Statistical inference for mixture GARCH models with financial application

Abstract

In this paper we consider mixture generalized autoregressive conditional heteroskedastic models, and propose a new iteration algorithm of type EM for the estimation of model parameters. The maximum likelihood estimates are shown to be consistent, and their asymptotic properties are investigated. More precisely, we derive simple expressions in closed form for the asymptotic covariance matrix and the expected Fisher information matrix of the ML estimator. Finally, we study the model selection and propose testing procedures. A simulation study and an application to financial real-series illustrate the results.

Appendix

Proof of Proposition 3.1

Let $f(y_t | \mathbf{z}_t, \mathbf{Y}_{t - 1}; {\varvec{\lambda }})$ denote the conditional density, and $Pr(\mathbf{z}_t | \mathbf{z}_{t - 1}; {\varvec{\lambda }})$ the conditional probability, for every $t = 1, \dots , T$.

Set

$$\begin{aligned} f(\mathbf{Y}_T | \mathbf{z}; {\varvec{\lambda }}) \, = \, \prod _{t = 1}^T\, f(y_t | \mathbf{z}_t, \mathbf{Y}_{t - 1}; {\varvec{\lambda }}) \end{aligned}$$

and

$$\begin{aligned} Pr(\mathbf{z} | {\varvec{\lambda }}) \, = \, \prod _{t = 1}^T \, Pr(\mathbf{z}_t | \mathbf{z}_{t - 1}; {\varvec{\lambda }}). \end{aligned}$$

Then the likelihood function can be written as

$$\begin{aligned} L({\varvec{\lambda }} | \mathbf{Y}_T) \, = \, \int \, f(\mathbf{Y}_T, \mathbf{z} | {\varvec{\lambda }})\, d\mathbf{z} \, = \, \int \, f(\mathbf{Y}_T | \mathbf{z}; {\varvec{\lambda }})\, Pr(\mathbf{z} | {\varvec{\lambda }}) \, d\mathbf{z} \end{aligned}$$

where the integration denotes summation over all possible values of $\mathbf{z} \, = \, \mathbf{z}_T \otimes \mathbf{z}_{T - 1} \otimes \cdots \otimes \mathbf{z}_1$. Here $\otimes$ is the usual Kronecker product. The derivation of the log likelihood function $\mathcal{L}({\varvec{\lambda }} | \mathbf{Y}_T) \, = \, {\text {ln}} \, L({\varvec{\lambda }} | \mathbf{Y}_T)$ with respect to ${\varvec{\lambda }}$ leads to the score function $S({\varvec{\lambda }})$.

We have

$$\begin{aligned} \begin{aligned} \frac{\partial \, {{{\mathcal {L}}}}({\varvec{\lambda }} | \mathbf{Y}_T)}{\partial \, {\varvec{\lambda }}}&\, = \, \frac{\partial \, {\text {ln}} \, L({\varvec{\lambda }} | \mathbf{Y}_T)}{\partial \, {\varvec{\lambda }}} \\&\, = \, \frac{1}{L({\varvec{\lambda }} | \mathbf{Y}_T)} \, \int \, \frac{\partial \, f(\mathbf{Y}_T | \mathbf{z}; {\varvec{\lambda }})}{\partial \, {\varvec{\lambda }}} \, Pr (\mathbf{z} | {\varvec{\lambda }}) \, d\mathbf{z} \\&\, = \, \frac{1}{L({\varvec{\lambda }} | \mathbf{Y}_T)} \, \int \, \frac{\partial \, {\text {ln}} \, f(\mathbf{Y}_T | \mathbf{z}; {\varvec{\lambda }})}{\partial \, {\varvec{\lambda }}} \, f(\mathbf{Y}_T | \mathbf{z}; {\varvec{\lambda }}) \, Pr (\mathbf{z} | {\varvec{\lambda }}) \, d\mathbf{z} \\&\, = \, \int \, \frac{\partial \, {\text {ln}} \, f(\mathbf{Y}_T | \mathbf{z}; {\varvec{\lambda }})}{\partial \, {\varvec{\lambda }}} \, Pr(\mathbf{z} | \mathbf{Y}_T ; {\varvec{\lambda }}) \, d\mathbf{z} \end{aligned} \end{aligned}$$

as

$$\begin{aligned} \frac{f(\mathbf{Y}_T | \mathbf{z}; {\varvec{\lambda }}) \, Pr (\mathbf{z} | {\varvec{\lambda }})}{L({\varvec{\lambda }} | \mathbf{Y}_T)} \, = \, \frac{f(\mathbf{Y}_T | \mathbf{z}; {\varvec{\lambda }}) \, Pr (\mathbf{z} | {\varvec{\lambda }})}{\int \, f(\mathbf{Y}_T, \mathbf{z} | {\varvec{\lambda }}) \, d\mathbf{z}} \, = \, Pr (\mathbf{z} | \mathbf{Y}_T; {\varvec{\lambda }}) \end{aligned}$$

from the Bayes theorem. In the second line of the above computation, we have used the fact that $Pr (\mathbf{z} | {\varvec{\lambda }}) \, = \, E(\mathbf{z} | {\varvec{\lambda }})$ is a constant function with respect to ${\varvec{\lambda }}$, hence its derivative vanishes.

