1. What is the unique stationary solution of the MGVAR model?
The unique stationary solution of the MGVAR model is given by the process (x it), defined as the ith component of x t. This solution is obtained when the top Lyapunov exponent, L(F), is strictly negative. The sufficient condition for strict stationarity is stated in Theorem 3.1. The solution is defined by the equation (7) x t = F t x t-1 + t (6) x t = t + + h=1 F t F t-1 F 1 ||. This solution is obtained by solving the MGVAR model equations (1) and (6) simultaneously for all x it, i = 0, ..., N. The solution is strictly stationary when the limit of products of infinitely many random matrices is estimated using Monte Carlo simulations based on equation (6).
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2. How does the EM-type algorithm perform in simulation experiments?
The EM-type algorithm's performance in simulation experiments is demonstrated through numerical examples using bivariate MGVAR(p i , q i ) processes. The algorithm estimates the unknown parameters and steady-state probabilities with mean values close to the true values. The algorithm's efficiency is attributed to avoiding numerical maximization of the likelihood at each iteration. The simulation results confirm the validity of theoretical results and highlight the algorithm's usefulness. The algorithm's performance is further supported by the roots mean standard errors (RMSE) and the comparison of estimated parameters in different states. Overall, the simulation experiments showcase the effectiveness of the proposed EM-type algorithm in estimating parameters and probabilities in bivariate MGVAR processes.
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