1 Introduction

Asset return distributions are typically characterized by fat tails, conditional heteroskedasticity, and nonlinear dependence. Regarding the latter, a frequent concern is that the dependence between assets increases in periods of market turbulence. This has important implications for portfolio and risk management, because it means that “the benefits of diversification are partly eroded when they are needed most” (Campbell, Koedijk & Kofman, 2002). An overview over the extensive literature studying this phenomenon and further evidence is provided, e.g. by Kasch and Caporin (2013), Mittnik (2014), and Gribisch (2016). For example, Kasch and Caporin (2013) develop a multivariate GARCH model with dynamic correlations being allowed to depend on conditional volatility and, for major stock markets, find that “turbulent periods coincide with an increase in cross-market comovement.”

Markov-switching models (MSMs) are able to capture all of the aforementioned stylized facts of asset return distributions, and their use is very popular in financial modeling because, in addition to their flexibility, “the idea of regime changes is natural and intuitive” (Ang & Timmermann, 2012). For example, in bearish markets, expected returns, conditional volatilities and their dynamics, and correlations can differ from their respective counterparts in more normal or bullish market periods. Regime-specific dynamics may also be related to various types of trading patterns, as represented by “information” and “feedback” traders (Dean & Faff, 2008). See Guidolin (2011) and Ang and Timmermann (2012) for an overview over the many applications of MSMs.

In this paper, we investigate the properties of a multivariate extension of the Markov-switching (MS) GARCH model of Haas, Mittnik, and Paolella (2004b), allowing for regime-specific volatility dynamics, leverage effects, and correlation structures. We derive conditions for strict and weak stationarity and provide expressions for the unconditional overall and regime-specific covariance and correlation matrices, and for the dynamic correlation structure of the absolute values of the process. The model we consider is an extension of Pelletier’s (2006) regime-switching model for dynamic correlations (RSDC) in that it combines constant conditional within-regime correlations with regime-dependent conditional variance dynamics. Among other convenient features, this property allows the derivation of a simple recursion for multi-step-ahead conditional covariance matrices for, e.g. mean-variance portfolio allocation. Alternative extensions of Pelletier (2006) have been proposed by Billio and Caporin (2005) and Otranto (2010), who allow for within-regime correlation dynamics à la Engle (2002), but do not allow for switching variance dynamics. To test whether the assumption of constant conditional within-regime correlations is justified empirically, we construct a test against within-regime conditional correlation dynamics, adopting Hamilton’s (1996) Lagrange Multiplier (LM) framework along with Tse’s (2000) test for constant conditional correlations in a single-regime GARCH model. The methodology is applied to global stock market and real estate equity returns. The empirical analysis highlights the importance of the conditional distribution in MS time series models. Namely, since the conditional distribution in MSMs with Gaussian regimes is a discrete mixture of normals and thus already thick-tailed, one might guess that use of a more flexible within-regime distribution is unnecessary in this framework. Indeed, as observed by Guidolin (2011), “it seems that most authors are still finding that traditional Gaussian mixture models are generally sufficient to the task assigned to MSMs.” However, in our application, specifications with Student’s t innovations dominate their Gaussian counterparts both in– and out-of-sample. In particular, as discussed in Section 4, the Gaussian specification turns out to suffer from its inability to correctly track the regime-switching process. The dominating specification appears to be a two-regime Student’s t process with correlations which are higher in the turbulent (high-volatility) regime.

The structure of the paper is as follows. In Section 2, we define the model and discuss its relation to the literature. Statistical properties are presented in Section 3. Section 4 provides an application to financial data, and Section 5 concludes. Proofs of theorems, calculations of (conditional and unconditional) moments, and details of the LM test against misspecification of conditional correlations are gathered in Appendices A, B, and C, respectively. Finally, Appendix D provides a brief discussion of the asymmetric multivariate normal mixture GARCH model of Bauwens, Hafner, and Rombouts (2007) and Haas, Mittnik, and Paolella (2009), which for comparison purposes has been included in the empirical application in Section 4.3.