Thus the score conditioned on a given regime vector $\mathbf{z}$ has the expression

$$\begin{aligned} S({\varvec{\lambda }}) \, = \, \frac{\partial \, \mathcal{L}({\varvec{\lambda }} | \mathbf{Y}_T)}{\partial \, {\varvec{\lambda }}} \, = \, \sum _{t = 1}^T \, S_t ({\varvec{\lambda }}) \end{aligned}$$

where

$$\begin{aligned} S_t ({\varvec{\lambda }}) \, = \, \frac{\partial \, \ell _t}{\partial \, {\varvec{\lambda }}} \, = \, \sum _{m = 1}^M \, \frac{\partial \, {\text {ln}} \, f(y_t | z_{tm}, \mathbf{Y}_{t - 1}; {\varvec{\lambda }})}{\partial \, {\varvec{\lambda }}} \, Pr(z_{tm} | \mathbf{Y}_t; {\varvec{\lambda }}). \end{aligned}$$

This proves (7) and (8).

Since

$$\begin{aligned} Pr (z_{tm} | \mathbf{Y}_t; {\varvec{\lambda }}) \, = \, E(z_{tm} | \mathbf{Y}_t; {\varvec{\lambda }}) \, = \, \tau _{tm | t}, \end{aligned}$$

we get the following relation

$$\begin{aligned} S_t ({\varvec{\lambda }}_m) \, = \, \frac{\partial \, \ell _t}{\partial \, {\varvec{\lambda }}_m} \, = \, \frac{\partial \, {\text {ln}} \, f(y_t | z_{tm}, \mathbf{Y}_{t - 1}; {\varvec{\lambda }})}{\partial \, {\varvec{\lambda }}_m} \, \tau _{tm | t} \end{aligned}$$

(19)

where

$$\begin{aligned} {\text {ln}}\, f(y_t | z_{tm}, \mathbf{Y}_{t - 1}; {\varvec{\lambda }}) \, = \, {\text {ln}}\, \pi _m \, - \, \frac{1}{2} \, {\text {ln}}\, h_{tm} \, - \, \frac{1}{2} \, \frac{(y_t \, - \, \mu _m)^2}{h_{tm}}. \end{aligned}$$

(20)

Deriving (20) with respect to ${\varvec{\theta }}_m$ gives

$$\begin{aligned} \begin{aligned} \frac{\partial \, {\text {ln}} \, f(y_t | z_{tm}, \mathbf{Y}_{t - 1}; {\varvec{\lambda }})}{\partial \, {\varvec{\theta }}_m}&\, = \, - \, \frac{1}{2} \, \frac{1}{h_{tm}} \, \frac{\partial \, h_{tm}}{\partial \, {\varvec{\theta }}_m} \, + \, \frac{1}{2} \, (y_t \, - \, \mu _m)^{2} \, h_{tm}^{- 2} \, \frac{\partial \, h_{tm}}{\partial \, {\varvec{\theta }}_m} \\&\, = \, -\, \frac{1}{2} \, \frac{\partial \,{\text {ln}}\, h_{tm}}{\partial \, {\varvec{\theta }}_m} \, + \, \frac{1}{2} \, \frac{(y_t \, - \, \mu _m)^{2}}{h_{tm}}\, \frac{\partial \, {\text {ln}} \, h_{tm}}{\partial \, {\varvec{\theta }}_m} \\&\, = \, \frac{1}{2} \, \left[ \frac{(y_t \, - \, \mu _m)^{2}}{h_{tm}} \, - \, 1\right] \, \frac{\partial \, {\text {ln}} \, h_{tm}}{\partial \, {\varvec{\theta }}_m}. \end{aligned} \end{aligned}$$

(21)

Substituting (21) into (19) yields formula (6) in Proposition 3.1.

Deriving (20) with respect to $\mu _m$ and $\pi _m$ produces

$$\begin{aligned} \frac{\partial \, {\text {ln}}\, f(y_t | z_{tm}, \mathbf{Y}_{t - 1}; {\varvec{\lambda }})}{\partial \, \mu _m} \, = \, - \, \frac{1}{2} \, \frac{2 (y_t \, - \, \mu _m)}{h_{tm}} \, ( - 1) \, = \, (y_t \, - \, \mu _m)\, h_{tm}^{- 1} \end{aligned}$$

(22)

and

$$\begin{aligned} \frac{\partial \, {\text {ln}}\, f(y_t | z_{tm}, \mathbf{Y}_{t - 1}; {\varvec{\lambda }})}{\partial \, \pi _m} \, = \, \pi _{m}^{- 1}. \end{aligned}$$

(23)

Substituting (22) resp. (23) into (19) gives formulae in (9). This completes the proof of Proposition 3.1.

Derivation of (10). From (8) and (9) we get

$$\begin{aligned} \frac{\partial \, J({\varvec{\lambda | \mathbf{Y}}_T})}{\partial \, \pi _m}\, = \, \frac{\partial \, {{{\mathcal {L}}}}({\varvec{\lambda }} | \mathbf{Y}_T)}{\partial \, \pi _m}\, - \, \lambda \, = \, \sum _{t = 1}^T \,\frac{\partial \, \ell _t}{\partial \, \pi _m}\, - \, \lambda \, = \, \sum _{t = 1}^T \, \pi _{m}^{- 1} \, \tau _{t m | t}\, - \, \lambda = 0 \end{aligned}$$

hence

$$\begin{aligned} \lambda \, = \, \sum _{t = 1}^T \, {{\hat{\pi }}}_{m}^{- 1} \, {\hat{\tau }}_{t m | t}. \end{aligned}$$

Note that the expectation of $\frac{\partial \, z_{t m}}{\partial \, \pi _m}$, conditional on $\mathbf{Y}_{t - 1}$, vanishes. Summing up over $m = 1, \dots , M$ produces

$$\begin{aligned} \lambda \, = \, \lambda \, \left( \sum _{m = 1}^M \, {{\hat{\pi }}}_m \right) \, = \, \sum _{m = 1}^M \, \sum _{t = 1}^T \, {\hat{\tau }}_{t m | t} \, = \, \sum _{t = 1}^T \, \left( \sum _{m = 1}^M \, {{\hat{\tau }}}_{t m | t}\right) \, = \, \sum _{t = 1}^T \, 1 \, = \, T. \end{aligned}$$

Thus relation (10) holds.

