Stochastic Modelling of Time Series - Hydrologic Time Series Analysis: Theory and Practice - page 86

the time series would not be stationary. For example, if there is only one
autoregressive model parameter (I 1 ) as the case of AR(1), then I 1 must lie
within ± 1 or -1< I 1 <+1. If there is more than one autoregressive model
parameter, similar kind of general restrictions on the parameter values can be
defined (Box and Jenkins, 1976; Montgomery et al., 1990).

5.1.3 Moving Average (MA) Process

Apart from the serial dependence of the data points as in case of autoregressive
process, each data point in the time series can also be affected by the past
random error (or random shock) that cannot be taken care of by the
autoregressive model. This can be expressed by moving average process as
given below (Box and Jenkins, 1976).

x t = PH T H T H T H

3) !

(3)

t 1 (t 1) 2 (t 2) 3 (t

where x t = data point of variable x at time t ; P = a constant or population mean;
T 1 , T 2 , T 3 = moving average model parameters; and

HH H H =
random error components of the data points at previous times t , t - 1, t - 2 and
t - 3, respectively.
According to Eqn. (3), each data point of the time series is made up of a
random error component or random shock and a linear combination of random
shocks involved in prior data points. For the population of a hydrologic variable,
expression given by Eqn. (3) represents an infinite moving average process.
However, Eqn. (3) can be rewritten for a sample by replacing population
mean with sample mean and reducing the order of the moving average process
from infinite to q , as shown (Box and Jenkins, 1976):

, , ,

t (t ) (t ) (t )

x t = PH T H T H T H ! (4)

t 1 t1 2 t 2 q t q

The order of the moving average process is defined by the highest value
of q , for which T q 0. Thus, for q = 1, the moving average (MA) process is
of the first order and for q = 2, the process is of the second order. The first and
second order moving average processes can be simply denoted as MA(1) and
MA(2), respectively. Similarly, the MA process of order q can be denoted as
MA( q ).
Invertibility Requirement: There is a duality between the moving average
process and the autoregressive process (Box and Jenkins, 1976; Montgomery
et al., 1990) such that a MA( q ) process is not uniquely determined by its
autocorrelation function. However, a unique relationship between moving
average process and their autocorrelation function is required, since the
coefficients of the MA( q ) process can only be estimated by empirical
autocorrelation function. Box and Jenkins (1976) resolved the problem by
introducing a concept of 'invertibility' condition. The 'invertibility' condition