We use the sample partial correlation function
to estimate the partial correlation function. It is obtained by replacing the true covariance function
(k) used in the Levinson-Durbin algorithm with the sample covariance function
. To get the sample partial correlation function from given data up to lag
h we can use the function
Again the function name is the same as that used to calculate the partial correlation function of models.
For an AR(
p) process, we know from Section
1.2.4 that
k, k=0 for
k>p. The sample partial correlation
of an AR(
p) process for large
n and
k>p has an asymptotic normal distribution with mean zero and variance given by
We can use
as a guide to decide if the sample partial correlation can be considered zero.
In the following example, we are given a set of stationary, zero-mean data of length 200 generated from the AR(3) model Xt=1.5Xt-1-1.0Xt-2+0.4Xt-3+Zt.
We first calculate and plot the sample correlation function of the given data.
Since there is no sharp cutoff in the plot of the correlation function, it is unlikely that the data are from a pure MA process. Next we calculate and plot the sample partial correlation function. The plot of the sample partial correlation function
versus the lag
k is sometimes referred to as the
partial correlogram; we display the partial correlogram of
data together with the bounds
.
We see that the sample partial correlation differs from zero significantly only at the first 3 lags. This provides evidence to support the hypothesis that the data are from an AR(3) process.