**3.3 Volatility spillover modelling**

Volatility and volatility transmission can be illustrated using most econometric models including VAR model. The formula for VAR model is:

$$y\_t = \mathfrak{c} + A\_1 y\_{t-1} + A\_2 y\_{t-2} + \dots + A\_p y\_{t-p} + \mathfrak{e}\_t \tag{1}$$

where *st* ¼ 0, 1 are still the Markovian state variables with the transition matrix (3a) and *ε<sup>t</sup>* are i.i.d. random variables with zero and variance *σ*<sup>2</sup>

model with a general *AR k*ð Þ dynamic structure and switching intercepts. For the

while *st* ¼ 0, 1 are still the Markovian state variables with the transition matrix (3a), *Bi*ð Þ *i* ¼ 1, … , *k* are matrices of parameters, and *ε<sup>t</sup>* are i.i.d. random vectors with zero and variance–covariance matrix ⅀0. Eq. (5) is a VAR model with switch intercepts. Although generalisation is easy but some parameters such as *d* variables

In presenting the empirical results, the article starts with VAR calculations. Thereafter, Markov regime-switching results are presented in order to explore if one can infer interdependence of volatilities regimes. In order to verify which VAR is suitable, the first and second order tests (i.e. residual serial correlation) testing validity are used. Thereafter, a lag-length criterion is used. All those tests confirmed the appropriateness of VAR (1) model. Further, in order to interpret the results Cholesky decomposition is used. Generally, when using Cholesky decomposition the order of VAR parameters order matters. The BRICS countries are inputted in alphabetical order because that order is consisted with normal writing order. Although, VAR results might be different when one inputs them in a different format, one views normal order as an appropriate one. It can be inferred from [31] alphabetical order modelling leads to better estimates. In **Tables 2** and **3** all variables highlighted in grey are statistically significant for VAR values as they are at least 2 irrespective of being negative or positive. The F-statistic is basically Anova values and one reads the in the following manner. Assume the following inequality *F*ð Þ¼ 2, 12 22*:*59, *p*< 0*:*05Þ, the 2 is the degrees of freedom numerator, 12 is total observations of freedom denominator, 22.59 is the calculated Anova value and 0.05 is alpha (i.e. significance level). This article assumes that both the degrees of freedom numerator and total observations of freedom denominator are infinities in order to illustrate the best case scenario. In the latter situation, the critical value is 1.22. Thus, F-statistic values highlighted in grey fall within the non-rejection (i.e. acceptable) regions while values which are not highlighted fall within rejection regions. That is, latter values exemplify autocorrelation for those VAR(1) model. The results panel 5 of **Table 2** illustrates that one-lag in Brazilian indexed volatility of bonds cause one-lag in Brazilian indexed volatility of bonds by 0.18 units. The letter statement is sensible given that what happens in one market should have similar effect in the short run-regimes show that regimes time is just over 2 weeks. Similarly, a one-lag in Brazilian indexed volatility of bond cause onelag in South African indexed volatility of bonds-this probably that of similarities between the two countries, i.e. ruling political parties stay in power much longer; historically, Brazil and South Africa have good trade relations and further, the BRICS formation is strengthening that relationship even more. The one-lag in Indian volatility of bonds cause one-lag in Indian volatility of bonds. The phenomenon is similar with the one of Brazil lags; however, Indian one is negative while Brazil one is positive. One possible explanation for India negative lag is that in India the government is highly involved in driving economic growth than in Brazil.

*zt* ¼ *α*<sup>0</sup> þ *α*1*st* þ *B*1*zt*�<sup>1</sup> þ … þ *Bkzt*�*<sup>k</sup>* þ *ε<sup>t</sup>* (6)

*d*-dimensional time series f g *zt* , Eq. (4) can be re-written as:

might be difficult to estimate.