2 Definition of the process

The multivariate Markov-switching (MS) GARCH process introduced in this section generalizes the univariate model proposed in Haas, Mittnik, and Paolella (2004b). For alternative approaches to MS GARCH, see, e.g. Gray (1996), Dueker (1997), Klaassen (2002), Augustyniak (2014), and Reher and Wilfling (2016), as well as the review in Haas and Paolella (2012). Liu (2007) extended the model of Haas, Mittnik, and Paolella (2004b) to allow for an asymmetric response of volatility to positive and negative shocks, which is also incorporated in the model discussed herein.

Let the M–dimensional time series {ϵ_t} satisfy

where {Δ_t} is a Markov chain with finite state space ℰ = {1, …, k} and irreducible and aperiodic transition matrix P,

where the transition probabilities p_ij = p(Δ_t = j|Δ_t−1 = i), i, j ∈ ℰ, and the stationary distribution of the chain is denoted by 𝝅_∞ = (π_1,∞, π_2,∞, …, π_k,∞)′. Matrix DΔt,t=diag(σΔt,t), where σjt=(σ1jt,…,σMjt)′∈RM, j ∈ ℰ, contains the regime-specific conditional standard deviations of the elements of ϵ_t. Moreover,

where Rj=(ρℓm,j)ℓ,m=1,…,M, j = 1, …, k, is a (regime-specific) correlation matrix, and {ξt} is a sequence of iid random vectors with zero mean and identity covariance matrix. In the applications below, we assume that 𝝃_t has a Student’s t distribution with ν > 2 degrees of freedom, with density given by (29) in Appendix C, i.e.

which includes normality as a limiting case (ν → ∞). It is assumed that {Δ_t} and {ξt} are independent.

The regime-specific conditional standard deviations follow simultaneous asymmetric absolute value GARCH(1,1) (AGARCH) processes, i.e.

where Z_t = diag(z_t), a matrix in absolute value bars means that the absolute value of each element is taken, 𝝎_j = (ω_1j, …, ω_Mj)′, and

Parameters γ_ℓm,j ∈ [−1, 1], ℓ, m = 1, …, M, j ∈ ℰ, allow the conditional standard deviations to react asymmetrically to positive and negative news of the same magnitude as in Ding, Granger, and Engle (1993). Equivalently, as in McAleer, Hoti, and Chan (2009) and Francq and Zakoïan (2012), we could write the asymmetric volatility process à la Glosten, Jagannathan, and Runkle (1993) as

where x⁺ = max{x, 0}, x⁻ = min{0, x}, and

Denoting the typical elements of matrices Aj+ and Aj− in (7) by aℓm,j+ and aℓm,j−, respectively, the relation between the parameters in (5) and (7) is

The model defined by (1)–(6) will be referred to as a k–component Markov-switching constant conditional correlation GARCH process, or, in short, MS(k) CCC-GARCH. It is an asymmetric multi-regime version of the extended CCC (ECCC) model studied by Jeantheau (1998), which itself generalizes the CCC of Bollerslev (1990) by allowing for volatility interactions, which are often of interest in finance and macroeconomics (e.g. Nakatani & Teräsvirta, 2009; and Conrad and Karanasos 2010, 2015). In many applications the diagonal model, with all A_j, B_j, and 𝚪_j being diagonal matrices, will be preferred for reasons of parsimony; an ARCH version of such a model was used by Ramchand and Susmel (1998).

The specification of the volatility dynamics (5) in terms of the conditional standard deviation instead of the conditional variance, as originally proposed by Taylor (1986), serves two purposes: First, empirically, it appears that it typically improves the fit as compared to the formulation in terms of the conditional variance, and is very often close to the MLE when the “power parameter” (as in Ding, Granger & Engle, 1993) is freely estimated from the data (e.g. Giot & Laurent, 2003; Lejeune, 2009; and Broda et al., 2013). Second, as noted by Pelletier (2006), this specification allows for closed-form calculation of multi-step-ahead conditional covariance matrices, as required, e.g. for mean-variance portfolio optimization over horizons longer than one period. The model suggested by Pelletier (2006), referred to as the regime-switching dynamic correlation (RSDC) model, is nested in (1)–(6) when only the conditional correlation matrices are subject to regime-switching, i.e. in (5), 𝝎₁ = ⋯ = 𝝎_k, A₁ = ⋯ = A_k, and B₁ = ⋯ = B_k. Covariance matrix forecasts for the RSDC are considered in Pelletier (2006) and Haas (2010), whereas a convenient scheme for forecasting the general model (1)–(6) is derived in Appendix B.3.

