Time Series: Partial Autocorrelation Function (PACF)

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.

royhaas said:

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.

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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!

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!