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

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> *<sup>ε</sup>*. This is a model with a general *AR k*ð Þ dynamic structure and switching intercepts. For the *d*-dimensional time series f g *zt* , Eq. (4) can be re-written as:

$$z\_t = a\_0 + a\_1 \mathbf{s}\_t + B\_1 \mathbf{z}\_{t-1} + \dots + B\_k \mathbf{z}\_{t-k} + \varepsilon\_t \tag{6}$$

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 might be difficult to estimate.