In (5), conditions have to be imposed to make sure that all elements of 𝝈_jt remain positive with probability 1, j = 1, …, k. As observed by He and Teräsvirta (2004), an obvious set of sufficient conditions is that

but these are not necessary when diagonality is not imposed (Nakatani & Teräsvirta, 2008; Conrad & Karanasos, 2010). For the diagonal model, which is of particular importance in the applications, conditions (8) are necessary, however.

Regarding the distribution of the innovations {𝝃_t}, note that (4) includes Gaussian innovations as a limiting case, when ν → ∞. Though normality is still the dominant distributional assumption in regime-switching models (cf. Guidolin, 2011), allowing for fat-tailed innovations can improve both in-sample fit and out-of-sample forecasting performance of MS GARCH models, as pointed out, e.g. by Klaassen (2002), Ardia (2009), and Shi and Feng (2016). Due to the dependence structure implied by the multivariate t distribution, this also holds for Pelletier’s (2006) model where volatility is regime-independent; see Section 4 for a detailed discussion and illustration.

3 Properties of the model: stationarity and moment structure

In this section, we discuss the statistical properties of the MS(k) CCC-GARCH process. In particular, we present conditions for strict stationarity and ergodicity and the existence of unconditional moments, with proofs of the theorems given in Appendix A. Explicit formulas for moments of frequent interest are provided (and derived) in Appendix B, namely the unconditional covariance matrix and the autocorrelations of the absolute values, which can be used to characterize the joint volatility dynamics. Moreover, a simple recursive scheme for obtaining multi-step-ahead covariance matrices is derived in Appendix B.3, fostering applications to mean-variance portfolio selection in environments with changing volatilities and correlations.

To set out the properties of the MS(k) CCC-GARCH process, we define the matrices

and B = blockdiag(B₁, …, B_k). This gives rise to the representation

where

and e_j is the jth unit vector in ℝ^k, j = 1, …, k.

We first provide a necessary and sufficient condition for the existence of a strictly stationary solution of the MS(k) CCC-GARCH process with the nonnegativity conditions (8) imposed. Theorem 1 generalizes results for the univariate MS GARCH process in Liu (2006 and 2007).^[1]

The condition in Theorem 1 may be inconvenient to check in practice. Theorem 2 offers an alternative criterion which is easier to handle and provides additional information about the moment structure of the process. To state this criterion, we define the matrices

Furthermore, we adopt the following notation from Francq and Zakoïan (2005): For any function f: ℰ ↦ M_n×n′(ℝ), where M_n×n′(ℝ) is the space of real n × n′ matrices, and ℰ = {1, …, k} is the state space of {Δ_t}, define the matrix

For example, the matrices required by Theorem 2 to check for the first moment are given by

where

To check the condition for covariance stationarity, we need the (regime-specific) second moment matrices of the absolute innovations, i.e.

the elements of which are provided by the result of Nabeya (1951) that, for bivariate standard normal x and y with correlation ρ, we have

Equation (13) continues to hold for a unit-variance bivariate Student’s t distribution, as detailed in the appendix of Haas (2010). Moreover, let

Then matrices C₂(j), j ∈ ℰ, are given by

E.g. in the practically important case of two regimes, checking for covariance stationarity involves inspecting the eigenvalues of the matrix

4 Application to financial data

We consider volatility and correlation dynamics of global stock market and real estate equity returns, using dollar-denominated weekly (Wednesday-to-Wednesday) returns of the MSCI world and the FTSE EPRA/NAREIT global indices from January 1990 to October 2011 (T = 1137 observations). The latter index represents the evolution of real estate equities.^[2] The analysis is based on continuously compounded percentage returns, i.e. r_it = 100 × log(I_it/I_i,t−1), i = 1, 2, where I_1t and I_2t are the MSCI and the FTSE EPRA/NAREIT index levels, respectively. Both the index levels and the returns are shown in the top and middle panels of Figure 1, reflecting the turbulent development of markets particularly since the beginning of the current millennium. Sample moments of the returns are reported in Table 1, along with the Jarque-Bera (JB) test for normality and Engle’s (1982) Lagrange Multiplier (LM) test for conditional heteroskedasticity.