Derivation of (11). From (8) and (9) we have

$$\begin{aligned} \frac{\partial \, J({\varvec{\lambda }} | \mathbf{Y}_T)}{\partial \, \mu _m} \, = \, \frac{\partial \, {{{\mathcal {L}}}}({\varvec{\lambda }} | \mathbf{Y}_T)}{\partial \, \mu _m} \, = \, \sum _{t = 1}^T \, S_t (\mu _m) \, = \, \sum _{t = 1}^T \, h_{t m}^{- 1} \, (y_t \, - \, \mu _m)\, {\tau }_{t m | t} \, = \, 0 \end{aligned}$$

hence

$$\begin{aligned} \sum _{t = 1}^T \, {{\hat{h}}}_{t m}^{- 1} \, y_t \, {{\hat{\tau }}}_{t m | t} \, = \, {{\hat{\mu }}}_m \, \sum _{t = 1}^T \, {{\hat{h}}}_{t m}^{- 1} \, {{\hat{\tau }}}_{t m | t}. \end{aligned}$$

Note that the expectation of $\frac{\partial \, z_{t m}}{\partial \, \mu _m}$, conditional on $\mathbf{Y}_{t - 1}$, vanishes. Then we have

$$\begin{aligned} {{\hat{\mu }}}_m \, = \left[ \sum _{t = 1}^T \, {{\hat{h}}}_{t m}^{- 1} \, {{\hat{\tau }}}_{t m | t} \right] ^{- 1} \, \left[ \sum _{t = 1}^T \, y_t \, {{\hat{h}}}_{t m}^{- 1} \, {{\hat{\tau }}}_{t m | t} \right] . \end{aligned}$$

But $y_t = {\mu }_m^{0} \, + \, \epsilon _t$ and $\epsilon _t \, = \, \sqrt{h_{t m}^{0}}\, \eta _t$, where ${\mu }_m^0$ is the true value of the intercept and $h_{tm}^0$ denote the function $h_{t m}$ evaluated at the true value ${\varvec{\lambda }}_{m}^{0}$. Substituting this relation into the above expression of ${\hat{\mu }}_m$ yields (11). $\square$

Proof of Proposition 3.2

The consistency of ${\hat{\pi }}_m$ follows immediately from (10).

Using (11) and (12), the consistency of ${{\hat{\mu }}}_m$ follows from the sequence

$$\begin{aligned} \begin{aligned} {\text {plim}}_{T \rightarrow \infty } \, {{\hat{\mu }}}_m \,&= \, {\mu }_m^0 \, + \, \frac{{\text {plim}}_{T \rightarrow \infty } \, T^{- 1} \, \sum _{t = 1}^T \, (y_t \, - \, \mu _m^0) \, {{\hat{h}}}_{t m}^{- 1} \, {{\hat{\tau }}}_{t m | t}}{{\text {plim}}_{T \rightarrow \infty } \, T^{- 1} \, \sum _{t = 1}^T \, {{\hat{h}}}_{t m}^{- 1} \, {{\hat{\tau }}}_{t m | t}}\\&= \, {\mu }_m^0 \, + \, \frac{E_{\infty }(\sqrt{{h}_{t m}^{0}} \, \eta _t \, {{\hat{h}}}_{t m}^{- 1} \, {{\hat{\tau }}}_{t m | t})}{E_{\infty } ({{\hat{h}}}_{t m}^{- 1} \, {{\hat{\tau }}}_{t m | t})}\\&= \, {\mu }_m^0 \, + \, E_{\infty } (\eta _t) \, \frac{E_{\infty }(\sqrt{{h}_{t m}^{0}} \, {{\hat{h}}}_{t m}^{- 1} \, {{\hat{\tau }}}_{t m | t})}{E_{\infty } ({{\hat{h}}}_{t m}^{- 1} \, {{\hat{\tau }}}_{t m | t})}\\&\, = \, {\mu }_m^0 \, + \, E_{\infty } (\eta _t) \, C \, = \, 0 \end{aligned} \end{aligned}$$

as $\eta _t$ is independent of $s_t$ and $h_{t m}$. Here C is a finite real constant. It remains to prove the consistency of ${\hat{\varvec{\theta }}}_m$ for all $m = 1, \dots , M$. Using (6) and the FOC condition

$$\begin{aligned} \frac{\partial \, J({\varvec{\lambda }} | \mathbf{Y}_T)}{\partial \, {\varvec{\theta }}_m} \, = \, \frac{\partial \, \mathcal{L}({\varvec{\lambda }} | \mathbf{Y}_T)}{\partial \, {\varvec{\theta }}_m} \, = \, \sum _{t = 1}^T \, S_t({\varvec{\theta }}_m)\, = \,{{\varvec{0}}} \end{aligned}$$

we have

$$\begin{aligned} \sum _{t = 1}^T \, \left[ \frac{1}{{{\hat{h}}}_{t m}}\, \frac{\partial \, h_{t m}}{\partial \, {\varvec{\theta }}_m} |_{{\varvec{\theta }}_m \, = \, {\hat{\varvec{\theta }}}_m} \, - \, \frac{ (y_t \, - \, \hat{\mu }_m)^{2}}{{{\hat{h}}}_{t m}^{2}}\, \frac{\partial \, h_{t m}}{\partial \, {\varvec{\theta }}_m} |_{{\varvec{\theta }}_m \, = \, {\hat{\varvec{\theta }}}_m} \right] \, {\hat{\tau }}_{t m | t} \, = \, {{\varvec{0}}} \end{aligned}$$

