The MS-GARCH model with constant within-regime correlations is attractive since it is analytically tractable. E.g. straightforward-to-check conditions for stationarity have been obtained, as well as a simple recursion for calculating multi-step conditional covariance matrices, which is crucial for mean-variance portfolio optimization. Moreover, in some simple cases, such as Pelletier’s (2006) model with regime-independent GARCH dynamics, estimation in high dimensions is feasible via a two-step procedure with an embedded EM algorithm. Despite its convenience, it is still desirable to test whether the assumption of constant within-regime correlation matrices is tenable, since otherwise further improvement of out-of-sample portfolio selection might be feasible by extending the model to allow for within-regime correlation dynamics as in Billio and Caporin (2005) and Otranto (2010). Using results of Hamilton (1996), we extend the Lagrange Multiplier (LM) test devised by Tse (2000) for constant conditional correlations in multivariate GARCH models. In Tse (2000) the LM test is derived under normality of the innovations, but he reports simulations indicating it being quite robust against nonnormality. However, in view of the discussion in Section 4.1, this cannot be expected to hold for MS-GARCH processes, and thus we derive the test allowing for Student’s *t* errors. The test under normality is then straightforwardly obtained if the degrees of freedom *ν* → ∞.

For the volatility dynamics, we assume that the conditional standard deviation of asset *i* in regime *j*, *σ*_{ijt}, is described by a standard (symmetric) AGARCH(1,1) process, i.e. ^{16}

$${\sigma}_{ijt}={\omega}_{ij}+{a}_{ij}|{\u03f5}_{i,t-1}|+{b}_{ij}{\sigma}_{ij,t-1},\phantom{\rule{1em}{0ex}}i=1,\dots ,M,\phantom{\rule{1em}{0ex}}j=1,\dots ,k.$$(25)

The conditional correlation matrix in regime *j* is

$${\mathit{R}}_{jt}=({\rho}_{i\ell ,jt}{)}_{i,\ell =1,\dots ,M},\phantom{\rule{1em}{0ex}}j=1,\dots ,k.$$

where, as in Tse (2000), correlations evolve according to^{17}

$${\rho}_{i\ell ,jt}={\overline{\rho}}_{i\ell ,j}+{\delta}_{i\ell ,j}{\u03f5}_{i,t-1}{\u03f5}_{\ell ,t-1},\phantom{\rule{1em}{0ex}}i=1,\dots ,M-1,\phantom{\rule{1em}{0ex}}\ell =i+1,\dots ,M,\phantom{\rule{1em}{0ex}}j=1,\dots ,k.$$(26)

The null hypothesis of constant conditional within-regime correlations corresponds to

$${{\textstyle \text{H}}}_{0}:{\delta}_{i\ell ,j}=0,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}i=1,\dots ,M-1,\phantom{\rule{1em}{0ex}}\ell =i+1,\dots ,M,\phantom{\rule{1em}{0ex}}j=1,\dots ,k.$$(27)

We distinguish between the following cases, which differ in the specification of the alternative hypothesis:

The conditional correlation dynamics are unrestricted across regimes. In this case, under (27), the LM test statistic is asymptotically distributed as *χ*^{2} with *kM*(*M* − 1)/2 degrees of freedom.

The conditional correlation dynamics are the same across regimes, i.e. *δ*_{iℓ,1} = *δ*_{iℓ,2} = ⋯ = *δ*_{iℓ,k}, *i* = 1, …, *M* − 1, ℓ = *i* + 1, …, *M*. In this case, under (27), the LM test statistic is asymptotically distributed as *χ*^{2} with *M*(*M* − 1)/2 degrees of freedom.

For the purpose of the current section, it is convenient to decompose the parameter vector of the model as $\mathit{\theta}=({\textstyle \text{vec}}(\mathit{P}{)}^{\mathrm{\prime}},{\mathit{\vartheta}}^{\mathrm{\prime}}{)}^{\mathrm{\prime}}$,^{18} where *ϑ* consists of the parameters of the conditional regime densities, i.e. $\mathit{\vartheta}=({\mathit{\vartheta}}_{1}^{\mathrm{\prime}},\dots ,{\mathit{\vartheta}}_{k}^{\mathrm{\prime}},\nu {)}^{\mathrm{\prime}}$, where *ν* is the (common) shape parameter of the *t* distribution and ${\mathit{\vartheta}}_{j}=({\mathit{\vartheta}}_{1j}^{\mathrm{\prime}},\dots ,{\mathit{\vartheta}}_{Mj}^{\mathrm{\prime}},{\mathit{\rho}}_{j}^{\mathrm{\prime}},{\mathit{\delta}}_{j}^{\mathrm{\prime}}{)}^{\mathrm{\prime}}$, *j* = 1, …, *k*, where ${\mathit{\vartheta}}_{ij}=({\omega}_{ij},{a}_{ij},{b}_{ij}{)}^{\mathrm{\prime}}$, and *𝝆*_{j} and *𝜹*_{j} are the *M*(*M* − 1)/2 vectors which stack, respectively, parameters ${\overline{\rho}}_{i\ell ,j}$ and *δ*_{iℓ,j} in Equation (26), i.e.

