Normality Tests for Latent Variables

We exploit the rationale behind the Expectation Maximization algorithm to derive simple to implement and interpret score tests of normality in the innovations to the latent variables in state space models against generalized hyperbolic alternatives, including symmetric and asymmetric Student ts. We decompose our tests into third and fourth moment components, and obtain one-sided likelihood ratio analogues, whose asymptotic distribution we provide. When we apply them to a cointegrated dynamic factor model which combines the expenditure and income versions of US aggregate real output to improve its measurement, we reject normality if the sample period extends beyond the Great Moderation.