The following SCA commands are used in the analysis of Example 8.6:
input x1,x2. file ‘m-gs1n3-5301.txt’ % Load data
--
r1=ln(x1) % Take log transformation
--
r2=ln(x2)
--
miden r1,r2. no ccm. arfits 1 to 8.
-- % Denote the model by v21.
mtsm v21. series r1,r2. model (i-p1*b-p2*b**2)series= @
c+(i-t1*b)noise.
--
mestim v21. % Initial estimation
--
p1(2,1)=0 % Set zero constraints
--
cp1(2,1)=1
--
p2(2,1)=0
--
cp2(2,1)=1
--
p2(2,2)=0
--
cp2(2,2)=1
--
t1(2,1)=0
--
ct1(2,1)=1
-- % Refine estimation and store residuals
mestim v21. method exact. hold resi(res1,res2)
--
miden res1,res2.
Exercises
8.1 Consider the monthly log stock returns, in percentages and including dividends, of Merck & Company, Johnson & Johnson, General Electric, General Motors, Ford Motor Company, and value-weighted index from January 1960 to December 2008; see the file m-mrk2vw.txt.
a. Compute the sample mean, covariance matrix, and correlation matrix of the data.
b. Test the hypothesis 
, where 
is the lag-i cross-correlation matrix of the data. Draw conclusions based on the 5% significance level.
c. Is there any lead–lag relationship among the six return series?
8.2 The Federal Reserve Bank of St. Louis publishes selected interest rates and U.S. financial data on its website: http://research.stlouisfed.org/fred2/.
Consider the monthly 1-year and 10-year Treasury constant maturity rates from April 1953 to October 2009 for 679 observations; see the file m-gs1n10.txt. The rates are in percentages.
a. Let ct = rt − rt−1 be the change series of the monthly interest rate rt. Build a bivariate autoregressive model for the two change series. Discuss the implications of the model. Transform the model into a structural form.
b. Build a bivariate moving-average model for the two change series. Discuss the implications of the model and compare it with the bivariate AR model built earlier.
8.3 Again consider the monthly 1-year and 10-year Treasury constant maturity rates from April 1953 to October 2009. Consider the log series of the data and build a VARMA model for the series. Discuss the implications of the model obtained.
8.4 Again consider the monthly 1-year and 10-year Treasury constant maturity rates from April 1953 to October 2009. Are the two interest rate series threshold cointegrated? Use the interest spread st = r10, t − r1, t as the threshold variable, where rit is the i-year Treasury constant maturity rate. If they are threshold cointegrated, build a multivariate threshold model for the two series.
8.5 The bivariate AR(4) model 
is a special seasonal model with periodicity 4, where 
is a sequence of independent and identically distributed normal random vectors with mean zero and covariance matrix 
. Such a seasonal model may be useful in studying quarterly earnings of a company. (a) Assume that 
is weakly stationary. Derive the mean vector and covariance matrix of 
. (b) Derive the necessary and sufficient condition of weak stationarity for 
. (c) Show that 
for ℓ > 0, where 
is the lag-ℓ autocovariance matrix of 
.
8.6 The bivariate MA(4) model 
is another seasonal model with periodicity 4, where 
is a sequence of independent and identically distributed normal random vectors with mean zero and covariance matrix 
. Derive the covariance matrices 
of 
for ℓ = 0, … , 5.
8.7 Consider the monthly U.S. 1-year and 3-year Treasury constant maturity rates from April 1953 to March 2004. The data can be obtained from the Federal Reserve Bank of St. Louis or from the file m-gs1n3-5304.txt (1-year, 3-year, dates). See also Example 8.6, which uses a shorter data span. Here we use the interest rates directly without the log transformation and define 
= (x1t, x2t)′, where x1t is the 1-year maturity rate and x2t is the 3-year maturity rate.
a. Identify a VAR model for the bivariate interest rate series. Write down the fitted model.
b. Compute the impulse response functions of the fitted VAR model. It suffices to use the first 6 lags.
c. Use the fitted VAR model to produce 1-step- to 12-step-ahead forecasts of the interest rates, assuming that the forecast origin is March 2004.
d. Are the two interest rate series cointegrated, when a restricted constant term is used? Use 5% significance level to perform the test.
e. If the series are cointegrated, build an ECM for the series. Write down the fitted model.
f. Use the fitted ECM to produce 1-step- to 12-step-ahead forecasts of the interest rates, assuming that the forecast origin is March 2004.
g. Compare the forecasts produced by the VAR model and the ECM.
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