5 Moving Average (Ma)

Let \(\{Z_t\}\) be a purely random process with mean \(\mu = 0\) and variance \(\sigma ^2\). Then \(\{X_t\}\) is a moving average process of order q written as \[X_t = \beta _0Z_t + \beta _1Z_{t-1} + \beta _2Z_{t-2} + ............... + \beta _qZ_{t-q}\] where \(\{\beta _{j's}\}\) are constants. We normally assume \(\beta _0 = 1\)

We write MA(q) to denote Moving Average of order q.
\(E(X_t) = 0\) since \(E(\beta _0Z_t) + E(\beta _1Z_{t-2}) + ......+ E(\beta _qZ_{t-q}) = 0 + 0 + \cdots + 0\)
\(var(X_t) = \sum \limits ^2_Z\sum \limits ^q_{j=0}\beta ^2_j\)

Since \(Z_{t's}\) are independent \begin {align*} \varphi (k) &= Cov(X_t,X_{t+k})\\ &= Cov(\beta _0Z_t + \beta _1Z_{t-1} +\cdots \cdots + \beta _qZ_{t-q},\beta _0Z_{t+k} + \beta _1Z_{t+k-1} +\cdots \cdots + \beta _qZ_{t+k-q})\\ \end {align*}

\[\implies \hspace {0.6cm}\varphi (k) = \begin {cases} 0, &k>q\\\\ \sigma ^2_Z\sum \limits ^{q-k}_{i=0}\beta _i\beta _{i+k}, &k = 0, 1, 2,\cdots \cdots , q\\\\ \varphi (-k),& k<0.\\ \end {cases} \]

\[\text {Since}\hspace {1cm} Cov(Z_s,Z_t) = \begin {cases} \sigma ^2_Z, &s=t\\ 0, &s\neq t\\ \end {cases} \]

As \(\varphi (k)\) does not depend on t and the mean is constant, the process is second order stationary for all values \(\{\beta _i\}\).

ac.f of MA(q) is \[ \rho (\mu ) = \begin {cases} 1, &k = 0\\\\ \dfrac {\sum \limits ^{q-k}_{i=0}\beta _i\beta _{i+k}}{\sum \limits ^q_{i=0}\beta ^2_i}, &k = 1, \cdots \cdots , q\\\\ \rho (-k), &k<0 \end {cases} \]

In particular the MA(1) with \(\beta _0 = 1\) is \(X_t = Z_t + \beta _1Z_{t-1}\). We have that \[ \rho (k) = \begin {cases} 1, &k = 0\\ \dfrac {\beta _1}{1 + \beta ^2_1}, &k = 1, 2,\cdots \cdots , q\\ 0, & \text {otherwise} \end {cases} \]

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