Why is VAR analysis all linear? Why not nonlinear? When doing VAR analysis, it is almost always a linear form that gets used. Is there any justifying reason (most DSGE models are non-linear unless linearized) to use a linear form?
First, it is important to note that there is a substantial literature both developing and using nonlinear VAR estimation (for example, see papers here, here, here, and here).

The reasons linear VARs are seen a lot, though, are similar to the reasons that least squares (in its varying forms) is seen a lot. The model is very transparent and analytically tractable. This is a benefit for both pedagogy (analysis can be done on linear VARs just using lag polynomials) and for interpretation of results. You also don't run in to the concavity and global vs. local solution issues you do with big nonlinear problems, which can make your results more credible in many circles.