Markov switching asymmetric GARCH model: stability and forecasting

Abstract

A new Markov switching asymmetric GARCH model is proposed where each state follows the smooth transition GARCH model, represented by Lubrano (Recherches Economiques de Louvain 67:257–287, 2001), that follows a logistic smooth transition structure between effects of positive and negative shocks. This consideration provides better forecasts than GARCH, Markov switching GARCH and smooth transition GARCH models, in many financial time series. The asymptotic finiteness of the second moment is investigated. The parameters of the model are estimated by applying MCMC methods through Gibbs and griddy Gibbs sampling. Applying the log return of some part of $S \& P\ 500$ indices, we show the competing performance of in sample fit and out of sample forecast volatility and value at risk of the proposed model. The Diebold–Mariano test shows that the presented model outperforms all competing models in forecast volatility.

Appendices

Appendix A

Proof of Lemma 3.1

As the hidden variables $\{Z_{t}\}_{t\ge 1}$ have Markov structure in MS-STARCH model, so

$$\begin{aligned} \alpha _j^{(t)}&=p(Z_t=j|{\mathcal {I}}_{t-1})=\sum _{m=1}^{K}{P(Z_t=j,Z_{t-1}=m|{\mathcal {I}}_{t-1})} \nonumber \\&=\sum _{m=1}^{K}{p(Z_t=j|Z_{t-1}=m,{\mathcal {I}}_{t-1})p(Z_{t-1}=m|{\mathcal {I}}_{t-1})} \nonumber \\&=\sum _{m=1}^{K}{p(Z_t=j|Z_{t-1}=m)p(Z_{t-1}=m|{\mathcal {I}}_{t-1})} \nonumber \\&=\frac{\sum _{m=1}^{K}{f({\mathcal {I}}_{t-1},Z_{t-1}=m)p_{m,j}}}{\sum _{m=1}^{K}{f({\mathcal {I}}_{t-1},Z_{t-1}=m)}} \nonumber \\&=\frac{\sum _{m=1}^{K}{f(y_{t-1}|Z_{t-1}=m,{\mathcal {I}}_{t-2})}\alpha _m^{(t-1)}p_{m,j}}{\sum _{m=1}^{K}{f(y_{t-1}|Z_{t-1}=m,{\mathcal {I}}_{t-2})\alpha _m^{(t-1)}}}. \end{aligned}$$

(7.1)

$\square$

Appendix B

Proof of Theorem 3.1

Let $E_{t}(.)$ denotes the expectation with respect to the information up to time t. Thus the second moment of the model can be calculated as, see Abramson and Cohen (2007):

$$\begin{aligned} E(y^2_t)= & {} E(H_{Z_t,t})=E_{Z_t}[E_{t-1}(H_{Z_t,t}|z_t)]\nonumber \\= & {} \sum _{z_t=1}^{K}{\pi _{z_t}E_{t-1}(H_{Z_t,t}|z_t)}. \end{aligned}$$

(7.2)

Also let $E(.|z_t)$ and $p(.|z_t)$ denote $E(.|Z_t=z_t)$ and $P(.|Z_t=z_t)$, respectively, where $z_t$ is the realization of the state at time t. Applying to the method of Medeiros and Veiga (2009), we find an upper bound of $E_{t-1}(H_{m,t}|z_t)$, for $m=1,2,\ldots ,K$ by the following

$$\begin{aligned} E_{t-1}(H_{m,t}|z_t)&= E_{t-1}(a_{0m}+a_{1m}y^2_{t-1}(1-w_{m,t-1}) +a_{2m}y^2_{t-1}w_{m,t-1}+b_mH_{m,t-1}|z_t) \nonumber \\&=\underbrace{a_{0m}}_{I}+\underbrace{a_{1m}E_{t-1}[y^2_{t-1}|z_t]}_{II}+\underbrace{(a_{2m}-a_{1m})E_{t-1}[y^2_{t-1}w_{m,t-1}|z_t]}_{III} \nonumber \\&\quad +\underbrace{b_mE_{t-1}[H_{m,t-1}|z_t]}_{IV}. \end{aligned}$$

(7.3)

