Vector Autoregressions and
Impulse Response Functions
The VAR
Introduction
• Assess the selection of the optimal lag
length in a VAR
• Evaluate the use of impulse response
functions with a VAR
• Assess the importance of variations on the
standard VAR
• Critically appraise the use of VAR s with
financial models.
• Assess the uses of VECMs
Lag Length in VAR
• When estimating VARs or conducting ‘Granger causality’ tests,
the test can be sensitive to the lag length of the VAR
• Sometimes the lag length corresponds to the data, such that
quarterly data has 4 lags, monthly data has 12 lags etc.
• A more rigorous way to determine the optimal lag length is to use
the Akaike or Schwarz-Bayesian information criteria.
• However the estimations tend to be sensitive to the presence of
autocorrelation, in this case following the use of information
criteria, if there is any evidence of autocorrelation, further lags
are added, above the number indicated by the information
criteria, until the autocorrelation is removed.
Information Criteria
• The main information criteria are the Schwarz-Bayesian
criteria and the Akaike criteria.
• They operate on the basis that there are two competing
factors from adding more lags to a model. More lags will
reduce the RSS, but also means a loss of degrees of
freedom (penalty from adding more lags).
• The aim is the minimise the information criteria, by
adding an extra lag, it will only benefit the model if the
reduction in the RSS outweighs the loss of degrees of
freedom.
• In general the Schwarz-Bayesian (SBIC) has a harsher
penalty term than the Akaike (AIC), which leads it to
indicate a parsimonious model is best.
The AIC and SBIC
• The two can be expressed as:
2k
AIC  ln( ˆ ) 
T
k
2
SBIC  ln( ˆ )  ln T
T
Where :
2
ˆ 2  residual variance
T - sample size, k - No. of parameters
Multivariate Information Criteria
• The multivariate version of the Akaike information
criteria is similar to the univariate:
MAIC  log ˆ  2k  / T ( Akaike)
ˆ  Variance  Co var iance matrix of the residuals. (This gives
the variances on the main diagonal and covariance s between
the residuals off the main diagonal of the matrix)
T  number of observatio ns
k   total number of regressors in all equations
Multivariate SBIC
• The multivariate version of the SBIC is:
k
ˆ
MSBIC  log   log( T )
T
ˆ  Variance  Co var iance matrix of the residuals
T  number of observatio ns
k   total number of regressors in all equations
The best criterion
• In general there is no agreement on which
criteria is best (Diebold for instance
recommends the SBIC).
• The Schwarz-Bayesian is strongly
consistent but not efficient.
• The Akaike is not consistent, generally
producing too large a model, but is more
efficient than the Schwarz-Bayesian
criteria.
VAR Models
• If we assume a 2 variable model, with a single
lag, we can write this VAR model as:
y1t  10  11 y1t 1  11 y2t 1  u1t
y2t   20   21 y2t 1   21 y1t 1  u2t
which can be rewritten as :
 y1t   10   11 11  y1t 1   u1t 

  
  
   

 y2t    20    21  21  y2t 1   u2t 
yt   0  1 yt 1  ut
g *1 g *1 g * gg *1 g *1(for a system of g variables )
Criticisms of Causality Tests
Granger causality test, much used in VAR
modelling, however do not explain some
aspects of the VAR:
• It does not give the sign of the effect, we
do not know if it is positive or negative
• It does not show how long the effect lasts
for.
• It does not provide evidence of whether
this effect is direct or indirect.
Impulse Response Functions
• These trace out the effect on the dependent variables in
the VAR to shocks to all the variables in the VAR
• Therefore in a system of 2 variables, there are 4 impulse
response functions and with 3 there are 9.
• The shock occurs through the error term and affects the
dependent variable over time.
• In effect the VAR is expressed as a vector moving
average model (VMA), as in the univariate case
previously, the shocks to the error terms can then be
traced with regard to their impact on the dependent
variable.
• If the time path of the impulse response function
becomes 0 over time, the system of equations is stable,
however they can explode if unstable.
Impulse Response Functions
• Given:
yt  A1 yt 1  ut
1 2 
Where : A1  