Figure 1:

The top panel shows the weekly index levels (left plot) and percentage log returns (right plot) of the MSCI world stock market index from January 1990 to October 2011. The middle panel is similar, but for the FTSE EPRA/NAREIT global index reflecting the evolution of real estate equities. The bottom panel shows conditional correlations implied by an exponentially weighted moving average (EWMA) covariance matrix estimator H_t with smoothing constant λ = 0.95, i.e. Ht=(1−λ)rt−1rt−1′+λHt−1, where the initial matrix H₁ is set equal to the sample covariance matrix.

From Table 1, we note that the return series exhibit a considerable correlation of 0.795, which reflects the common finding that real estate equities display much more similarity to the general stock market than direct real estate investments (e.g. Morawski, Rehkugler & Füss, 2008; Heaney & Sriananthakumar, 2012). Moreover, the bottom panel of Figure 1 shows conditional correlations implied by an exponentially weighted moving average (EWMA) estimator (cf. Alexander, 2008, Ch. 3.8), which hints at time-varying conditional correlations with a particularly strong degree of comovement both at the beginning and the end of the sample, with the latter being also characterized by an outburst of unprecedented volatility.^[3] Results for versions of Tse’s (2000) Lagrange Multiplier (LM) test for constant conditional correlations in a multivariate GARCH model are reported in Table 2 and also provide support for time-varying conditional correlations.

4.1 Fitting MS CCC-GARCH(1,1) processes

The evidence for time-varying correlations coupled with periods of low and high volatility makes the MS CCC-GARCH model defined in Section 2 a candidate for modeling these series. We fit the model with k = 1, 2, and 3 regimes,^[4] where we confine ourselves to the diagonal model, with A_j, B_j, and 𝚪_j in (5) being diagonal matrices, j = 1, …, k. In addition, we restrict the asymmetry parameters to be regime-independent, i.e.

Γ1=Γ2=⋯=Γk=:Γ.

There are no clear-cut signs of conditional mean dynamics in the data, and thus we specify the model for return vector r_t as

rt=μ+ϵt,

where 𝝁 is the constant conditional mean, and ϵ_t is generated by an MS(k) CCC-GARCH process as described in Section 2. We compare the fit of models with different k by means of the Bayesian information criterion (BIC) of Schwarz (1978), which, from results of Keribin (2000) and Francq, Roussignol, and Zakoïan (2001), can be expected to have favorable properties for this purpose. Results are reported in Table 3 for both Gaussian and Student’s t innovations {𝝃_t} in (3). In both cases, models with two components are preferred, as is a conditional t distribution. Thus we focus on two-component models in the following discussion. Both normal and Student’s t innovations are considered in order to highlight the role of the conditional distribution.

Table 3:

Likelihood-based goodness-of-fit of MS(k) CCC-GARCH models.

	Gaussian innovations			Student’s t innovations
	k = 1	k = 2	k = 3	k = 1	k = 2	k = 3
K	11	20	31	12	21	32
log L	−4371.5	−4288.0	−4261.0	−4323.7	−4260.0	−4247.6
BIC	8820.4	8716.6	8740.1	8731.7	8667.7	8720.3

1. Reported are likelihood-based goodness-of-fit measures for diagonal MS(k) CCC-GARCH models fitted to the MSCI world and FTSE EPRA/NAREIT global returns. The number of regimes is denoted by k, and k = 1 corresponds to the single-regime CCC of Bollerslev (1990). In all models, the asymmetry parameters are restricted to be constant across regimes, i.e. in (5), 𝚪₁ = ⋯ = 𝚪_k. K is the number of parameters of a model, log L is the value of the maximized log-likelihood, and BIC is the Bayesian information criterion of Schwarz (1978), i.e. BIC = −2 × log L + K log T, where T is the sample size. Smaller values of BIC are preferred.

