**3.2 Dynamic connectedness based time-varying vector autoregressive (TVPVAR) model**

We use the "Time-Varying Parameter Vector-Autoregressive (TVP-VAR) Model" in our analysis because we aim to investigate the return-spillovers between the series. The brief literature review and the explanation of the TVP-VAR Model is given below.

This chapter applies the asymmetric dynamic interconnectedness approach to the South African financial time series data. This analysis is an extension of the empirical works of Diebold and Yılmaz [6–23]. These authors wrote the pioneering work regarding the development of "dynamic connectedness measures". In addition, we rely on the works of Antonakakis et al. [4] and, Kahyaoglu Bozkus and Kahyaoglu [24]. Antonakakis et al. [4] proposes a measure for dynamic connectedness, developed to evaluate time-varying parameter VAR models. In this way, they manage to cope with the drawbacks of the standard rolling-windows dynamic approach [25].

It is important to capture the asymmetries in the financial time analysis. In this respect, we calculate all the absolute returns both in positive and negative signs for our dataset. This calculation is essential for the TVP-VAR based connectedness approach [25]. In our case, we use TVP-VAR (1). This is suggested based on the Bayesian Information Criterion (BIC). Our approach is defined in Eqs. (1) and (2) as follows:

*Asymmetric TVP-VAR Connectedness Approach: The Case of South Africa DOI: http://dx.doi.org/10.5772/intechopen.107248*

$$z\_t = B\_t z\_{t-1} + u\_t \qquad u\_t \sim N\left(0, \sum\_t \right) \tag{1}$$

$$\text{vec}(B\_t) = \text{vec}(B\_{t-1}) + \upsilon\_t \qquad \upsilon\_t \sim N(\mathbf{0}, R\_t) \tag{2}$$

Where:


According to the "Wold Representation Theorem" introduced by Pesaran and Shin [26], it is necessary to transform the estimated TVP-VAR model into its TVP-VMA process by using Eq. (3):

$$z\_{l} = \sum\_{i=1}^{p} B\_{il} z\_{l-i} + u\_{l} = \sum\_{j=0}^{\infty} A\_{jl} u\_{l-j} \tag{3}$$

The index for "pairwise directional connectedness", "total directional connectedness", "NET total directional connectedness" and "total connectedness index (TCI)" from *j* to *i* are formulated and obtained via R program based on the recent work of [27]. These authors define "influence variable (H) as forecast horizon" as follows:

$$\tilde{\rho}\_{ij,t}^{\mathfrak{g}}(H) = \frac{\tilde{\rho}\_{ij,t}^{\mathfrak{g}}(H)}{\sum\_{j=1}^{k} \mathcal{Q}\_{ij,t}^{\mathfrak{g}}(H)} \tag{4}$$

In this respect, we begin our analysis by determining the average connectedness measure, i.e., without considering asymmetry. It should be noted that all figures relate concurrently to both negative and positive returns and the symmetric connectedness measures. In other words, "off-diagonal factors" in the figures indicate the interaction between the variables in the system. On the other hand, the "elements in the main diagonal" match to "idiosyncratic shocks", namely, "own-innovations" for the series. The process is followed by applying Eq. (4) to calculate different cases based on the work of Kahyaoglu, Bozkus and Kahyaoglu [24] such that:

