Semi-parametric quantile estimation for double threshold autoregressive models with heteroskedasticity

Abstract

Compared to the conditional mean or median, conditional quantiles provide a more comprehensive picture of a variable in various scenarios. A semi-parametric quantile estimation method for a double threshold auto-regression with exogenous regressors and heteroskedasticity is considered, allowing representation of both asymmetry and volatility clustering. As such, GARCH dynamics with nonlinearity are added to a nonlinear time series regression model. An adaptive Bayesian Markov chain Monte Carlo scheme, exploiting the link between the quantile loss function and the asymmetric-Laplace distribution, is employed for estimation and inference, simultaneously estimating and accounting for nonlinear heteroskedasticity plus unknown threshold limits and delay lags. A simulation study illustrates sampling properties of the method. Two data sets are considered in the empirical applications: modelling daily maximum temperatures in Melbourne, Australia; and exploring dynamic linkages between financial markets in the US and Hong Kong.

Appendix

Derivation of the standardized SL density and corresponding likelihood expression in (5) is given below. Consider that a random variable $z$ follows a SL distribution with density

$$\begin{aligned} f_\tau (z)=\frac{\tau (1-\tau )}{\delta }\exp \left\{ -\frac{z-\mu }{\delta }(\tau -I(\frac{z-\mu }{\delta }<0)) \right\} , \end{aligned}$$

where $\mu$ is the mode and $\delta$ is a scale parameter. The mean and variance of a $SL(0,\delta ,\tau )$ are

$$\begin{aligned} E(z)=\frac{1-2\tau }{\tau (1-\tau )}\delta ,\quad Var(z)=\frac{1-2\tau +2\tau ^2}{\tau ^2(1-\tau )^2}\delta ^2. \end{aligned}$$

Suppose that the model error $\varepsilon _t$ is equal to the standardized $z$, i.e.

$$\begin{aligned} \varepsilon _t=\frac{z}{\sqrt{Var(z)}}, \end{aligned}$$

The density function for $\varepsilon _t$ is then:

$$\begin{aligned} g_\tau (\varepsilon _t)&= |J|f_\tau (\varepsilon _t\sqrt{ Var(z)}) = \left(\frac{\sqrt{1-2\tau +2\tau ^2}}{\tau (1-\tau )}\delta \right) \\&\times \frac{\tau (1-\tau )}{\delta }\exp \left\{ -\frac{\varepsilon _t\sqrt{ Var(z)}}{\delta }\left(\tau \!-\!I\left(\frac{\varepsilon _t\sqrt{ Var(z)}}{\delta }\!<\!0\right)\right) \right\} \\&= \sqrt{1-2\tau +2\tau ^2}\exp \left\{ -\frac{\varepsilon _t\frac{\sqrt{1-2\tau +2\tau ^2}}{\tau (1-\tau )}\delta }{\delta }\left(\tau -I\left(\frac{\varepsilon _t\frac{\sqrt{1-2\tau +2\tau ^2}}{\tau (1-\tau )}\delta }{\delta }<0\right)\right) \right\} \\&= \sqrt{1-2\tau +2\tau ^2}\exp \left\{ -\varepsilon _t\frac{\sqrt{1-2\tau +2\tau ^2}}{\tau (1-\tau )}(\tau -I(\varepsilon _t<0)) \right\} \\&= \sqrt{1-2\tau +2\tau ^2}\exp \left\{ \varepsilon _t\frac{\sqrt{1-2\tau +2\tau ^2}}{\tau -I(\varepsilon _t\ge 0))} \right\} , \end{aligned}$$

where $|J|=|\frac{\partial z}{\partial \varepsilon _t}|=\frac{\sqrt{1-2\tau +2\tau ^2}}{\tau (1-\tau )}\delta$

Moreover, since $\varepsilon _t=(y_t-\acute{Y_{t-1}}\phi (\tau )-\acute{X_{t-1}}\psi (\tau ))/(\sqrt{h_t})$ and $\frac{\partial \varepsilon _t}{\partial y_t}=(\sqrt{h_t})^{-1}$, the likelihood function for this model becomes as given in (5).

$$\begin{aligned}&{\mathcal L}({\varvec{Y}}\mid {\varvec{\Phi }}) \propto \prod _{t=s+1}^T \frac{1}{\sqrt{h_{t}}} \\&\quad \left\{ \exp \left( \sum _{t=s+1}^{T}\sum ^{2}_{j=1} \frac{\sqrt{1-2\tau +2\tau ^{2}}(y_{t}-\mathbf Y _{t-1}^{\prime }{\varvec{\phi }} ^{(j)}(\tau )-\mathbf X _{t-1}^{\prime }{\varvec{\psi }}^{(j)}(\tau ))}{\sqrt{h_{t}}(\tau -I(y_{t}\ge \mathbf Y _{t-1}^{\prime }{\varvec{\phi }}^{(j)}(\tau )+\mathbf X _{t-1}^{\prime }{\varvec{\psi }}^{(j)}(\tau )))}\right.\right.\nonumber \\&\quad \qquad \times \left.\left.I( z_{t-d(\tau )}\in S_j) \right) \right\} . \end{aligned}$$