$$\begin{array}{rll}{\mathit{\rho}}_{j}& =& ({\overline{\rho}}_{12,j},{\overline{\rho}}_{13,j},\dots ,{\overline{\rho}}_{1M,j},{\overline{\rho}}_{23,j},\dots ,{\overline{\rho}}_{2M,j},\dots ,{\overline{\rho}}_{M-1,M,j}{)}^{\mathrm{\prime}},\\ {\mathit{\delta}}_{j}& =& ({\delta}_{12,j},{\delta}_{13,j},\dots ,{\delta}_{1M,j},{\delta}_{23,j},\dots ,{\delta}_{2M,j},\dots ,{\delta}_{M-1,M,j}{)}^{\mathrm{\prime}}.\end{array}$$

The log-likelihood of the model for a sample of size *T* is given by

$$\begin{array}{rll}\mathrm{log}L(\mathit{\theta})& =& \sum _{t=1}^{T}\mathrm{log}f({\u03f5}_{t}|{\mathrm{\Omega}}_{t-1};\mathit{\theta})\\ & =& \sum _{t=1}^{T}\mathrm{log}\left\{\sum _{j=1}^{k}p({\mathrm{\Delta}}_{t}=j|{\mathrm{\Omega}}_{t-1};\mathit{\theta})f({\u03f5}_{t}|{\mathrm{\Omega}}_{t-1},{\mathrm{\Delta}}_{t}=j;{\mathit{\vartheta}}_{j},\nu )\right\},\end{array}$$(28)

where ${\mathrm{\Omega}}_{t}=\{{\mathit{\u03f5}}_{t},{\mathit{\u03f5}}_{t-1},\dots ,{\mathit{\u03f5}}_{0}\}$, $p({\mathrm{\Delta}}_{t}=j|{\mathrm{\Omega}}_{t-1};\mathit{\theta})$ are the one-step predicted regime inferences (cf. Hamilton, 1994, Ch. 22), and the conditional regime densities are

$$\begin{array}{rll}f({\u03f5}_{t}|{\mathrm{\Omega}}_{t-1},{\mathrm{\Delta}}_{t}=j;{\mathit{\vartheta}}_{j},\nu )& =& \frac{\mathrm{\Gamma}\left(\frac{\nu +M}{2}\right)}{\mathrm{\Gamma}(\nu /2)(\pi (\nu -2){)}^{M/2}|{\mathit{R}}_{jt}{|}^{1/2}\prod _{i=1}^{M}{\sigma}_{ijt}}{\left\{1+\frac{{d}_{jt}^{2}}{\nu -2}\right\}}^{-(\nu +M)/2}\\ & =:& {f}_{jt}({\mathit{\vartheta}}_{j},\nu ),\end{array}$$(29)

where the squared Mahalanobis distance

$${d}_{jt}^{2}={\mathit{\u03f5}}_{t}^{\mathrm{\prime}}{\mathit{D}}_{jt}^{-1}{\mathit{R}}_{jt}^{-1}{\mathit{D}}_{jt}^{-1}{\mathit{\u03f5}}_{t}={\mathit{\u03f5}}_{jt}^{{\star}^{\mathrm{\prime}}}{\mathit{R}}_{jt}^{-1}{\mathit{\u03f5}}_{jt}^{\star}={\mathit{\u03f5}}_{jt}^{{\star}^{\mathrm{\prime}}}{\stackrel{~}{\mathit{\u03f5}}}_{jt},$$

with

$${\mathit{\u03f5}}_{jt}^{\star}={\mathit{D}}_{jt}^{-1}{\mathit{\u03f5}}_{t},\phantom{\rule{1em}{0ex}}{\stackrel{~}{\mathit{\u03f5}}}_{jt}={\mathit{R}}_{jt}^{-1}{\mathit{\u03f5}}_{jt}^{\star},\phantom{\rule{1em}{0ex}}j=1,\dots ,k.$$(30)

