# VAR model interpretation: Coef vs Impulse response functions

In courses such as time series analysis, we learned that the relationships derived from impulse response functions or Granger causalties are more interesting than the estimation output.

I was wondering why and whether some academic literature is available.

• Is it because the estimation output is based upon a reduced form VAR, whereas the IRF are determined on SVAR (+ assumptions e.g. choleski ) – Olivier Thierie May 14 '16 at 3:02
• Are impulse responses really based on SVAR? I think I have seen plenty examples of impulse-response analysis without any SVAR, just the VAR. – Richard Hardy May 21 '16 at 16:16
• As far as I know, the structural VAR can only be estimated by solving the identification problem (e.g. cholesky decomposition). By doing this, we can set the order of which error term affect each other (e.g a cholesky decomposition: media coverage, consumer sentiment , economic activity, indicates that the error term of media coverage is not affected by any other error term in period t, consumer sentiment only by the error term in period t of media coverage and eco. Act. By both. In this way we can study the effect of an independent shck of media, which is not the case in a reduced var – Olivier Thierie May 23 '16 at 7:24
• Thank you. The answer contrasts structural impulse responses (for SVAR) to orthogonal impulse responses (for VAR), I guess that is what I meant. – Richard Hardy May 23 '16 at 8:13

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)
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: