Inverse Gamma

Parameters

shape (\(a\)) and scale (\(b\)). An inverse gamma is the reciprocal of a gamma. The special case is the scaled inverse chi-squared with parameters \(\nu\) (degrees of freedom) and \(\tau ^2\) (scale parameter) which has \(a = \nu / 2\) and \(b = \nu \tau ^2 / 2\)

Kernel

\(x^{-(a+1)}\exp \left( -{b}{x^{-1}}\right) \)

Integrating Constant

\({b^a}/\Gamma (a)\)

Support

\((0,\infty)\)

Mean

\(\frac{b}{{(a - 1)}}\) if \(a > 1\)

Variance

\(\frac{b^2}{{{{(a - 1)}^2}(a - 2)}}\) if \(a > 2\)

Main Uses

Prior, exact and approximate posterior for the variance of residuals or other shocks in a model. For these purposes, it's usually simpler to directly use the scaled inverse chi-squared variation.

Density Function

%LOGINVGAMMADENSITY(x,a,b)

Random Draws

You can use b/%rangamma(a). A draw from a scaled inverse chi-squared is typically done with nu*tausqr/%ranchisqr(nu)

Moment Matching

%InvGammaParms(mean,sd) (external function) returns the 2-vector of parameters ( \((a,b)\) parameterization) for the parameters of an inverse gamma with the given mean and standard deviation. If sd is the missing value, this will return \(a=2\), which is the largest value of \(a\) which gives an infinite variance.