Spectral representation of Markov-switching bilinear processes

This article establishes a method for deriving the spectral representation of an inherently Markov-switching bilinear (MS-BL) process. The procedure is based on the application of the Riesz-Fisher theorem, which states that the spectral density can be obtained as the Fourier transform of the covariance function. We provide sufficient conditions for the second-order stationarity of MS-BL models, expressed in terms of the spectral radius of a specific matrix that involves the model's coefficients. The exact form of the spectral density function demonstrates that it is impossible to distinguish between an MS-ARMA and an MS-BL model solely based on the second-order properties of the process.