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CPS1158 Varun A. et al.
th
th
lag considered and rij is the threshold parameter of the j regime of the i
break point constitute a partition on the real line and satisfy
− = r i0 r i1 r i ik = . The locations of structural breaks are
0 = T T T = T that partition the time series into TAR models. The
m
1
0
number of structural breaks (m) having partition in such a way that each break
segment having ki regimes TAR model with delay parameter di. The error term
is independent and identically normal distributed random variable with
(ij
)
e
t
mean zero and unknown variance . Also, TAR model coefficients and
2
(ij
)
ij
l
lags order are also different with respect to change in threshold and delay
p
ij
parameter as well as the presence of break points. In time series, sequential
observations is valuable for making effective statistical inference so
reorganized the MB-TAR generating observations in a group of each regimes
for a particular break interval. This separation appears in an order form and
th
does not need to know the threshold value. Let πij be the time index of the j
th
smallest observation in i break point in this series. Then, each regime has (sij-
sij-1) observations from (yp+1-d, yp+2-d,…, yTi-d) where p=max(pij; i=1,2,…,m;
j=1,2,…,ki). Therefore, we can rewrite the model (1) as given by
p ij ( ij) ( ij) (2)
ij)
(
=
+
y w + d i l =1 l y w + d i − l + e w + d i ij s −1 w ij ij s i T −1 ij s i T (2)
ij
ij
ij
Given the first p-observations, likelihood function of the model (2) can be
obtained as
s ij −s ij −1 2
m k i
( L y | ) (2 ij 2 ) − 2 exp − 1 s ij y − ( ij) − p ij l ( ij) y −l
=
= i 1 = j 1 2 ij 2 w ij =s ij −1 +1 w ij +d i = l 1 w ij +d i
n ij
m k i 2 − 1 ( ij) ' ( ij)
ij − ( Y ij − X ij )( Y ij − X ij ) (3)
( ) 2 exp
i 1 2 ij 2
= = j 1
where sij satisfy y r y s , = T s , = T , nij =sij-sij-1;
ij s ij ij s + 1 i0 1 - i ik i i
1 y y
ij s −1 +1 +d i −1 ij s −1 +1 +d i − p ij
1 y +d −1 y +d − p
X ij = ij s −1 +2 i ij s −1 +2 i ij
1 y y
ij s +d i −1 ij s +d i − p ij Y ij = y s +1 +d i , y s +2 +d i , , y s +d i ;
ij −1 ij −1 ij
3. Bayesian Inference
In general, Bayesian inference provides some additional information about
the unknown parameter rather than existing data information named as prior
information. Selection of a prior is a main assignment in Bayesian study
because on the basis of suitable prior, distinguish characteristic of the
unknown parameters. However, present study targets only for estimation, so
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