or, equivalently,

$$\begin{aligned} \sum _{t = 1}^T \, \left[ \frac{{{\hat{h}}}_{t m} \, - \, (y_t \, - \, {{\hat{\mu }}}_m)^2}{{{\hat{h}}}_{t m}^{2}}\, \frac{\partial \, h_{t m}}{\partial \, {\varvec{\theta }}_m} |_{{\varvec{\theta }}_m \, = \, {\hat{\varvec{\theta }}}_m}\right] \, {{\hat{\tau }}}_{t m | t} \, = \, {{\varvec{0}}}. \end{aligned}$$

Note that the expectation of $\frac{\partial \, z_{t m}}{\partial \, {\varvec{\theta }}_m}$, conditional on $\mathbf{Y}_{t - 1}$, vanishes. Substituting $y_t = \mu _{m}^0 \, + \, \epsilon _t = \mu _{m}^0 \, + \sqrt{h_{t m}^{0}} \, \eta _t$ into the last equation yields

$$\begin{aligned} \sum _{t = 1}^T \, \left[ \frac{{{\hat{h}}}_{t m} \, - \, ( \mu _{m}^0 \, + \sqrt{h_{t m}^{0}} \, \eta _t \, - \, {{\hat{\mu }}}_m)^2}{{{\hat{h}}}_{t m}}\, \frac{\partial \, {\text {ln}} \, h_{t m}}{\partial \, {\varvec{\theta }}_m} |_{{\varvec{\theta }}_m \, = \, {\hat{\varvec{\theta }}}_m}\right] \, {{\hat{\tau }}}_{t m | t} \, = \, {{\varvec{0}}} \end{aligned}$$

hence

$$\begin{aligned} \begin{aligned} \sum _{t = 1}^T \,&\Bigg [\frac{{{\hat{h}}}_{t m} \, - \, h_{t m}^{0}\, \eta _{t}^{2} \, - \, (\mu _{m}^0 \, - \, {{\hat{\mu }}}_m)^2 \, - \, 2 \, \sqrt{h_{t m}^{0}} \, \eta _t \, (\mu _{m}^0 \, - \, {{\hat{\mu }}}_m)}{{{\hat{h}}}_{t m}}\, \Bigg .\\&\quad \times \,\Bigg .\frac{\partial \, {\text {ln}} \, h_{t m}}{\partial \, {\varvec{\theta }}_m} |_{{\varvec{\theta }}_m \, = \, {\hat{\varvec{\theta }}}_m}\Bigg ] \, {{\hat{\tau }}}_{t m | t} \, = \, {{\varvec{0}}}. \end{aligned} \end{aligned}$$

Taking the first Taylor expansion of ${{\hat{h}}}_{t m}$ around $h_{t m}^{0}$, i.e.,

$$\begin{aligned} {{\hat{h}}}_{t m} \, = \, h_{t m}^{0} \, + \, \frac{\partial \, h_{t m}}{\partial \, {\varvec{\theta }}_{m}^{'}} |_{{\varvec{\theta }}_m \, = \, {\varvec{\theta }}_{m}^{0}}\, ( {\hat{\varvec{\theta }}}_m \, - \, {\varvec{\theta }}_{m}^{0}) \end{aligned}$$

and substituting into the previous equation yields

$$\begin{aligned} \begin{aligned} \sum _{t = 1}^T \, \{&\left[ h_{t m}^{0} \, + \, \frac{\partial \, h_{t m}}{\partial \, {\varvec{\theta }}_{m}^{'}} |_{{\varvec{\theta }}_m \, = \, {\varvec{\theta }}_{m}^{0}}\, ( {\hat{\varvec{\theta }}}_m \, - \, {\varvec{\theta }}_{m}^{0}) \, - \, h_{t m}^{0}\, \eta _{t}^{2} \, - \, (\mu _{m}^{0} \, - \, {{\hat{\mu }}}_m)^2 \, \right. \\&\left. \quad - \, 2 \, \sqrt{h_{t m}^{0}} \, {\eta }_t \, (\mu _{m}^{0} \, - \, {{\hat{\mu }}}_m)\right] \, \frac{1}{{{\hat{h}}}_{t m}}\, \frac{\partial \, {\text {ln}} \, h_{t m}}{\partial \, {\varvec{\theta }}_m} |_{{\varvec{\theta }}_m \, = \, {\hat{\varvec{\theta }}}_m} \} \, {{\hat{\tau }}}_{t m | t} \, = \, {{\varvec{0}}}. \end{aligned} \end{aligned}$$

