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

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
construct a VECM model in differences (with error correction terms)
apply the Johansen test to the VECM model in differences to find out the number of cointegration (r) (none or Atmost)
if r = 0, estimate VAR in differences
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
Test the variables for stationarity using the usual ADF tests.
If all the variables are I(1) include in the cointegrating relationship.
Use the AIC or SIC to determine the number of lags in the cointegration test (order of VAR)
Use the trace and maximal eigenvalue tests to determine the number of cointegrating vectors present.
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
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.
We reject null hypo and probability values are also less than 0.05
(In opposite case we run unrestricted VAR)
We perform and estimate the table for vector error correction model and then find the equations for our model.
From equations we derive the residuals for cointegration eq. for dependent variables.
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
First coefficient indicate the speed of adjustment either towards or move away from equilibrium in long run
(negative coefficient sign is good for bring back the whole system) p va;ue must be less than 0.05 for significance)
T value if it is greater than 2 it is significant
Then after we perform wald test for short run causality
From ols table we go to coefficient diagnostic for performing WALD test
We use following null hypo equation for performing wald test
C(3)=C(4)=0
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. 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. Impulse response sample estimation and interpretation
 R. of DCPI:
 Period
RGDP
DCPI
DNIR
DREER  1      2        3      4    5    6        7        8      9  
 10    
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)

37. Variance decomposition estimation and interpretation
 VD of DCPI:
 Period
S.E.
RGDP
DCPI
DNIR
DREER  1          2            3            4            5            6            7            8            9          
 10          
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

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