I'd like to know if, when running a VAR model in R (package "vars"), the cholesky decomposition matrix is upper triangular (so I should order my variables from more endogenous to less endogenous), or lower triangular (so I should order my variables from less endogenous to more endogenous).


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According to page 6 of the documentation for vars, the Cholesky decomposition matrix is lower triangular:

The long-run impact matrix is the lower-triangular Choleski decomposition of the above matrix and the contemporaneous impact matrix is equal to:

$$(I_K − A_1 − \dots − A_p)Q$$

where $Q$ assigns the lower-triangular Cholesky decomposition.

In the theoretical literature, Cholesky factorization is traditionally defined as lower-triangular; that is, as $A = L L^{T}$. There is a function is base R, chol(), that returns the upper-triangular Cholesky decomposition of a matrix $X = R^{T}R$.


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