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We provide a formal definition of an  M -state multivariate Markov switchingtrend, describe its asymptotic distribution, and consider vector autoregressive processes withtrends which contain either unit roots or a stationary part. Then, we estimate the coefficients of such models via ordinary least squares, and determine the asymptotic distributions ofestimators in terms of functionals on a multivariate Brownian motion.

Estimation and asymptotics for vector autoregressive models with unit roots and Markov switching trends