Multiplying the last equation by $T^{- 1}$ and taking the limit when T goes to infinity, we have

$$\begin{aligned} \begin{aligned}&{\text {plim}}_{T \rightarrow \infty } \, T^{- 1} \, \left[ \sum _{t = 1}^T \, \frac{\partial \, {\text {ln}} \, h_{t m}}{\partial \, {\varvec{\theta }}_m} |_{{\varvec{\theta }}_m \, = \, {\hat{\varvec{\theta }}}_m} \, \frac{\partial \, {\text {ln}} \, h_{t m}}{\partial \, {\varvec{\theta }}_{m}^{'}} |_{{\varvec{\theta }}_m \, = \, {\hat{\varvec{\theta }}}_m} \, {{\hat{\tau }}}_{t m | t}\right] \\&\quad \times \, {\text {plim}}_{T \rightarrow \infty } ( {\hat{\varvec{\theta }}}_m \, - \, {\varvec{\theta }}_{m}^{0})\\&\quad \, = \, {{{\mathcal {I}}}}_m \, {\text {plim}}_{T \rightarrow \infty } ( {\hat{\varvec{\theta }}}_m \, - \, {\varvec{\theta }}_{m}^{0})\, = \, {{\varvec{0}}} \end{aligned} \end{aligned}$$

as $E(\eta _t^2) = 1$ and ${\text {plim}}_{T \rightarrow \infty } \, {{\hat{\mu }}}_m \, = \, {\mu }_m^0$, for all $m = 1, \dots , M$. By assumption (13) the $r \times r$ matrix ${{{\mathcal {I}}}}_m$ is finite and nonsingular. It follows that

$$\begin{aligned} {\text {plim}}_{T \rightarrow \infty } {\hat{\varvec{\theta }}}_m \, = \, {\varvec{\theta }}_{m}^{0}, \end{aligned}$$

which proves the consistency of ${\hat{\varvec{\theta }}}_m$. $\square$

Proof of Proposition 3.3

Set ${\varvec{\lambda }}_m^0 = {\varvec{\lambda }}_m$ to simplify notation. The asymptotic variance of ${{\hat{\pi }}}_m$ is given by

$$\begin{aligned} {\text {var}}_{\infty }({{\hat{\pi }}}_m) = E({{\hat{\pi }}}_m^2) \, - \, [E({{\hat{\pi }}}_m)]^2 \, = \, \pi _m \, - \, \pi _m^2 \, = \, \pi _m (1 - \pi _m). \end{aligned}$$

The aymptotic variance of ${{\hat{\mu }}}_m$ is given by

$$\begin{aligned} \begin{aligned} {\text {var}}_{\infty }[\sqrt{T} \, ({{\hat{\mu }}}_m \, - \, \mu _m^o)]&= T \, E[({{\hat{\mu }}}_m \, - \, \mu _m^o)^2]\\&= \, T \, E\{[\sum _{t = 1}^T \, {{\hat{h}}}_{t m}^{- 1}\, {{\hat{\tau }}}_{t m | t}]^{- 2}\, [\sum _{t = 1}^T \, \epsilon _t \, {{\hat{h}}}_{t m}^{- 1}\, {{\hat{\tau }}}_{t m | t}]^{ 2}\}\\&= T \, E\{[T^{- 1} \, \sum _{t = 1}^T \, {{\hat{h}}}_{t m}^{- 1}\, {{\hat{\tau }}}_{t m | t}]^{- 2}\, [T^{- 1} \, \sum _{t = 1}^T \, \epsilon _t \, {{\hat{h}}}_{t m}^{- 1}\, {{\hat{\tau }}}_{t m | t}]^{ 2}\}\\&= T \, E\{[T^{- 1} \, \sum _{t = 1}^T \, {{\hat{h}}}_{t m}^{- 1}\, {{\hat{\tau }}}_{t m | t}]^{- 2}\, [T^{- 2} \, \sum _{t = 1}^T \, \epsilon _t^2 \, {{\hat{h}}}_{t m}^{- 2}\, {{\hat{\tau }}}_{t m | t}^{ 2}]\}\\&= E\{[T^{- 1} \, \sum _{t = 1}^T \, {{\hat{h}}}_{t m}^{- 1}\, {{\hat{\tau }}}_{t m | t}]^{- 2}\, [T^{- 1} \, \sum _{t = 1}^T \, {{\hat{h}}}_{t m}\, \eta _t^2 \, {{\hat{h}}}_{t m}^{- 2}\, {{\hat{\tau }}}_{t m | t}^{ 2}]\}\\&= E\{[T^{- 1} \, \sum _{t = 1}^T \, {{\hat{h}}}_{t m}^{- 1}\, {{\hat{\tau }}}_{t m | t}]^{- 2}\, [T^{- 1} \, \sum _{t = 1}^T \, {{\hat{h}}}_{t m}^{- 1}\, {{\hat{\tau }}}_{t m | t}^{ 2}]\}\\&= E[T^{- 1} \, \sum _{t = 1}^T \, {{\hat{h}}}_{t m}^{- 1}\, {\hat{\tau }}_{t m | t}]^{- 1} \, = \, \gamma _m^{- 1} \end{aligned} \end{aligned}$$

as $E(\eta _t^2) = 1$ and $E({{\hat{\tau }}}_{t m | t}^{ 2}) = E({\hat{\tau }}_{t m | t})$. To compute the asymptotic variance of ${\hat{\varvec{\theta }}}_{m}$ we set

