Forecast Error Term Analysis for Reduced Form VAR: Statistical Proofs

A previous post described how the primitive VAR equations violate the Classical Linear Regression Model (CLRM) because of correlation of the explanatory variable with the error term.

https://espin086.wordpress.com/2011/01/17/understanding-multivariable-relationships-across-time-introduction-to-the-theory-of-vector-autoregressionvar/

It is argued that transforming the primitive system through matrix algebra will eliminate the theoretical violation of the CLRM.  This post will present and prove some key assumptions about the reduced for VAR equations derived in a previous post.

Property 1: The error terms for a reduced for VAR have an expected value of zero.

Proof:

Property 2: The error terms for a reduced for VAR have a constant variance.

Proof:

Property 3: The error terms are not serially correlated in either equation.

Proof:

Similarly for properties about the second equations error terms.

Property 4: The error terms in each equation are correlated with each other.

The error terms are correlated with each other, but indirectly through the primitive equations error terms.


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