VAR Model

 

SVAR, as its name undoubtedly suggests, is concerned with the statistical analysis of vector autoregressive (VAR) models. In this context, a VAR model is defined very broadly, but a maintained assumption is that the underlying residual process is Gaussian.

 

 

Definition of the VAR Model

 

Let yt be an n dimensional process represented by the system:

 

 

where x1,t is a q1 dimensional exogenous I(1) process, x0,t is q0 dimensional exogenous I(0) process, Dt is a d dimensional deterministic process, and

 

 

The parameter r represents the cointegration rank, with r ∈ {0,1,...,n}. If the model does not involve any exogenous I(1) variables, then q1=0. Similarly, if there are no exogenous I(0) variables in the model, then q0=0. Furthermore, k is the lag order for the endogenous variables, and SVAR requires that this parameter is at least 1. The lag order for any exogenous stochastic variables is denoted by m and is at least 0. Notice, however, that setting m=0 in a model with exogenous I(1) variables actually means that there is 1 lag for these variables since they (i) appear in first differences, and (ii) are allowed to appear in levels, lagged one period, in the cointegration relations.

 

The vector Dt contains deterministic variables automatically generated by SVAR, such as a constant, a linear trend (if asked for), and centered seasonal dummies (if asked for). If you wish to add deterministic variables of your own (impulse dummies and other silly variables), these will be appended to the deterministic variables created by SVAR. At a minimum, SVAR always includes a constant.

 

The VAR model above is written on vector error correction (VEC) form. This may seem restrictive, especially if you don't care much for cointegration, but is just a question of choosing a parameterization for the VAR model. With r=n, the VAR model above is not subject to any unit root restrictions. By default, SVAR does not impose any unit root restrictions, and, in fact, you can choose not to see any cointegration analysis at all. One important idea behind the Structural VAR project is that you should be allowed to analyse the type of VAR model that you prefer, not the type that I would want!

 

Cointegration analysis concerns the Π matrix, and the choice of restrictions on α and β. For models where there is either a constant or a linear trend restricted to the cointegration space, the vector Yt is appended with either a constant or a linear trend, while this deterministic variable is deleted from the Dt vector.

 

The only time when SVAR deviates from the above parameterization when presenting its output is when you have selected (i) no cointegration estimation, and (ii) you've opted for Bayesian analysis. In that case, the VAR model is entirely written in levels form.

 

The matrix B is of interest when analysing a structural form of the VAR model. When SVAR is concerned with reduced form analysis, the matrix B is ignored and the residual are given by εt = Bηt. from the above definitions, it follows that εt ~ N(0,Ω), where Ω is assumed to be positive definite.

 

 

Choice of Statistical Approach

 

SVAR allows you to choose between analysing the above VAR model using either classical or Bayesian methods. Since this is often viewed as a philosophical issue, I decided to make this an option rather than a model feature. The difference between these in the lingo of SVAR is simply that options are stored in the file var.ini, while model features are stored in model files, i.e., files with the extension .var. Options are therefore "permanent" in the sense that they apply to all models, while model features vary across models. You can, of course, change the options so that for a given model you can first perform a classical analysis, and then a Bayesian. This is done on the Estimation tab of the Preferences dialog.