Variance decomposition for correlated shocks

[jpfeifer]

No, that is not a contradiction. The FEVD just tells you that most of the effect of A observed was most likely caused by fluctuations in another shock B that was correlated with A (because you assigned causality that way)

[ZBCPA]

If you want to only consider the effect of the shock B ordered last, you need to purge the residual of that equation from the effect of shock A that affects both equations.
Say the observed residuals are given by
[res_1; res_2]=[A;B]*R
where R is the upper Cholesky matrix. Then you can back out the shocks A and B by postmultiplying the left hand side by R^{-1}.

Dear Johannes,

Could I ask you further on this? Could you please take a look at the following example?

Code: Select all

var Gov, Tfp;
varexo g, z;
parameters rho;
rho   = 0.95;

model;
Gov = rho*Gov(-1) + g;
Tfp = rho*Tfp(-1) + z;
end;

shocks;
var g; stderr 0.0039;
var z; stderr 0.0086;
corr e, u = 0.3868;
end;

stoch_simul(order=1);

The innovation g and z are correlated with correlation=0.4
Then the covariance matrix is A=[0.0039^2 0.0039*0.0086*0.3868; 0.0039*0.0086*0.3868 0.0086^2];

R=chol(A)
inv(R)=[ 256.4103 -107.5509; 0 126.0937]

[obs_g;obs_z]=[g;z]*R

So I need use the observed smoothed innovations timing inv(R) to back out [g;z].

However, you see, inv(R) has very large scale, will this generate any problem since [g;z] then will have much different scale from smoothed ones?


Many thanks!

Kind regards,
Huan

[jpfeifer]

I am not sure I am still following our discussion. If you mulitply the two correlated shock processes with the inverse of the Cholesky decomposition of the covariance matrix, you will end up with two uncorrelated standard normal shock series. The difference in size you are talking about is thus required to bring the variances to 1.