$$\begin{aligned} {{{\mathcal {I}}}}_{m T} \, = \, T^{- 1}\, \sum _{t = 1}^T \, \frac{\partial \, {\text {ln}} \, h_{t m}}{\partial \, {\varvec{\theta }}_m}|_{ {\varvec{\theta }}_m = {\hat{\varvec{\theta }}}_{m}}\, \frac{\partial \, {\text {ln}} \, h_{t m}}{\partial \, {\varvec{\theta }}_{m}^{'}}|_{ {\varvec{\theta }}_m = {\hat{\varvec{\theta }}}_{m}} \, {{\hat{\tau }}}_{t m | t} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} {J}_{m T}&= \sum _{t = 1}^T \, (\eta _t^2 \, - \, 1)\, \frac{h_{t m}^0}{{{\hat{h}}}_{t m}} \, \frac{\partial \, {\text {ln}} \, h_{t m}}{\partial \, {\varvec{\theta }}_m}|_{ {\varvec{\theta }}_m = {\hat{\varvec{\theta }}}_{m}}\, {{\hat{\tau }}}_{t m | t}\\&\quad + \, (\mu _m^0 \, - \, {{\hat{\mu }}}_m)^2 \, \sum _{t = 1}^T \, \frac{1}{{{\hat{h}}}_{t m}} \, \frac{\partial \, {\text {ln}} \, h_{t m}}{\partial \, {\varvec{\theta }}_m}|_{ {\varvec{\theta }}_m = {\hat{\varvec{\theta }}}_{m}}\, {{\hat{\tau }}}_{t m | t}\\&\quad + \, 2\, (\mu _m^0 \, - \, {{\hat{\mu }}}_m)\, \sum _{t = 1}^T \, \frac{\sqrt{h_{t m}^0} \, \eta _t}{{{\hat{h}}}_{t m}} \, \frac{\partial \, {\text {ln}} \, h_{t m}}{\partial \, {\varvec{\theta }}_m}|_{ {\varvec{\theta }}_m = {\hat{\varvec{\theta }}}_{m}}\, {{\hat{\tau }}}_{t m | t}. \end{aligned} \end{aligned}$$

Then we have

$$\begin{aligned} {\text {plim}}_{T \rightarrow \infty } \, {{{\mathcal {I}}}}_{m T} \, = \, \mathcal{I}_m \qquad {\text {plim}}_{T \rightarrow \infty } \, T^{- 1} \, {J}_{m T} \, {J}_{m T}^{'}\, = \, 2 \, {{{\mathcal {I}}}}_m. \end{aligned}$$

So it follows

$$\begin{aligned} \begin{aligned} {\text {var}}_{\infty } [\sqrt{T} \, ({\hat{\varvec{\theta }}}_m \, - \, {\varvec{\theta }}_m^0)]&= T E\left[ ({\hat{\varvec{\theta }}}_m \, - \, {\varvec{\theta }}_m^0)\, ({\hat{\varvec{\theta }}}_m \, - \, {\varvec{\theta }}_m^0)^{'}\right] \\&= E\left[ {{{\mathcal {I}}}}_{m T}^{- 1}\, T^{- 1} \, J_{m T}\, J_{m T}^{'} \, {{{\mathcal {I}}}}_{m T}^{- 1}\right] \\&= \, 2 \, {{{\mathcal {I}}}}_{m}^{- 1} \, {{{\mathcal {I}}}}_m \, {{{\mathcal {I}}}}_{m}^{- 1} \, = \, 2 \, {{{\mathcal {I}}}}_{m}^{- 1}. \end{aligned} \end{aligned}$$

Derivation of (14). The derivative $\frac{\partial \, S_t ({\varvec{\theta }}_m)}{\partial \, {\varvec{\theta }}_{m}^{'}}$ is given by

$$\begin{aligned} \begin{aligned} \frac{\partial \, S_t ({\varvec{\theta }}_m)}{\partial \, {\varvec{\theta }}_{m}^{'}} \,&= \, \frac{1}{2} \, \left[ - \, \epsilon _t^2 \, h_{t m}^{- 2}\, \frac{\partial \, h_{t m}}{\partial \, {\varvec{\theta }}_m}\right] \, \frac{\partial \, {\text {ln}} \, h_{t m}}{\partial \, {\varvec{\theta }}_{m}^{'}}\, {\tau }_{t m | t}\\&\quad + \, \frac{1}{2} \, \left[ \frac{\epsilon _t^2}{h_{t m}}\, - \, 1\right] \, \frac{\partial ^2 \, {\text {ln}} \, h_{t m}}{\partial \, {\varvec{\theta }}_m \, \partial \, {\varvec{\theta }}_{m}^{'}}\, {\tau }_{t m | t}\\&\quad + \, \frac{1}{2} \, \left[ \frac{\epsilon _t^2}{h_{t m}}\, - \, 1\right] \, \frac{\partial \, {\text {ln}} \, h_{t m}}{\partial \, {\varvec{\theta }}_m}\, \frac{\partial \, \tau _{t m | t}}{ \partial \, {\varvec{\theta }}_{m}^{'}}. \end{aligned} \end{aligned}$$

Now we can approximate $\frac{\partial ^2 \, {{{\mathcal {L}}}}({\varvec{\lambda }})}{\partial \, {\varvec{\theta }}_m \, \partial \, {\varvec{\theta }}_{m}^{'}}$ by taking the expectation of $\frac{\partial \, S_t ({\varvec{\theta }}_m)}{\partial \, {\varvec{\theta }}_{m}^{'}}$ conditional on $\mathbf{Y}_{t - 1}$. Recall that conditional on $\mathbf{Y}_{t - 1}$ the magnitudes of $h_{t m}$, $\tau _{t m | t}$, and $\frac{\partial \, \tau _{t m | t}}{ \partial \, {\varvec{\theta }}_{m}^{'}}$ are nonstochastic. Furthermore, we have

$$\begin{aligned} E(\epsilon _t^2 | \mathbf{Y}_{t - 1}) \, = \, E(h_{t m}\, \eta _t^2 | \mathbf{Y}_{t - 1}) \, = \, E(h_{t m} | \mathbf{Y}_{t - 1}) \, = \, h_{t m}. \end{aligned}$$