The diagonal MS(k) CCC-GARCH model without further restrictions is rather flexible in that it allows the variances as well as the correlations to be regime-dependent. The contribution of both of these features to the overall improvement over the single-regime specification documented in Table 3 is not clear a priori. It is thus of interest to test various restricted models against the unrestricted specification. Specifically, we consider Pelletier’s (2006) RSDC model where the switching applies to the conditional correlation matrix only, i.e. conditional volatilities are constant across regimes. The second constrained specification represents the opposite of Pelletier’s (2006) model, namely the case where volatility can switch but R₁ = R₂ in (3). The results reported in Table 4 show that, although both restrictions are rejected against the full model by means of likelihood ratio tests, allowance for regime-specific correlations appears to be more important than switching in the univariate GARCH dynamics, and particularly so for the (generally preferred) models with Student’s t innovations.

Table 4:

Likelihood ratio tests (LRT) of restricted MS(2) CCC-GARCH specifications against the full (diagonal) model

	Gaussian innovations			Student’s t innovations
	Full	Pelletier	R1 = R2	Full	Pelletier	R1 = R2
		(RSDC)			(RSDC)
K	20	14	19	21	15	20
log L	−4288.0	−4311.7	−4313.0	−4260.0	−4272.6	−4310.9
LRT	−	47.5***	50.1***	−	25.2***	101.9***

1. The table reports likelihood ratio tests (LRT) for restricted versions of the two-component diagonal MS(2) CCC-GARCH model. The unrestricted specification, denoted as “full”, is the model introduced in Section 2 with k = 2, and where the matrices A_j, B_j, j = 1, 2, and 𝚪₁ = 𝚪₂ in (5) are diagonal. “Pelletier” refers to Pelletier’s (2006) regime-switching dynamic correlation (RSDC) model where only the correlation matrix is subject to regime-switching, i.e. the additional restrictions 𝝎₁ = 𝝎₂, A₁ = A₂, and B₁ = B₂ are imposed in (5). The third model restricts the correlation to be the same in both regimes, i.e. R₁ = R₂ in (3). log L is the value of the maximized log-likelihood, and K is the number of parameters of a model. The associated likelihood ratio test statistics, denoted as LRT, have 6 and 1 degrees of freedom, respectively. Asterisks ^*** indicate significance at the 1% level.

Several characteristics of the estimated MS(2) CCC-GARCH models with Gaussian and Student’s t innovations are reported in Table 5, where the single regime CCC-GARCH models are included for comparison. In Table 5, the regimes are ordered such that π_1,∞ > π_2,∞. Both two-regime models have in common that Regime 1 is a low-volatility regime with moderate correlation (relative to the unconditional correlation), and Regime 2 is a high-volatility regime with rather high correlation, i.e. the diversification potential deteriorates in turbulent market periods. As reported in the bottom part of Table 5, the unconditional moments implied by the single-regime models are close to those of the two-regime specifications and are in between their regime-specific counterparts documented in the top and middle parts of the table for Regimes 1 and 2, respectively. All estimated models are covariance stationary, since λ(PC2)<1 for all estimated specifications (cf. Theorem 2).

Table 5:

Characteristics of estimated MS CCC-GARCH(1,1) models.

