With nothing solved out this is \((I - B)^{\top}V^{-1}(I - B)\) over every cell. Once a series has an sd of zero \(V\) is singular and that product does not exist. Substituting the solved cells into the remaining rows leaves \(S(x_o - \mu_o) = \varepsilon_o\) with $$S = (I - B)_{oo} - (I - B)_{ou}(I - B)_{uu}^{-1}(I - B)_{uo},$$ so those cells are normal with precision \(S^{\top}V_{oo}^{-1}S\). Building it rather than reading the density off the innovations is what accounts for the determinant of \(S\), which is not one once the arrows loop within a year.
Arguments
- IminusB
Sparse \(I - B\), from
get_dsem_matrices.- parts
The rest of that output, read for \(V^{-1}\), \(V\) or the sds.
- solve_mat
Output of
get_dsem_solve_mat.- dsem_cells
Output of
get_dsem_cells.
Details
The middle term is
\(\tilde{V}_{oo} = V_{oo} + C V_{uo} + V_{ou}C^{\top}\) with
\(C = M_{ou}M_{uu}^{-1}\), the two cross terms covering a covariance line
between a solved cell and one that keeps its innovation. That line is
refused by read_dsem_arrows, so both terms are always zero and
only \(V_{oo}\) is formed.