In the VAR analysis at here, we have
discussed how we analysis and run the VAR model. The most popular method using
the VAR is the causality test which is also been discussed at here and here.
In this section, we discussed another method
which is also using the VAR and it is called Impulse response function (IRFs).
An IRFs show the adjustment or
time path of the variables explained in a VAR model, when one of the variables
in the model is “shocked”.
We get a “picture” of how the variable in question respond to the shock
over several period of time.
An IRFs is essentially type of conditional forecast. It’s a messy
function of the estimated coefficients in the VAR model, and the data. So, it’s
a really just a point estimate, period by period.
Just as an AR has a moving average representation, a VAR can be written
as a vector moving average (VMA) representation in that the variables (lets
say, \({{y}_{t}}\) and \({{z}_{t}}\)
are expressed in term of current and past values of the two types of shocks
(lets say, \({{e}_{1t}}\)
and \({{e}_{2t}}\) .
The
VMA representation is an essential feature of Sim’s (1980) methodology in that
allows us to trace out the time path of the various shocks on the variables
contained in the VAR systems.
Consider
the two equations in matrix form;
or;
(refer Ender (1995, p 297)
Eq(2)
express \({{y}_{t}}\)
and \({{z}_{t}}\) in term of the \(\left\{ {{e}_{1t}} \right\}\)
and \(\left\{ {{e}_{2t}}
\right\}\) sequences. However, it is insightful to rewrite Eq(2) in term
of \(\left\{ {{\varepsilon
}_{1t}} \right\}\) and \(\left\{ {{\varepsilon }_{2t}}
\right\}\) sequences. The vector
error can be written as;
So that Eq(2) and (3) can be combined to form
Since the notation is getting unwieldy, we can simplify by defining 2 X
2 matrix \({{\phi }_{\iota
}}\) with the elements \({{\phi }_{jk}}\left( i \right)\):
Hence, the moving average representation of Eq(4) and Eq(5) can be
written in terms of the \(\left\{ {{\varepsilon }_{1t}} \right\}\) and \(\left\{ {{\varepsilon }_{2t}} \right\}\) sequences.
or
more compactly
\({{x}_{t}}=\mu
+\sum\nolimits_{i=0}^{\infty }{{{\phi }_{i}}{{\varepsilon }_{t-i}}}\) (7)
The moving average representation is an especially useful tool to
examine between the \(\left\{
{{y}_{t}} \right\}\) and \(\left\{ {{z}_{t}} \right\}\) sequences. The
coefficients of \({{\phi
}_{\iota }}\) can
be used to generate the effects of \({{\varepsilon }_{yt}}\) and \({{\varepsilon }_{zt}}\) shocks on the entire time path of the \(\left\{ {{y}_{t}} \right\}\)
and
\(\left\{ {{z}_{t}}
\right\}\) sequences.
It
should be clear that the four elements \({{\phi }_{jk}}\left( 0 \right)\) are impact multiplier. For
example, the coefficients \({{\phi
}_{12}}\left( 0 \right)\) is the
instantaneous impact of a one unit change in \({{\varepsilon }_{zt}}\) on \({{y}_{t}}\). In the same way, the elements \({{\phi }_{11}}\left( 1 \right)\) and \({{\phi }_{12}}\left( 1 \right)\) are the one
period responses of unit changes in \({{\varepsilon }_{yt-1}}\) and \({{\varepsilon }_{zt-1}}\) on \({{y}_{t}}\)
respectively.
Updating
one period indicate that \({{\phi
}_{11}}\left( 1 \right)\) and \({{\phi }_{12}}\left( 1 \right)\)
also represent the effects of unit changes in \({{\varepsilon }_{yt}}\) and \({{\varepsilon }_{zt}}\)
on \({{y}_{t+1}}\) .The
accumulated effects of unit impulses in \({{\varepsilon }_{yt}}\) and/or \({{\varepsilon }_{zt}}\)
can be obtained by the appropriate summation of the coefficients of the IRFs. For
example, after \(n\)
periods,the cumulated sun of effects of \({{\varepsilon }_{zt}}\) on the \({{y}_{t+1}}\) is \({{\phi }_{12}}\left( n \right)\).
Thus,
after
periods, the cumulated sun of the effects of \({{\varepsilon }_{zt}}\)
on the \(\left\{ {{y}_{t}}
\right\}\) sequence is;
\(\sum\nolimits_{i=0}^{n}{{{\phi
}_{12}}\left( n \right)}\) (8)
Letting \(n\)
approach infinity yields the long-run
multiplier.
