Vector Error Correction model restricted VAR model

1. Assoc Prof Dr Ergin Akalpler
VECM -Restricted VAR Model
Impulse Response and
Variant Decomposition used

2. VAR Model
 VECTOR auto-regressive (VAR) integrated model
comprises multiple time series and is quite a useful
tool for forecasting. It can be considered an
extension of the auto-regressive (AR part of
ARIMA) model.

3. VAR Model
 VAR model involves multiple independent variables and
therefore has more than one equations.
 Each equation uses as its explanatory variables lags of all
the variables and likely a deterministic trend.
 Time series models for VAR are usually based on applying
VAR to stationary series with first differences to original
series and because of that, there is always a possibility of
loss of information about the relationship among integrated
series.

4. VAR model
 Differencing the series to make them stationary is
one solution, but at the cost of ignoring possibly
important (“long run”) relationships between the
levels. A better solution is to test whether the levels
regressions are trustworthy (“cointegration”.)

5. VAR Model
 The usual approach is to use Johansen’s method for
testing whether or not cointegration exists. If the answer is
“yes” then a vector error correction model (VECM),
which combines levels and differences, can be estimated
instead of a VAR in levels. So, we shall check if VECM is
been able to outperform VAR for the series we have.

6. How to determine Restricted VAR –VECM- or
Unrestricted VAR
 If all variables converted to first difference then they become
stationary (integrated in same order)
 Null hypo: variables are stationary
 Alt Hypo: Variables are not stationary
 If the variables are cointegrated and have long run association
then we run restricted VAR (that is VECM),
 But if the variables are not cointegrated we cannot run VECM
rather we run unrestricted VAR.

7. What is the difference between VAR and
VECM model?
 Through VECM we can interpret long term and short term
equations. We need to determine the number of co-integrating
relationships. The advantage of VECM over VAR is that the
resulting VAR from VECM representation has more efficient
coefficient estimates.

When to use VAR/VECM?
You should use VECM if 1) your variables are nonstationary
and 2) you find a common trend between the variables
(cointegration).

8. UNRESTRICTED VAR
 After performing cointegration test results will
shows following estimations:
 Trace STATS < TCV
 Null: there is no cointegration
 Alt: There is cointegration
 When the Trace stats is less than TCV we cannot
reject null hypo there is no cointegration
 Probability values are more than 0.05
 >

9. RESTRICTED VAR -VECM
 After performing cointegration test results will
shows following estimations:
 Trace STATS > TCV
 Null: there is no cointegration
 Alt: There is cointegration
 When the Trace stats is more than TCV we can
reject null hypo there is cointegration
 Probability values are less than 0.05

10. According to Engle and Granger (1987), two I(1) series are said to be co-
integrated if there exists some linear combination of the two which
produces a stationary trend [I(0)].
Any non-stationary series that are co-integrated may diverge in the short-
run, but they must be linked together in the longrun.
Moreover, it has been proven by Engle and Granger (1987) that if a set of
series are co-integrated, there always exists a generating mechanism,
called “error-correction model”, which forces the variables to move
closely together over time, while allowing a wide range of short-run
dynamics.

11. Introduction
The basics of the vector autoregressive model.
We lay the foundation for getting started with this crucial multivariate time
series model and cover the important details including:
•What a VAR model is.
•Who uses VAR models.
•Basic types of VAR models.
•How to specify a VAR model.
•Estimation and forecasting with VAR models.

12. To determine whether VAR model in levels is possible or not, we need to transform
VAR model in levels to a VECM model in differences (with error correction terms),
to which the Johansen test for cointegration is applied.
In other words, we take the following 4 steps
1. construct a VECM model in differences (with error correction terms)
2. apply the Johansen test to the VECM model in differences to find out the
number of cointegration (r) (none or Atmost)
3. if r = 0, estimate VAR in differences
4. if r > 0, estimate VECM model in differences or VAR in levels (at least one
cointegration equation exist)

13. Its identification depends on the number of cointegration in the following
way.
(none) or 0, r = 0 (no cointegration)
In the case of no cointegration, since all variables are non-stationary in level,
the above VECM model reduces to a VAR model with growth variables.
At most 1, r = 1 (one cointegrating vector)
At most 2, r = 2 (two cointegrating vectors)
At most 3) r = 3 (full cointegration)
In the case of full cointegration, since all variables are stationary, the above
VECM model reduces to a VAR model with level variables.

