Bayesian inference of multiple structural change models with asymmetric GARCH errors

Abstract

Structural change in any time series is practically unavoidable, and thus correctly detecting breakpoints plays a pivotal role in statistical modelling. This research considers segmented autoregressive models with exogenous variables and asymmetric GARCH errors, GJR-GARCH and exponential-GARCH specifications, which utilize the leverage phenomenon to demonstrate asymmetry in response to positive and negative shocks. The proposed models incorporate skew Student-t distribution and prove the advantages of the fat-tailed skew Student-t distribution versus other distributions when structural changes appear in financial time series. We employ Bayesian Markov Chain Monte Carlo methods in order to make inferences about the locations of structural change points and model parameters and utilize deviance information criterion to determine the optimal number of breakpoints via a sequential approach. Our models can accurately detect the number and locations of structural change points in simulation studies. For real data analysis, we examine the impacts of daily gold returns and VIX on S&P 500 returns during 2007–2019. The proposed methods are able to integrate structural changes through the model parameters and to capture the variability of a financial market more efficiently.

Appendix

We assume ${\varvec{\phi }}_{1}$ $\sim$ N$({\varvec{\phi }}_{10},{\varvec{\varSigma }}_1)$. Let ${\varvec{\theta }}_{-{\phi }}$ be the parameter vector excluding the element ${\varvec{\phi }}$. The conditional posterior distribution of ${\varvec{\phi }}_{1}$ is :

$$\begin{aligned} P({\varvec{\phi }}_{1} |\varvec{r},{\varvec{\theta }}_{-{\phi }_{1}}) \propto \mathcal{L}(\varvec{r}|{\varvec{\theta }})\times P({\varvec{\phi }}_{1}). \end{aligned}$$

In this case, when $\varepsilon _t$ is a normal distribution, we can write the conditional posterior as:

$$\begin{aligned}&P({\varvec{\phi }}_{1} | \varvec{r},{\varvec{\theta }}_{-{\phi }_{1}}) \\&\quad \propto |{\varvec{H}}|^{-\frac{1}{2}} {\text{ exp }} \left\{ -\frac{1}{2}(\varvec{r}-{\varvec{Z}}{\varvec{\phi }}_{1})^T{\varvec{H}}^{-1} ({\varvec{r}}-{\varvec{Z}}{\varvec{\phi }}_{1}) -\frac{1}{2}({\varvec{\phi }}_{1}-{\varvec{\phi }}_{10})^T{{\varvec{\varSigma }}}_i^{-1}({\varvec{\phi }}_{1}-{\varvec{\phi }}_{10}) \right\} \\&\quad \propto |{\varvec{H}}|^{-\frac{1}{2}} {\text{ exp }} \left\{ -\frac{1}{2}({\varvec{\phi }}_{1}-{\varvec{\phi }}^*)^T {{\varvec{\varSigma }}}^{*-1}({\varvec{\phi }}_{1}-{\varvec{\phi }}^*) \right\} , \end{aligned}$$

(8)

and

$$\begin{aligned} {\varvec{\phi }}^*={\varvec{\varSigma }}^*({\varvec{Z}}^T{\varvec{H}}^{-1}\varvec{r}+{\varvec{\varSigma }}_1^{-1}{\varvec{\phi }}_{10}), \, \, {\varvec{\varSigma }}^*= ({\varvec{Z}}^T{\varvec{H}}^{-1}{\varvec{Z}}+{\varvec{\varSigma }}_1^{-1})^{-1}. \end{aligned}$$

(9)

The definitions of ${\varvec{Z}}$, $\varvec{r}$, and ${\varvec{H}}$ for regime I are as follows.

$$\begin{aligned}&{\varvec{Z}}={ \left[ \begin{array}{cccc} 1 &{} r_{1} &{} x_{1,1} &{}x_{2,1} \\ \vdots &{} \vdots &{} \vdots &{}\vdots \\ \vdots &{} \vdots &{} \vdots &{}\vdots \\ 1 &{} r_{T_{1}-1} &{} x_{1,T_{1}-1} &{}x_{2,T_{1}-1} \end{array} \right] },\\&{\varvec{r}}= \left[ {\begin{array}{l} r_{2}\\ \vdots \\ \vdots \\ r_{T_{1}} \end{array}} \right] \quad {\text{and}} \quad {\varvec{H}} = \left[ {\begin{array}{llll} h_{2} &{}0 &{}\cdots &{}0\\ 0 &{}h_{3} &{}\cdots &{}0\\ \vdots &{}\vdots &{}\ddots &{}0\\ 0 &{}0 &{}\cdots &{}h_{T_{1}} \end{array}} \right] . \end{aligned}$$

The conditional posterior in Eq. (8) is not a standard form. We employ it to obtain new estimates. In other words, we draw estimates ${\varvec{\phi }}_{1}$ from a truncated Gaussian density $N_{B_1}({\varvec{\phi }}^*,{\varvec{\varSigma }}^*)$, where $B_1=\{ |\phi _1^{(1)}|<1\}$, to get new iterates and apply the random walk MH algorithm to decide whether or not to update the previous estimates. One can usually select a suitable value with good convergence properties by having an acceptance rate of 25–50%, as Gelman et al. (1996) indicate, which represents good convergence performance.