The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Stationary Time Series

  • E. J. Hannan
Reference work entry


The concept of a stationary time series was, apparently, formalized by Khintchine in 1932. An infinite sequence y(t), t = 0, ±1,…, of random variables is called stationary if the joint probability law of y(t1), y(t2),…, y(tn), is the same as that of y(t1 + t),…, y(tn + t), for any integers, t1, t2,…, tn, t and any n. Thus the stochastic mechanism generating the sequence is not changing. In the natural sciences approximately stationary phenomena abound, but the continuing social evolution of man makes such phenomena rarer in social science. Nevertheless, stationary time series models have been widely used in econometrics since they may fit the data well over periods of time that are not too long and thus may provide a basis for short term predictions. The notions of trend, cycle, seasonal are closely related to a frequency decomposition of a series, with the trend corresponding to very low frequencies, and the spectral decomposition of a stationary series [see (2) below] is therefore of interest to economists. Finally, models have also been used where the observed series is regarded as the output of an evolving mechanism whose input is stationary.

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  1. A good general reference for the structure theory of stationary time series is Rozananov (1967). General treatments of this theory and of the statistical methods are in Priestley (1981). There is historical material in Wold (1934) and Doob (1953). For recent material, for example on robust methods and nonlinear models, see Brillinger and Krishnaiah (1983) and Hannan, Krishnaiah and Rao (1985).Google Scholar
  2. Akaike, H. 1969. Fitting autoregressive models for prediction. Annals of the Institute of Statistical Mathematics 21: 243–247.CrossRefGoogle Scholar
  3. Birkhoff, G.D. 1931. Proof of the ergodic theorem. Proceedings of the National Academy of Sciences of the United States of America 17: 556–600.Google Scholar
  4. Blackman, R.B., and J.W. Turkey. 1958. The measurement of power spectra. New York: Dover.Google Scholar
  5. Box, G.E.P., and G.M. Jenkins. 1971. Time series analysis, forecasting and control. San Francisco: Holden-Day.Google Scholar
  6. Brillinger, D.R., and P.R. Krishnaiah (eds.). 1983. Handbook of statistics 3, time series in the frequency domain. New York: North-Holland.Google Scholar
  7. Cramer, H. 1942. On harmonic analysis in certain functional spaces. Arkiv för Matematik, Astronomi och Fysik 28B, no. 12.Google Scholar
  8. Doob, J.L. 1953. Stochastic processes. New York: Wiley.Google Scholar
  9. Friedlander, B. 1982. Lattice filters for adaptive processing. Proceedings of the IEEE 70: 830–867.Google Scholar
  10. Hannan, E.J. 1971. The identification problem for multiple equation systems with moving average errors. Econometrica 39: 751–766.CrossRefGoogle Scholar
  11. Hannan, E.J., and L. Kalvalieris. 1984. Multivariate linear time series models. Advances in Applied Probability 16: 492–561.CrossRefGoogle Scholar
  12. Hannan, E.J., P.R. Krishnaiah, and M.M. Rao. 1985. Time series in the time domain. New York: North Holland.Google Scholar
  13. Kalman, R.E. 1960. A new approach to linear filtering and prediction problems. Transactions of the American Society of Mechanical Engineers. Journal of Basic Engineering, Series D 82: 35–45.CrossRefGoogle Scholar
  14. Khintchine, A. 1934. Korrelationstheorie de stationären stochastiischen Prozesse. Mathematische Annalen 109: 604–615.CrossRefGoogle Scholar
  15. Kolmogoroff, A.N. 1939. Sur I’interpolation et extrapolation des suites stationnaires. Comptes Rendus. Académie des Sciences (Paris) 208: 2043–2045.Google Scholar
  16. Priestley, M.B. 1981. Spectral analysis and time series. New York: Academic Press.Google Scholar
  17. Rozanov, Yu.A. 1967. Stationary random processes. San Francisco: Holden-Day.Google Scholar
  18. Whittle, P. 1951. Hypothesis testing in time series analysis. Uppsala: Almqvist and Wiksell.Google Scholar
  19. Wiener, N. 1945. Extrapolation, interpolation and smoothing of stationary time series. New York: Wiley.Google Scholar
  20. Wold, H. 1934. A study in the analysis of stationary time series. Stockholm: Almqvist and Wiksell.Google Scholar
  21. Yule, G.U. 1927. On a method of investigating periodicities in disturbed series with special reference to Woolfer’s sunspot numbers. Philosophical Transactions of the Royal Society of London. Series A 226: 267–298.CrossRefGoogle Scholar

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© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • E. J. Hannan
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