In this subsection, we state the GARCH (p,q) model and highlight its main properties. The model is given by the following equation:

h t 2 = ω + Σ i = 1 q α i ε t - i 2 + Σ j = 1 p β j h t - j 2 (1)

Where each h t 2 is obtained in a recursive way by taking an initial value of the variance at time t = 0, with a backcast methodology:

h 0 2 = ϵ 0 2 = λ T h ^ t = 0 2 + 1 - λ Σ j = 0 T λ T - j - 1 ϵ T - j 2 (2)

In order to find the optimal vector θ ∈ (ω, α₁,…,α_q,β₁,…,β_p) of the parameters defined in equation (1), the most used algorithm of optimization is that from BHHH (Berndt, Hall, Hall and Hausman, Berndt et al.,1974), which maximizes a likelihood function, as in equation (3).⁸ The iterative optimization method from BHHH for each step is given by:

θ i + 1 = θ i + E ∂ 2 L T ( i ) ∂ θ ∂ θ ' - 1 ∂ L T i ∂ θ (3)

Where the likelihood function to be maximized, assuming a normal distribution for the error term, satisfies:⁹

L t = - 1 2 log ⁡ 2 π - 1 2 log ⁡ h t 2 - 1 2 ε t 2 h t 2 (4)

On the other hand, by considering a system of n variables, we can express the error term as:

ε t Φ t - 1 ~ N 0 , D t R t D t (5)

ε t = μ t θ + ϵ t . (6)

The error term vector is then modeled as follows:

ε t = H t 1 2 θ z t . (7)

Here H t 1 2 θ is a positive definite matrix of order n × n ; and represents the conditional variance of ε t .

When studying volatility of diverse variables, the analysis is usually performed with a single equation of the GARCH family.¹⁰ However, in order to explore how volatility jointly affects the MFIs’ returns, we use a Dynamic Conditional Correlation (DCC) methodology that allows us to evaluate the impact on their performance (in terms of contagion). This research will now examine the relationship that exists between the conditional correlations and the conditional variances of the returns of the stock prices of the MFIs under study.¹¹

In this work, we focus on the methodology proposed by Engle (2002). Other authors as Bauwens et al. (2003), Wang and Tsay (2013), and Ling and McAleer (2003) propose other extensions of the multivariate-GARCH model with a system of non-related variables. Initially the BEKK model (a Multi-Variable GARCH model) was introduced under a bivariate representation by Engle and Kroner (1995). Under this framework, a generalized multivariate ARCH model can be stated as follows:

H t = C 0 ' C 0 + Σ k = 1 K Σ i = 1 q A i k ' ε t - i ε t - i ' A i k + Σ k = 1 K Σ i = 1 q G i k ' H t - i G i k (8)

Recently, Bauwens et al. (2012) have replaced the BEKK model by other specifications, a DCC model. In this case, it is possible to specify the model (in two steps) in order to obtain a covariance matrix. In this regard, two main dynamic coefficients of correlation were discussed by Engle (2002). On one hand, one of them is the rolling correlation estimator for returns with mean zero, which is defined by:

ρ ^ 12 , t = Σ s = t - n - 1 t - 1 r 1 , s r 2 , s Σ s = t - n - 1 t - 1 r 1 , s 2 Σ s = t - n - 1 t - 1 r 2 , s 2 (9)

On the other hand, the coefficient of correlation of exponential smoothing is defined as:

ρ ^ 12 , t = Σ s = 1 t - 1 λ t - j - 1 r 1 , s r 2 , s Σ s = 1 t - 1 λ t - j - 1 r 1 , s 2 Σ s = 1 t - 1 λ t - j - 1 r 2 , s 2 (10)

The DCC is defined from the covariance matrix H_t, as follows:¹²

H t = E ϵ t ϵ t ' | I t - 1 (11)

The matrix H_t can be decomposed, from the following expression:

H t = D t R t D t (12)

