Stochastic Modelling of Time Series - Hydrologic Time Series Analysis: Theory and Practice - page 92

exact maximum likelihood method proposed by Melard (1984) may become
inefficient when used to estimate the model parameters with long seasonal
lags (e.g., annual lags). A general recommendation is to first use the approximate
maximum likelihood method in order to establish initial parameter estimates
that may be close to the real final values and then the exact maximum likelihood
method may be employed to get final estimates of the model parameters with
certainly a few iterations.
The ARIMA models may also include a constant in addition to the standard
parameters of AR and MA models. However, interpretation of the statistically
significant constant depends on the type of the model that is to be fit. When
the AR parameters are not present in the ARIMA model, the expected value
of the constant simply represents mean of the series. Whereas, if the
autoregressive parameters are present in the series, the constant represents the
intercept. If the series is differenced, the constant represents the mean or
intercept of the differenced series. Thus, if the series is differenced once, and
there are no autoregressive parameters in the model, the constant represents
the mean of the differenced series.

5.2.3 ARIMA: Evaluation of the Model
When the first two steps of ARIMA modelling are complete then orders p and
q of the model and respective AR and MA parameters are known in order to
model an ARIMA( p , d , q ) process underlying the data. Before proceeding to
make forecasting by using the ARIMA model, it is essential to apply diagnostic
check of the model. The approximate values of the t test-statistics are computed
from the parameter standard errors. If the test-statistics are not found significant,
the respective parameter can in most cases be dropped from the model without
affecting substantially the overall fit of the model.
Another straightforward way for evaluating the reliability of the selected
ARIMA model is to check the accuracy of generated forecasts. A comparison
of the forecasts with the observed (measured) data points can reveal how
efficient the model is in making forecasts. A good model should not only
provide sufficiently accurate forecasts, but it should also be parsimonious and
produce statistically independent residuals that contain only noise and no
systematic components. The correlogram of the residuals should not reveal
any serial dependencies. One more approach is to plot the residuals of the
original (observed) series and inspect them for any systematic trends, and to
examine the autocorrelogram of residuals. There should not be any serial
dependency between residuals.
The portmanteau lack-of-fit test is generally applied to evaluate the model
fitness. The portmanteau lack-of-fit test-statistic , Q is defined as follows (Box
and Pierce, 1970):

L

Ç

k

Q = nr () K (9)

k 1