Free Access
RAIRO-Oper. Res.
Volume 11, Number 2, 1977
Page(s) 129 - 143
Published online 06 February 2017
  • 1. On these and related points, see G. Box and G. M. JENKINS, Time Series Analysis, Forecasting, and Control, Holden-Day, San Francisco, 1970; [MR: 272138] [Zbl: 0249.62009] [Google Scholar]
  • 1. C. R. NELSON, Applied Time Series Analysis, Holden-Day, San Francisco, 1973; [Zbl: 0271.62113] [Google Scholar]
  • 1. Seminar on Time Series Analysis, in: The Statistician, vol. 17, No. 3, 1967; [Google Scholar]
  • 1. W. L. YOUNG, Exponential Smoothing, Seasonality, and Projection Sensitivity: the Case of Exports, Bull. Econ. Res. Yorkshire, vol. 26, No. 1, May 1974, [Google Scholar]
  • 1. and, by the same author, Seasonality, Autoregression, and Exponential Smoothing: the Case of Economic Time Series, Metron, International Statistical Journal, Gini Institute, Rome, vol. 31, n° 1-4, December 1973; [Google Scholar]
  • 1. C. CHATFIELD and D.L. PROTHERO, Box-Jenkins Seasonal Forecasting: Problems in a Case Study, J. Roy.Stat. Soc A, No. 136, 1973, p. 295; [Google Scholar]
  • 1. T. H. NAYLOR, T. G. SEAKS and D. W. WICHERN, Box-Jenkins Models: an Alternative to Econometric Models, Int. Stat. Rev., vol. 40, 1972, p. 123; [Zbl: 0239.62062] [Google Scholar]
  • 1. J. M. BATES and C. W. GRANGER, The Combination of Forecasts, Op. Res. Q., vol. 20, 1969. [Google Scholar]
  • 2. The constant ?o can take on a nonzero value, for example, in cases which require "differencing". This introduces a polynomial of degree d, which is, by nature, deterministic, into the forecast function eventually generated. On this point,see Box and JENKINS, ibid., pp. 91-94 and 194-195. [Google Scholar]
  • 3. See NELSON, op. cit, Chapter 2, 3 ff. [Google Scholar]
  • 4. Ibid, Chapter 5 ff. [Google Scholar]
  • 5. An intuitive explanation regarding the nature of the partial: autocorrelation function could be given as follows: if, for example, x1, x2,..., xt is a time series, then the partial autocorrelation function at, say, lag j would be the auto correlation of xt and xt +?, under the condition that we already know the values of the observations xt + l, xt + 2,..., xt + ? - 1. [Google Scholar]
  • 6. It should be noted that the minimum sample for reliable Box-Jenkins analysesis about fifty observations. In addition, given a time series of length T, the autocorrelation and partial autocorrelation functions should be computed only to about K ? T/4 lags, due to the fact that for larger values of K, or as it approaches T, these estimates become quite bad. On these and related points, see R. L. ANDERSON, Distribution of the Serial Correlation Coefficient, Annals Math. Stats., vol. 13, 1942, p. 1; [Google Scholar]
  • 6. M. S. BARTLETT, On the Theoretical Specification of Sampling Properties of Autocorrelated Time Series, J. Roy. Stat. Soc, B, No. 8, 1946, p. 27; [MR: 18393] [Zbl: 0063.00228] [Google Scholar]
  • 6. J. DURBIN, The Fitting of Time Series Models, Rev. Int. Inst. Stats., vol. 28, 1960, p. 223; [Zbl: 0101.35604] [Google Scholar]
  • 6. M. H. QUENOUILLE, Approximate Test of Correlation in Time Series, J. Roy. Stat. Soc. B, vol. 11, 1949, p. 68. [MR: 32176] [Zbl: 0035.09201] [Google Scholar]
  • 7. Box and JENKINS, op. cit; D. W. MARQUARDT, An Algorithm for Least Squares Estimation of Non-Linear Parameters, J. Soc. Ind. Appl. Math., vol. 2, 1963, p. 431. [Zbl: 0112.10505] [Google Scholar]
  • 8. C. R. NELSON, op. cit, pp. 82 ff. [Google Scholar]
