TS Module 11: simulated and actual time series HW
(The attached PDF file has better formatting)
Homework assignment: Partial autocorrelations
[Partial autocorrelations are covered in Module 10, along with sample autocorrelations.]
A stationary ARMA process has ñ_{2} = 0.20.
ñ_{1} ranges from 0.2 to 0.7 in units of 0.1.
A. Graph the partial autocorrelation of lag 2 (ö_{22}) as a function of ñ_{1}.
B. Explain why the partial autocorrelation is positive for low ñ_{1} and negative for high ñ_{1}.
ben, you are correct in your assumption for part A. For part B, examine equation 6.2.3 remembering that rho_{2} is a constant. The question should answer itself at this point. Let me know if you have any further questions.
RDH
I know it's obvious what's going on with phi_22 from the formula, but I'm having trouble answering in a more conceptual, intuitive way. I guess I'm thinking the question is more like, why should the Corr(Y_t, Y_t-2|Y_t-1) become negative when Corr(Y_t, Y_t-1) increases? Am I making this more complicated than it is?
[NEAS: When the correlation for lag 1 increases, more of the correlation for lag 2 is explained by the correlation for lag 1.]
what's meaning , more correltion of lag 2 is explained by lag 1? because of being explained by lag, the phi 22 becomes more negative? I couldn't figure out the relationship between these two consequences.
My guess - Since autocorrelation lag 2 is constant, as autocorrelation lag 1 increases we would expect autocorrelation lag 2 to increase since more and more of it should rely on autocorrelation lag 1. However, since autocorrelation lag 2 is constant, it must be that the autocorrelation of lag 2 removing the effect of intervening variable must be decreasing to make up the difference for the larger effect of autocorrelation lag 1.
I'm unsure if I explained my thought process very well, but that is the general idea I think they are trying to get.
Let me know if you guys come up with a more tangible answer.