Triangularize VAR

[fan]

Hi Tom. I am trying to triangularize a VAR(1) system in a way that the innovations in DIV,TREM,DEF,and TBILL are orthogonal to the innovation in Rm. In addition, I need to scale all the innovations to have the same variance as the innovation in the Rm. Could you please kindly give me some tips about how to do the triangularization and scaling? Thank you for your help.

[TomDoan]

I'm not really sure what you mean. A Cholesky factorization makes the shocks to the last four uncorrelated with the first and makes all the shocks the same size (1). However

(a) I'm pretty sure that's not what you want
(b) That's not uniquely defined, since any change in the ordering of 2-5 will result in a factorization which does the same thing.

[fan]

I'm not really sure what you mean. A Cholesky factorization makes the shocks to the last four uncorrelated with the first and makes all the shocks the same size (1). However

(a) I'm pretty sure that's not what you want
(b) That's not uniquely defined, since any change in the ordering of 2-5 will result in a factorization which does the same thing.

Hi Tom, Thank you for your reply. Actually, I am trying to replicate what Petkova (2006) did in her paper. According to the author, she triangularize the VAR system in the way that the innovation in the Rm,t is unaffected , the orthologonalized innovation in the DIV is the component of the original DIV innovation orthogonal to the Rm,t and so on. The orthogonalized innovation to DIV is a change in the dividend/price ratio with no change in the market return; therefore, it can be interpreted as a shock to the dividend. Similarly, shocks to the term spread, default spread, and short-term rate are all orthogonal to the contemporaneous stock market return (Rm,t). As in Campbell (1996), the author also scale all innovations to have the same variance as the innovation in the Rm,t.

My way to remove the effect of Rm,t on other variables is to run the VAR system and treating the Rm,t and its lags as exogenous variables in the system. It will be helpful if you could kindly help me to learn how Petkova did exactly in her study. Many Thanks.

[TomDoan]

Petkova is behind a pay wall, but that's a description of a Cholesky factorization. Just generate the structural residuals and rescale to make the variances match with what you want.

[fan]

Petkova is behind a pay wall, but that's a description of a Cholesky factorization. Just generate the structural residuals and rescale to make the variances match with what you want.

Thank you for the reply. If it is possible, could you please kindly take a look my code?

Code: Select all

system(model=varmodel)
vars mktrf1 divyield1 term1 default1  tbill1
lags 1
end(system)
estimate(outsigma=sigma,resid=resids)
@StructResids(factor=%decomp(sigma)) resids / residsnew
dec vector  rescale(%nvar)

[TomDoan]

It doesn't look like you finished what you're doing. RESIDSNEW has the structural residuals. Now what?

[fan]

It doesn't look like you finished what you're doing. RESIDSNEW has the structural residuals. Now what?

Now, I need to rescale the structural residuals to make the variances are the same. Can I do rescale in this way?

Code: Select all

set residsnew(2) = (residsnew(1)/residsnew(2))*residsnew(2)
set residsnew(3) = (residsnew(1)/residsnew(3))*residsnew(3)
set residsnew(4) = (residsnew(1)/residsnew(4))*residsnew(4)

[TomDoan]

That really doesn't work. That makes them all copies of residsnew(1).

Aren't they already variance one series (if computed with a T divisor)? By construction, they should be.

[fan]

That really doesn't work. That makes them all copies of residsnew(1).

Aren't they already variance one series (if computed with a T divisor)? By construction, they should be.

Hi Tom, Thank you for the reply. Do you mean the factorization has already set all the residuals to have the same variance ?

[TomDoan]

You're welcome to check that out, but yes, they are mean zero variance one. If you do a STATISTICS, the variances will be slightly smaller than one because STATISTICS uses a different divisor than ESTIMATE, but they'll all be the same.