*The Independence of Indexed Volatilities DOI: http://dx.doi.org/10.5772/intechopen.90240*

**4. Analysis**

**141**

*<sup>ε</sup>*. This is a

Where the *<sup>l</sup>*-periods back observation *yt*�<sup>1</sup> is called the *<sup>l</sup>*-th lag of *<sup>l</sup>*-th lag of *<sup>y</sup>*, *<sup>c</sup>* is a *k* ∗ 1 vector of constants (intercepts), *Aj* is the time-invariant *k* ∗ *k* matrix and e*<sup>t</sup>* is a *k* ∗ 1 vector of error terms satisfying Eð Þ¼ e*<sup>t</sup>* 0, every error term has mean zero. E e*t*e<sup>0</sup> *t* � � <sup>¼</sup> <sup>Ω</sup>, the contemporaneous covariance matrix of error terms is <sup>Ω</sup> (a *<sup>k</sup>* <sup>∗</sup> *<sup>k</sup>* positive-semidefinite matrix. E e*t*e<sup>0</sup> *t*�*k* � � <sup>¼</sup> 0, for any non-zero *<sup>k</sup>*, there is no correlation across time; in particular, no serial correlation in individual error terms. In order to have a deeper insight in volatility spills, this article proposes using regime-switching model in order to capture different spills regimes. The common model used for regime-switching variables is Markov switching model. The simple Markov model of conditional mean presented when *st* denotes an unobservable state variable assuming the value one or zero. A simple switching model for the variable *zt* involves two AR specifications:

$$\mathbf{z}\_{t} = \begin{cases} \mathbf{z}\_{0} + \beta \mathbf{z}\_{t} + \mathbf{e}\_{t}, \mathbf{s}\_{t} = \mathbf{0}, \\\\ \mathbf{a}\_{0} + \mathbf{a}\_{1} + \beta \mathbf{z}\_{t} + \mathbf{e}\_{t}, \mathbf{s}\_{t} = \mathbf{1}, \end{cases} \tag{2}$$

where j j *<sup>β</sup>* <sup>&</sup>lt;1 and *<sup>ε</sup><sup>t</sup>* are i.i.d. random variables with mean zero and variance *<sup>σ</sup>*<sup>2</sup> *ε*. This is a statitionay AR (1) process with the mean *<sup>α</sup>*<sup>0</sup> <sup>1</sup>�*<sup>β</sup>* when *st* <sup>¼</sup> 0, and it switches to another stationary AR (1) process with mean *<sup>α</sup>*0þ*α*<sup>1</sup> <sup>1</sup>�*<sup>β</sup>* when *st* <sup>¼</sup> 1. If *<sup>α</sup>*<sup>1</sup> 6¼ 0 then the model admits two dynamic structures at different levels, depending on the value of the state variable *st*. In this case, *zt* are governed by two regimes with distinct means, and *st* determines switching between two different regimes. The transition matrix for the Markov is:

$$\mathcal{P} = \begin{bmatrix} IP(\mathfrak{s}\_t = \mathbf{0} | \mathfrak{s}\_{t-1} = \mathbf{0}) IP(\mathfrak{s}\_t = \mathbf{1} | \mathfrak{s}\_{t-1} = \mathbf{0}) \\ IP(\mathfrak{s}\_t = \mathbf{0} | \mathfrak{s}\_{t-1} = \mathbf{1}) IP(\mathfrak{s}\_t = \mathbf{1} | \mathfrak{s}\_{t-1} = \mathbf{1}) \end{bmatrix} \tag{3}$$

and

$$= \begin{bmatrix} p\_{00} \ p\_{01} \\ p\_{10} \ p\_{11} \end{bmatrix},\tag{4}$$

where *pij* ð Þ *i*, *j* ¼ 0, 1 denote the transition probabilities of *st* ¼ *j* given that *st* ¼ *i*. The transition probabilities satisfy *pi*<sup>0</sup> þ *pi*<sup>1</sup> ¼ 1. The matrix governs the random behavior of the state variable, and it contains two parameters *<sup>p</sup>*<sup>00</sup> and *<sup>p</sup>*<sup>11</sup> � �. One can extend model (5) such that a more general dynamic structure is captured. Then model (2) is extended into:

$$z\_t = a\_0 + a\_1 \mathbf{s}\_t + \beta\_1 \mathbf{z}\_{t-1} + \dots + \beta\_k \mathbf{z}\_{t-k} + \varepsilon\_t \tag{5}$$
