The SIMNORMAL Procedure

Conditional Simulation

For a conditional simulation, this distribution of

\[ \bY = \left[ \begin{array}{c} Y_1\\ Y_2\\ \vdots \\ Y_ k \end{array} \right] \]

must be conditioned on the values of the CONDITION variables. The relevant general result concerning conditional distributions of multivariate normal random variables is the following. Let $\bX \sim N_ m(\bmu ,\bSigma )$, where

\[ \bX = \left[ \begin{array}{c} \bX _1\\ \bX _2\\ \end{array} \right] \]
\[ \bmu = \left[ \begin{array}{c} \bmu _1\\ \bmu _2\\ \end{array} \right] \]
\[ \Sigma = \left( \begin{array}{cc} \bSigma _{11} & \bSigma _{12}\\ \bSigma _{21} & \bSigma _{22}\\ \end{array} \right) \]

and where $bX_1$ is $k\times 1$, $X_2$ is $n\times 1$, $\bSigma _{11}$ is $k\times k$, $\bSigma _{22}$ is $n\times n$, and $\bSigma _{12}=\bSigma _{21}’$ is $k\times n$, with $k+n=m$. The full vector $\bX $ has simply been partitioned into two subvectors, $\bX _1$ and $\bX _2$, and $\bSigma $ has been similarly partitioned into covariances and cross covariances.

With this notation, the distribution of $\bX _1$ conditioned on $\bX _2=x_2$ is $N_ k(\tilde{\bmu },\tilde{\bSigma })$, with

\[ \tilde{\bmu }=\bmu _1+\bSigma _{12}\bSigma _{22}^{-1}(\mb{x}_2-\bmu _2) \]

and

\[ \tilde{\bSigma } = \bSigma _{11}-\bSigma _{12}\bSigma _{22}^{-1}\bSigma _{21} \]

See Searle (1971, pp. 46–47) for details.

Using the SIMNORMAL procedure corresponds with the conditional simulation as follows. Let $Y_1,\cdots ,Y_ k$ be the VAR variables as before (k is the number of variables in the VAR list). Let the mean vector for $\bY $ be denoted by $\bmu _1=\mr{E}(\bY )$. Let the CONDITION variables be denoted by $C_1,\cdots ,C_ n$ (where n is the number of variables in the COND list). Let the mean vector for $\bC $ be denoted by $\bmu _2=\mr{E}(\bC )$ and the conditioning values be denoted by

\[ \mb{c} = \left[ \begin{array}{c} c_1\\ c_2\\ \vdots \\ c_ n\\ \end{array} \right] \]

Then stacking

\[ \bX = \left[ \begin{array}{c} \bY \\ \bC \end{array} \right] \]

the variance of $\bX $ is

\[ \bV = \mr{Var}(\bX ) = \bSigma = \left( \begin{array}{cc} \bV _{11} & \bV _{12}\\ \bV _{21} & \bV _{22}\\ \end{array} \right) \]

where $\bV _{11} = \mr{Var}(\bY )$, $\bV _{12} = \mr{Cov}(\bY ,\bC )$, and $\bV _{22} = \mr{Var}(\bC )$. By using the preceding general result, the relevant covariance matrix is

\[ \tilde{\bV } = \bV _{11}-\bV _{12}\bV ^{-1}_{22}\bV _{21} \]

and the mean is

\[ \tilde{\bmu } = \bmu _1 + \bV _{12}\bV ^{-1}_{22}(\mb{c}-\bmu _2) \]

By using $\tilde{\bV }$ and $\tilde{\bmu }$, simulating $(\bY |\bC =\mb{c}) \sim N_ k(\tilde{\bmu },\tilde{\bV })$ now proceeds as in the unconditional case.