Derive the vector error correction model

Question

Question 1:

a. Suppose  that E(u_t|u_t-1,u_t-2,...) = 0, that var (u_t|u_t-1,u_t-2,...) follows the ARCH(1) model σ²t = α_o + α₁u²_t-1,and  that  the process for u_t, is stationary. Show  that var(u_t) = α_o(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 Y_t = θX_t v_1t and Xt = X_t-1 + v_2t where v_1t and v_2t are mean zero serially  uncorrelated  random  variables with E(v_1tv_2j) = 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 Y_t:ln(GDP_t), R_t denote  the 3-month  Tieasury bill rate, and π_t^CPI and π_t^PCE 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 ΔY_t and ΔR_t.

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 ΔY_t+2/t = (ΔY_t + ΔY_t-1 + ΔY_{t-2 +} ΔY_t-3)/4.

d. Which model has  the smallest  root mean  souared  forecast error?

Attachment:- econometrics.zip