As the name suggest, VAR is a system of equation where a vector of time-series variables is regressed on its own lags. To distinguish with univariate AR process, the vector consists of two or more variables. Moreover, VAR requires the series to be stationer. For non-stationary variables, a vector error correction (VEC) model may be more appropriate.

VAR is an attempt to take account a complex nature of dynamic relationships amongst (economic) variables that are often neglected in the simultaneous equation regression. It is based on the philosophy to let the data ‘speak’. Economic theory is somehow being abstracted.

For technical detail, you may refer to textbooks such as:

Green, W.H (2003), Econometric Analysis (5th edition). Prentice Hall.

Heij, C., de Boer, P., Franses, P.H., Kloek, T, and van Dijk, H.K. (2004), Econometric Methods with Applications in Business and Economics. Oxford University Press.

Hill, R.C., Griffiths, W.E., Lim, G.C. (2011), Principles of Econometrics (4th edition). John Willey and Sons.

Johnston, J and DiNardo, J (1997). Econometric Methods (4th edition). McGraw-Hill.

Judge, G.G., Griffiths, W.E., Hill, R.C., Lutkepohl, H., Lee, T.C. (1985). The Theory and Practice of Econometrics (2nd edition). John Wiley and Sons.

Lutkepohl, H. (2005), New Introduction to Multiple Time Series Analysis. Springer.

## VAR estimation

Users can estimate a VAR in i-Regand simply by doing the following steps:

Step 1. Tap on the main menu and select estimate VAR

Step 2. Specify endogenous variables, exogenous variables, and lag length

Step 3. Run and see the results

Note that constant term is not included automatically in the estimation. You can include constant term in the list of exogenous variables. It is also possible to include time trend as exogenous variables. You can also include other variables as exogenous, but you have to make sure that they are stationary.

It may be more desirable to use Granger causality/exogeneity test to determine which variable should be classified as exogenous or endogenous. Alternatively, users can always refer to economic theory to do the classification. For instance, using a small country assumption, international interest rate should be treated as exogenous for a VAR describing macroeconomic interrelations of a small country. It does not make sense to use the exogeneity test in such obvious case, though the test shows otherwise. Bear in mind to always subscribe to a relevant theory.

The regression results are presented by equation. The header should indicate which equation is being presented. If the lag length is more than zero, there is a label ahead of a variable indicating lag terms. For instance, L1 is the first lag, or L3 is the third lag.

Summary statistics are presented for each equation, just like in the single equation. A VAR is actually estimated equation by equation, sequentially. The same summary is also provided for the whole equation system.

## Lag length test

Users need to specify appropriate lag length included in the estimation. The Apps can help users to specify the optimal lags based on Akaike information criterion (AIC), Schwarz criterion (SC), Hannan-Quinn criterion (HQC). Users can always pick one of the criterion (minimum value), but beware of the (dis)advantages of each criterion. Please consult to reliable references or text books.

The input setup is pretty much similar to the VAR estimation, except the specification of maximum lag. The Apps calculates the criteria by running VAR regression for each lag length, from zero to maximum lag, and by holding the number of observation constant.

The pictures below show how to do lag length test. You will find it very easy.

## Granger Non causality/exogeneity test

There at least three issues that can be addressed by the Granger causality test. First, it can distinguish which variables are endogenous or exogenous. The test can indicate which variable to be included or excluded in the equations. However, as discussed before, economic theory should always be the main guide. The test can be very useful in a situation where there are conflicting arguments (e.g. classical versus Keynesian).

The above discussion rises the second issue. Granger causality can indicate which argument is supported by empirical data. There are issues whether monetary policies can effectively alter aggregate output or not. The other example would be monetary policy transmission channels.

Third, some analysis needs exact variable ordering, but researchers often unsure about the appropriate order. For example, the results of Cholesky decomposition will depend on how you order the variables. Granger causality can be used as a guide for variable ordering.

The menu setup is exactly the same as VAR estimation. Once you familiar with one of them, you can easily understand how to use the other. The results of Granger causality are presented by equation. You can check the causality of one variable on each equation simply by assessing the significance of the statistics. There are also information about joint significance test. Here are the examples.