D t = d i a g   h i i t 1 / 2 ⋯ h N N t 1 / 2 (13)

In this way, it is possible to obtain the dynamic conditional correlation(R_t) from expression (12). The M-variable likelihood maximization can be applied in two steps by GMM optimization (Newey and MacFaden, 1994) according to the optimization methodology proposed by Engle (2002); assuming normally distributed errors as in equations (15) and (16). Hence, the two-step optimization method expressed in aggregated form is:

L θ , ϕ = L V θ + L C θ , ϕ (14)

The first step considers the following objective function:

L V θ = - 1 2 Σ t n l o g 2 π + l o g D t 2 + r t ' D t - 2 r t (15)

In the second step, we have

L C θ , ϕ = - 1 2 Σ t l o g R t + ϵ t ' R t - 1 ϵ t - ϵ t ' ϵ t (16)

We present now the results obtained with a system of non-related variables for each MFI (with benchmark variables) for every stock market analyzed. The selection of the GARCH model and the error specification, to obtain the DCCs in each market, are chosen according to the less explosive parameters, see Table 6. The estimations are shown in Table 7. Subsequently, we present the DCCs for each MFIs and their benchmark variables, see Figures 4 and 5. Finally, in Table 7 we calculate the arithmetic and geometric maens for each estimated DCC considering the period of analysis.

Table 6

Table 7

Figure 1 Source: own elaboration with the use of the R software.

Figure 2 Source: own elaboration with the use of the R software.

Figure 3 Source: own elaboration with the use of the R software.

Figure 4 Source: own elaboration with the use of the R software.

Table 7 shows the estimated coefficients for each MFI and its benchmark variables (based on the criteria in Table 6). The obtained results show that the specification in each equation successfully captures the estimated variances with persistence in both series ε i t - 1 2   a n d   h i t - 1 2 respectively. However, in the case of the parameters ω i , some specifications of the GARCH models capture that persistence in a smaller proportion; see columns (1)-(5) in Table 6. It can also be observed (in the same table) that for the estimated ω i in the columns (1) and (2) it is not possible to obtain a long-term variance.

It can be also observed in Figures 1 and 2 that it is not possible to appreciate a pattern (or contagion) common in the DCCs in each system of the MFIs. In other words, the particular specifications for each model of the GARCH family, according to the criteria in Table 5, do not capture a common contagion in the volatilities for each MFI. Contagion in volatilities can only be seen in some peaks with high volatility clusters, but not within the entire analysis period. Furthermore, we can see, in Table 8, that the arithmetic and geometric means do not show common patterns in the behavior of the DCCs in the analyzed stock markets.

Table 8

In order to analyze the contagion (emphasizing in the analysis of external sources of contagion) in the volatilities of the returns of the MFIs that are listed in the stock market, we use a global reference index, particularly the All Country World Index (ACWI). The latter captures the sources of capital return for 23 emerging markets and 23 developed markets. In the same order of ideas, we can observe, in Table 8, the results obtained with respect to h i t 2 for each equation. These results show that the series ε i t - 1 2 and h i t - 1 2 acceptably capture persistence in volatilities. However, for the case of the !_i parameters in some of the estimated GARCH family models, especially model (3) of Table 8, the results suggest that it is not possible to obtain a long-term variance. It is important to notice that in the estimated models, in columns (1), (2), (4) and (5) of Table 9, the long-term variances can be partially obtained.

Table 9

Finally, we can see in Figures 3 and 4, considering the global index (ACWI) as a point of reference, that it is not possible to observe a common pattern (in terms of contagion) in the DCC for each MFI system. It is worth mentioning that the previous results are similar to those obtained in the preceding section. In this way, the DCC obtained under the specification of the GARCH models do not capture a common contagion in the volatilities of the returns for each MFI. Basically, contagion in volatilities can only be seen in periods of high volatility. Complementing the previous results, it can be observed, in Table 9, that the arithmetic and geometric means do not have common patterns among the DCC of the studied markets.

Table 10