The regime-specific log-density for observation *t* is

$$\begin{array}{rll}\mathrm{log}{f}_{jt}({\mathit{\vartheta}}_{j},\nu )& =& -\frac{M}{2}(\mathrm{log}\pi +\mathrm{log}(\nu -2))+\mathrm{log}\mathrm{\Gamma}\left(\frac{\nu +M}{2}\right)-\mathrm{log}\mathrm{\Gamma}\left(\frac{\nu}{2}\right)-\frac{1}{2}\mathrm{log}|{\mathit{R}}_{jt}|\\ & & -\sum _{i=1}^{M}\mathrm{log}{\sigma}_{ijt}+\frac{\nu +M}{2}\mathrm{log}\left(1+\frac{{d}_{jt}^{2}}{\nu -2}\right)\\ & =& -\frac{M}{2}\mathrm{log}\pi +\mathrm{log}\mathrm{\Gamma}\left(\frac{\nu +M}{2}\right)-\mathrm{log}\mathrm{\Gamma}\left(\frac{\nu}{2}\right)+\frac{\nu}{2}\mathrm{log}(\nu -2)-\frac{1}{2}\mathrm{log}|{\mathit{R}}_{jt}|\\ & & -\sum _{i=1}^{M}\mathrm{log}{\sigma}_{ijt}-\frac{\nu +M}{2}\mathrm{log}(\nu -2+{d}_{jt}^{2}),\phantom{\rule{1em}{0ex}}j=1,\dots ,k.\end{array}$$(31)

The partial derivatives of (31) are obtained as

$$\begin{array}{rll}\frac{\mathrm{\partial}\mathrm{log}{f}_{jt}({\mathit{\vartheta}}_{j},\nu )}{\mathrm{\partial}\nu}& =& \frac{1}{2}\left[\psi \left(\frac{\nu +M}{2}\right)-\psi \left(\frac{\nu}{2}\right)\right]-\frac{1}{2}\mathrm{log}\left(1+\frac{{d}_{jt}^{2}}{\nu -2}\right)\\ & & +\frac{1}{2}\left[\frac{\nu}{\nu -2}-\frac{\nu +M}{\nu -2+{d}_{jt}^{2}}\right],\\ \frac{\mathrm{\partial}\mathrm{log}{f}_{jt}({\mathit{\vartheta}}_{j},\nu )}{\mathrm{\partial}{\mathit{\vartheta}}_{ij}}& =& \frac{\mathrm{\partial}{\sigma}_{ijt}}{\mathrm{\partial}{\mathit{\nu}}_{ij}}\frac{1}{{\sigma}_{ijt}}\left(\frac{{\u03f5}_{ijt}^{\star}{\stackrel{~}{\u03f5}}_{ijt}(\nu +M)}{\nu -2+{d}_{jt}^{2}}-1\right),\phantom{\rule{1em}{0ex}}i=1,\dots ,M,\\ \frac{\mathrm{\partial}\mathrm{log}{f}_{jt}({\mathit{\vartheta}}_{j},\nu )}{\mathrm{\partial}{\mathit{\rho}}_{j}}& =& \frac{1}{2}\mathit{U}\left(\frac{\nu +M}{\nu -2+{d}_{jt}^{2}}({\stackrel{~}{\mathit{\u03f5}}}_{jt}\otimes {\stackrel{~}{\mathit{\u03f5}}}_{jt})-{\textstyle \text{vec}}{\mathit{R}}_{jt}^{-1}\right),\\ \frac{\mathrm{\partial}\mathrm{log}{f}_{jt}({\mathit{\vartheta}}_{j},\nu )}{\mathrm{\partial}{\mathit{\delta}}_{j}}& =& \frac{1}{2}\mathit{U}\left\{\left(\frac{\nu +M}{\nu -2+{d}_{jt}^{2}}({\stackrel{~}{\mathit{\u03f5}}}_{jt}\otimes {\stackrel{~}{\mathit{\u03f5}}}_{jt})-{\textstyle \text{vec}}{\mathit{R}}_{jt}^{-1}\right)\odot \left({\mathit{\u03f5}}_{t-1}\otimes {\mathit{\u03f5}}_{t-1}\right)\right\},\end{array}$$(32)

