I have so far discovered three different ways of utilizing the Cholesky decomposition for calculating the OIRFs of a VAR(k). These seem contradictory so I would like some input where I am making the mistake. Cholesky decomposition decomposes the contemporanous variance covariance matrix of the error term into PP'.

First method is to estimate the generic 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 way is the same, but only the right side is multiplied by P. Example in a previous question: SVAR, Cholesky decomposition and impulse-response function in R

The third way is to use the VMA form (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?