## TS Module 11: simulated and actual time series HW

 Author Message NEAS Supreme Being Group: Administrators Posts: 4.2K, Visits: 1.2K 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 ñ2 = 0.20.            ñ1 ranges from 0.2 to 0.7 in units of 0.1.   A.     Graph the partial autocorrelation of lag 2 (ö22) as a function of ñ1.B.     Explain why the partial autocorrelation is positive for low ñ1 and negative for high ñ1.    Attachments TS Module 11 partial autocorrelations HW.pdf (1.4K views, 36.00 KB) benjaminttp Forum Newbie Group: Forum Members Posts: 6, Visits: 1 part A, so the graph is like showing rho on x-axis and phi on y-axi?part B, what is this about? RayDHIII Forum Member Group: Forum Members Posts: 39, Visits: 138 ben, you are correct in your assumption for part A.  For part B, examine equation 6.2.3 remembering that rho2 is a constant.  The question should answer itself at this point.  Let me know if you have any further questions.RDH CalLadyQED Forum Guru Group: Forum Members Posts: 62, Visits: 2 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.] minnie53053 Junior Member Group: Forum Members Posts: 11, Visits: 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. moo5003 Junior Member Group: Forum Members Posts: 10, Visits: 60 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. chrisdacoolman Forum Newbie Group: Awaiting Activation Posts: 2, Visits: 40 Using this definition I found online:“A partial autocorrelation is the amount of correlation between a variable and a lag of itself that is not explained by correlations at all lower-order-lags. The autocorrelation of a time series Y at lag 1 is the coefficient of correlation between Y(t) and Y(t-1), which is presumably also the correlation between Y(t-1) and Y(t-2). But if Y(t) is correlated with Y(t-1), and Y(t-1) is equally correlated with Y(t-2), then we should also expect to find correlation between Y(t) and Y(t-2). (In fact, the amount of correlation we should expect at lag 2 is precisely the square of the lag-1 correlation.) Thus, the correlation at lag 1 "propagates" to lag 2 and presumably to higher-order lags. The partial autocorrelation at lag 2 is therefore the difference between the actual correlation at lag 2 and the expected correlation due to the propagation of correlation at lag 1. “And the hint given by NAES “When the correlation for lag 1 increases, more of the correlation for lag 2 is explained by the correlation for lag 1“ B) Because the partial auto correlation is the amount of correlation between a variable and a lag of itself that is NOT explained by correlations at all lower-order-lags, a low ρ1 means that less of the correlation for the lag 2 is explained by the correlation for lag 1. Thus less of the partial correlation is explained by the lower-order-lag and more is explained by the variable and a log of itself, making the partial auto correlation greater/positive at lower values of ρ1. At higher values of ρ1 the opposite is true and less correlation is explained by the variable and a log of itself, which makes the partial auto correlation negative.
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