, Volume 17, Issue 3, pp 567–584 | Cite as

Partial sums of lagged cross-products of AR residuals and a test for white noise

  • Jan G. De Gooijer
Original Paper


Partial sums of lagged cross-products of AR residuals are defined. By studying the sample paths of these statistics, changes in residual dependence can be detected that might be missed by statistics using only the total sum of cross-products. Also, a test statistic for white noise is proposed. It is shown that the limiting distribution of the test statistic converges weakly to a vector Brownian motion with independent elements under the null hypothesis of no residual autocorrelation. An indication of the circumstances under which the asymptotic results apply in finite-sample situations is obtained through a simulation study. Some considerations are given to the empirical size and power of the test statistic vis-à-vis the Ljung–Box (Biometrika 65:297–303, 1978) portmanteau statistic, and a diagnostic test statistic proposed by Peña and Rodriguez (J. Am. Stat. Assoc. 97:601–610, 2002). An empirical example illustrates the importance of examining partial sums of time series residuals when inadequacies in model fit are anticipated due to a change in autocorrelation structure.


Brownian motion Noncentral chi-square Partial sums Portmanteau diagnostic check Time series 

Mathematics Subject Classification (2000)

62F03 62M10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bai J (1993) On the partial sums of residuals in autoregressive and moving average models. J Time Ser Anal 14:247–260 zbMATHCrossRefMathSciNetGoogle Scholar
  2. Box GEP, Jenkins GM (1976) Time series analysis: forecasting and control, 2nd edn. Holden-Day, San Francisco zbMATHGoogle Scholar
  3. Brillinger DR (1973) Estimation of the mean of a stationary time series by sampling. J Appl Probab 10:419–431 zbMATHCrossRefMathSciNetGoogle Scholar
  4. Csörgő M, Horváth L (1997) Limit theorems in change-point analysis. Wiley, Chicester Google Scholar
  5. De Gooijer JG, MacNeill IB (1999) Lagged regression residuals and serial-correlation tests. J Bus Econ Stat 17:236–247 CrossRefGoogle Scholar
  6. Kulperger R, Yu H (2005) High moment partial sum processes of residuals in GARCH models and their application. Ann Stat 33:2395–2422 zbMATHCrossRefMathSciNetGoogle Scholar
  7. Kwan ACC, Wu Y (2003) A re-examination of the finite-sample properties of Peña and Rodriguez’s portmanteau test of lack of fit for time series. Report 157, Department of Economics, The Chinese University of Hong Kong Google Scholar
  8. Lee S, Wei C-Z (1999) On residual empirical processes of stochastic regression models with applications to time series. Ann Stat 27:237–261 zbMATHCrossRefMathSciNetGoogle Scholar
  9. Lin J-W, McLeod AI (2006) Improved Peña–Rodriguez portmanteau test. Comput Stat Data Anal 51:1731–1738 zbMATHCrossRefMathSciNetGoogle Scholar
  10. Ljung GM, Box GEP (1978) On a measure of lack of fit in time series models. Biometrika 65:297–303 zbMATHCrossRefGoogle Scholar
  11. MacNeill IB (1978) Limit processes for sequences of partial sums of regression residuals. Ann Probab 6:695–698 zbMATHCrossRefMathSciNetGoogle Scholar
  12. MacNeill IB, Jandhyala VK (1985) The residual process for non-linear regression. J Appl Probab 22:957–963 zbMATHCrossRefMathSciNetGoogle Scholar
  13. Monti AC (1994) A proposal for a residual autocorrelation test in linear models. Biometrika 81:776–780 zbMATHCrossRefMathSciNetGoogle Scholar
  14. Patnaik PB (1949) The non-central λ 2 and F-distributions and their applications. Biometrika 36:202–232 zbMATHMathSciNetGoogle Scholar
  15. Peña D, Rodriguez J (2002) A powerful test of lack of fit for time series. J Am Stat Assoc 97:601–610 zbMATHCrossRefGoogle Scholar
  16. Rothman ED, Woodroofe M (1972) A Cramér von-Mises type statistic for testing symmetry. Ann Math Stat 43:2035–2038 zbMATHCrossRefMathSciNetGoogle Scholar
  17. Sanjel D, Provost SB, MacNeill IB (2005) On approximating the distribution of an alternative statistic for detecting serial correlation at a given lag. J Probab Stat Sci 3:229–239 Google Scholar
  18. Tyssedal JS, Tjøstheim D (1988) An autoregressive model with suddenly changing parameters and an application to stock market prices. Appl Stat 37:353–369 CrossRefGoogle Scholar
  19. Wichern DW, Miller RB, Hsu D-A (1976) Changes of variances in first-order autoregressive time series models—with an application. Appl Stat 25:248–256 CrossRefGoogle Scholar
  20. Yu H (2007) High moment partial sum processes of residuals in ARMA models and their applications. J Time Ser Anal 28:72–91 zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2007

Authors and Affiliations

  1. 1.Department of Quantitative EconomicsAmsterdamThe Netherlands

Personalised recommendations