Equality Restricted Random Variables: Densities and Sampling Algorithms

Many common statistical models can be specified as linear models with restrictions imposed on the parameters. A large amount of these models impose restrictions which do not allow for the analytical construction of the probability density function (pdf) of the parameters given the restrictions. This is often implicitly assumed which leads to an inconsistency as the pdf of the parameters of the linear specification under the imposed restrictions is then not nested within the assumed pdf of the unrestricted linear specification. The paper shows how these restrictions need to be incorporated by constructing the pdfs incorparating the restrictions and algorithms to sample from these pdfs. We show how these methods are applied to some common statistical models, i.e. ARMA, cointegration and simultaneous equation models.