Then the conditional expectation of the second and third summand in $\frac{\partial \, S_t ({\varvec{\theta }}_m)}{\partial \, {\varvec{\theta }}_{m}^{'}}$ vanish. So the conditional expectation of the first summand reduces to

$$\begin{aligned} \begin{aligned} E\left[ \frac{\partial \, S_t ({\varvec{\theta }}_m)}{\partial \, {\varvec{\theta }}_{m}^{'}} | \mathbf{Y}_{t - 1}\right]&= - \frac{1}{2} \, E\left[ \frac{\epsilon _t^2}{h_{t m}} \, \frac{\partial \, {\text {ln}} \, h_{t m}}{\partial \, {\varvec{\theta }}_m} \, \frac{\partial \,{\text {ln}} \, h_{t m}}{\partial \, {\varvec{\theta }}_{m}^{'}} \, { \tau }_{t m | t} | \mathbf{Y}_{t - 1}\right] \\&= \, - \, \frac{1}{2} \, E\left( \frac{\partial \, {\text {ln}} \, h_{t m}}{\partial \, {\varvec{\theta }}_m} \, \frac{\partial \,{\text {ln}} \, h_{t m}}{\partial \, {\varvec{\theta }}_{m}^{'}} \, { \tau }_{t m | t} | \mathbf{Y}_{t - 1}\right) \\&\approx \, - \, \frac{1}{2 T} \, \sum _{t = 1}^T \, \frac{\partial \, {\text {ln}} \, h_{t m}}{\partial \, {\varvec{\theta }}_m} \, \frac{\partial \,{\text {ln}} \, h_{t m}}{\partial \, {\varvec{\theta }}_{m}^{'}} \, { \tau }_{t m | t}. \end{aligned} \end{aligned}$$

Derivation of (15). The derivative $\frac{\partial \, S_t ({\pi }_m)}{\partial \, {\pi }_{m}}$ is given by

$$\begin{aligned} \begin{aligned} \frac{\partial \, S_t ({\pi }_m)}{\partial \, {\pi }_{m}}&= \, - \, \pi _{m}^{- 2} \, \tau _{t m | t} \, + \, \pi _{m}^{- 1} \, \frac{\partial \, \tau _{t m | t}}{\partial \, \pi _m} \\&= - \, \pi _{m}^{- 2} \, \tau _{t m | t} \, + \, \pi _{m}^{- 1} \, [\pi _{m}^{- 1}\, \tau _{t m | t} \, - \, \pi _{m}^{- 1}\, \tau _{t m | t}^{2}] \, = \, \, - \, \pi _{m}^{- 2} \, \tau _{t m | t}^2. \end{aligned} \end{aligned}$$

Hence it follows

$$\begin{aligned} - \, \frac{\partial ^2 \, {{{\mathcal {L}}}}({\varvec{\lambda }} | \mathbf{Y}_T)}{\partial ^2 \, \pi _m} \approx E\left[ - \, \frac{\partial ^2 \, {{{\mathcal {L}}}}({\varvec{\lambda }} | \mathbf{Y}_T)}{\partial ^2 \, \pi _m} | \mathbf{Y}_{t - 1}\right] \approx \, \pi _{m}^{- 2} \, \frac{1}{T} \, \sum _{t = 1}^T \, \tau _{t m | t}^2. \end{aligned}$$

Derivation of (16). The derivative $\frac{\partial \, S_t ({\mu }_m)}{\partial \, {\mu }_{m}}$ is given by

$$\begin{aligned} \frac{\partial \, S_t ({\mu }_m)}{\partial \, {\mu }_{m}} = \frac{\partial \,}{\partial \, \mu _m} \left[ h_{t m}^{- 1} \, (y_t \, - \, \mu _m) \, \tau _{t m | t}\right] \approx - \, h_{t m}^{- 1}\, \tau _{t m | t}, \end{aligned}$$

hence

$$\begin{aligned} - \, \frac{\partial ^2 \, {{{\mathcal {L}}}}({\varvec{\lambda }} | \mathbf{Y}_T)}{\partial ^2 \, \mu _m} \approx E\left[ - \, \frac{\partial \, S_t ({\mu }_m)}{\partial \, {\mu }_{m}} | \mathbf{Y}_{t - 1}\right] \approx \, \frac{1}{T} \sum _{t = 1}^T\, h_{t m}^{- 1}\, \tau _{t m | t}. \end{aligned}$$

Derivation of (18). The last term of

$$\begin{aligned} {{{\mathcal {L}}}}({\hat{\varvec{\lambda }}} | \mathbf{Y}_T) \, = \, \sum _{t = 1}^T \, \sum _{m = 1}^M \, \left[ {\text {ln}}\, {{\hat{\pi }}}_m \, - \, \frac{1}{2} \, {\text {ln}}\, {{\hat{h}}}_{t m} \, - \, \frac{1}{2} \, \frac{{{\hat{\epsilon }}}_t^2}{{{\hat{h}}}_{t m}}\right] \, {{\hat{\tau }}}_{t m | t} \end{aligned}$$

can be approximated by T/2 for T sufficiently large. In fact, we have

$$\begin{aligned} \begin{aligned} \frac{1}{2} \, \sum _{t = 1}^T \, \sum _{m = 1}^M \, \frac{{{\hat{\epsilon }}}_t^2}{{{\hat{h}}}_{t m}} \, {{\hat{\tau }}}_{t m | t}&= \frac{1}{2} \, \sum _{t = 1}^T \, \sum _{m = 1}^M \, \eta _t^2 \, {{\hat{\tau }}}_{t m | t} \\&= \frac{1}{2} \, \sum _{t = 1}^T \, \eta _t^2 \, \sum _{m = 1}^M \, {{\hat{\tau }}}_{t m | t}\\&\, = \, \frac{1}{2} \, \sum _{t = 1}^T \, \eta _t^2 \\&\, = \, \frac{1}{2} \, T \, \frac{1}{T} \, \sum _{t = 1}^T \, \eta _t^2 \end{aligned} \end{aligned}$$

