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Suppose that {X1, X2,..., Xn} is a finite sample from a stationary process with autocovariance function

INSTRUCTIONS TO CANDIDATES
ANSWER ALL QUESTIONS

3. Suppose that {X1, X2,..., Xn} is a finite sample from a stationary process with autocovariance function {C; Cov(Xi, Xi+j), j = 0, ±1, ±2,...}, so that Co = Var(X;) = o, and autocorrelation function {pj = Cj/oz, j = 0, ±1, ±2,...}. Let Xn and S2 be the sample mean and sample variance, respectively. Show that

and then use the latter equation and the expression for the Var(X) on slide 8 to obtain the relationship

1 Var(x), where a2 = 1+2 (1-4) pj.

Hint: Start with the expression for S2, write each (X; - Xn)2 = [(X¡ − μ) − (Xn – μ)]2, expand the squares inside the summation, and then take expectations...

4. Suppose that the ɛ; are i.i.d. random variables from the normal distribution with mean 0 and standard deviation 4. Define the process {X;} by

where a is a constant.

(a) Find the marginal mean and variance of X¿. (b) Compute the Cov(Xi, Xi+j) for all i and j

= ±1, ±2,.... Argue that the process {X;} is

weakly stationary by showing that the Cov(Xi, Xi+j) depends only on the lag j.

(c) Is the process {X;} (strictly) stationary? Give a concise answer.

(d) Find the autocorrelation function p; = Corr(Xi, Xi+j), j = 0, ±1, ±2,....

(e) Find an expression for the Var(Xn) using parts (b) and (d).

(f) Compute the asymptotic variance o2 = limn→∞ n Var(Xn).

(g) Use Problem 3 above to compare E(S2/n) and Var(Xn).

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