Impulse responses and variance
decompositions
Impulse response functions show the effects of shocks on the adjustment path of the variables. Forecast
error variance decompositions measure the contribution of each type of shock to the forecast error
variance. Both computations are useful in assessing how shocks to economic variables reverberate
through a system.
Impulse response functions (IRFs) and forecast error variance decompositions (FEVD) can be
produced after using the varbasic command. The results can be presented in a table or a graph. In this
example we illustrate both. After the varbasic command, we use the irf table command to generate
IRFs and FEVDs:
use fred, clear
varbasic D.lc D.ly, lags(1/1) step(12) nograph
valmar
Sample: 1960q3 Log likelihood =
FPE
=
Det(Sigma_ml) =
Equation
2009q4
1400.444
2.62e-09
2.46e-09
Parms
D_lc
D_ly
3
3
Coef.
No. of obs
AIC
HQIC
SBIC
RMSE
.006575
.008562
R-sq
chi2
P>chi2
0.1205
0.1118
27.12459
24.92656
0.0000
0.0000
=
198
= -14.0853
= -14.04496
= -13.98565
Std. Err.
z
P>|z|
[95% Conf. Interval]
D_lc
lc
LD.
.2156068
.0741801
2.91
0.004
.0702164
.3609972
ly
LD.
.1493798
.0572953
2.61
0.009
.0370832
.2616765
_cons
.0052776
.0007516
7.02
0.000
.0038046
.0067507
lc
LD.
.4754276
.0965863
4.92
0.000
.286122
.6647332
ly
LD.
-.2171679
.0746013
-2.91
0.004
-.3633839
-.070952
_cons
.0060367
.0009786
6.17
0.000
.0041187
.0079547
D_ly
. varlmar
Lagrange-multiplier test
lag
chi2
df
1
2
9.5086
5.6784
4
4
Prob > chi2
0.04957
0.22449
H0: no autocorrelation at lag order
Pick lag lengths using varsoc.
. varsoc D.lc D.ly, maxlag(4)
Selection-order criteria
Sample: 1961q2 - 2009q4
lag
0
1
2
3
4
LL
LR
1355.02
1379.09
1383.92
1388.24
1391.6
Endogenous:
Exogenous:
48.129
9.6655*
8.6379
6.7149
Number of obs
df
p
4
4
4
4
0.000
0.046
0.071
0.152
FPE
3.2e-09
2.6e-09
2.6e-09
2.6e-09*
2.6e-09
AIC
-13.8772
-14.083
-14.0915
-14.0948*
-14.0882
=
HQIC
-13.8636
-14.0422*
-14.0235
-13.9996
-13.9659
195
SBIC
-13.8436
-13.9823*
-13.9237
-13.8598
-13.7861
D.lc D.ly
_cons
Only the FPE picks a lag longer than one. Now run the VAR again with a single lag and limit the step
forecasts to 12 (that is plenty for these data).
qui varbasic D.lc D.ly, lags(1/1) step(12)
irf table irf
irf table fevd
The syntax for irf table is:
irf table [stat] [, options]
where stat can be any of the following:
irf
oirf
dm
cirf
coirf
cdm
fevd
sirf
sfevd
impulse-response function
orthogonalized impulse-response function
dynamic-multiplier function
cumulative impulse-response function
cumulative orthogonalized impulse-response function
cumulative dynamic-multiplier function
Cholesky forecast-error variance decomposition
structural impulse-response function
structural forecast-error variance decomposition
The results from the IRF and FEVD tables are:
. irf table irf
Results from varbasic
(1)
irf
step
0
1
2
3
4
5
6
7
8
9
10
11
12
1
.215607
.117506
.025224
.013808
.002951
.001623
.000345
.000191
.00004
.000022
4.7e-06
2.6e-06
(3)
irf
step
0
1
2
3
4
5
6
7
8
9
10
11
12
95%
(1)
(2)
(3)
(4)
0
.14938
-.000233
.017604
-.000055
.002075
-9.7e-06
.000244
-1.5e-06
.000029
-2.2e-07
3.4e-06
-3.2e-08
lower and
irfname =
irfname =
irfname =
irfname =
(1)
Lower
1
.070216
.042463
-.014994
-.003806
-.004713
-.001478
-.000905
-.000295
-.000148
-.000049
-.000022
-7.4e-06
(3)
Lower
0
.037083
-.027858
-.000946
-.006566
-.001028
-.001161
-.000241
-.000182
-.000043
-.000027
-6.9e-06
-3.8e-06
(1)
Upper
(2)
irf
1
.360997
.192549
.065442
.031422
.010615
.004723
.001595
.000676
.000228
.000094
.000032
.000013
0
.475428
-.000742
.056027
-.000175
.006603
-.000031
.000778
-4.9e-06
.000092
-7.2e-07
.000011
-1.0e-07
(3)
Upper
(4)
irf
0
.261676
.027392
.036153
.006456
.005177
.001141
.00073
.000179
.000101
.000026
.000014
3.7e-06
1
-.217168
.118181
-.025776
.013967
-.003059
.001651
-.000363
.000195
-.000043
.000023
-5.1e-06
2.7e-06
upper bounds reported
varbasic, impulse = D.lc,
varbasic, impulse = D.lc,
varbasic, impulse = D.ly,
varbasic, impulse = D.ly,
and
and
and
and
response
response
response
response
(2)
Lower
0
.286122
-.088665
.007234
-.020898
-.002593
-.003694
-.000719
-.000581
-.000134
-.000086
-.000022
-.000012
(4)
Lower
=
=
=
=
1
-.363384
.043031
-.066356
-.003818
-.010845
-.001506
-.001642
-.000303
-.000237
-.000051
-.000033
-7.7e-06
(2)
Upper
0
.664733
.087181
.104819
.020548
.015799
.003632
.002275
.000571
.000317
.000084
.000043
.000012
(4)
Upper
1
-.070952
.193332
.014804
.031752
.004726
.004807
.000916
.000693
.000151
.000097
.000023
.000013
D.lc
D.ly
D.lc
D.ly
The results appear in four quadrants and a key is given at the bottom of the table. In quadrant one
(northwest) is the response in ln(RCPE) to a shock in itself. You’ll remember that this series is stationary
and therefore shocks are not persistent; their effects eventually die out. Shocks to ln(RPDI) are not
persistent either. More interesting is how ln(RPDI) responds to shocks in the ln(RCPE), and vice versa.
