By Hay J.L., Pettitt A.N.

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**Example text**

8. Let 1, h=O, 0 otherwise. (a) For what values of a is p(h) the autocorrelation function of a stationary time series? (b) Compute the partial autocorrelation function associated with p(h), given that a is such that p(h) is an autocorrelation function. 9. rrt + e, - for f = 3,4, . . , where e, NI(0,l) and -M(O, 1) denotes normally and independently distributed with zero mean and unit variance. Compute the partial autocorrelation function of U, for h = 1,2,3,4. 10. 4) is zero, then i s also zero.

7e,- ,. z adjusted for X,-,is not zero. The correlation between XI and Xt-l is p ( l ) = ( l + Cy2)-la,which is the same as that between X,-l and Xg-2. Therefore, the partial correlation between X, and X,-2 adjusted for X I - , is - COV{X, - cxI-I,xf-z CX,-J var(x,- cx,- 11 -- - a' 1+ a z + Cy4 * where c = (1 f a2)-'(u. 1. 4. 2. 5. 15, = 0, = h=1, h=2, hL3. Once again the nature of the correlation can be observed from the plot. 2. 7e,- ,. observations are often on opposite sides of the mean, indicating that the first order autocorrelation is negative.

2. 22,. )} =Value of a randomly chosen observation from a n o d ] t odd distribution with mean $ and variance = 1 if toss of a true coin results in a head t even = 0 if toss of a true coin results in a tail + I (b) {X,:t E (0,tl,-t-2,. )} is a time series of independent identicidly distributed random variables whose distribution function is that of Student’s t-distribution with one degree of freedom. 1) random variables. 3. 1,2,.. )}, where the e, am independent identically distributed (0,l) random variables and a , , a, are fixed real numbers?