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 | Zbl

1. C. R. Nelson, Applied Time Series Analysis, Holden-Day, San Francisco, 1973; | Zbl

1. Seminar on Time Series Analysis, in: The Statistician, vol. 17, No. 3, 1967;

1. W. L. Young, Exponential Smoothing, Seasonality, and Projection Sensitivity: the Case of Exports, Bull. Econ. Res. Yorkshire, vol. 26, No. 1, May 1974,

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;

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;

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

1. J. M. Bates and C. W. Granger, The Combination of Forecasts, Op. Res. Q., vol. 20, 1969.

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.

3. See Nelson, op. cit, Chapter 2, 3 ff.

4. Ibid, Chapter 5 ff.

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.

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;

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 | Zbl

6. J. Durbin, The Fitting of Time Series Models, Rev. Int. Inst. Stats., vol. 28, 1960, p. 223; | Zbl

6. M. H. Quenouille, Approximate Test of Correlation in Time Series, J. Roy. Stat. Soc. B, vol. 11, 1949, p. 68. | MR | Zbl

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

8. C. R. Nelson, op. cit, pp. 82 ff.

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 | Zbl

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;

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

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.

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;

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).

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.

13. On the principle of "overfitting" and "parameter redundancy", see Nelson, op. cit, p. 114.

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.

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

15. G. Box and G. M. Jenkins, Some Recent Advances in Forecasting and Control, Applied Stats., 17, 1968; | MR

15. C. Granger and P. Newbold, Forecasting Transformed Series, J. Roy. Stat. Soa, B, No. 38 1976. | MR | Zbl

16. G. V. Glass et al., Design and Analysis of Time Series Experiments, Colorado Associated Univ. Press; Boulder, Colo., 1975;

G. Glass, Estimating the Effects of Intervention Into a Non-Stationary Time Series, Amer. Educ. Res. J., vol. 9, No. 3, 1972;

G. Box and G. Tiao, A Change in a Level of a Non-Stationary Time Series, Biometrika, vol. 52, June 1965; | MR | Zbl

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 | Zbl