where $\psi (x)={\textstyle \text{d}}\mathrm{log}\mathrm{\Gamma}(x)/{\textstyle \text{d}}x$ is the logarithmic derivative of the gamma function (the digamma function); ${\u03f5}_{ijt}^{\star}$ and $\stackrel{~}{\u03f5}$_{ijt} are, respectively, the *i*th elements of vectors ${\mathit{\u03f5}}_{jt}^{\star}$ and ${\stackrel{~}{\mathit{\u03f5}}}_{jt}$ defined in (30) *i* = 1, …, *M*; and the *M*(*M* − 1)/2 × *M*^{2} matrix *U* is defined as in Silvennoinen and Teräsvirta (2009a), i.e. with its [(*i* − 1)*M* − *i*(*i* + 1)/2 + ℓ]th row given by

$$\text{vec}({\stackrel{~}{\mathit{e}}}_{i}{\stackrel{~}{\mathit{e}}}_{\ell}^{\mathrm{\prime}}+{\stackrel{~}{\mathit{e}}}_{\ell}{\stackrel{~}{\mathit{e}}}_{i}^{\mathrm{\prime}}{)}^{\mathrm{\prime}},\phantom{\rule{1em}{0ex}}i=1,\dots ,M-1,\phantom{\rule{1em}{0ex}}\ell =i+1,\dots ,M,$$

where ${\stackrel{~}{\mathit{e}}}_{i}$ is the *i*th column of the *M*–dimensional identity matrix. The derivative in (32) is

$$\frac{\mathrm{\partial}{\sigma}_{ijt}}{\mathrm{\partial}{\mathit{\vartheta}}_{ij}}={\mathit{\eta}}_{ij,t-1}+{b}_{ij}\frac{\mathrm{\partial}{\sigma}_{ij,t-1}}{\mathrm{\partial}{\mathit{\vartheta}}_{ij}},\phantom{\rule{1em}{0ex}}t=2,\dots ,T,$$(33)

where ${\mathit{\eta}}_{ijt}=(1,|{\u03f5}_{it}|,{\sigma}_{ijt}{)}^{\mathrm{\prime}}$, and the starting value in recursion (33) is

$$\frac{\mathrm{\partial}{\sigma}_{ij,t=1}}{\mathrm{\partial}{\mathit{\vartheta}}_{ij}}=(1,|{\u03f5}_{i0}|,{\sigma}_{ij0}{)}^{\mathrm{\prime}},$$(34)

where we initialize all regime-specific conditional standard deviations with the sample standard deviation, i.e. in (34),

$${\sigma}_{ij0}=\sqrt{\frac{1}{T-1}\sum _{t=1}^{T}{\u03f5}_{it}^{2}},\phantom{\rule{1em}{0ex}}i=1,\dots ,M,\phantom{\rule{1em}{0ex}}j=1,\dots ,k.$$

The score of the *t*th observation is given by the derivative of the conditional log-density of ${\mathit{\u03f5}}_{t}$ as given in (28) and (29),

$$\frac{\mathrm{\partial}\mathrm{log}f({\u03f5}_{t}|{\mathrm{\Omega}}_{t-1};\mathit{\theta})}{\mathrm{\partial}\mathit{\theta}}=\frac{\mathrm{\partial}\mathrm{log}\left\{\sum _{j=1}^{k}p({\mathrm{\Delta}}_{t}=j|{\mathrm{\Omega}}_{t-1};\mathit{\theta}){f}_{jt}({\mathit{\vartheta}}_{j},\nu )\right\}}{\mathrm{\partial}\mathit{\theta}}.$$(35)

Hamilton (1996) has shown that the derivatives in (35) involving elements of *ϑ* can be evaluated as

$$\begin{array}{rll}\frac{\mathrm{\partial}\mathrm{log}f({\u03f5}_{t}|{\mathrm{\Omega}}_{t-1};\mathit{\theta})}{\mathrm{\partial}\mathit{\vartheta}}& =& \sum _{j=1}^{k}\frac{\mathrm{\partial}\mathrm{log}{f}_{jt}({\mathit{\vartheta}}_{j},\nu )}{\mathrm{\partial}\mathit{\vartheta}}p({\mathrm{\Delta}}_{t}=j|{\mathrm{\Omega}}_{t};\mathit{\theta})\\ & & +\sum _{\tau =1}^{t-1}\sum _{j=1}^{k}\frac{\mathrm{\partial}\mathrm{log}{f}_{j\tau}({\mathit{\vartheta}}_{j},\nu )}{\mathrm{\partial}\mathit{\vartheta}}\left[p({\mathrm{\Delta}}_{\tau}=j|{\mathrm{\Omega}}_{t};\mathit{\theta})-p({\mathrm{\Delta}}_{\tau}=j|{\mathrm{\Omega}}_{t-1};\mathit{\theta})\right],\\ t& =& 1,\dots ,T,\end{array}$$(36)

