Time Series Analysis - ARIMA models - ARMA(1,1) process

h. The ARMA(1,1) process

On combining an AR(1) and a MA(1) process one obtains an ARMA(1,1) model which is defined as

(V.I.1-154)

where W_t is a stationary time series, e_t is a white noise error component, and F_t is the forecasting function.

Note that the model of (V.I.1-154) may alternatively be written as

(V.I.1-155)

such that

(V.I.1-156)

in y-weight notation.

The y-weights can be related to the ARMA parameters on using

(V.I.1-157)

such that the following is obtained

(V.I.1-158)

Also the p-weights can be related to the ARMA parameters on using

(V.I.1-159)

such that the following is obtained

(V.I.1-160)

From (V.I.1-158) and (V.I.1-160) it can be clearly seen that an ARMA(1,1) is in fact a parsimonious description of either an AR or a MA process with an infinite amount of weights. This does not imply that all higher order AR(p) or MA(q) processes may be written as an ARMA(1,1). Though, in practice an ARMA process (c.q. a mixed model) is, quite frequently, capable of capturing higher order pure-AR p-weights or pure-MA y-weights.

On writing the ARMA(1,1) process as

(V.I.1-161)

(which is a difference equation) we may multiply by W_t-k and take expectations. This gives

(V.I.1-162)

In case k > 1 the RHS of (V.I.1-162) is zero thus

(V.I.1-163)

If k = 0 or if k = 1 then

(V.I.1-164)

Hence we obtain

(V.I.1-165)

The theoretical ACF is therefore

(V.I.1-166)

The theoretical ACF and PACF patterns for the ARMA(1,1) are illustrated in figures (V.I.1-7), (V.I.1-8), and (V.I.1-9).

(figure V.I.1-7)

(figure V.I.1-8)

(figure V.I.1-9)