as ${{\hat{\epsilon }}}_t^2 \, = \, {{\hat{h}}}_{t m} \, \eta _t^2$ and $\sum _{m = 1}^M \, {{\hat{\tau }}}_{t m | t} = 1$. For T large the term $T^{- 1} \, \sum _{t = 1}^T \, \eta _t^2$ converges to $E(\eta _t^2) = 1$, hence $\frac{1}{2} \, \sum _{t = 1}^T \, \sum _{m = 1}^M \, \frac{{{\hat{\epsilon }}}_t^2}{{{\hat{h}}}_{t m}} \, {\hat{\tau }}_{t m | t}$ can be approximated by T/2 for T sufficiently large. By using (10), we get

$$\begin{aligned} \begin{aligned} {{{\mathcal {L}}}}({\hat{\varvec{\lambda }}} | \mathbf{Y}_T) \,&= \, \sum _{t = 1}^T \, \sum _{m = 1}^M \, {{\hat{\tau }}}_{t m | t}\, {\text {ln}}\, {{\hat{\pi }}}_m \, - \, \frac{1}{2} \, \sum _{t = 1}^T \, \sum _{m = 1}^M \, {{\hat{\tau }}}_{t m | t}\, {\text {ln}} \, {{\hat{h}}}_{t m} \, - \, \frac{T}{2} \\&= \sum _{m = 1}^M \, \left[ \sum _{t = 1}^T {{\hat{\tau }}}_{t m | t}\right] \, {\text {ln}}\, {{\hat{\pi }}}_m \, - \, \frac{1}{2}\, \sum _{t = 1}^T \sum _{m = 1}^M {{\hat{\tau }}}_{t m | t} \,{\text {ln}} \, {{\hat{h}}}_{t m} \, - \, \frac{T}{2} \\&= T \, \sum _{m = 1}^M \left[ \frac{1}{T} \, \sum _{t = 1}^T {{\hat{\tau }}}_{t m | t}\right] \, {\text {ln}} \, {{\hat{\pi }}}_m \, - \, \frac{1}{2}\, \sum _{t = 1}^T \, \sum _{m = 1}^M\, {{\hat{\tau }}}_{t m | t}\,{\text {ln}} \, {{\hat{h}}}_{t m} \, - \, \frac{T}{2} \\&= T \sum _{m = 1}^M \, {{\hat{\pi }}}_m\, {\text {ln}} \, {\hat{\pi }}_m \, - \, \frac{1}{2}\, \sum _{t = 1}^T \, \sum _{m = 1}^M\, {{\hat{\tau }}}_{t m | t}\,{\text {ln}} \, {{\hat{h}}}_{t m} \, - \, \frac{T}{2} \end{aligned} \end{aligned}$$

for T sufficiently large. Then formula (18) follows.

Representing a MGARCH by a MS ARMA model. A MGARCH(M; p, q) model can be represented by a M-state Markov switching ARMA(r, p) process, where $r = \max (p, q)$. Set $\epsilon _t^2 = h_{t, s_t} + v_t$. Then we have

$$\begin{aligned} \begin{aligned} \epsilon _t^2&\, = \, h_{t, s_t} \, + \, v_t = \omega _{s_t} \, + \, \sum _{i = 1}^q\, \alpha _{i, s_t}\, \epsilon _{t - i}^{2} \, + \, \sum _{j = 1}^p\, \beta _{j, s_t}\, h_{t - j, s_t} \, + \, v_t \\&= \omega _{s_t} \, + \, \sum _{i = 1}^q\, \alpha _{i, s_t}\, \epsilon _{t - i}^{2} \, + \, \sum _{j = 1}^p\, \beta _{j, s_t}\, (\epsilon ^{2}_{t - j} \, - \, v_{t - j}) \, + \, v_t \\&= \omega _{s_t} \, + \, \sum _{k = 1}^r\, \delta _{k, s_t}\, \epsilon _{t - k}^{2} \, + \, v_t \, - \, \sum _{j = 1}^p\, \beta _{j, s_t}\, v_{t - j} \end{aligned} \end{aligned}$$

where $\delta _{k, s_t} = \alpha _{k, s_t} \, + \, \beta _{k, s_t}$. Here we set $\alpha _{k, s_t} = 0$ for $k > q$ and $\beta _{k, s_t} = 0$ for $k > p$. For t sufficiently large, the disturbance process $v_t$ can be approximately driven by a normal distribution with zero mean and variance

$$\begin{aligned} \begin{aligned} E(v_{t}^2)&\, = \, E[(\epsilon _t^2 \, - \, h_{t, s_t})^2] \, = \, E[(h_{t, s_t} \, \eta _t^2 \, - \, h_{t, s_t})^2] \\&\, = \, E[h_{t, s_t}^{2} \, (\eta _t^2 \, - \, 1)^2] \, = \, E(h_{t, s_t}^{2})\, E[(\eta _t^2 \, - \ 1)^2]\\&\, = \, 2\, E(h_{t, s_t}^{2}) \end{aligned} \end{aligned}$$

as $E[(\eta _t^2 \, - \, 1)^2] = E(\eta _t^4) \, - \, 2\, E(\eta _t^2) \, + \, 1 \, = \, 2$ for $\eta _t \sim N(0, 1)$. The explicit computation of unconditional moments of the powers of $\epsilon _t^2$ and $h_{t, s_t}$ can be found in Francq and Zakoïan (2008). $\square$