I have so far discovered three different ways of utilizing the Cholesky decomposition for calculating the OIRFs of a VAR(k). The different methods seem contradictory so I would like some input on where I am making the mistake. First, all methods use the Cholesky decomposition, which decomposes the contemporaneous variance covariance matrix of the error term into PP'.

First method to solve the OIRFS is to first estimate the reduced VAR(1) equation (for simplicity):

$Y_t = A_1 Y_{t-1} + e_t $

Premultiply the equation by the P and the OIRFs can be solved recursively.

Second method is the same, but only the right side is multiplied by P. The following question contains an example and more details: SVAR, Cholesky decomposition and impulse-response function in R

The third method is to use the VMA form of the VAR (thanks to hejseb for the equation): $ y_t=\sum_{s=0}^\infty\Psi_se_{t-s}=\sum_{s=0}^\infty\Psi_sPP^{-1}e_{t-s}=\sum_{s=0}^\infty\Psi_s^*v_{t-s}. $

Where $\Psi_s^*$ is the matrix of orthogonalized impulse responses, $\Psi_s$ being the matrix of simple IRFs. More details on the method: How to calculate the impulse response function of a VAR(1)? (With example)

Two questions:

1. As is apparent, the equation used for the first and second method are not the same, there is an added P on the left side in the first case. Yet they are supposed to provide the same result as far as I am aware. How is this possible? To me the 2nd way seems just wrong as there doesn't even appear to be a contemporaneous effect in it.

2. In the VMA representation the newly created uncorrelated error term is $P^{-1}\epsilon_{t}$ In the VAR representations it is $P\epsilon_{t}$. Isn't this a contradiction?