where the second line in (36) is set to zero for *t* = 1. For *τ* = *t* and *τ* < *t*, the regime probabilities $p({\mathrm{\Delta}}_{\tau}=j|{\mathrm{\Omega}}_{t};\mathit{\theta})$ in (36) are known as *filtered* and *smoothed* regime inferences, respectively; see, e.g. Hamilton (1994, Ch. 22) for their recursive calculation. We initialize these recursions by assuming that Δ_{1} is drawn from the stationary distribution of the chain, i.e.

$$\begin{array}{rll}p({\mathrm{\Delta}}_{1}=1|\mathit{\theta})& =& {\pi}_{1,\mathrm{\infty}}=\frac{1-{p}_{22}}{2-{p}_{11}-{p}_{22}},\\ p({\mathrm{\Delta}}_{1}=2|\mathit{\theta})& =& {\pi}_{2,\mathrm{\infty}}=\frac{1-{p}_{11}}{2-{p}_{11}-{p}_{22}}.\end{array}$$(37)

Note that, in (36), many terms are zero for parameters that appear in only one regime; e.g. if all parameters except *ν* are regime-specific, then

$$\begin{array}{rll}\frac{\mathrm{\partial}\mathrm{log}f({\u03f5}_{t}|{\mathrm{\Omega}}_{t-1};\mathit{\theta})}{\mathrm{\partial}{\mathit{\vartheta}}_{j}}& =& \frac{\mathrm{\partial}\mathrm{log}{f}_{jt}({\mathit{\vartheta}}_{j},\nu )}{\mathrm{\partial}{\mathit{\vartheta}}_{j}}p({\mathrm{\Delta}}_{t}=j|{\mathrm{\Omega}}_{t};\mathit{\theta})\\ & & +\sum _{\tau =1}^{t-1}\frac{\mathrm{\partial}\mathrm{log}{f}_{j\tau}({\mathit{\vartheta}}_{j},\nu )}{\mathrm{\partial}{\mathit{\vartheta}}_{j}}\left[p({\mathrm{\Delta}}_{\tau}=j|{\mathrm{\Omega}}_{t};\mathit{\theta})-p({\mathrm{\Delta}}_{\tau}=j|{\mathrm{\Omega}}_{t-1};\mathit{\theta})\right]\\ t& =& 1,\dots ,T,\phantom{\rule{1em}{0ex}}j=1,\dots ,k.\end{array}$$

For the score with respect to the parameters of *P*, see Hamilton (1996).^{19}

Now let *S* be the *T* × *N* matrix (where *N* is the dimension of *𝜽*) the *t*th row of which is given by (the transpose of) the score (35), *t* = 1, …, *T*, and let $\hat{\mathit{S}}$ be *S* evaluated at ${\hat{\mathit{\theta}}}^{c}$ the constrained MLE of *𝜽* under (27). Then the LM test statistic for H_{0} given by (27) can be calculated as in the outer gradient product form as (cf. Hamilton, 1996)

$$\text{LM}}={\mathbf{1}}_{T}^{\mathrm{\prime}}\hat{\mathit{S}}({\hat{\mathit{S}}}^{\mathrm{\prime}}\hat{\mathit{S}}{)}^{-1}{\hat{\mathit{S}}}^{\mathrm{\prime}}{\mathbf{1}}_{T}\stackrel{d}{\to}{\chi}^{2}({c}_{0}),$$(38)

where **1**_{T} is a *T*–dimensional column of ones, and *c*_{0} is the number of parameter constraints under H_{0}, see the discussion following Equation (27). Note that the elements of ${\mathbf{1}}_{T}^{\mathrm{\prime}}\hat{\mathit{S}}$ corresponding to unrestricted parameters are zero, so that a slight simplification of the LM statistic (38) can be obtained (cf. Tse, 2000).

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