[Gretl-users] Re: VECM long-run impact matrix
On Sun, Sep 14, 2025 at 6:29 AM Riccardo (Jack) Lucchetti
<p002264(a)staff.univpm.it> wrote:
On 13/09/2025 14:52, Artur T. wrote:
Hi all,
I would like to retrieve the long-run impact matrix, called "C" in
Johansen's jargon.
After estimating a vecm, there is item "C". Is this object the matrix I am
looking for (I could not find anything in the docs)?
No, that's the Cholesky factor of the residual covariance matrix.
Furthermore, in a recent paper by Juselius, she reports standard errors for C. Is this
information also stored in the system bundle?
That matrix is referred to with different letters in different sources. For example,
Kilian & Lütkepohl use the letter $\Xi$. Its calculation is not very difficult, given
the information contained in the $system bundle you get out of the "vecm"
command. Its asymptotic distribution, however, is a bit more intricate. It was derived by
Paolo Paruolo in https://www.jstor.org/stable/pdf/3532701.pdf., but you can also find it
in Søren's 1995 OUP book (section 13.5).
In gretl git, I've added a couple of elements to the vecm_info bundle
inside the $system bundle that can be retrieved after estimation of a
cointegrated VAR.
* "Gamma": the matrix formed as (identity matrix minus the summation
of the per-lag \Gamma_i matrices in the short-run dynamics) -- see
Johansen (1995) page 45 -- and
* "JC": the matrix that Johansen refers to as 'C' (and that Artur was
interested in), namely C = \beta_{\perp} (\alpha_{\perp}' \Gamma
\beta_{\perp})^{-1} \alpha_{\perp}' (1995, page 49).
I'd appreciate it if anyone can check the correctness of these
results! In particular I'm assuming that when \beta has more rows than
\alpha (because some deterministic and/or exogenous terms are
restricted to the cointegration space), the "beta" that enters the
"JC" formula should just be the first p rows of what we're calling
beta in gretl (where p is the number of endogenous variables).
Otherwise the formula Johansen gives for C doesn't work.
Allin