Vector Autoregression Model

3.5 VAR Model
Vector autoregression model( Sims,1980) commonly used in the analysis of multivariate time series. it is using the model to characterize the number of relationships between vectors. In another word, a VAR is the extension of the autoregressive model to the case in which there is more than one variable under study .Different with AR model which involved one depended variable and depended only on lags of itself, a VAR has more then one dependent variable and thus has more than one equation. Each equation uses as its explanatory variables lags of all the variables under study.

However, When use VAR model in time series, the first step is to determine the lag order. that is to say, before fitting the VAR model, it is necessary to determine the optimal choice of the lag period p. The reason is that, if the lag is too small, there will be a strong autocorrelation in error term, thus it will cause consistent parameter estimation.On the other hand, if the lag is too big, it will cause a decrease of the degree of freedom. and further affect the validity of the model parameter estimation. This step will be materialized in the empirical part.

In which including akaike information criterion(AIC), schwarz criterion(SBIC) and hannan-quinnlnformation criterion(HQIC)..And the three variable’s calculation formula is as follows: AIC=-2 ln(L) + 2 k BIC=-2 ln(L) + ln(n)*k HQIC=-2 ln(L) + ln(ln(n))*k
Where L is the maximum likelihood, n is the number of the data, K is the number of variables. Frankly speaking,when the values of these indicators are small enough, meaning that it corresponds to the lag order is the best one.

3.6 Granger causality test based on VAR