time series

piecewise linear model
to get a better approximation of the conditional mean structure, motivated by asymmetry in rising and decline pattern.

For time series $y_t$, it is said to follow TAR$(g;p_1,\cdots,p_g)$ with $y_{t-d}$ as a threshold variable if
\begin{eqnarray} y_t = \phi_0^{(k)} + \sum_{i=1}^{p_k} \phi_i^{(k)} y_{t-i} + a_t^{(k)}, ~~ r_{k-1} \leq y_{t-d} < r_k, \text{ for } k=1, \cdots, g \end{eqnarray}

where
g: number of regime,
$\{a_t^{(k)}\}$ : innovation, i.i.d., $\sim N(0, \sigma_k^2)$
$d$: threshold lag, positive integer,
$r_j$: threshold variable, real, $-\infty=r_0<r_1<\cdots<r_g = \infty$

For $a_t$, innovation or shock for time series $y_t$, it is said to follow SV model if
\begin{eqnarray} a_t = \sqrt{h_t} \epsilon_t, ~~~ log{h_t} = \alpha_0 + \sum_{i=1}^{p} \alpha_i \log{h_{t-i}} + \eta_t \label{SV} \end{eqnarray}

where
$\epsilon_t$: i.i.d. $\sim N(0,1)$
$\eta_t$ : i.i.d. $\sim N(0, \sigma^2)$
$\{\epsilon_t\}$ and $\{\eta_t\}$ are independent.

Adding $\eta_t$, the innovation, considerably increase the flexibility of the model in describing the $h_t$ compared to other volatility models.

combine TAR (\ref{TAR}) and SV (\ref{SV}).
Consider following model.
\begin{eqnarray} y_t &=& \left\{ \begin{matrix} \phi_0^{(1)}+\phi_1^{(1)}y_{t-1}+a_t &, ~~ y_{t-d} \leq r \nonumber \\ \phi_0^{(2)}+\phi_1^{(2)}y_{t-1}+a_t &, ~~ y_{t-d}>r \end{matrix}\right. \nonumber \\ a_t &=& \sqrt{h_t}\epsilon_t, ~~ \epsilon_t \overset{i.i.d.}{\sim} N(0,1) \nonumber \\ \log{h_t} &=& \alpha_0 + \alpha_1 \log{h_{t-1}} + \eta_t, ~~ \eta_t \overset{i.i.d.}{\sim} N(0, \sigma^2) \label{TARSV} \end{eqnarray}

where
$\{\epsilon_t\}$ and $\{\eta_t\}$ are independent.
d: delay lag
r: threshold value

We are interested in estimating unknown parameter $\theta = (\boldsymbol{\phi}, \boldsymbol{\alpha}, \sigma^2, r, d)$ based on observation $\boldsymbol{y} = (y_1, \cdots, y_n)$. (where $\boldsymbol\phi = (\phi_0^{(1)}, \phi_1^{(1)}, \phi_0^{(2)}, \phi_1^{(2)})$, $\boldsymbol\alpha = (\alpha_0, \alpha_1)$)

In this problem, maximum likelihood method is not applicable because of the existence of latent variables $\boldsymbol{h} = (h_1, \cdots, h_n)$. By using data augmentation(\cite{Tanner:1987}) in the Bayesian framework, we can overcome this difficulty.

Applying data augmentation strategy, the parameter space is augmented to $(\theta, \boldsymbol{h})$. Conditioning on $\boldsymbol{h}$, likelihood $p(\boldsymbol{y}|\theta, \boldsymbol{h})$ has closed form.