Time Series: Partial Autocorrelation Function (PACF)

[kingwinner]

 

[royhaas]

Subtract the known constant from each term of the series and proceed. The level of the time series does not enter into the relationship between lags.

 

[kingwinner]

Subtract the known constant from each term of the series and proceed. The level of the time series does not enter into the relationship between lags.

Let Y[sub:1muxiod4]t[/sub:1muxiod4]=X[sub:1muxiod4]t[/sub:1muxiod4]-20
Then Y[sub:1muxiod4]t[/sub:1muxiod4]-Y[sub:1muxiod4]t-1[/sub:1muxiod4]+0.3Y[sub:1muxiod4]t-2[/sub:1muxiod4]=a[sub:1muxiod4]t[/sub:1muxiod4]

I believe I know how to transform it into a zero-mean process {Y_t}, but if we find the PACF from the last equation, it will be the PACF between {Y_t} at different lags. But we actually want the PACF between {X_t} at different lags. I just don't see the exact relationship linking the PACF of {Y_t} and the PACF of {X_t}. Are they the same? Why or why not?

Thanks!

 

[kingwinner]

OK, since we know Var(X+c)=Var(X) and Cov(X+a,Y+b)=Cov(X,Y), the variances, autocovariances, and autocorrelations are NOT going to be affected by the constant term (which leads to non-zero mean) in the ARMA model. So (i) and (ii) in my original post will have the same variance, autocovariance, and autocorrelation.

But I still don't see exactly why (i) and (ii) will have the same partial autocorrelation function. How can we prove this?

Thanks!