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CPS1158 Varun A. et al.
model is discontinuous. Liu et al. (2011) considered the limiting distributions
of the least-squares estimators for non-stationary TAR model and Li et al.
(2011) extended Liu et al. (2011) works for TARMA model. Rather than classical
methodology, it is also attracting researchers under Bayesian framework. Chen
and Lee (1995) obtained its Bayesian estimation for two regimes through
Gibbs sampler and M-H algorithm. Chen (1998) and Charif (2003) constructed
a Bayesian framework for generalized TAR model and TMA model,
respectively. Yu (2012) obtained the likelihood based inference in threshold
regression model that allow heteroskedasticity and threshold effect in both
mean and variance. Xia et al. (2012) generalized TARMA with explanatory
variables model and considered iterative least square and two stage MCMC
methods for estimation. Pan et al. (2017) discussed the multiple thresholds
autoregressive model and obtained the threshold dependent sequence using
stochastic search selection method. TAR model is also extended with
structural break when change in state space parallel occurs with change in
time domain. This is very serious problem in time series which is not much
explored by researchers. Yau et al. (2015) derived a minimum descriptive
length principle to estimate the TAR parameters and detecting the
breakpoints. Gao and Ling (2018) considered the least square estimation of
TAR model with structural break and shows that threshold and break points
are n-consistent and converge weakly to a Poisson process and two-sided
random walk respectively.
Above literature shows that inference about TAR model with structural
change was done in classical approach but no one have perform this under
Bayesian framework. So, the main objective of this paper is to develop a
Bayesian setup of multiple regimes TAR model with multiple structural breaks.
A conditional posterior distribution is obtained for parameter estimation form
a Bayesian point of view. For building better inference, we used various
symmetric and asymmetric loss functions. A simulation and empirical study is
carried out using Gibbs sampler and N-H algorithm techniques to record the
performance of the Bayesian perspective of multiple breaks and multiple
regimes TAR model.

2. Threshold Autoregressive model with Structural Break
Let {yt ; t=1,2,…,T } be a stochastic process having a multiple unknown
change points and each change point interval follows a multiple-regimes
threshold autoregressive (TAR) model. Then, the mathematical form of m-
breaks and k-regimes TAR model (MB-TAR(m,k)) is obtained as
p ij
y =  ( ij) +  l ( ij) y t l − + e t ( ij) r ij−1  y t−  r ij T i−1  t  T i
t
l =1 d i (1)

th
for i=1,2,…,m; j=1,2,…,ki, where di is the i delay parameter of the TAR model
acquire some positive integer value from  ,2,1  ,d 0 i , d is the maximum delay
0 i
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