 1 2 
Given a unit shock to y1t at time t  0
u10  1 
y0  
 

u20  0
An Impulse Response Function
Shock
1.2
1
y
0.8
0.6
Shock
0.4
0.2
0
0
1
2
3
4
5
Time
6
7
8
9
10
VARs and SUR
• In general the VAR has all the lag lengths
of the individual equations the same size.
• It is possible however to have different lag
lengths for different equations, however
this involves another estimation method.
• When lag lengths differ, the seemingly
unrelated regression (SUR) approach can
be used to estimate the equations, this is
often termed a ‘near-VAR’.
Alternative VARs
• It is possible to include contemporaneous terms in a
VAR, however in this case the VAR is not identified.
• It is also possible to include exogenous variables in
the VAR, although they do not have separate
equations where they act as a dependent variable.
They simply act as extra explanatory variables for all
the equations in the VAR.
• It is worth noting that the impulse response functions
can also produce confidence intervals to determine
whether they are significant, this is routinely done by
most computer programmes.
VECMs
• Vector Error Correction Models (VECM) are the
basic VAR, with an error correction term
incorporated into the model.
• The reason for the error correction term is the
same as with the standard error correction
model, it measures any movement away from
the long-run equilibrium.
• These are often used as part of a multivariate
test for cointegration, such as the Johansen ML
test.
VECMs
• However there are a number of differing
approaches to modelling VECMs, for
instance how many lags should there be
on the error correction term, usually just
one regardless of the order of the VAR
• The error correction term becomes more
difficult to interpret, as it is not obvious
which variable it affects following a shock
VECM
• The most basic VECM is the following first-order
VECM:
y1t   11 
u1t 
y     y1t 1  y2t 1   u 
 2t   21 
 2t 
Where  11 and  22 are the error correction
terms.
Criticisms of the VAR
• Many argue that the VAR approach is lacking in theory.
• There is much debate on how the lag lengths should be
determined
• It is possible to end up with a model including numerous
explanatory variables, with different signs, which has
implications for degrees of freedom.
• Many of the parameters will be insignificant, this affects
the efficiency of a regression.
• There is always a potential for multicollinearity with many
lags of the same variable
Stationarity and VARs
• Should a VAR include only stationary
variables, to be valid?
• Sims argues that even if the variables are
not stationary, they should not be firstdifferenced.
• However others argue that a better
approach is a multivariate test for
cointegration and then use first-differenced
variables and the error correction term
VAR Example
• The basic theory behind this following model is
simply that we believe there is a relationship
between short-term (TBILL) and long-term (R10)
interest rates. We believe the VAR approach is
the best model in this case.
• Following the Akaike and Schwarz-Bayesian
criteria, we find a VAR of order 1 is most
appropriate, this produces the following results
(edited) :
VAR Result
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OLS estimation of a single equation in the Unrestricted VAR
******************************************************************************
Dependent variable is TBILL
127 observations used for estimation from 1960Q2 to 1991Q4
Regressor
Coefficient
Standard Error
T-Ratio[Prob]
TBILL(-1)
.96200
.067845
14.1795[.000]
R10(-1)
-.015333
.068439
-.22404[.823]
K
.36563
.23386
1.5635[.120]
R-Squared
.90159 R-Bar-Squared
.90000
Akaike Info. Criterion -165.9593 Schwarz Bayesian Criterion -170.22
Serial Correlation*CHSQ( 4)= 22.3179[.000]
Dependent variable is R10
******************************************************************************
Regressor
Coefficient
Standard Error
T-Ratio[Prob]
TBILL(-1)
.11106
.039920
2.7821[.006]
R10(-1)
.87432
.040269
21.7117[.000]
K
.26981
.13760
1.9608[.052]
R-Squared
.96507 R-Bar-Squared
.96451
Akaike Info. Criterion
-98.6049 Schwarz Bayesian Criterion -102.8712
Serial Correlation*CHSQ( 4)= 8.6481[.071]
Granger Causality Test
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******************************************************************************
Dependent variable is R10
List of the variables deleted from the regression: TBILL (-1)
127 observations used for estimation from 1960Q2 to 1991Q4
******************************************************************************
Regressor
Coefficient
Standard Error
T-Ratio[Prob]
R10(-1)
.97627
.017142
56.9508[.000]
K
.20365
.13914
1.4637[.146]
******************************************************************************
Joint test of zero restrictions on the coefficients of deleted variables:
F Statistic
F( 1, 124)= 7.7400[.006]
Dependent variable is TBILL
List of the variables deleted from the regression: R10(-1)
Regressor
Coefficient
Standard Error
T-Ratio[Prob]
TBILL(-1)
.94817
.028025
33.8328[.000]
K
.33727
.19589
1.7217[.088]
******************************************************************************
Joint test of zero restrictions on the coefficients of deleted variables:
F Statistic
F( 1, 124)= .050192[.823]
*****************************************************************************
Impulse Response Function for
TBILL
1.2
1
TBILL
0.8
TBILL
0.6
R10
0.4
0.2
0
0
1
2
3
4
5
6
Lags
7
8
9
10
11
12
Summary of Results
• When TBILL is the dependent variable, only
TBILL(-1) is significant, but the model has
serious autocorrelation
• When R10 is the dependent variable, both
variables are significant.
• TBILL ‘Granger causes’ R10, but not vice versa
• Impulse Response Functions show most of the
effect from a unit shock comes through the
lagged dependent variable, but the shock falls
away to zero fairly quickly.
Granger Causality Tests Continued
• According to Granger, causality can be further subdivided into long-run and short-run causality.
• This requires the use of error correction models or
VECMs, depending on the approach for determining
causality.
• Long-run causality is determined by the error correction
term, whereby if it is significant, then it indicates
evidence of long run causality from the explanatory
variable to the dependent variable.
• Short-run causality is determined as before, with a test
on the joint significance of the lagged explanatory
variables, using an F-test or Wald test.
Long-run Causality
• Before the ECM can be formed, there first has to be
evidence of cointegration, given that cointegration
implies a significant error correction term, cointegration
can be viewed as an indirect test of long-run causality.
• It is possible to have evidence of long-run causality, but
not short-run causality and vice versa.
• In multivariate causality tests, the testing of long-run
causality between two variables is more problematic, as
it is impossible to tell which explanatory variable is
causing the causality through the error correction term.
Causality Example
• The following basic ECM was produced,
following evidence of cointegration between s
and y:
yt  0.3  0.1yt 1  0.9st 1  0.7( y  s)t 1
(0.1) (0.3)
(0.9)
DW  1.87, R 2  0.15
(0.1)
Causality Example
• In the previous example, there is long-run
causality between s to y, as the error
correction term is significant (t-ratio of 7).
• There is no evidence of short-run causality
as the lagged differenced explanatory
variable is insignificant (t-ratio of 1, an Ftest would also include insignificance).
Conclusion
• VARs have a number of important uses,
particularly causality tests and forecasting
• To assess the affects of any shock to the
system, we need to use impulse response
functions and variance decomposition
• VECMs are an alternative, as they allow firstdifferenced variables and an error correction
term.
• The VAR has a number of weaknesses, most
importantly its lack of theoretical foundations