Estimating Markov-Switching ARMA Models with Extended Algorithms of Hamilton

This paper proposes two innovative algorithms to estimate a general class of N-state Markov-switching autoregressive moving-average (MS-ARMA) models with a sample of size T. To resolve the problem of NT possible routes induced by the presence of MA parameters, the first algorithm is built on Hamilton’s (1989) method and Gray’s (1996) idea of replacing the lagged error terms with their corresponding conditional expectations. We thus name it as the Hamilton-Gray (HG) algorithm. The second method refines the HG algorithm by recursively updating the conditional expectations of these errors and is named as the extended Hamilton-Gray (EHG) algorithm. The computational cost of both algorithms is very mild, because the implementation of these algorithms is very much similar to that of Hamilton (1989). The simulations show that the finite sample performance of the EHG algorithm is very satisfactory and is much better than that of the HG counterpart. We also apply the EHG algorithm to the issues of dating U.S. business cycles with the same real GNP data employed in Hamilton (1989). The turning points identified with the EHG algorithm resemble closely to those of the NBER’s Business Cycle Dating Committee and confirm the robustness of the findings in Hamilton (1989) about the effectiveness of Markov-switching models in dating U.S. business cycles.Markov-switching, ARMA process

Quantum Critical Spin-2 Chain with Emergent SU(3) Symmetry

We study the quantum critical phase of a SU(2) symmetric spin-2 chain obtained from spin-2 bosons in a one-dimensional lattice. We obtain the scaling of the entanglement entropy and finite-size energies by exact diagonalization and density-matrix renormalization group methods. From the numerical results of the energy spectrum, central charge, and scaling dimension we identify the conformal field theory describing the whole critical phase to be the SU(3)$_1$ Wess-Zumino-Witten model. We find that while in the whole critical phase the Hamiltonian is only SU(2) invariant, there is an emergent SU(3) symmetry in the thermodynamic limit