Stochastic Modelling of Time Series - Hydrologic Time Series Analysis: Theory and Practice - page 93

where r k (K) = autocorrelation coefficient of the residual series at lag k , and L
= maximum lag considered. If K t is independent, then Q , which is approximately
chi-squared distributed with L - p - q degrees of freedom, should be less than
F 2 ( L - p - q ).
The portmanteau lack-of-fit test checks whether the estimated residuals
ˆ H t = 1, 2, …, n , behave approximately like realizations from a white noise
process.
The major concern here is that the residuals are systematically distributed
across the series (e.g., they could be negative in the first part of the series and
approach zero in the second part) or that they contain some serial dependency
which may suggest that the ARIMA model is inadequate. The analysis of
ARIMA residuals constitutes an important test of the model. The estimation
procedure assumes that the residuals are not autocorrelated and that they are
normally distributed.

5.2.4 ARIMA: Forecasting

When the selected ARIMA model successfully passes the evaluation step, the
estimated model parameters are then used in the last stage of forecasting to
compute new values of the time series and confidence intervals for the predicted
values. Usually, the forecasts are made for future such that these computed
new values are beyond the data points included in the input time series. It is
worth mentioning that if the estimation process is performed on transformed
or differenced time series, then the series needs to be integrated before the
forecasts are generated. Integration is the inverse process of differencing,
which is performed in order to express the forecasts in values compatible with
the input time series data. This integration feature is represented by the letter
'I' in the name of the model (ARIMA = Autoregressive Integrated Moving
Average).

REFERENCES

Bails, D.G. and Peppers, L.C. (1982). Business Fluctuations: Forecasting Techniques
and Applications. Englewood Cliffs, Prentice-Hall, NJ.
Box, G.E.P. and Jenkins, G.M. (1976). Time Series Analysis: Forecasting and Control.
Holden-Day, San Francisco, 575 pp.
Box, G.E.P. and Pierce, D.A. (1970). Distribution of the residual autocorrelations in
autoregressive integrated moving average time series models. Journal of the
American Statistical Association , 65: 1509-1526.
Brockwell, P.J. and Davis, R.A. (1991). Time Series: Theory and Methods. 2 nd Edition,
Springer Series in Statistics, Springer, New York.
Chatfield, C. (1980). The Analysis of Time Series: An Introduction. 2 nd Edition,
Chapman and Hall, London, U.K.