Neas-Seminars

TS Module 11: simulated and actual time series HW


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By NEAS - 12/4/2009 6:27:52 AM

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.


 

 

 
By chrisdacoolman - 8/27/2013 12:13:02 PM

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. “

http://people.duke.edu/~rnau/411arim3.htm

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.