Chapter 11: Problem 2
Let \(\left\\{e_{t}: t=-1,0,1, \ldots\right\\}\) be a sequence of independent, identically distributed random variables with mean zero and variance one. Define a stochastic process by $$x_{t}=e_{t}-(1 / 2) e_{t-1}+(1 / 2) e_{t-2}, t=1,2, \ldots$$ i. Find \(\mathrm{E}\left(x_{t}\right)\) and \(\operatorname{Var}\left(x_{t}\right) .\) Do either of these depend on \(t ?\) ii. Show that \(\operatorname{Corr}\left(x_{t}, x_{t+1}\right)=-1 / 2\) and \(\operatorname{Corr}\left(x_{t}, x_{t+2}\right)=1 / 3 .\) (Hint: It is easiest to use the formula in Problem \(1 .\) ) iii. What is \(\operatorname{Corr}\left(x_{t}, x_{t+h}\right)\) for \(h>2 ?\) iv. Is \(\left\\{x_{t}\right\\}\) an asymptotically uncorrelated process?
Short Answer
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Key Concepts
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