The term (II) in (7.3) can be interpreted as follows:

$$\begin{aligned} E_{t-1}[y^2_{t-1}|z_t]= & {} \sum _{z_{t-1}=1}^{K} {\int _{S_{{\mathcal {I}}_{t-1}}}{y^2_{t-1}p({\mathcal {I}}_{t-1}|z_t,z_{t-1})p(z_{t-1}|z_t)d{\mathcal {I}}_{t-1}}}\nonumber \\= & {} \sum _{z_{t-1}=1}^{K}{p(z_{t-1}|z_t)E_{t-2}[H_{Z_{t-1},t-1}|z_{t-1}]}, \end{aligned}$$

(7.4)

where $S_{{\mathcal {I}}_{t-1}}$ is the support of ${\mathcal {I}}_{t-1}=(y_1,\ldots ,y_{t-1})$. $\square$

Upper bound for III in (7.3): Let $0<M<\infty$ be a constant, so

$$\begin{aligned} E_{t-1}[y^2_{t-1}w_{m,t-1}|z_t]=&E_{t-1}[y^2_{t-1}w_{m,t-1}I_{|y_{t-1}|<M}|z_t]\\&+E_{t-1}[y^2_{t-1}w_{m,t-1}I_{|y_{t-1}|\ge M}|z_t] \end{aligned}$$

in which

$$\begin{aligned} I_{x<a}=\left\{ \begin{array}{ll} 1 &{} \hbox { if}\ x<a\\ 0 &{} \text {otherwise.} \end{array} \right. \end{aligned}$$

As by (2.4), $0<w_{m,t-1}<1$ and so

$$\begin{aligned} E_{t-1}[y^2_{t-1}w_{m,t-1}|z_t]\le M^2+E_{t-1}[y^2_{t-1}w_{m,t-1}I_{|y_{t-1}|\ge M}|z_t], \end{aligned}$$

also

$$\begin{aligned} E_{t-1}[y^2_{t-1}w_{m,t-1}I_{|y_{t-1}|\ge M}|z_t]&=\int _{S_{{\mathcal {I}}_{t-2}},y_{t-1}\le -M}{y^2_{t-1}[w_{m,t-1}]p({\mathcal {I}}_{t-1}|z_t)d{\mathcal {I}}_{t-1}}\\&\quad +\int _{S_{{\mathcal {I}}_{t-2}},y_{t-1}\ge M}{y^2_{t-1}[w_{m,t-1}]p({\mathcal {I}}_{t-1}|z_t)d{\mathcal {I}}_{t-1}}, \end{aligned}$$

by (2.4),

$$\begin{aligned} \lim _{y_{t-1}\rightarrow +\infty }w_{m,t-1}=1 \quad \lim _{y_{t-1}\rightarrow -\infty }w_{m,t-1}=0. \end{aligned}$$

(7.5)

So for any fixed positive small number $\delta >0$, we can consider $M>0$ so large that for $y_{t-1}\ge M$, $|w_{m,t-1}-1|\le \delta$ and for $y_{t-1}\le -M$, $|w_{m,t-1}|\le \delta$. Hence

$$\begin{aligned} E_{t-1}[y^2_{t-1}w_{m,t-1}I_{|y_{t-1}|\ge M}|z_t]&\le \delta \int _{S_{{\mathcal {I}}_{t-2}},y_{t-1}\le -M}{y^2_{t-1}p({\mathcal {I}}_{t-1}|z_t)d{\mathcal {I}}_{t-1}}\\&\quad + (\delta +1) \int _{S_{{\mathcal {I}}_{t-2}},y_{t-1}\ge M}{y^2_{t-1}p({\mathcal {I}}_{t-1}|z_t)d{\mathcal {I}}_{t-1}}. \end{aligned}$$

Since the distribution of the $\{\varepsilon _{t}\}$ is symmetric, then

$$\begin{aligned} \delta \int _{S_{{\mathcal {I}}_{t-2}},y_{t-1}\le -M}{y^2_{t-1} p({\mathcal {I}}_{t-1}|z_t)d{\mathcal {I}}_{t-1}}&\le \delta \int _{S_{{\mathcal {I}}_{t-2}},-\infty<y_{t-1}<0}{y^2_{t-1}p({\mathcal {I}}_{t-1}|z_t)d{\mathcal {I}}_{t-1}}\\&=\delta \frac{ E_{t-1}[y^2_{t-1}|z_t]}{2} \end{aligned}$$

and

$$\begin{aligned} (\delta +1)\int _{S_{{\mathcal {I}}_{t-2}},y_{t-1}\ge M}{y^2_{t-1}p({\mathcal {I}}_{t-1}|z_t)d{\mathcal {I}}_{t-1}}&\le (\delta +1)\int _{S_{{\mathcal {I}}_{t-2}},0<y_{t-1}<\infty }{y^2_{t-1} p({\mathcal {I}}_{t-1}|z_t)d{\mathcal {I}}_{t-1}}\\&=(\delta +1)\frac{ E_{t-1}[y^2_{t-1}|z_t]}{2}. \end{aligned}$$