Estimated characteristic	Gaussian innovations		Student’s t innovations
	k = 1	k = 2	k = 1	k = 2
ρ12,1	0.766 (0.013)	0.636 (0.025)	0.769 (0.014)	0.655 (0.024)
E(ϵ1t2|Δt=1)	4.265	3.691	3.847	3.161
E(ϵ2t2|Δt=1)	5.528	3.540	4.713	3.318
Corr(ϵ1t,ϵ2t|Δt=1)	0.754	0.617	0.756	0.636
p11	1	0.985 (0.007)	1	0.996 (0.003)
π1,∞	1	0.610 (0.151)	1	0.563 (0.109)
(1−p11)−1	∞	66.54a	∞	233.8a
ρ12,2	–	0.929 (0.009)	–	0.921 (0.009)
E(ϵ1t2|Δt=2)	–	6.080	–	5.846
E(ϵ2t2|Δt=2)	–	7.867	–	7.785
Corr(ϵ1t,ϵ2t|Δt=2)	–	0.916	–	0.912
p22	0	0.976 (0.012)	0	0.994 (0.005)
π2,∞	0	0.390 (0.151)	0	0.437 (0.109)
(1−p22)−1	–	42.49a	–	181.8a
E(ϵ1t2)	4.265	4.622	3.847	4.336
E(ϵ2t2)	5.528	5.226	4.713	5.272
Corr(ϵ1t, ϵ2t)	0.754	0.779	0.756	0.805
δ = p11 + p22 − 1	–	0.961 (0.017)	–	0.990 (0.006)
ν	–	–	7.368 (1.025)	8.584 (1.370)
γ11	0.680 (0.157)	0.738 (0.168)	0.575 (0.181)	0.627 (0.180)
γ22	0.401 (0.105)	0.635 (0.147)	0.273 (0.116)	0.472 (0.152)
λ(PC2)	0.903	0.933	0.921	0.953

1. Standard errors are given in parentheses. ρ_12,j is the constant conditional correlation in Regime j; π_j,∞ is the stationary probability and (1 − p_jj)⁻¹ is the expected duration of the jth regime, j = 1, 2; δ = p₁₁ + p₂₂ − 1 is a measure for the persistence of the regime process; γ₁₁ and γ₂₂ are the asymmetry parameters in the volatility equation (5) for the MSCI and the FTSE/NAREIT, respectively; λ(PC2) is the largest eigenvalue of matrix PC2 defined in (11) and (12) with l = 2, and with λ(PC2)<1 being the condition for covariance stationarity (cf. Theorem 2).

2. ^aAsymptotic standard errors for the expected duration of regime j, (1 − p_jj)⁻¹, j = 1, 2, could, at least in principle, also be calculated via the delta method. However, with p^_jj rather close to unity, as in the case under consideration, the normal approximation would be basically useless and thus we abstain from reporting them. For example, with the estimates in the table above, the asymptotic standard error of (1 − p^₁₁)⁻¹ in the Student’s t switching model would be estimated as Var^(p^11)/(1−p^11)2=0.0028/(1−0.9957)2=151.4.

Comparing the regime-switching models with Gaussian and Student’s t innovations, we observe that both models are characterized by fairly persistent regimes, but the persistence is more pronounced with Student’s t innovations, where both ”staying probabilities” p₁₁ and p₂₂ are rather close to unity. To illustrate the differences in estimated persistence, expected regime durations implied by estimated parameters, given by (1 − p^_jj)⁻¹, j = 1, 2, are also reported in Table 5. With Gaussian regimes, expected duration of the low (high)–volatility regime is slightly longer (shorter) than 1 year, whereas it is almost five (four) years with Student’s t regimes.^[5] This pattern, which is also discussed in Bulla (2011), Haas (2010), and Haas and Paolella (2012), is due to the tendency of a model with Gaussian regime densities to signal a regime shift whenever an untypically large (small) observation occurs within an otherwise calm (turbulent) regime.^[6] Such untypical observations are easier accommodated within a given regime when the regime densities are allowed to be leptokurtic, i.e. display fatter tails and higher peaks than the normal. The same logic applies to Pelletier’s model where only the correlations are subject to regime-switching, since, for fixed correlation, simultaneous extreme realizations of both variables are more likely with Student’s t innovations.^[7]

The upper panel of Figure 2 illustrates the models’ inferred switching activity by means of the smoothed regime probabilities of the high-volatility/correlation regime under both types of innovation distribution. Both models indicate a switch to the high-correlation regime at the end of the sample beginning with the financial turmoil in the wake of the burst of the housing bubble. Implications for forecasting are depicted in the lower panel of Figure 2. The left plot of the lower panel of Figure 2 shows conditional correlations as implied by a Gaussian regime-switching model for the two situations where we know for certain that at the forecast origin we are either in the first or second regime.^[8] As a function of the forecast horizon, the conditional correlation of the Gaussian model rapidly converges to its unconditional value, whereas forecasts are much more persistently affected by the current state of the world in the Student’s t model, as shown in the bottom right graph of Figure 2.

