Impulse Response and Variance Decompositions
As in the univariate case, a VAR(p) process can be represented in the form of a vector moving average (VMA) process.
where the k x k moving average matrices tys are determined recursively using (6.1.3).
The elements of coefficient matrices tys mean effects of ut_s shocks on Yt. That is, the (i j)-th element, rtpfj, of the matrix ^s is interpreted as the impulse response
Sets of coefficients (s) = ijilj, i j = 1 T are called the impulse response functions.
It is possible to decompose the ^-step-ahead forecast error variance into the proportions due to each shock Ujt.
The forecast variance decomposition determines the proportion of the variation Yjt due to the shock Ujt versus shocks of other variables uit for i = j.
VAR in EViews
As an example of VAR estimation in EViews, consider two time series of returns of monthly IBM stocks and the market portfolio returns from Fama-French database (data is contained in IBM1.wf1).
There are several ways to estimate VAR model in EViews. The first one is through the main menu. Clicking on View/Estimate VAR... will open a dialog window for VAR model estimation.
Figure 6.1: VAR model estimation dialog window
We choose Unrestricted VAR and in the Endogenous Variables box we
have to specify the list of endogenous time series variables to be included in the VAR model. We consider two excess return series of the IBM stock IBM_ex and the market portfolio Mkt_ex.
In the Lag Intervals for Endogenous we have to specify the order of the model, that is interval of lags to be included in the model. If we want to build a model with only two lags, we write 1 2. This means, we include all lags beginning from the first one and ending with the lag of order 2. We do not specify any exogenous variables apart from the intercept term c.
Another way of calling the VAR estimation dialog window is to select both endogenous variables in the workfile and in the context menu (right button click) choose Open/as VAR.... The Endogenous Variables box will be filled in automatically.
Finally, we can estimate VAR model from the command line. There is a separate object, called var, to declare the VAR model. The estimation of the above mentioned example will look like
Here ibm2 is a name of the var-object which will be saved in the workfile, Is indicates the estimation method; in this case it is OLS estimation method of the unrestricted VAR model. Then, specifications of the lags pairs and the list of endogenous variables follow. If one wishes to include exogenous variables besides the intercept, it can be done by typing a symbol @ followed by a list of exogenous variables. For example,
Click OK and EViews produces an estimation output for the specified VAR model.
Figure 6.2: Output for the VAR model estimation
Two columns correspond to two equation in the VAR model. The only significant coefficient besides the intercept one is at the second lag of the market portfolio returns in the IBM equation. As expected, there is a unidirectional dynamic relationship from the market portfolio returns to the IBM returns, Thus, the IBM return is affected by the past movements of the market while past movements of IBM stock returns do not affect the market portfolio returns. The second equation (for market portfolio) is not significant as suggested by the F-statistics. This means that the estimated model cannot explain variation in the market portfolio returns. This can happen because we possibly omitted some important exogenous variables or the order of the model is inappropriately selected. EViews provides a tool to choose the most suitable lag order. In the workfile menu choose View/Lag Structure/Lag Length Criteria... to determine the optimal model structure. In the appeared Lag Specification window we choose pmax = 8 (maximal lag order).
All criteria indicate that the optimal lag order of the model is 0. This means that the VAR model is inappropriate model to explain IBM and market portfolio returns. Indeed, we know from the CAPM that market portfolio returns affect the stock returns contemporaneously and are not in lag relationship. Thus, either additional exogenous factors should be found to include in the model or another structure of the model should be employed in this case.
Figure 6.3: Output for the lag length selection procedure
Lag selection can be programmed manually in the same way as it is done for ARMA model (see Chapter 3). There are some command references given below which can be used to assess various statistic values in the VAR analysis in EViews.