Towards Identification of Shocks in Linear State‐Space Models: Application to Stochastic Volatility Model

ABSTRACT State‐space models are widely used by statisticians because they allow a useful interpretation of some components of interest. Their efficient estimation, as well as the computation of forecasts or nonlinear functions of the observables, depends crucially on the correct specification of the error terms. Gaussianity is a common assumption explicitly used in the Kalman filter recursion, but departing from Gaussianity is of particular interest in fields such as finance, where there is a need for leptokurtic and/or asymmetric distributions to capture some features of the data. We introduce an approach based on the characteristic function or Laplace transform of the observed process and show that, for a large class of state‐space models (with finite second‐order moments and non‐zero higher‐order cumulants), it is possible to recover the cumulants of the structural shocks and the measurement errors from the cumulants and cross‐cumulants of the observed process and the first‐order parameters. This allows the statistician to design specification tests related to the properties of the structural shocks or measurement errors, separately or jointly, or of the data generating process (DGP) of the observed time series. In a non‐Gaussian framework, we design a test for the property that the DGP of the observed time series is a state‐space model with different shocks versus an ARMA DGP with only a single innovation process, but with the same second‐order properties. We illustrate the size and power properties of this test applied to a simple stochastic volatility model.