Quadrant 2 (northeast) shows a shocks to ln(RCPE) affects ln(RPDI) for one period, but dies out very
quickly. Shocks to the ln(RRDI) create a smaller, but significant response in ln(RCPE) (quadrant 3),
though once again if falls to zero very quickly.
A separate command called irf graph basically does the same thing as irf table, except the
results appear as a graph rather than in tabular form.
irf graph
varbasic, D.lc, D.lc
varbasic, D.lc, D.ly
varbasic, D.ly, D.lc
varbasic, D.ly, D.ly
1
.5
0
-.5
1
.5
0
-.5
0
5
10
15
0
5
10
15
step
95% CI
impulse response function (irf)
Graphs by irfname, impulse variable, and response variable
This is merely the data produced by irf table in graphical form. It should also be noted that these
graphs are actually produced automatically by varbasic. Recall that we used the nograph option, which
suppresses the graphs. As this shows, the graphs in memory can be summoned later if desired.
Like the impulse responses, the FEVDs can be produced in either tabular or graphical form. The
commands are, respectively:
irf table fevd
irf graph fevd
The results are arranged in the same way, by quadrant with a key at the bottom.
. irf table fevd
Results from varbasic
step
0
1
2
3
4
5
6
7
(1)
fevd
0
1
.97297
.973298
.972967
.972972
.972967
.972967
(1)
Lower
(1)
Upper
0
1
.932729
.933619
.93271
.932723
.932709
.932709
0
1
1.01321
1.01298
1.01322
1.01322
1.01323
1.01323
(2)
fevd
0
.198552
.2438
.243198
.243752
.243743
.24375
.24375
(2)
Lower
(2)
Upper
0
.099067
.146908
.145308
.145393
.145351
.145347
.145346
0
.298037
.340692
.341087
.34211
.342135
.342153
.342154
(3)
fevd
step
0
1
2
3
4
5
6
95%
(1)
(2)
(3)
(4)
0
0
.02703
.026702
.027033
.027028
.027033
lower and
irfname =
irfname =
irfname =
irfname =
(3)
Lower
(3)
Upper
0
0
-.013212
-.012977
-.013225
-.01322
-.013225
0
0
.067271
.066381
.06729
.067277
.067291
upper bounds reported
varbasic, impulse = D.lc,
varbasic, impulse = D.lc,
varbasic, impulse = D.ly,
varbasic, impulse = D.ly,
and
and
and
and
(4)
fevd
0
.801448
.7562
.756802
.756248
.756257
.75625
response
response
response
response
=
=
=
=
(4)
Lower
(4)
Upper
0
.701963
.659308
.658913
.65789
.657865
.657847
0
.900933
.853092
.854692
.854607
.854649
.854653
D.lc
D.ly
D.lc
D.ly
And graph
varbasic, D.lc, D.lc
varbasic, D.lc, D.ly
varbasic, D.ly, D.lc
varbasic, D.ly, D.ly
1
.5
0
1
.5
0
0
5
10
15
0
5
10
15
step
95% CI
fraction of mse due to impulse
Graphs by irfname, impulse variable, and response variable
The tabulation and graphing functions in Stata for IRFs and FEVDs are really very flexible. You can
combine tables, overlay graphs, and do many other things using either the language or the dialog boxes.
Choose Statistics > Multivariate time series > IRF and FEVD analysis to reveal the final box:
Here you have a number of choices including the ones we’ve just discussed. However, you can also use
these dialogs to overlay graphs, combine graphs, or select specific portions of the tables or graphs to use
in your output. Feel free to experiment!