Reference no: EM131119902
Homework 2- Econometrics III - Spring 2007
Throughout, Z = {. . . , 1, 0, 1, . . .}, N = {1, 2, . . .} ⊂ Z.
A. Let Xt and Yt be two white noise processes. Is their sum, Zt = Xt + Yt, also white noise?
B. Define a random walk as Y0 = 0 and Yt = Yt-1 + εt, t ∈ N, where is εt is white noise with variance s. Show that Yt, t ∈ N, is not covariance stationary.
C. Let wt, t ∈ Z, be a bounded sequence and consider the following difference equation in yt:
yt = 0.9yt-1 - 0.2yt-2 + wt, t ∈ Z.
1. Find a solution of this equation in terms of the infinite history by inverting the appropriate lag polynomial.
2. Prove that this solution is bounded.
3. Find an explicit expression for ∂y/∂wt-j.
4. Show that for any A1, A2 ∈ R, the solution you found in part 1 plus A1(0.4)t + A2(0.5)t is also a solution, and that this new solution is unbounded over t ∈ Z.
D. Consider a stationary AR(1) process yt = ?yt-1 + εt, t ∈ Z.
1. Show that if εt is iid, then E(Yt|Yt-1) = ?Yt-1.
2. Show that if εt is a martingale difference sequence, then E(Yt|Yt-1) = ?Yt-1.
3. Show that E(Yt|Yt-1) = ?Yt-1 does not hold if εt = zt/2 if t is even and εt = (z(t-1)/2)2 - 1 if t is odd where zt is a Gaussian white noise process.
E. Let X, Y be random variables with finite variance.
1. Solve minα,β E (Y - (α + βX))2 for (α∗, β∗).
2. Give an example where E (Y|X) ≠ α∗ + β∗X.
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