# Time Series: Partial Autocorrelation Function (PACF)

1. Aug 1, 2011

### kingwinner

Consider a stationary AR(2) process:
Xt - Xt-1 + 0.3Xt-2 = 6 + at
where {at} is white noise with mean 0 and variance 1.
Find the partial autocorrelation function (PACF).

I searched a number of time series textbooks, but all of them only described how to find the PACF for an ARMA process with mean 0 (i.e. without the constant term). So if the constant term "6" above wasn't there, then I know how to find the PACF, but how about the case WITH the constant term "6" as shown above?

I'm guessing that (i) and (ii) below would have the same PACF, but I'm just not so sure. So do they have the same PACF? Can someone explain why?
(i) Xt - Xt-1 + 0.3Xt-2 = 6 + at
(ii) Xt - Xt-1 + 0.3Xt-2 = at

Any help would be much appreciated! :)

2. Aug 1, 2011

### pmsrw3

Yes, (i) and (ii) have the same autocorrelation functions. Correlation coefficients are defined based on mean-centered deviations, so changes in the means only of the correlated values have no effect on the correlation.

3. Aug 1, 2011

### kingwinner

I see. How about the PARTIAL autocorrelation functions of (i) and (ii)? Are they the same? Why or why not?

http://en.wikipedia.org/wiki/Partial_autocorrelation_function
http://fedc.wiwi.hu-berlin.de/xplore/tutorials/sfehtmlnode59.html

Thanks!

4. Aug 1, 2011

### pmsrw3

I wasn't familiar with this, but based on those links, it should be the same. In particular, if you look at the second, the correction from ACF to PACF is calculated from the covariance matrix. Covariances, like correlations, are mean-corrected.