The partial autocorrelation function
The partial autocorrelation function, or pacf (denoted Tkk), measures the correlation between an observation k periods ago and the current observation, after controlling for observations at intermediate lags (i.e. all lags < k) - i.e. the correlation between yt and yt-k, after removing the effects of yt-k+1, yt-k+2,..., yt-1. For example, the pacf for lag 3 would measure the correlation between yt and yt-3 after controlling for the effects of yt-1 and yt-2.
At lag 1, the autocorrelation and partial autocorrelation coefficients are equal, since there are no intermediate lag effects to eliminate. Thus, r11 = r1, where r1 is the autocorrelation coefficient at lag 1. At lag 2
where x\ and t2 are the autocorrelation coefficients at lags 1 and 2, respectively. For lags greater than two, the formulae are more complex and hence a presentation of these is beyond the scope of this book. There now proceeds, however, an intuitive explanation of the characteristic shape of the pacf for a moving average and for an autoregressive process.
In the case of an autoregressive process of order p, there will be direct connections between yt and yt-s for s < p, but no direct connections for s > p. For example, consider the following AR(3) model yt = 0o + 0i yt-i + 02 yt-2 + 03 yt-3 + ut (5.118)
There is a direct connection through the model between yt and yt-1, and between yt and yt-2, and between yt and yt-3, but not between yt and yt-s, for s > 3. Hence the pacf will usually have non-zero partial autocorrelation coefficients for lags up to the order of the model, but will have zero partial autocorrelation coefficients thereafter. In the case of the AR(3), only the first three partial autocorrelation coefficients will be non-zero.
What shape would the partial autocorrelation function take for a moving average process? One would need to think about the MA model as being transformed into an AR in order to consider whether yt and yt-k, k = 1, 2,..., are directly connected. In fact, so long as the MA(g) process is invertible, it can be expressed as an AR(to). Thus a definition of invertibility is now required.
5.5.1 The invertibility condition
An MA(q) model is typically required to have roots of the characteristic equation e(z) = 0 greater than one in absolute value. The invertibility condition is mathematically the same as the stationarity condition, but is different in the sense that the former refers to MA rather than AR processes. This condition prevents the model from exploding under an AR(d) representation, so that e-1(L) converges to zero. Box 5.2 shows the invertibility condition for an MA(2) model.
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