Bayesian analysis of multiple thresholds autoregressive model

Abstract

Bayesian analysis of threshold autoregressive (TAR) model with various possible thresholds is considered. A method of Bayesian stochastic search selection is introduced to identify a threshold-dependent sequence with highest probability. All model parameters are computed by a hybrid Markov chain Monte Carlo method, which combines Metropolis–Hastings algorithm and Gibbs sampler. The main innovation of the method introduced here is to estimate the TAR model without assuming the fixed number of threshold values, thus is more flexible and useful. Simulation experiments and a real data example lend further support to the proposed approach.

Appendices

Appendix 1: The proof of Theorem 1

The conditional posterior distribution of $(\gamma ,d)$ given $(Y,\sigma ^2)$ is

$$\begin{aligned} f\left( \gamma ,d|Y,\sigma ^2\right)\propto & {} \int f\left( \Theta ,\gamma ,\sigma ^2,d|Y\right) d\Theta \nonumber \\\propto & {} \pi \left( \sigma ^2|\gamma ,d\right) \pi (\gamma |d)\int L\left( \Theta ,\gamma ,\sigma ^2,d|Y\right) \pi (\Theta |\gamma ,d)d\Theta ,\qquad \quad \end{aligned}$$

(5.1)

where $L(\Theta ,\gamma ,\sigma ^2,d|Y)$ is defined by (2.4) and

$$\begin{aligned}&L\left( \Theta ,\gamma ,d|Y\right) \pi (\Theta |\gamma ,d)\nonumber \\&\quad \propto \prod \limits _{k=1}^{K_{\gamma }} \left( \sigma _k^2\right) ^{-n_k/2} \exp \left\{ -{1\over 2\sigma _k^2} \left( Y_k^*-X_k^{*}\Theta _k\right) ' \left( Y_k^*-X_k^{*}\Theta _k\right) \right\} \nonumber \\&\qquad \times \prod \limits _{k=1}^{K_{\gamma }}(2\pi )^{-\left( q_k+1\right) /2}|V_k|^{1\over 2}\exp \left\{ -{1\over 2}\left( \Theta _k-\Theta _{k0}\right) 'V_k\left( \Theta _k-\Theta _{k0}\right) \right\} . \end{aligned}$$

(5.2)

Let $\Theta _k^*,V_k^*$ and $\hat{\Theta }_k$ be defined as before, then we have

$$\begin{aligned}&{1\over \sigma _k^2} \left( Y_k^*-X_k^{*}\Theta _k\right) ' \left( Y_k^*-X_k^{*}\Theta _k\right) + \left( \Theta _k -\Theta _{k0}\right) 'V_k \left( \Theta _k-\Theta _{k0}\right) \nonumber \\&\quad = \left[ Y_k^*-X_k^{*}\hat{\Theta }_k+X_k^{*} \left( \hat{\Theta }_k-\Theta _k \right) \right] ' \left[ Y_k^*- X_k^{*}\hat{\Theta }_k+X_k^{*}\left( \hat{\Theta }_k-\Theta _k\right) \right] \nonumber \\&\qquad + \,\left( \Theta _k-\Theta _{k0}\right) 'V_k\left( \Theta _k-\Theta _{k0}\right) \nonumber \\&\quad ={1\over \sigma _k^2} \left[ \left( Y_k^*-X_k^*\hat{\Theta }_k\right) ' \left( Y_k^*-X_k^*\hat{\Theta }_k\right) + \left( \hat{\Theta }_k-\Theta _k\right) ' \left( X_k^{*'} X_k^{*}\right) \left( \hat{\Theta }_k-\Theta _k\right) \right] \nonumber \\&\qquad +\,\left( \Theta _k-\Theta _{k0}\right) 'V_k\left( \Theta _k-\Theta _{k0}\right) \nonumber \\&\quad ={1\over \sigma _k^2}\left[ \left( Y_k^*-X_k^*\hat{\Theta }_k\right) ' \left( Y_k^*-X_k^*\hat{\Theta }_k\right) +\hat{\Theta }_k'X_k^{*'} X_k^{*}\hat{\Theta }_k\right] +\Theta _{k0}'V_k\Theta _{k0}\nonumber \\&\qquad +\,\left( \Theta _k-\Theta _k^*\right) 'V_k^* \left( \Theta _k-\Theta _k^*\right) -\Theta _k^{*'}V_k^*\Theta _k^*\nonumber \\&\quad ={1\over \sigma _k^2}Y_k^{*'}Y_k^*-\Theta _k^{*'}V_k^*\Theta _k^*+ \Theta _{k0}'V_k\Theta _{k0}+ \left( \Theta _k-\Theta _k^*\right) ' V_k^*\left( \Theta _k-\Theta _k^*\right) \nonumber \\&\quad =S_k^{*2}+\Theta _{k0}'V_k\Theta _{k0}+ \left( \Theta _k-\Theta _k^*\right) ' V_k^*\left( \Theta _k-\Theta _k^*\right) , \end{aligned}$$

