Markov Switching GARCH Models: Filtering, Approximations and Duality

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.

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,

$$\displaystyle{\begin{array}{rcl} p_{t\vert t-1,t}& =&p(s_{t}\vert s_{t-1},\varPsi _{t}) = p(s_{t}\vert s_{1},\ldots,s_{t-1},\varPsi _{t}) = p(s_{t}\vert s_{1},\ldots,s_{t-1},\epsilon _{t},\varPsi _{t-1}) \\ & =&\frac{f(\epsilon _{t}\vert s_{1},\ldots,s_{t},\varPsi _{t-1})p(s_{t}\vert s_{1},\ldots,s_{t-1},\varPsi _{t-1})} {f(\epsilon _{t}\vert s_{1},\ldots,s_{t-1},\varPsi _{t-1})} \\ & =&\frac{f(\epsilon _{t}\vert s_{1},\ldots,s_{t},\varPsi _{t-1})p(s_{t}\vert s_{t-1},\varPsi _{t-1})} {f(\epsilon _{t}\vert s_{1},\ldots,s_{t-1},\varPsi _{t-1})} = \frac{f(\epsilon _{t}\vert s_{1},\ldots,s_{t},\varPsi _{t-1})p_{t\vert t-1,t-1}} {f(\epsilon _{t}\vert s_{1},\ldots,s_{t-1},\varPsi _{t-1})} \end{array} }$$

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

$$\displaystyle{\begin{array}{rcl} B_{t\vert t-1}^{(i,j)} & =&E(B_{t}\vert \varPsi _{t-1},s_{t} = j,s_{t-1} = i) = E(\mu _{s_{t}} + F_{s_{t}}B_{t-1} + Gv_{t}\vert \varPsi _{t-1},s_{t} = j,s_{t-1} = i) \\ & =&\mu _{j} + F_{j}E(B_{t-1}\vert \varPsi _{t-1},s_{t} = j,s_{t-1} = i) =\mu _{j} + F_{j}B_{t-1\vert t-1}^{i}.\end{array} }$$

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

$$\displaystyle{\begin{array}{rcl} P_{t\vert t-1}^{(i,j)} & =&E[(B_{t} - B_{t\vert t-1}^{(i,j)})(B_{t} - B_{t\vert t-1}^{(i,j)})^{'}\vert \varPsi _{t-1},s_{t} = j,s_{t-1} = i] \\ & =&E[(F_{j}(B_{t-1} - B_{t-1\vert t-1}^{i}) + Gv_{t})(F_{j}(B_{t-1} - B_{t-1\vert t-1}^{i}) + Gv_{t})^{'}\vert \varPsi _{t-1},s_{t} = j,s_{t-1} = i] \\ & =&F_{j}E[(B_{t-1} - B_{t-1\vert t-1}^{i})(B_{t-1} - B_{t-1\vert t-1}^{i})^{'}\vert \varPsi _{t-1},s_{t-1} = i]F_{j}^{'} + GE(v_{t}^{2}\vert s_{t} = j)G^{'} \\ & =&F_{j}P_{t-1\vert t-1}^{i}F_{j}^{'} + GG^{'}\sigma _{vj}^{2}\end{array} }$$

and

$$\displaystyle{\begin{array}{rcl} \eta _{t\vert t-1}^{(i,j)} & =&y_{t} - y_{t\vert t-1}^{(i,j)} = y_{t} - E(y_{t}\vert \varPsi _{t-1},s_{t} = j,s_{t-1} = i) = y_{t} - E(HB_{t}\vert \varPsi _{t-1},s_{t} = j,s_{t-1} = i) \\ & =&y_{t} - HE(B_{t}\vert \varPsi _{t-1},s_{t} = j,s_{t-1} = i) = y_{t} - HB_{t\vert t-1}^{(i,j)}.\end{array} }$$

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

$$\displaystyle{\begin{array}{rcl} f_{t\vert t-1}^{(i,j)} & =&E[(\eta _{t\vert t-1}^{(i,j)})^{2}\vert \varPsi _{t-1},s_{t} = j,s_{t-1} = i)] \\ & =&E[(H(B_{t} - B_{t\vert t-1}^{(i,j)}))(H(B_{t} - B_{t\vert t-1}^{(i,j)}))^{'}\vert \varPsi _{t-1},s_{t} = j,s_{t-1} = i] \\ & =&HE[(B_{t} - B_{t\vert t-1}^{(i,j)})(B_{t} - B_{t\vert t-1}^{(i,j)})^{'}\vert \varPsi _{t-1},s_{t} = j,s_{t-1} = i]H^{'} = HP_{t\vert t-1}^{(i,j)}H^{'}.\end{array}}$$

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

$$\displaystyle{\begin{array}{rcl} \varSigma _{12} & =&cov(Z_{1},Z_{2}) = E[(B_{t} - B_{t\vert t-1}^{(i,j)})\eta _{t\vert t-1}^{(i,j)}\vert \varPsi _{t-1},s_{t} = j,s_{t-1} = i] \\ & =&E[(B_{t} - B_{t\vert t-1}^{(i,j)})(B_{t} - B_{t\vert t-1}^{(i,j)})^{'}H^{'}\vert \varPsi _{t-1},s_{t} = j,s_{t-1} = i] = P_{t\vert t-1}^{(i,j)}H^{'} =\varSigma _{ 21}^{'}.\end{array}}$$

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):

$$\displaystyle{\begin{array}{rcl} B_{t\vert t}^{j}& =&\frac{\sum _{i=1}^{M}B_{ t\vert t}^{(i,j)}p(s_{ t-1}=i,s_{t}=j\vert Y _{t})} {p(s_{t}=j\vert Y _{t})} =\sum _{ i=1}^{M}\frac{p(s_{t-1}=i,s_{t}=j\vert Y _{t})} {p(s_{t}=j\vert Y _{t})} B_{t\vert t}^{(i,j)} \\ & =&\sum _{i=1}^{M}p(s_{t-1} = i\vert s_{t} = j,Y _{t})\,\,B_{t\vert t}^{(i,j)} =\sum _{ i=1}^{M}p_{i,t-1\vert t,t}\,\,B_{t\vert t}^{(i,j)}. \end{array} }$$