Modelling Stationary Time Series: the ARMA Approach

Abstract

When analysing a time series using formal statistical methods, it is often useful to regard the observations (x₁,x₂,…,x _T ) on the series, which we shall denote generically as x _t , as a particular realisation of a stochastic process.¹ In general, a stochastic process can be described by a T-dimensional probability distribution, so that the relationship between a realisation and a stochastic process is analogous to that between the sample and population in classical statistics. Specifying the complete form of the probability distribution, however, will typically be too ambitious a task and we usually content ourselves with concentrating attention on the first and second moments: the T means

$$E({x_1}),E({x_2}), \ldots ,E({x_T})$$

T variances

$$V({x_1}),V({x_2}), \ldots ,V({x_T})$$

and T(T-1)/2 covariances

$$Cov({x_i},{x_j}),\quad i < j$$

If we could assume joint normality of the distribution, this set of expectations would then completely characterise the properties of the stochastic process. Such an assumption, however, is unlikely to be appropriate for every economic and financial series we might wish to analyse.