Thoeretical varicance-covariance matrix

[Byron]

Dear all,
My question is that I have got the evolution function of a simple RBC model as in the picture, with c is consumption, k is capital and a is technology, and e is white noise, (all variables are log linearized). and thus I have got the policy function. So how do I calculate the theoretical covariance matrix based on these equations? Thanks ahead!

[jpfeifer]

Usually, that involves computing the fixed point of the Lyapunov equation. There exists an explicit but slow formula for that.

[Byron]

Yep, I first calculate the covariance matrix of (kt , at) by solving a matrix equation, and then I calculate the covariance of ct and (kt, at). But I have a new question, how do I calculate the autoregression matrix of these variables?

[jpfeifer]

You mean the first order autocorrelation? That should be straightforward based on the state space representation.

[Byron]

This is of great help to me! Thanks jpfeifer! BTW, can you share the rest of the slides?

[jpfeifer]

Chapter_2.pdf (718.3 KB)

[Byron]

Thanks again!

[JC-Granados]

If the formula above is the solution to your model (endogenous as a function of states, a, k), then you can obtain the covariance matrix of the states, and the corresponding variance of your decision variables vector implied by the parameters you got: