Inverse Chi-Squared (scaled)

Parameters	Degrees of freedom ($\nu$) and scale ($\tau ^2$). An inverse chi-squared is the reciprocal of a chi-squared combined with scaling factor which represents a "target'' variance that the distribution is intended to represent. (Note that the mean is roughly $\tau ^2$ for large degrees of freedom.)
Kernel	$x^{-(a+1)}\exp \left( -{b}{x^{-1}}\right) $
Support	$\left[ 0,\infty \right) $
Mean	$\frac{{\nu {\tau ^2}}}{{\nu - 2}}$ if $\nu > 2$
Variance	$\frac{{2{\nu ^2}{\tau ^4}}}{{{{\left( {\nu - 2} \right)}^2}(\nu - 4)}}$ if $\nu > 4$
Main Uses	Prior, exact and approximate posterior for the variance of residuals or other shocks in a model. The closely-related inverse gamma can be used for that as well, but the scaled inverse chi-squared tends to be more intuitive.
Density Function	%LogInvChiSqrDensity(x,nu,tausq)
Moment Matching	%InvChisqrParms(mean,sd) (external function) returns the 2-vector of parameters ( $(\nu,\tau ^2)$ in that order) for the parameters of an inverse chi-squared with the given mean and standard deviation. If sd is the missing value, this will return $\nu=4$, which is the largest value of $\nu$ which gives an infinite variance.
Random Draws	You can use (nu*tausq)/%ranchisqr(nu). Note that you divide by the random chi-squared.

Parameters	Degrees of freedom (\(\nu\)) and scale (\(\tau ^2\)). An inverse chi-squared is the reciprocal of a chi-squared combined with scaling factor which represents a "target'' variance that the distribution is intended to represent. (Note that the mean is roughly \(\tau ^2\) for large degrees of freedom.)
Kernel	\(x^{-(a+1)}\exp \left( -{b}{x^{-1}}\right) \)
Support	\(\left[ 0,\infty \right) \)
Mean	\(\frac{{\nu {\tau ^2}}}{{\nu - 2}}\) if \(\nu > 2\)
Variance	\(\frac{{2{\nu ^2}{\tau ^4}}}{{{{\left( {\nu - 2} \right)}^2}(\nu - 4)}}\) if \(\nu > 4\)
Main Uses	Prior, exact and approximate posterior for the variance of residuals or other shocks in a model. The closely-related inverse gamma can be used for that as well, but the scaled inverse chi-squared tends to be more intuitive.
Density Function	%LogInvChiSqrDensity(x,nu,tausq)
Moment Matching	%InvChisqrParms(mean,sd) (external function) returns the 2-vector of parameters ( \((\nu,\tau ^2)\) in that order) for the parameters of an inverse chi-squared with the given mean and standard deviation. If sd is the missing value, this will return \(\nu=4\), which is the largest value of $\nu$ which gives an infinite variance.
Random Draws	You can use (nu*tausq)/%ranchisqr(nu). Note that you divide by the random chi-squared.