[SOLVED] Econometrics ii homework 7 spring 2025. p0

1.
a) Let xt = α0 + 5 ∗ ut + ut−1 ,
where ut is white noise.
Find the autocovariances for xt in terms of σu2 (the variance of ut).
b) Given the stationary AR(1) process
xt = 3 + .5 ∗ xt−1 + ut
where Eu2t = 3. Find the variance of xt, and the first 3 autocovariances and autocorrelations.
2. Given the AR(1) process:
et = aet−1 + ut ,
with a = 0.5. Let E be the vector (e1,e2,e3).
a) Assuming e0 is a fixed numbert, find the variance matrix Ω = var(e).
b) Assuming et is stationary, find the variance matrix Ω = var(e).
3. Define the lag polynomials a(L) = a0 + a1 L and b(L) = b0 + b1L + b2L2. (Notice: in the notes, and in class, it is often assumed a0 = 1 and b0 = 1. This is just for simplification and doesn’t matter for any results since you can always re-scale the data and the lag-polynomial such that the first coefficient becomes unity (write a(L) as a0 a′(L) where the lag polynomial and similarly for b(L)). The constant a0 will not affect the properties of the lag-polynomial that we care about.)
Assume a0 = 1, a1 = −2, b0 = 3, b1 = −.3, and b2 = .5 .
i) If xt = 3,xt−1 = −3,xt−2 = −2, xt−3 = 9, and xt−4 = 9, what is a(L)xt? and b(L)xt? (This should be a number.) ii) Find the roots of a(L) and b(L). iii) What is c(L) = a(L)b(L)? What are the roots of c(L)?
iv) Find the coefficients to the constant (identify), L, and L2 in the lag-polynomial b−1(L).
4. Computer question (continuation of previous homeworks). In Matlab, regress real per capita U.S.
data consumption growth on income growth and the interest rate using the posted dataset.
(This is the what you did in homework 1.)
a) Calculate the residuals e. Regress et on et−1? Is there evidence of autocorrelation (Use t-tests.)