Since the \(\left\{ {{y}_{t}} \right\}\) and \(\left\{ {{z}_{t}} \right\}\)
sequences are assumed to be stationary, it must be the case that for all \(j\) and \(k\) ,
\(\sum\nolimits_{i=0}^{\infty
}{\phi _{jk}^{2}\left( i \right)}\) is finite. (9)
The four sets of coefficients \({{\phi }_{11}}\left( i \right)\),
\({{\phi }_{12}}\left( i
\right)\), \({{\phi
}_{21}}\left( i \right)\) and \({{\phi }_{22}}\left( i \right)\) are called the impulse response functions. Plotting
the impulse response function (i.e.
plotting the coefficients of \({{\phi
}_{jk}}\left( i \right)\) against
\(\left( i \right)\)
is a practical way to visually represent
the behavior of the \(\left\{
{{y}_{t}} \right\}\) and \(\left\{ {{z}_{t}} \right\}\) series in response to the various
shock.
With
such knowledge, it would be possible to trace out the time path of the effects
of pure \({{\varepsilon }_{yt}}\) and \({{\varepsilon }_{zt}}\) shock.
In Stata, IRFs can be produced after using the varbasic command. The results can be presented in a table
or a graph.
Estimating
Using Stata
To create the IRFs, we use the command irf
create and it will estimates the five type of
IRFs; simple IRFs, orthogonalized IRFs,
cumulative IRFs, cumulative orthogonalized IRFs and structural IRFs.
Before we
analyze the IRFs, we need the fit model, then use irf
create
to estimate the IRFs and store them in file, and
finally use irf graph or any other irf analysis commands to examine results.
We use again the data file Data09.dta.
The VAR variable we want to estimate is log for gross fixed capital
formation (lrgrossinv) , log for
real household consumption expenditure (lrconsump)
and log for real gross domestic products (lrgdp).
Before we run the IRFs, we need to know the optimal lag will be chosen
for our VAR model;
varsoc lrgrossinv lrconsump lrgdp, max (12)
The analysis in VAR show that based on AIC, the model is fit with VAR(7).
Estimate again the VAR model;
quietly
var lrgrossinv lrconsump lrgdp,lags(1/7)dfk >small
Then, to create the IRFs ;
irf create order1, step(10) set(myirf1) replace
Mulitiple sets of IRFs can be placed in the same file, with each set of
results in file bearing a distinct name. The
irf create command above
created file myirf1.irf and put one set of results in it, named order1.
The order1 results include estimates of the simple
IRFs, orthogonalized IRFs, cumulative IRFs, cumulative orthogonalized IRFs and
Cholesky FEVDs.
Now, we will create the graph for IRFs using the command irf graph.
To create the IRFs graph with standard statistics IRFs;
irf graph oirf, impulse(lrgrossinv lrconsump lrgdp) response(lrgrossinv
lrconsump lrgdp) yline (0,lcolor(black))
xlabel(0(4)20) byopts(yrescale)
The IRFs
graph places one impulse in each row and one response variable in each column.
The horizontal axis for each graph is in the unit of time that our VAR is
estimated in, in this case is quarters. Hence, IRFs graph shows the effect of a
shocks over a 10-quarter period. The vertical axis is in unit of the variables
in the VAR; in this case, everything is measured on percentage points, so the
vertical units in all panels are percentage point changes.
The first
row shows the effect of a one-standard-deviation impulse to the consumption
equation. The household consumption decrease until 5 period and then elevated. The
gross domestic product rises for about 4 periods, peaking about nearly 0.01
percentage point increases , before declining slowly. The pattern for gross
fixed capital also share the same pattern as gross domestic product, rises for
about 4 period with peaking at 0.01 and
then decreases slowly after the impulse to consumption.
The
second row shows the impact of the shock to the gross domestic equation. The
gross domestic product is decrease slightly until at 5 period and then recover
to increase slowly. Both the household
consumption and gross fixed capital decrease until 5 period and then elevated
after the impulse to gross domestic shocks.
Finally, the
third row shows the impact to a shock to the gross fixed capital equation. An
impulse to the gross fixed capital causes highly persistent decrease for the
both household consumption and the gross domestic product .
