Question 1:
a. Suppose that E(ut|ut-1,ut-2,...) = 0, that var (ut|ut-1,ut-2,...) follows the ARCH(1) model σ2t = αo + α1u2t-1,and that the process for ut, is stationary. Show that var(ut) = αo(1 - α1).
b. Extend the result in (a) to the ARCH(p) model.
c. Show that i=1∑p α1< 1 for a stationary ARCH(p) model.
d. Extend the result in (a) to the GARCH(1,1) model.
e. Show that α1+ Φ1 < 1 for a stationary GARCH(1,1) model.
Question 2:
Consider the cointegrated model Yt = θXt v1t and Xt = Xt-1 + v2t where v1t and v2t are mean zero serially uncorrelated random variables with E(v1tv2j) = 0 for all t and j. Derive the vector error correction model for X and Y.
Question 3:
These exercises are based on data series in the data files USMacro_Quarterly and USMacro_Monthly described in the Empirical Exercises. Let Yt:ln(GDPt), Rt denote the 3-month Tieasury bill rate, and πtCPI and πtPCE denote the inflation rates from the CPI and Personal Consumption Expenditures (PCE) Deflator, respectively.
Question 4:
Using quarterly data from 1955:1 through 2009:4,estimate aVAR(a) (aVAR with four lags) for ΔYt and ΔRt.
a. Does ΔR Granger-cause ΔY? Does ΔY Granger-cause ΔR?
b. Should the VAR include more than four lags?
Question 5:
In this exercise you will compute pseudo out-of-sample two-quarter-ahead forecasts for ΔY beginning in 1989:4 through the end of the sample. (That is, you will compute ΔY1990:2/1989:4, Δ1990:3/1990:1, and so forth.)
a. Construct iterated two-quarter-ahead pseudo out-of-sample forecasts using an AR(1) model.
b. Construct iterated two-quarter-ahead pseudo out-of-sample forecasts using aVAR(4) model for ΔY and ΔR.
c. Construct iterated two-quarter-ahead pseudo out-of-sample forecasts using the naive forecast ΔYt+2/t = (ΔYt + ΔYt-1 + ΔYt-2 + ΔYt-3)/4.
d. Which model has the smallest root mean souared forecast error?
Attachment:- econometrics.zip