14. Johansen Test for Cointegration
The rank equals the number of its non-zero eigenvalues and the Johansen test
provides inference on this number. There are two tests for the number of co-
integration relationships.
The first test is the trace test whose test statistic is
H0 : cointegrating vectors ≤ r
H1 : cointegrating vectors ≥ r + 1
The second test is the maximum eigenvalue test whose test statistic is given by
H0 : There are r cointegrating vectors
H1 : There are r + 1 cointegrating vectors

15.  RESTRICTED VAR (VECM)
 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 VARs with financial models.
 Assess the uses of VECMs

16. Lets start with the RESTRICTED VAR- VECM
what was the guideline
 After performing cointegration test results will
shows following estimations:
 Trace STATS > TCV
 Null: there is no cointegration
 Alt: There is cointegration
 When the Trace stats is more than TCV we can
reject null hypo there is cointegration
 Probability values are less than 0.05

17. How to do the Estimation Multivariate
Cointegration and VECMs
1) Test the variables for stationarity using the usual ADF tests.
2) If all the variables are I(1) include in the cointegrating
relationship.
3) Use the AIC or SIC to determine the number of lags in the
cointegration test (order of VAR)
4) Use the trace and maximal eigenvalue tests to determine the
number of cointegrating vectors present.
5) When the Trace stats is more than TCV we can reject null hypo
there is at least one cointegration eq. and our variables have
long run association in the long run they move together

18. How to do the Estimation Multivariate
Cointegration and VECMs cont.1
1) This implies we can run restricted VAR VECM because trace and
maximum eigen values are more that TCV and there is at least one
cointegration equation.
2) We reject null hypo and probability values are also less than 0.05
3) (In opposite case we run unrestricted VAR)
4) We perform and estimate the table for vector error correction
model and then find the equations for our model.
5) From equations we derive the residuals for cointegration eq. for
dependent variables.
6) We use the least square method to find long run effects of
variables.

19. How to do the Estimation Multivariate
Cointegration and VECMs cont.2
1) First coefficient indicate the speed of adjustment either towards or
move away from equilibrium in long run
2) (negative coefficient sign is good for bring back the whole system) p
va;ue must be less than 0.05 for significance)
3) T value if it is greater than 2 it is significant
4) Then after we perform wald test for short run causality
5) From ols table we go to coefficient diagnostic for performing WALD
test
6) We use following null hypo equation for performing wald test
7) C(3)=C(4)=0
8) P values must be less than 0.05 for significance

20. What is Wald test
 The Wald statistic explains the short run causality
between variables whiles the statistics provided by the
lagged error correction terms explain the intensity of the
long run causality effect.
 Short run Granger causalities are determined by Wald
statistic for the significance of the coefficients of the
series.

21. Vector Error Correction Models (VECM) are the basic VAR, with an
error correction term incorporated into the model and as with
bivariate cointegration, multivariate cointegration implies an
appropriate VECM can be formed.
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 test, having found evidence of
cointegration of some I(1) variables, we can then assess the short
run and potential Granger causality with a VECM.

22. The finding that many macro time series may contain a unit root has spurred
the development of the theory of non-stationary time series analysis.
Engle and Granger (1987) pointed out that a linear combination of two or more
non-stationary series may be stationary.
If such a stationary, or I(0), linear combination exists, the non-stationary (with
a unit root), time series are said to be cointegrated.
The stationary linear combination is called the cointegrating equation and may
be interpreted as a long-run equilibrium relationship between the variables.
For example, consumption and income are likely to be cointegrated. If they
were not, then in the long-run consumption might drift above or below
income, so that consumers were irrationally spending or piling up savings.

23. A vector error correction (VEC) model is a restricted VAR that has
cointegration restrictions built into the specification, so that it is
designed for use with nonstationary series that are known to be
cointegrated.
The VEC specification restricts the long-run behavior of the
endogenous variables to converge to their cointegrating
relationships while allowing a wide range of short-run dynamics.
The cointegration term is known as the error correction term
since the deviation from long-run equilibrium is corrected
gradually through a series of partial short-run adjustments.

24. 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 -Maximum likelihood
test.

25. 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

26. VECM
 The most basic VECM is the following first-
order VECM:

27. VECM
First we test if the variables are stationary, i.e. I(0).
If not, they are assumed to have a unit root and are
I(1).
If a set of variables are all I(1), they should not be
estimated using OLS as there may be one or more
long-run equilibrium relationships,
i.e. cointegration. We can estimate how many
"cointegration vectors" exist between variables using
the Johansen technique.

28. VECM
 If a set of variables is found to have one or more
cointegration vectors, a suitable estimation technique is a
VECM (Vector Error Correction Model) that adjusts for both
short-term changes in variables and deviations from
equilibrium.

