Abstract
This paper is devoted to show duality in the estimation of Markov Switching (MS) GARCH processes. It is well-known that MS GARCH models suffer of path dependence which makes the estimation step unfeasible with usual Maximum Likelihood procedure. However, by rewriting the model in a suitable state space representation, we are able to give a unique framework to reconcile the estimation obtained by filtering procedure with that coming from some auxiliary models proposed in the literature. Estimation on short-term interest rates shows the feasibility of the proposed approach.
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Appendix
Appendix
A1 We show that p t | t−1, t = p(s t | s t−1, Ψ t ) can be expressed in terms of p t | t−1, t−1 and the conditional density of ε t which depends on the current regime s t and the past regimes, i.e, f(ε t | s 1, …, s t , Ψ t−1). In fact,
where \(f(\epsilon _{t}\vert s_{1},\ldots,s_{t-1},\varPsi _{t-1}) =\sum _{ s_{t}=1}^{M}\,f(\epsilon _{t}\vert s_{1},\ldots,s_{t},\varPsi _{t-1})p(s_{t}\vert s_{t-1},\varPsi _{t-1}) =\sum _{ s_{t}=1}^{M}\,f(\epsilon _{t}\vert s_{1},\ldots,s_{t},\varPsi _{t-1})\,p_{t\vert t-1,t-1}.\)
A2 We present explicit derivation of the filter for MS GARCH as given in Sect. 3. The prediction step is obtained as follows
In particular, we have \(B_{t} - B_{t\vert t-1}^{(i,j)}\vert _{s_{t}=j}\) = F j (B t−1 − B t−1 | t−1 i) + Gv t . Then
and
Hence, \(\eta _{t\vert t-1}^{(i,j)}\vert _{\varPsi _{t-1},s_{t}=j,s_{t-1}=i} = H(B_{t} - B_{t\vert t-1}^{(i,j)}) + v_{t}\) and
Furthermore, the updating step is derived as follows. Define Z 1 = B t and Z 2 = η t | t−1 (i, j) = y t − y t | t−1 (i, j). Then μ 1 = E[Z 1 | Ψ t−1, s t = j, s t−1 = i] = B t | t−1 (i, j), μ 2 = E[Z 2 | Ψ t−1, s t = j, s t−1 = i] = 0, Σ 11 = P t | t−1 (i, j) and Σ 22 = f t | t−1 (i, j). We have
Thus \(Z_{1}\vert _{Z_{2},\varPsi _{t-1},s_{t}=j,s_{t-1}=i}\) is given by μ 1 | 2 = μ 1 + Σ 12 Σ 22 −1(Z 2 −μ 2), that is, B t | t (i, j) = B t | t−1 (i, j) + P t | t−1 (i, j) H ′[ f t | t−1 (i, j)]−1 η t | t−1 (i, j). Further, we have Σ 11 | 2 = Σ 11 −Σ 12 Σ 22 −1 Σ 21, hence P t | t (i, j) = P t | t−1 (i, j) − K t (i, j) HP t | t−1 (i, j), where K t (i, j) = P t | t−1 (i, j) H ′[ f t | t−1 (i, j)]−1 is the Kalman gain.
A3 Here we derive the approximation on the line of [20] applied to model in (8), which is Eq. (9):
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Billio, M., Cavicchioli, M. (2017). Markov Switching GARCH Models: Filtering, Approximations and Duality. In: Corazza, M., Legros, F., Perna, C., Sibillo, M. (eds) Mathematical and Statistical Methods for Actuarial Sciences and Finance . Springer, Cham. https://doi.org/10.1007/978-3-319-50234-2_5
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