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• Under general conditions the test is valid also asymptotically
50 Impulse Responses
•
• • •
• VAR models are often difficult to interpret: one solution is to construct
the impulse responses and variance decompositions.
Impulse responses trace out the responsiveness of the dependent variables
in the VAR to shocks to the error term. A unit shock is applied to each
variable and its effects are noted.
Consider for example a simple bivariate VAR(1): A change in u1t will immediately change y1. It will change change y2 and
also y1 during the next period.
We can examine how long and to what degree a shock to a given equation
has on all of the variables in the system.
51 Impulseresponse function: response of yi ,t + s to onetime impulse in
y jt with all other variables dated t or earlier held constant. ∂yi ,t + s
∂u jt ψ ij 1 2 3 = ψ ij s
52 ⎡σ 12 σ 12 ⎤
⎡ y1t ⎤ ⎡φ11 φ12 ⎤ ⎡ y1t −1 ⎤ ⎡u1t ⎤
⎥
⎢ ⎥=⎢
⎥ ⎢ y ⎥ + ⎢ ⎥; Σ a = ⎢
2
⎢σ 12 σ 2 ⎥
⎣ y2t ⎦ ⎣φ 21 φ 22 ⎦ ⎣ 2t −1 ⎦ ⎣u 2t ⎦
⎣
⎦
t<0 y1t = y 2t = 0 t = 0 u 20 = 1 ( y2t increases by 1 unit)
(no more shocks occur) Reaction of the system ⎡ y10 ⎤ = ⎡ 0 ⎤
⎢y ⎥
⎣ 20 ⎦
⎡ y11 ⎤
⎢y ⎥ =
⎣ 21 ⎦ ⎢1 ⎥
⎣⎦ (impulse) ⎡φ11
⎢φ
⎣ 21 φ12 ⎤ ⎡ 0 ⎤ ⎡φ12 ⎤
=
φ 22 ⎥ ⎢1 ⎥ ⎢φ 22 ⎥
⎦⎣ ⎦ ⎣ ⎦ ⎡ y12 ⎤ ⎡φ11
⎢ y ⎥ = ⎢φ
⎣ 22 ⎦ ⎣ 21
M φ12 ⎤ ⎡ y11 ⎤ ⎡φ11 φ12 ⎤ ⎡ 0 ⎤
=
φ 22 ⎥ ⎢ y 21 ⎥ ⎢φ 21 φ 22 ⎥ ⎢1 ⎥
⎦⎣ ⎦ ⎣
⎦⎣⎦ ⎡ y1 s ⎤ ⎡φ11
⎢ y ⎥ = ⎢φ
⎣ 2 s ⎦ ⎣ 21 φ12 ⎤
φ 22 ⎥
⎦ 2 s ⎡0⎤
s ⎡0 ⎤
⎢1 ⎥ = Φ 1 ⎢1 ⎥
⎣⎦
⎣⎦
53 Variance Decompositions
• Variance decompositions offer a slightly different method of
examining VAR dynamics. They give the proportion of the
movements in the dependent variables that are due to their
“own” shocks, versus shocks to the other variables.
• This is done by determining how much of the sstep ahead
forecast error variance for each variable is explained
innovations to each explanatory variable (s = 1,2,…).
• The variance decomposition gives information about the
relative importance of each shock to the variables in the VAR.
54 Impulse Responses and Variance Decompositions:
The Ordering of the Variables • But for calculating impulse responses and variance
decompositions, the ordering of the variables is important.
• The main reason for this is that above, we assumed that the VAR
error terms were statistically independent of one another.
• This is generally not true, however. The error terms will typically
be correlated to some degree.
• Therefore, the notion of examining the effect of the innovations
separately has little meaning, since they have a common
component.
• What is done is to “orthogonalize” the innovations.
• In the bivariate VAR, this problem would be approached by
attributing all of the effect of the common component to the first
of the two variables in the VAR.
• In the general case where there are more variables, the situation is
more complex but the interpretation is the same.
55...
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This note was uploaded on 09/20/2013 for the course FINA 5170 taught by Professor Janebargers during the Summer '13 term at Greenwich School of Management.
 Summer '13
 JaneBargers
 Financial Markets

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