We exploit the rationale behind the Expectation Maximization algorithm to derive simple to implement and interpret LM normality tests for the innovations of the latent variables in linear 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 our tests to a common trend 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.

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We analyze a model for N different measurements of a persistent latent time series when measurement errors are mean-reverting, which implies a common trend among measurements. We study the consequences of overdifferencing, finding potentially large biases in maximum likelihood estimators of the dynamics parameters and reductions in the precision of smoothed estimates of the latent variable, especially for multiperiod objects such as quinquennial growth rates. We also develop an R 2 measure of common trend observability that determines the severity of misspecification. Finally, we apply our framework to US quarterly data on GDP and GDI, obtaining an improved aggregate output measure.

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