Time Series: Partial Autocorrelation Function (PACF)

[kingwinner]

Consider a stationary AR(2) process:
X_t - X_t-1 + 0.3X_t-2 = 6 + a_t
where {a_t} 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) X_t - X_t-1 + 0.3X_t-2 = 6 + a_t
(ii) X_t - X_t-1 + 0.3X_t-2 = a_t

Any help would be much appreciated! :)

 

[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.

 

[kingwinner]

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.

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!

 

[pmsrw3]

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

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.

 

[blue_raver22]

I would like to clarify that the PACF is a statistical tool used to measure the correlation between a time series and its lagged values, while taking into account the effects of other variables. It is often used to identify the order of an autoregressive (AR) process.

In the given stationary AR(2) process, we have a constant term "6" added to the equation. This constant term does not affect the order of the process, which is still AR(2). Therefore, the PACF for both (i) and (ii) would be the same.

The reason for this is that the constant term does not have any effect on the correlation between the time series and its lagged values. The PACF only takes into account the correlation between the time series and its lagged values, not the overall mean or level of the series.

In conclusion, the PACF for both (i) and (ii) would be the same, as the constant term does not alter the order of the AR process.