Stochastic Modelling of Time Series - Hydrologic Time Series Analysis: Theory and Practice - page 88

5.2 Methodology for ARIMA Model Application
Methodology for applying any of the four stochastic models (AR, MA, ARMA
and ARIMA) is almost the same. This chapter describes methodology for
applying stochastic models to time series with reference to the ARIMA model,
since this model contains fundamental parameters of other stochastic models
(AR, MA and ARMA). In addition, the ARIMA( p , d , q ) model can be easily
transformed to AR, MA and ARMA models by adjusting the model parameters.
The methodology for stochastic modelling of the time series involves four
basic steps (Box and Jenkins, 1976): (i) identification or selection of model,
(ii) estimation of model parameters/coefficients, (iii) evaluation or diagnostic
check of the model, and (iv) forecasting. It is necessary for a time series to be
stationary in nature, free from any kind of trends, and adjusted for seasonality
before proceeding to stochastic modelling. The four steps of the methodology
are elucidated in subsequent sections. An application-oriented methodology
of ARIMA models without mathematical descriptions can be found in
McDowall et al. (1980).
5.2.1 ARIMA: Identification of the Model
Number of Differencing Passes: The input time series for an ARIMA model
needs to be stationary, i.e., the time series should have a constant mean,
variance, and autocorrelation through time. A non-stationary time series is
first required to be made stationary. The most common way of making time
series stationary is simply differencing the series repetitively till it becomes
stationary. Sometimes, the time series is transformed for stabilizing the variance
of the series by applying suitable transformations; mostly logarithmic
transformation is applied. The number of times the series is differenced to
attain stationarity is known as number of differencing passes and is indicated
by the parameter ' d ' of ARIMA( p , d , q ) model. To get an idea of the expected
number of differencing passes to make the time series stationary, time plot
and autocorrelogram of the series can be critically examined. Significant
changes in level (strong upward or downward) suggests that the time
series require first-order non-seasonal (lag = 1) differencing. However,
strong changes of slope suggest that the time series require second-order
non-seasonal differencing. If there are seasonal patterns, the time series
usually require respective seasonal differencing. In an autocorrelogram,
when the estimated autocorrelation coefficients decline slowly at longer lags,
first-order differencing of the time series is usually needed. It is suggested for
the newly practising analysts to avoid unnecessary differencing of the time
series as sometimes the time series may require little or no differencing. An
over-differenced time series may produce less stable coefficient/parameter
estimates.