Therefore

$$\begin{aligned} E_{t-1}[y^2_{t-1}w_{m,t-1}|z_t]\le M^2+(\delta +\frac{1}{2})E_{t-1}[y^2_{t-1}|z_t]. \end{aligned}$$

Upper bound for IV in (7.3):

$$\begin{aligned} b_{m}E_{t-1}(H_{m,t-1}|z_t)= & {} b_{m}{\int _{S_{{\mathcal {I}}_{t-1}}}{H_{m,t-1}p({\mathcal {I}}_{t-1}|z_t)d{\mathcal {I}}_{t-1}}}\nonumber \\= & {} b_{m}\sum _{z_{t-1}=1}^{K} {p(z_{t-1}|z_t)E_{t-2}(H_{m,t-1}|z_{t-1})}. \end{aligned}$$

(7.6)

By replacing the obtained upper bounds and relations (7.4) in (7.3), the upper bound for $E_{t-1}(H_{m,t}|z_t)$ is obtained by:

$$\begin{aligned} E_{t-1}(H_{m,t}|z_t)&\le a_{0m}+|a_{2m}-a_{1m}|M^2 \nonumber \\&\quad +\sum _{z_{t-1}=1}^{K}{{[a_{1m}+|a_{2m}-a_{1m}|(\delta +\frac{1}{2})] p(z_{t-1}|z_t)}E_{t-2}[H_{z_{t-1},t-1}|z_{t-1}]}\nonumber \\&\quad +\sum _{z_{t-1}=1}^{K}{b_mp(z_{t-1}|z_t)E_{t-2}(H_{m,t-1}|z_{t-1})}, \end{aligned}$$

(7.7)

in which by Bayes’ rule

$$\begin{aligned} p(z_{t-i}|z_t)=\frac{\pi _{z_{t-i}}}{\pi _{z_t}}\{P_{z_{t-i}z_t }\}, \end{aligned}$$

where P is the transition probability matrix.

Let $A_t(j,k)=E_{t-1}[H_{j,t}|Z_{t}=k]$, ${A}_t=[A_t(1,1),A_t(2,1),\ldots ,A_t(K,1),A_t(1,2),\ldots ,A_t(K,K)]$ be a $K^2$-by-1 vector and consider $\dot{ { {\Omega }}}=( { {\Omega }}^{\prime },\ldots , { {\Omega }}^{\prime })^{\prime }$ be a vector that is made of K vector ${ {\Omega }}$. By (7.23)–(7.26), the following recursive inequality is attained,

$$\begin{aligned} \mathbf{A }_t\le \dot{ { {\Omega }}}+ \mathbf{{C}}\mathbf{A }_{t-1},\quad t\ge 0. \end{aligned}$$

(7.8)

with some initial conditions $\mathbf{A }_{-1}.$ The relation (7.8) implies that

$$\begin{aligned} {A}_t\le \dot{{ {\Omega }}}\sum _{i=0}^{t-1}{{ {C}}^i}+{ {C}}^{t} {A}_{0}:= {B}_{t}. \end{aligned}$$

(7.9)

Following the matrix convergence theorem Lancaster and Tismenetsky (1985), the necessary condition for the convergence of ${B}_{t}$ when $t\rightarrow \infty$ is that $\rho ({ {C}}) <1$. Under this condition, ${ {C}}^t$ converges to zero as t goes to infinity and $\sum _{i=0}^{t-1}{{ {C}}^i}$ converges to $(I-{ {C}})^{-1}$ provided that matrix $(I-{ {C}})$ is invertible. So if $\rho ({ {C}}) <1$,

$$\begin{aligned} \lim _{t\rightarrow \infty } {A}_t\le (I-{ {C}})^{-1}\dot{{ {\Omega }}}. \end{aligned}$$

By (7.2) the upper bound for the asymptotic behavior of unconditional variance is given by

$$\begin{aligned} \lim _{t\rightarrow \infty }E(y^2_t)\le { {\Pi ^\prime (I-C)^{-1}}}\dot{{ {\Omega }}}. \end{aligned}$$