Time Series Regression with Periodically Correlated Errors and Missing Data

  • W. Dunsmuir
Part of the Lecture Notes in Statistics book series (LNS, volume 25)


Many problems, especially those related to environmental issues, are concerned with estimation of, and testing the significance of, a time trend in a series possibly when explanatory variables are included in the model. As is well known, if there is correlation in the residuals, and this is not accounted for, then incorrect inferences about the size of trend may be made. Regressions, of the type mentioned above, often contain not only correlated residuals but may have the more complicated form of periodic autocorrelation. In addition data may be missing. How much of an effect that these two problems have on trend assessment, if not accounted for, is not entirely obvious. This paper describes how to perform correct inferences in regression models in which both of these problems exist and the emphasis is on easy to implement techniques in order that an assessment, of the impact of these two problems, can be made relatively easily. Application of the techniques is given to monthly average salinity measurements and to daily average carbon monoxide measurements. Both examples include explanatory variables and the emphasis is on assessing the significance of time trends. Evidence exists in both of these examples that the residuals are periodically correlated but the analysis shows that accounting for this more complicated error structure has little additional impact over a non-periodically correlated error structure. However, for assessing crossing probabilities, e.g.,, it is important to model the residuals correctly. Theoretical justification will be given for the methodology presented. This provides theoretical justification for the familiar Cochrane-Orcutt method (to correct for autoregressive errors in regressions) when there are missing data.


Ordinary Little Square Central Limit Theorem Ordinary Little Square Regression Theoretical Justification Stationary Time Series 
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  1. Cunningham, R.B. and Morton, R. (1982). “A statistical method for the estimation of trend in salinity in the River Murray”.Google Scholar
  2. Dunsmuir, W. (1981). “Estimation for stationary time series when data are irregularly spaced or missing”, in D.F. Findley, ed., Applied Time Series II (Academic Press, New York).Google Scholar
  3. Dunsmuir, W. (1982a). “Estimation and testing in periodically stationary time series”, Biometrika (to appear).Google Scholar
  4. Dunsmuir, W. (1982b). “Testing for periodic departures from normality in time series”, J. Amer. Stat. Assoc. (to appear).Google Scholar
  5. Dunsmuir, W. (1983). “A central limit theorem for estimation in Gaussian stationary time series observed in unequally spaced times”, Stochastic Processes and their Applications, 14, 1–17.MathSciNetCrossRefGoogle Scholar
  6. Hannan, E.J. (1970). Multiple Time Series, New York: Wiley.zbMATHCrossRefGoogle Scholar
  7. Harvey, A.C. (1981). The Econometric Analysis of Time Series, Oxford: Philip Allan Publishers.zbMATHGoogle Scholar
  8. Jones, R.J. (1980). “Maximum likelihood fitting of ARMA models to time series with missing observations”. Technometrics, 22, 389–395.MathSciNetzbMATHGoogle Scholar
  9. Martin, M.K. (1977). Analysis of trends in ambient air quality, Master of Science Thesis, Massachusetts Institute of Technology.Google Scholar
  10. Pagano, M. (1978). “On periodic and multiple autoregressions”. Ann. Statist., 6, 1310–1317.MathSciNetzbMATHCrossRefGoogle Scholar
  11. Troutman, B.M. (1979). “Some results on periodic autoregressions”. Biometrika, 66, 219–228.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • W. Dunsmuir
    • 1
  1. 1.SIROMATH Pty LtdAustralia

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