Cholesky decomposition

[farrel]

To staff:

What triangle (upper or lower) is restricted in VARs regarding cholesky decomposition?
For example, A or B variant:

Code: Select all

a11 0 0 A = a21 a22 0 a31 a32 a33 b11 b12 b13 B = 0 b22 b23 0 0 b33

[EViews Glenn]

Choleskys are lower in EViews.

[farrel]

Choleskys are lower in EViews.

So, for impulse responses Variant B is appropriate. But regarding Variance decomposition which one is true?
I'd like to suppose that this is the same, but citing User's guide II (p.356):
"As with the impulse responses, the variance decomposition ... . For example,
the first period decomposition for the first variable in the VAR ordering is completely due to
its own innovation."
I understood that Variant A is appropriate, since only its own shock affects the first variable.
So, is it true? Maybe I misunderstood something ...

[EViews Glenn]

I'm not the VAR expert around here, but I don't quite understand the premise of the question.

My understanding is that the structural decomposition for the first (ordered) series formed by post-multiplying the first row of the inverse Cholesky by the matrix of reduced form coefficients? (c.f., Hamilton, p. 329) Or equivalently, by orthogonalizing the system of equations using the Cholesky and re-estimating. Since the Cholesky is lower triangular, the first equation does not have any other endogenous variables from the system, the second equation has the first two, the third has the first three, etc. Thus, the first has only its own innovation, while the second has (implicitly) the first coming from the inclusion of the contemporaneous first variable...

So I believe that the interpretation of the effects of the orthogonalization carry over from the impulse response to the variance decomposition without additional issues (other than the central one of order dependence).

[farrel]

I'm not the VAR expert around here, but I don't quite understand the premise of the question.

My understanding is that the structural decomposition for the first (ordered) series formed by post-multiplying the first row of the inverse Cholesky by the matrix of reduced form coefficients? (c.f., Hamilton, p. 329) Or equivalently, by orthogonalizing the system of equations using the Cholesky and re-estimating. Since the Cholesky is lower triangular, the first equation does not have any other endogenous variables from the system, the second equation has the first two, the third has the first three, etc. Thus, the first has only its own innovation, while the second has (implicitly) the first coming from the inclusion of the contemporaneous first variable...

So I believe that the interpretation of the effects of the orthogonalization carry over from the impulse response to the variance decomposition without additional issues (other than the central one of order dependence).

Hmm, probably I misunderstood something ...
So, I'll ask the question once again:
In my first post which VARIANT A or B is Cholesky decomposition in Eviews?

[EViews Glenn]

As I wrote in my first response, lower triangular.

My last post was in answer to your reply implying that the interpretation of the variance decomposition differed from the interpretation of the impulse responses. To quote:

So, for impulse responses Variant B is appropriate. But regarding Variance decomposition which one is true?
I'd like to suppose that this is the same, but citing User's guide II (p.356):
"As with the impulse responses, the variance decomposition ... . For example,
the first period decomposition for the first variable in the VAR ordering is completely due to
its own innovation."
I understood that Variant A is appropriate, since only its own shock affects the first variable.
So, is it true? Maybe I misunderstood something ..

For those following at home, I think we got mixed up on our A's and B's. Variant A is the lower and variant B is the upper. I'm not certain why my original response was interpreted as "Variant B is appropriate," since I said "lower" it should have been A, but given that it was, I now understand why we have the confusion about the implications for variance decomposition.

And for anyone who wants context regarding what we're debating. The variance decomposition is essentially a rotation of the entire system to triangularity and diagonal errors. If you do so by pre-multiplying the estimated reduced form VAR coefficients by the inverse of the lower triangular Cholesky of the residual covariance, the first equation has only it's own contemporaneous error, the second equation has it's own error and the implicit error from the contemporaneous first equation, etc... This matches the interpretation of the impulse responses and what is written in the manual.

If the Cholesky were upper, then the last equation would only have it's own errors, the second to last would have its own and the last equation's errors, etc...

I hope this clarifies things.

[farrel]

Thanks Glenn
Now everything is clear
Sorry, I just misunderstood, in some way I thought that "zeros" are restrictions ... :)

[sschaffrath]

I'm sorry Glenn, but just to be sure I understand what you mean, the "most endogenous" variable should be placed on the left when you have a lower-triangular B matrix, isn't it? Thanks a lot!

[farrel]

Assuming economic activity determines prices (not always of course), then prices are more "endogenous" than GDP, only because GDP also endogenous by itself.