Figure 2:

The upper panel shows the smoothed probabilities of Regime 2 (high-volatility/correlation) implied by the MS(2) CCC-GARCH process with Gaussian (left plot) and Student’s t innovations (right plot). The lower panel shows conditional correlations under the assumption that we either start in the low– or high-volatility/correlation regime, as represented by the solid and dash-dotted lines, respectively. Conditional standard deviations have been initialized with appropriate unconditional expectations (cf. Footnote 8). As in the upper panel, the left and right graphs are for Gaussian and Student’s t innovations, respectively.

4.2 Testing for within-regime correlation dynamics and comparison with other models

One of the most popular approaches to time-varying conditional correlations is the dynamic conditional correlation (DCC) model of Engle (2002). In the DCC, conditional correlations are driven by standardized shocks rather than by discrete regime shifts as in the Markov-switching processes studied herein. Both models can be combined to produce an even more flexible structure which allows the conditional correlations in each regime to be driven by DCC-type dynamics (e.g. Billio & Caporin, 2005; Otranto, 2010). However, the MS CCC-GARCH model has several advantages over its DCC-type generalization, since it is easier to estimate and admits the computation of multi-step-ahead conditional covariance matrices. In view of these advantages, it is desirable to have at one’s disposal a simple test of the regime-switching CCC against the alternative of within-regime correlation dynamics. To this end, we extend Tse’s (2000) Lagrange Multiplier (LM) test for constant conditional correlations to the multi-regime framework and allowing for fat-tailed (Student’s t) innovations.^[9] The details of this test, which fits into the general framework described by Hamilton (1996), are developed in Appendix C. Results are reported in Table 6 for two conditional volatility specifications under the null hypothesis, that is, both the “full” model from Table 4 as well as Pelletier’s model, and both specifications are considered with Gaussian and Student’s t innovations. Under the alternative, within-regime correlation dynamics are either regime-dependent (case (a) in Table 6) or regime-independent (case (b)). Overall, the results in Table 6 show no clear-cut sign of within-regime correlation dynamics, i.e. the switching between low– and high-correlation periods appears to capture most of the time-variation in conditional correlations.

Table 6:

Lagrange Multiplier (LM) tests for constant within-regime correlations.

	Specification of alternative hypothesis
	Case (a)	Case (b)
Gaussian regime densities
Pelletier (RSDC)	4.73*	0.32
MS CCC	0.46	0.33
Student’s t regime densities
Pelletier (RSDC)	1.46	0.12
MS CCC	4.48	0.00

1. Reported are Lagrange Multiplier (LM) tests for constant conditional within-regime correlations in two-regime MS CCC-GARCH models, as described in Appendix C. “Pelletier” is Pelletier’s (2006) regime-switching dynamic correlation (RSDC) model where only the correlation is subject to regime-switching. Note that the tests reported here are based on symmetric volatility processes, i.e. Γ₁ = Γ₁ = 0 in (5). Cases (a) and (b) are distinguished by means of the alternative hypothesis as described in Appendix C:

   • Case (a) refers to the situation where, under the alternative, the correlation dynamics may be different in both regimes, i.e. in (26), both δ_12,j, j = 1, 2, may be nonzero and different.

   • In case (b), it is assumed under the alternative that correlation dynamics are regime-independent, i.e. δ_12,1 = δ_12,2 in (26).

   Under the null hypothesis of constant conditional correlations in both regimes, the test statistic for case (a) is asymptotically distributed as χ²(2), whereas in case (b) the limiting distribution is χ²(1). Asterisk ^* denotes significance at the 10% level (the p–value is 0.094).