(5.3)

where $S_k^{*2}$ is defined by (2.12). Thus the conditional posterior distribution (5.1) reduces to

$$\begin{aligned} f\left( \gamma ,d|Y,\sigma ^2\right)\propto & {} \prod \limits _{k=1}^{K_{\gamma }} \left( \sigma _k^2\right) ^{-n_k/2}(2\pi )^{-\left( q_k+1\right) /2}|V_k|^{1\over 2}\exp \left\{ -{1\over 2}\left( S_k^{*2}+\Theta _{k0}'V_k\Theta _{k0}\right) \right\} \nonumber \\&\times \,\pi \left( \sigma ^2|\gamma ,d\right) \pi (\gamma |d)\nonumber \\\propto & {} \prod \limits _{k=1}^{K_{\gamma }}\left( \sigma _k^2\right) ^{-\left( n_k+\upsilon _k+2\right) /2}(2\pi )^{-\left( q_k+1\right) /2}|V_k|^{1\over 2}\nonumber \\&\times \exp \left\{ -{1\over 2}\sum \limits _{k=1}^{K_{\gamma }} \left( S_k^{*2}+\Theta _{k0}'V_k\Theta _{k0}+{\upsilon _k\lambda _k\over \sigma _k^2}\right) \right\} \left( \lambda \over 1-\lambda \right) ^{K_{\gamma }}\nonumber \\\propto & {} \exp \left\{ -U\left( \gamma ,d|Y,\sigma ^2\right) \right\} , \end{aligned}$$

(5.4)

where

$$\begin{aligned} U(\gamma ,d|Y)={1\over 2}\sum \limits _{k=1}^{K_{\gamma }} \left( S_k^{*2}+\omega _k\right) +\beta K_{\gamma }, \end{aligned}$$

in which $\omega _k$ and $\beta$ are defined as in (2.14). The proof of Theorem 1 is completed.

Appendix 2: The proof of equation (2.17)

If $\sigma _1^2=\cdots =\sigma _k^2=\sigma ^2$ and the prior distribution of $\sigma ^2$ is $IG(\upsilon /2,\upsilon \lambda /2)$, then conditional posterior distribution of $(\gamma ,d)$ given $(Y,\sigma ^2)$ is

$$\begin{aligned} f\left( \gamma ,d|Y,\sigma ^2\right)\propto & {} \int f\left( \Theta ,\gamma ,\sigma ^2,d|Y\right) d\Theta \nonumber \\\propto & {} \pi (\gamma |d)\int L\left( \Theta ,\gamma ,\sigma ^2,d|Y\right) \pi (\Theta |\gamma ,d)d\Theta , \end{aligned}$$

(5.5)

where $L(\Theta ,\gamma ,\sigma ^2,d|Y)$ is defined by (2.15), $\pi (\Theta |\gamma ,d)$ is as before. Then it follows from (5.3) the conditional posterior distribution (5.5) reduces to

$$\begin{aligned} f\left( \gamma ,d|Y,\sigma ^2\right)\propto & {} \prod \limits _{k=1}^{K_{\gamma }} (2\pi )^{-\left( q_k+1\right) /2}|V_k|^{1\over 2}\exp \left\{ -{1\over 2}\left( S_k^{*2}+\Theta _{k0}'V_k\Theta _{k0}\right) \right\} \pi (\gamma |d)\nonumber \\\propto & {} \prod \limits _{k=1}^{K_{\gamma }}(2\pi )^{-\left( q_k+1\right) /2}|V_k|^{1\over 2}\exp \left\{ -{1\over 2}\left( S_k^{*2}+\Theta _{k0}'V_k\Theta _{k0}\right) \right\} \left( \lambda \over 1-\lambda \right) ^{K_{\gamma }}\nonumber \\\propto & {} \exp \left\{ -U\left( \gamma ,d|Y,\sigma ^2\right) \right\} , \end{aligned}$$

(5.6)

where

$$\begin{aligned} U(\gamma ,d|Y)={1\over 2}\sum \limits _{k=1}^{K_{\gamma }} \left( S_k^{*2}+\omega _k\right) +\beta K_{\gamma }, \end{aligned}$$

in which $S_k^{*2}={1\over \sigma ^2}Y_k^{*'}Y_k^*-\Theta _k^{*'}V_k^*\Theta _k^*$, $\omega _k$ is defined by (2.17) and $\beta$ is defined as in (2.14). The proof is completed.