1. Time Series Econometrics
Didar Erdinc, Ph.D.
Associate Professor of Economics
American University in Bulgaria

2. Vector Autoregression (VAR)
Introduction
VAR resembles a SEM modeling – we consider several endogenous variables together. Each endogenous variables is explained by its lagged values and the lagged values of all other endogenous variables in the model.
In the SEM model, some variables are treated as endogenous and some are exogenous (predetermined). In estimating SEM, we have to make sure that the equation in the system are identified – this is achieved by assuming that some of the predetermined variables are present only in some equation (which is very subjective) – and criticized by Christopher Sims.
If there is simultaneity among set of variables, they should all be treated on equal footing, i.e., there should not be any a priori distinction between endogenous and exogenous variables.

3. Vector Autoregression (VAR)
Its Uses
Forecasting
VAR forecasts extrapolate expected values of current and future values of each of the variables using observed lagged values of all variables, assuming no further shocks
Impulse Response Function (IRFs)
IRFs trace out the expected responses of current and future values of each of the variables to a shock in one of the VAR equations

4. Vector Autoregression (VAR)
Its Uses
Forecast Error Decomposition of Variance (FEDVs)
FEDVs provide the percentage of the variance of the error made in forecasting a variable at a given horizon due to specific shock. Thus, the FEDV is like a (partial) R2 for the forecast error
Granger Causality Tests
Granger-causality requires that lagged values of variable A are related to subsequent values in variable B, keeping constant the lagged values of variable B and any other explanatory variables

5. Vector Autoregression (VAR)
Mathematical Definition
[Y]t = [A][Y]t-1 + … + [A’][Y]t-k + [e]t or
where:
p = the number of variables be considered in the system
k = the number of lags be considered in the system
[Y]t, [Y]t-1, …[Y]t-k = the 1x p vector of variables
[A], … and [A'] = the p x p matrices of coefficients to be estimated
[e]t = a 1 x p vector of innovations that may be contemporaneously correlated but are uncorrelated with their own lagged values and uncorrelated with all of the right-hand side variables.

6. Vector Autoregression (VAR)
Example
Consider a case in which the number of variables n is 2, the number of lags p is 1 and the constant term is suppressed. For concreteness, let the two variables be called money, mt and output, yt .
The structural equation will be:

7. Vector Autoregression (VAR)
Example
Then, the reduced form is

8. Vector Autoregression (VAR)
Example
Among the statistics computed from VARs are:
Granger causality tests – which have been interpreted as testing, for example, the validity of the monetarist proposition that autonomous variations in the money supply have been a cause of output fluctuations.
Variance decomposition – which have been interpreted as indicating, for example, the fraction of the variance of output that is due to monetary versus that due to real factors.
Impulse response functions – which have been interpreted as tracing, for example, how output responds to shocks to money (is the return fast or slow?).

9. Vector Autoregression (VAR)
Granger Causality
In a regression analysis, we deal with the dependence of one variable on other variables, but it does not necessarily imply causation. In other words, the existence of a relationship between variables does not prove causality or direction of influence.
In our GDP and M example, the often asked question is whether GDP  M or M GDP. Since we have two variables, we are dealing with bilateral causality.
Given the previous GDP and M VAR equations:

10. Vector Autoregression (VAR)
Granger Causality
We can distinguish four cases:
Unidirectional causality from M to GDP
Unidirectional causality from GDP to M
Feedback or bilateral causality
Independence
Assumptions:
Stationary variables for GDP and M
Number of lag terms
Error terms are uncorrelated – if it is, appropriate transformation is necessary

11. Vector Autoregression (VAR)
Granger Causality – Estimation (t-test)
A variable, say mt is said to fail to Granger cause another variable, say yt, relative to an information set consisting of past m’s and y’s if: E[ yt | yt-1, mt-1, yt-2, mt-2, …] = E [yt | yt-1, yt-2, …].
mt does not Granger cause yt relative to an information set consisting of past m’s and y’s iff 21 = 0.
yt does not Granger cause mt relative to an information set consisting of past m’s and y’s iff 12 = 0.
In a bivariate case, as in our example, a t-test can be used to test the null hypothesis that one variable does not Granger cause another variable. In higher order systems, an F-test is used.

12. Vector Autoregression (VAR)
Granger Causality – Estimation (F-test)
1. Regress current GDP on all lagged GDP terms but do not include the lagged M variable (restricted regression). From this, obtain the restricted residual sum of squares, RSSR.
2. Run the regression including the lagged M terms (unrestricted regression). Also get the residual sum of squares, RSSUR.
3. The null hypothesis is Ho: i = 0, that is, the lagged M terms do not belong in the regression.
5. If the computed F > critical F value at a chosen level of significance, we reject the null, in which case the lagged m belong in the regression. This is another way of saying that m causes y.

13. Vector Autoregression (VAR)
Variance Decomposition
Our aim here is to decompose the variance of each element of [Yt] into components due to each of the elements of the error term and to do so for various horizon. We wish to see how much of the variance of each element of [Yt] is due to the first error term, the second error term and so on.
Again, in our example:
The conditional variance of, say mt+j, can be broken down into a fraction due to monetary shock, mt and a fraction due to the output shock, yt .

14. Vector Autoregression (VAR)
Impulse Response Functions
Here, our aim is to trace out the dynamic response of each element of the [Yt] to a shock to each of the elements of the error term. Since there are n elements of the [Yt], there are n2 responses in all.
From our GDP and money supply example:
We have four impulse response functions:

15. Vector Autoregression (VAR)
Pros and Cons
Advantages
The method is simple; one does not have to worry about determining which variables are endogenous and which ones exogenous. All variables in VAR are endogenous
Estimation is simple; the usual OLS method can be applied to each equation separately
The forecasts obtained by this method are in many cases better than those obtained from the more complex simultaneous-equation models.

16. Vector Autoregression (VAR)
Pros and Cons
Some Problems with VAR modeling
A VAR model is a-theoretic because it uses less prior information. Recall that in simultaneous equation models exclusion or inclusion of certain variables plays a crucial role in the identification of the model.
Because of its emphasis on forecasting, VAR models are less suited for policy analysis.
Suppose you have a three-variable VAR model and you decide to include eight lags of each variable in each equation. You will have 24 lagged parameters in each equation plus the constant term, for a total of 25 parameters. Unless the sample size is large, estimating that many parameters will consume a lot of degree of freedom with all the problems associated with that.

17. Vector Autoregression (VAR)
Pros and Cons
Strictly speaking, in an m-variable VAR model, all the m variables should be (joint) stationary. If they are not stationary, we have to transform (e.g., by first-differencing) the data appropriately. If some of the variables are non-stationary, and the model contains a mix of I(0) and I(1), then the transforming of data will not be easy.
Since the individual coefficients in the estimated VAR models are often difficult to interpret, the practitioners of this technique often estimate the so-called impulse response function. The impulse response function traces out the response of the dependent variable in the VAR system to shocks in the error terms, and traces out the impact of such shocks for several periods in the future.