Find a solution of this equation in terms of the infinite

Question

Homework 2- Econometrics III - Spring 2007

Throughout, Z = {. . . , 1, 0, 1, . . .}, N = {1, 2, . . .} ⊂ Z.

A. Let X_t and Y_t be two white noise processes. Is their sum, Z_t = X_t + Y_t, also white noise?

B. Define a random walk as Y₀ = 0 and Y_t = Y_t-1 + ε_t, t ∈ N, where is ε_t is white noise with variance s. Show that Y_t, t ∈ N, is not covariance stationary.

C. Let w_t, t ∈ Z, be a bounded sequence and consider the following difference equation in y_t:

y_t = 0.9y_t-1 - 0.2y_t-2 + w_t, 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/∂w_t-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 y_t = ?y_t-1 + ε_t, t ∈ Z.

1. Show that if ε_t is iid, then E(Y_t|Y_t-1) = ?Y_t-1.

2. Show that if ε_t is a martingale difference sequence, then E(Y_t|Y_t-1) = ?Y_t-1.

3. Show that E(Y_t|Y_t-1) = ?Y_t-1 does not hold if ε_t = z_t/2 if t is even and ε_t = (z_(t-1)/2)² - 1 if t is odd where z_t is a Gaussian white noise process.

E. Let X, Y be random variables with finite variance.

1. Solve min_α,β E (Y - (α + βX))² for (α^∗, β^∗).

2. Give an example where E (Y|X) ≠ α^∗ + β^∗X.