dalla et al. supplementary material

Dalla et al. supplementary material

Download Dalla et al. supplementary material(PDF)
PDF 594.3 KB