Derive the vector error correction model


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=1p 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?


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Microeconomics: Derive the vector error correction model
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