Stochastic Modelling of Time Series - Hydrologic Time Series Analysis: Theory and Practice - page 89

Order of the ARIMA Model: Number of autoregressive ( p ) and moving average
( q ) parameters (i.e., order of the model) are also decided in identification
step. The order of the model is selected in such a way that the model
should be effective and parsimonious. A parsimonious model will have
the fewest parameters and the greatest number of degrees of freedom
among all the stochastic models that fit to the time series. It is observed that
the number of AR and MA parameters hardly exceed two in most of the
studies.
In addition to help deciding required number of differencing passes, the
time plots of the data series, correlograms of autocorrelation function (ACF),
and partial autocorrelation function (PACF) can also assist analysts in selecting
order of a stochastic model. Though the decision cannot be straightforward
and requires not only vast experience but also a good deal of testing with
alternative stochastic models and their parameters. Autocorrelation function
and partial autocorrelation functions are discussed below.
Autocorrelation refers to the correlation of a time series with its own past
and future data points. Autocorrelation is also sometimes called as 'lagged
correlation' or 'serial correlation', which refers to the correlation between
members of a series of numbers arranged in time. Positive autocorrelation
might be considered as a specific form of persistence , a tendency for a system
to remain in the same state from one data point to the next. Autocorrelation
analysis has been discussed in Chapter 4. The autocorrelation function (ACF)
is expressed as Eqns (84) and (85) in Chapter 4 for 'population' and 'sample'
of the time series, respectively.
Partial autocorrelation function (PACF) is the partial correlation
coefficients between the time series and lags of the time series over time. The
partial autocorrelation at lag k is the autocorrelation between x t and x t-k that is
not accounted for by lags 1 through k - 1. The partial autocorrelation of an
AR( p ) process is zero at lag more than or equal to ( p + 1). Detailed algorithm
and mathematical expressions for computing the PACF can be found in Box
and Jenkins (1976) and Brockwell and Davis (1991).
Pankratz (1983) formulated general guidelines for identifying one of the
five basic stochastic models based on the shape/characteristics of
autocorrelogram (ACF) and partial autocorrelogram (PACF) (Table 5.1). A
majority of time series patterns can be satisfactorily approximated using one
of the five basic models mentioned in Table 5.1. Further details and suggestions
for selecting order of the stochastic model can be found in Box and Jenkins
(1976), Hoff (1983), McCleary and Hay (1980), McDowall et al. (1980), and
Vandaele (1983).
The selection of the correct orders p and q of an ARIMA model is fairly
challenging. In this situation, few criteria have been proposed in the literature
to select such a pair ( p , q ) of the parameters minimizing some function, which
is based on the variance estimate (

V ) of the estimated model parameters.

p,q