impulse responseinterpretationmacroeconomicstime seriesvector-autoregression
I have some issues while discussing and interpreting this impulse response function (the graphics analysis). What do they mean and represent economically? What can the conclusions be?
Basically initially I had the time series of government spending (first column), GDP growth rate (second column) and inflation rate (third one).
Can you please help me understanding it?
When you conduct VAR all variables should be on the same scale or same variable transformation basis (or as close as possible). It makes perfect sense that when you multiply your original variables by a 100, the IRF graph also reflects responses that are 100 times greater than in the original. The revised graph proportionally has not changed the response (visually the graphs will look identical). You are just using a different scale (i.e. 1 instead of 1% or something similar).
An IRF indicates what is the impact of an upward unanticipated one-unit change in the "impulse" variable on the "response" variable over the next several periods (typically 10).
IRFs do not have coefficients. The original regressions as you specified them have the coefficients. The IRFs has three main outputs: the expected level of the shock in a given period surrounded by a 95% Confidence Interval (a low estimate and a high estimate). And, all those also generate the IRF graphs.
Interpretability is another issue. While you are of course right that structural responses are generally of more interest, even an orthogonal impulse response generally is more useful than the estimated VAR coefficients simply because it is easier to see the dynamic response of the variables to a shock in one variable.
Here is an example from the vars package:
library(vars)
data(Canada)
var.3c <- VAR(Canada, p = 3, type = "const")
var.3c
plot(irf(var.3c, boot = FALSE))
The estimated coeffcients are
VAR Estimation Results:
=======================
Estimated coefficients for equation e:
======================================
Call:
e = e.l1 + prod.l1 + rw.l1 + U.l1 + e.l2 + prod.l2 + rw.l2 + U.l2 + e.l3 + prod.l3 + rw.l3 + U.l3 + const
e.l1 prod.l1 rw.l1 U.l1 e.l2 prod.l2 rw.l2 U.l2 e.l3 prod.l3
1.75274409 0.16961948 -0.08260010 0.09951924 -1.18385358 -0.10574096 -0.02438546 -0.05077361 0.58725218 0.01053871
rw.l3 U.l3 const
0.03823877 0.34138928 -150.68737459
Estimated coefficients for equation prod:
=========================================
Call:
prod = e.l1 + prod.l1 + rw.l1 + U.l1 + e.l2 + prod.l2 + rw.l2 + U.l2 + e.l3 + prod.l3 + rw.l3 + U.l3 + const
e.l1 prod.l1 rw.l1 U.l1 e.l2 prod.l2 rw.l2 U.l2 e.l3 prod.l3
-0.14879583 1.14798569 0.02359443 -0.65814244 -0.18164920 -0.19627478 -0.20337023 0.82236693 0.57494977 0.04414683
rw.l3 U.l3 const
0.09336521 0.40078042 -195.86984902
Estimated coefficients for equation rw:
=======================================
Call:
rw = e.l1 + prod.l1 + rw.l1 + U.l1 + e.l2 + prod.l2 + rw.l2 + U.l2 + e.l3 + prod.l3 + rw.l3 + U.l3 + const
e.l1 prod.l1 rw.l1 U.l1 e.l2 prod.l2 rw.l2 U.l2 e.l3 prod.l3
-4.715930e-01 -6.499785e-02 9.090532e-01 -7.940803e-04 6.667031e-01 -2.164497e-01 -1.456573e-01 -3.013740e-01 -1.288947e-01 2.139588e-01
rw.l3 U.l3 const
1.901601e-01 1.506129e-01 -1.166855e+01
Estimated coefficients for equation U:
======================================
Call:
U = e.l1 + prod.l1 + rw.l1 + U.l1 + e.l2 + prod.l2 + rw.l2 + U.l2 + e.l3 + prod.l3 + rw.l3 + U.l3 + const
e.l1 prod.l1 rw.l1 U.l1 e.l2 prod.l2 rw.l2 U.l2 e.l3 prod.l3
-0.61773366 -0.09778145 0.01454884 0.65976287 0.51811384 0.08798974 0.06993062 -0.08098673 -0.03005992 -0.01092231
rw.l3 U.l3 const
-0.03909215 0.06684284 114.36732138
I dare say you won't find it easy to see "what's going on".
In contrast, the plot of an impulse response function is more amenable to interpretation:
Best Answer
I would strongly encourage you to read a an introduction in VAR estimation, e.g. Lütkepohl. The visual impulse response analysis is quite simple: The columns always indicate the reaction to one shock. The first column gives the reaction to an one time expansive fiscal policy (GS-Shock). We see that GS increases up to 45 in period 0 and then decreases slowly in the following periods. As a reaction to this shock GDP and Inflation increase, the IRF is above 0 in the first periods. That is in line with the economic literature. The blue areas desrcibe the confidence intervals of your estimates. If the x axis is not included in your confidence interval you could conclude that your IRF is significantly different from 0.
Again please have a look in the literature. -> When you use the cholesky decomposition the ordering of your variables matter. Your ordering differs from previous literature. Normally one would assume that inflation and GDP react only with some delay.