29. Granger causality
 Granger causality tests whether a variable is “helpful”
for forecasting the behavior of another variable.
 It’s important to note that Granger causality only allows
us to make inferences about forecasting capabilities --
not about true causality.

30. Granger-causality statistics
As we previously discussed, Granger-causality statistics test whether
one variable is statistically significant when predicting another variable.
The Granger-causality statistics are F-statistics that test if the
coefficients of all lags of a variable are jointly equal to zero in the
equation for another variable.
As the p-value of the F-statistic decreases, evidence that a variable is
relevant for predict another variable increases.

31. The Granger causality
 The Granger causality test were use when the variables are
cointegrated.
 Engle and Granger (1987) warned that if the variables are
stationary after first differencing in the existence of
cointegration the application of VAR to the analysis will be
spurious.
 The outcome of the stationarity test using ADF revealed
that our variables are I (1)

32.  For example, in the Granger-causality test of X on Y, if the p-
value is 0.02
 we would say that X does help predict Y at the 5% level.
 However, if the p-value is 0.3
 we would say that there is no evidence that X helps predict Y.

33. Impulse Response and Variance
decomposition
 the impulse responses are the relevant tools for
interpreting the relationships between the variables
 Variance decompositions examine how important each of
the shocks is as a component of the overall
(unpredictable) variance of each of the variables over
time.

34. Impulse response functions

35.  The impulse response function traces the dynamic path of variables in the system
to shocks to other variables in the system. This is done by:
• Estimating the VAR model.
• Implementing a one-unit increase in the error of one of the variables in the model,
while holding the other errors equal to zero.
• Predicting the impacts h-period ahead of the error shock.
• Plotting the forecasted impacts, along with the one-standard-deviation confidence
intervals.

36.  The results show IR (Impulse
response) to dependent variables. Only
for NIR IR function is illustrated on the
table and
 as on the table seen only NIR has
positive response to CPI. But against to
this all other variables have negative
response to NIR
 Impulse Response positive values
have positive negative values have
negative effects on dependent (here
CPI)
R. of
DCPI:
Period RGDP DCPI DNIR DREER
1 -3.870022 10.52160 0.000000 0.000000
2 4.350339 0.388418 0.635650 -3.964539
3 2.581088 -0.057747 1.343376 -0.210536
4 -1.406336 0.760648 0.709599 -0.485223
5 -1.189040 0.131412 0.477037 -0.098667
6 0.043845 -0.346002 0.243500 0.050212
7 0.401353 -0.000346 0.078936 0.053059
8 -0.003204 0.089603 0.006877 -0.037810
9 -0.052022 0.019648 -0.044851 -0.027014
10 -0.032278 -0.017211 0.007166 0.004444
Impulse response sample estimation and interpretation

37. Variance decomposition estimation and interpretation
 On the table, the variance
decomposition results for CPI
illustrated.
 RGDP and REER affects CPI
more than NIR.
 Higher values have more
effects than smaller values
VD of
DCPI:
Period S.E. RGDP DCPI DNIR DREER
1 11.21076 11.91672 88.08328 0.000000 0.000000
2 12.68381 21.07330 68.90575 0.251152 9.769804
3 13.01512 23.94694 65.44426 1.303893 9.304905
4 13.14111 24.63526 64.53047 1.570594 9.263682
5 13.20444 25.21040 63.92289 1.686082 9.180623
6 13.21138 25.18501 63.92429 1.718280 9.172418
7 13.21782 25.25268 63.86205 1.720173 9.165098
8 13.21818 25.25131 63.86316 1.720107 9.165417
9 13.21840 25.25202 63.86125 1.721200 9.165528
10 13.21845 25.25241 63.86091 1.721216 9.165466

38.  Forecast error decomposition separates the forecast error variance into
proportions attributed to each variable in the model.
 Intuitively, this measure helps us judge how much of an impact one
variable has on another variable in the VAR model and how intertwined
our variables' dynamics are.
 For example, if X is responsible for 85% of the forecast error variance of Y,
it is explaining a large amount of the forecast variation in X.
 However, if X is only responsible for 20% of the forecast error variance
of Y, much of the forecast error variance of Y is left unexplained by X.

39. Forecast error decomposition

40. How to Identify possible the Structural
Shocks?
 Shock run restriction?
 Long run restriction?
 Sign restriction?
 Available convention: for example Ex rate
Exchange rate shock from flexible to peg should increase crisis
probability;
Capital Account Liberalization shock from less to more free
capital flow should increase crisis probability
What are their effects on output?

42.  Thank You
 erginakalpler@csu.edu.tr