# Lower Triangular Factorization with Additional Zero Restriction

Consider the the following VAR(1) regression \begin{align*} \mathbf{y}_t = \mathbf{B}\mathbf{y}_{t-1} + \mathbf{\varepsilon}_t \end{align*} with the corresponding residual variance-covariance matrix \begin{align*} \mathbf{\Sigma} = \begin{bmatrix} \sigma_{1,1} & \sigma_{1,2} & \sigma_{1,3} \\ \sigma_{2,1} & \sigma_{2,2} & \sigma_{2,3} \\ \sigma_{3,1} & \sigma_{3,2} & \sigma_{3,3}\end{bmatrix} \end{align*} The standard Choleski decomposition with $$\mathbf{\Sigma}=\mathcal{C}\mathcal{C}'$$ is given as \begin{align*} \mathcal{C} = \begin{bmatrix} c_{1,1} & 0 & 0 \\ c_{2,1} & c_{2,2} & 0 \\ c_{3,1} & c_{3,2} & c_{3,3}\end{bmatrix} \end{align*}

What i want to do is to achieve that the structural shocks in the first equation do not appear contemporaneously in the third equation, i.e. i want to restrict $$c_{3,1}$$ to zero (keeping the other restrictions untouched).

Is there a way to achieve that? I mean i can manually calculate the values for the $$c$$'s but is there a general way for the general case with $$n$$ variables?