  • 9. G. Box and D. A. PIERCE, Distribution of Residual Autocorrelations in Autoregressive Integrated Moving Average Time Series Models, J. Amer. Stat. Assn., vol. 65, 1970, p. 1509. [MR: 273762] [Zbl: 0224.62041] [Google Scholar]
  • 10. For a classification of this type of forecast and forecasting types in general, see W. L. YOUNG, Forecasting Types and Forecasting Techniques: a Taxonomic Approach in Quality and Quantity, Europ. Amer. J. Method., Elsevier, 1976. Also see NELSON, op. cit, Ch. 6 ff; [Google Scholar]
  • 10. P. NEWBOLD and C. GRANGER, Experience with Forecasting Univariate Time Series and the Combination of Forecasts, J. Roy. Stats. Soc, A, No. 137, 1974. [MR: 451583] [Google Scholar]
  • 11. Practically speaking, however, seasonal modeling using the Box-Jenkins approach can prove difficult. This is due to the fact that at present, we do not know much about the theoretical behaviour of seasonal autocorrelation and partial autocorrelation functions. On these and related points, see CHATFIELD and PROTHERO, op. cit; NELSON, Ibid, Ch. 7 ff; T. F. SMITH, A Comparison of Some Models for Predicting Time Series Subject to Seasonal Variation, in: Seminar on Time Series Analysis, op. cit. [Google Scholar]
  • 12. On these and related points, see R. J. ALLARD, An Economie Analysis of the Effects of Regulating Hire Purchase, H. M. Treasury, Gov't. Economic Service Occasional Papers, 9, London: HMSO, 1974; [Google Scholar]
  • 12. R. J. ALLARD, Hire Purchase Controls and Consumer Durable Purchases, Queen Mary College, Univ. of London, Dept. of Economics Discussion Paper, March 1975 (25-39-75). [Google Scholar]
  • 12. Also see R. J. BALL and P. S. DRAKE, Impact of Credit Controlon Consumer Durable Goods Spending in the United Kingdom, 1957-1961, Rev. Econ. Stud. vol. 30, No. 3, 1963, for an alternative view of the problem. [Google Scholar]
  • 13. On the principle of "overfitting" and "parameter redundancy", see NELSON, op. cit, p. 114. [Google Scholar]
  • 14. R. J. ALLARD, op. cit, Appendix H, 1974, pp. 95 ff. The large forecasting error for 1971 III is due to the fact that hire-purchase credit controls were abolished in July, 1971, and were also distorted by strikes earlier that year, in addition to the fact that an increasing number of automobiles were purchased with credit from sources other than finance houses, e. g. personal bank loans and overdrafts, etc. [Google Scholar]
  • 15. Box and JENKINS, op. cit; G. Box and D. R. Cox, An Analysis of Transformations, J. Roy. Stat. Soc, B, No. 26, 1964; [Zbl: 0156.40104] [Google Scholar]
  • 15. G. Box and G. M. JENKINS, Some Recent Advances in Forecasting and Control, Applied Stats., 17, 1968; [MR: 234593] [Google Scholar]
  • 15. C. GRANGER and P. NEWBOLD, Forecasting Transformed Series, J. Roy. Stat. Soa, B, No. 38 1976. [MR: 445749] [Zbl: 0344.62076] [Google Scholar]
  • 16. G. V. GLASS et al., Design and Analysis of Time Series Experiments, Colorado Associated Univ. Press; Boulder, Colo., 1975; [Google Scholar]
  • G. GLASS, Estimating the Effects of Intervention Into a Non-Stationary Time Series, Amer. Educ. Res. J., vol. 9, No. 3, 1972; [Google Scholar]
  • G. Box and G. TIAO, A Change in a Level of a Non-Stationary Time Series, Biometrika, vol. 52, June 1965; [MR: 208788] [Zbl: 0142.15901] [Google Scholar]
  • G. Box and G. TIAO, Intervention Analysis with Applications to Economic and Environmental Problems, J. Amer. Stat. Assn., vol. 70, No. 349, March 1975, pp. 70 ff, 72. [MR: 365957] [Zbl: 0316.62045] [Google Scholar]

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