corresponding matrices
Problem 1: Consider the following two-
variable Blanchard-Quah model:
ptptpttttt ryryrbby ++++++−= −−−− 12111121111210 …………….
rtptpttttt ryryybbr ++++++−= −−−− 22211221212110 …………….
Here ty is the nominal interest rate and zr is the ex-ante real interest rate. pt is expected inflation rate
shock and rt is ex-ante real interest rate shock. Assume that te1 and te2 are the error terms of the
reduced-form versions of the above two equations. Consider the VAR(1) version of the above mentioned
model.
(a) Write down a reduced form representation of the model (with its corresponding matrices).
The relationship between te1 and te2 and pt , rt can be represented by the following two equations:
rtptt cce )0()0( 21111 +=
rtptt cce )0()0( 22122 +=
Clearly, if we can estimate the four parameters: ),0(),0(),0(),0( 22211211 cccc then we can use the above
two equations to recover the original shocks pt , rt from the estimated reduced-form residuals: te1 , te2 .
(b) Derive the residuals as functions of the expected inflation shocks pt and real interest rate
shocks rt . What do you have to do to identify the shocks from the residuals in this model?
(c) At least how many restrictions do you need in a VAR model to identify structural shocks from the
residuals? Explain how do you transform the contemporaneous coefficients matrix.
(d) Find the variance-covariance matrix of the reduced-from residuals.
We can write down the moving average representation of ty and zr as follows:
= −
= − +=
0 120 11 )()(
k krtk kptt kckcy
= −
= − +=
0 220 21 )()(
k krtk kptt kckcr
(a) Using the VMA representation of the model, show the effect of a one-unit shock in pt on ty and
zr at time t+2. Include the restrictions that are needed to identify the shocks from the residuals,
(for your answer choose only one restriction but mention all possible restrictions).
(b) How do you interpret c12(k) and c21(k) in general? What do these parameters represent for k=8?
What would the cumulative sum
=0 12 )(
k kc represent?
(c) Describe what is variance decomposition? Write down the proportions of the forecast error
variance of yt+4 (σy 2 (4) ) due to shocks in the rt and pt sequences. And how do we interpret
this result? If rt explained none of the forecast error variance of {yt} what can be said about
{yt}?
Problem 2: Consider the following two-variable VAR model.
ytptpttttt zyzyzbby ++++++−= −−−− 12111121111210 …………….
ztptpttttt zyzyybbz ++++++−= −−−− 22211221212110 …………….
Here yt and zt are normally distributed with mean zero and variances 2
y and 2
z respectively.
They are also mutually uncorrelated and serially uncorrelated.
(d) Explain why you cannot directly estimate these two equations in order to plot the impulse
responses of ty and tz due to shocks in yt , and zt .
Problem 3: Consider six variable VAR proposed in Sims (1986). The variables included in the study
are real GNP (y), real business fixed investment (i), the GNP deflator (p), the money supply measured
by M1 (m), unemployment (u) and the treasury bill rate (r). An unrestricted VAR model was
estimated with four lags of each variable and a constant term.
(e) Sims obtained the 36 impulse responses using a Choleski decomposition with the ordering y → I
→ p → m → u → r. Interpret the identifying restriction of this ordering.
Problem 4: Consider a three variable standard monetary VAR . The variables included in the study
are real GDP (dy) growth, the GDP deflator (pi) inflation rate, and the federal funds rate (r). An
unrestricted VAR model was estimated with four lags of each variable and a constant term. I am
attaching the variance decomposition analyses. Interpret the graphs.
0
20
40
60
80
100
5 10 15 20 25 30 35 40
Percent DY var iance due to DY
0
20
40
60
80
100
5 10 15 20 25 30 35 40
Percent DY variance due to PI
0
20
40
60
80
100
5 10 15 20 25 30 35 40
Percent DY var iance due to FEDFUNDS
0
20
40
60
80
100
5 10 15 20 25 30 35 40
Percent PI var iance due to DY
0
20
40
60
80
100
5 10 15 20 25 30 35 40
Percent PI var iance due to PI
0
20
40
60
80
100
5 10 15 20 25 30 35 40
Percent PI var iance due to FEDFUNDS
0
20
40
60
80
100
5 10 15 20 25 30 35 40
Percent FEDFUNDS variance due to DY
0
20
40
60
80
100
5 10 15 20 25 30 35 40
Percent FEDFUNDS variance due to PI
0
20
40
60
80
100
5 10 15 20 25 30 35 40
Percent FEDFUNDS variance due to FEDFUNDS
Variance Decomposition