In view of these results, we compare conditional correlations implied by the MS CCC and the DCC model of Engle (2002).^[10] These correlations are shown in Figure 3 for both Gaussian (top panel) and Student’s t innovations (bottom panel). Comparing the upper with the lower panel of Figure 3, MS CCC-implied correlations are smoother with Student’s t than with Gaussian innovations. The DCC-implied correlations depend much less on the innovation distribution and, as already observed by Pelletier (2006), are less smooth than their regime-switching CCC counterparts.^[11] Roughly, however, both types of models contain similar information about low– and high-correlation periods in the data. In particular, they agree with regard to the jump in correlation at the onset of the recent financial crisis.

Figure 3:

The upper panel shows conditional correlations as implied by the MS CCC-GARCH model and the DCC process with Gaussian innovations. The lower panel repeats this, but for models with Student’s t innovations. The one-step-ahead conditional correlations of the MS CCC-GARCH models are extracted from the conditional covariance matrix (23) for d = 1.

Multi-step ahead conditional correlations for both types of models are illustrated in Figure 4, which resembles the lower part of Figure 3 but additionally includes conditional correlations implied by the Student’s t DCC model, as calculated by simulation.^[12] Initial values for the conditional correlation matrix and the conditional standard deviations in the DCC were selected such that they match those of the Student’s t MS CCC in the respective regimes. The long-run correlation of the DCC model is a bit lower than those implied by estimated MS CCC processes. However, with regard to the persistence of multi-step correlations, the DCC is more like the Student’s t rather than the Gaussian MS CCC-GARCH. This is in line with DCC parameter estimates, as reported in Table 7. Interpreting a^ + b^ as an estimate of the persistence of conditional correlations, i.e. the equivalent of δ in Table 5, then the persistence in conditional correlations implied by estimated DCC models is close to the value of δ^ = 0.99 of the Student’s t MS CCC model.

Figure 4:

Shown are conditional correlations similar to the lower part of and as explained in the legend of Figure 3. The curves for the MS CCC-GARCH models reproduce those in the bottom part of Figure 3. Initial values for the conditional correlation matrix and the conditional standard deviations in the DCC model were determined such that they match those of the Student’s t MS CCC in the respective (low– and high correlation/variance) regimes.

Table 7:

Parameter estimates for correlation dynamics in DCC models.

Innovations	a^	b^	a^ + b^
Gaussian	0.044 (0.014)	0.942 (0.022)	0.986 (0.009)
Student’s t	0.047 (0.015)	0.941 (0.021)	0.988 (0.008)

1. Shown are estimates of the parameters driving the correlation dynamics in Gaussian and Student’s t DCC models (Engle, 2002), with standard errors given in parentheses. The evolution of the conditional correlation matrix R_t is described by

   Qt=(1−a−b)S+azt−1zt−1′+bQt−1
   Rt=(I⊙Qt)−1/2Qt(I⊙Qt)−1/2,

   where the z_t are the standardized (“degarched”) residuals, and S is estimated via their sample correlation matrix.

5 Concluding remarks

We conclude with a remark referring to the frequently contemplated “curse of dimensionality” problem. An advantage of the (diagonal) CCC, DCC, and Pelletier’s (2006) RSDC models is that, via two-step estimation, application to high-dimensional time series is feasible.^[14] This property is not shared by the model studied herein with both regime-specific correlations and variance dynamics. We do not deem this to be a disadvantage, since there are many applications where the advantage of a more flexible dynamic structure may very well outweigh the benefits of parsimony as long as the dimension of the problem is sufficiently low. Studies of the dynamics of broadly defined asset classes, as illustrated in Section 4, are a typical example, where a richer specification can lead to a better understanding and potentially improved forecasts of the joint process under study. As another recent example from the literature somewhat related to the application in Section 4, Case, Guidolin, and Yildrim (2014), using monthly data from 1972 to 2009, find that even a four-regime MS model is required to appropriately describe the evolution of the joint conditional distribution of REIT, stock, and bond returns, since in particular the bond market regimes fail to be synchronized with those of the other two markets. For higher-dimensional systems, Pelletier’s (2006) model, which is nested in the general specification of this paper, appears to provide a reasonable balance between flexibility on the one hand and parsimony and tractability on the other. This is suggested in particular since the results in Section 4 (cf. Table 4) revealed that allowing for regime-switching correlations is of greater value than doing the same for the dynamics of individual volatilities.