">C B


ˆ
var2
t
Y
A



11)

2
2
1
t
Y






,
AE

2
2
2
t
t
Y
Y






,
BE

4
2
2
t
t
Y
CE
Y






Since this asymptotic variance of ˆ
still involves the
unknarameter own p
, Ling [5] further proposed to use

1
2
2
1
2
varˆ
2
0ˆˆˆ
A
0
ˆ
ˆ
n
t
tt
y
y
A
CB




,







te of
as an estima
ˆ
var
, where

2
2
1
1
ˆn
t
A
ˆt
y

,

2
2
2
1
ˆˆ
n
t
tt
y
B
y

,

4
2
2
1
ˆˆ
n
t
tt
y
C
y

.
Then inference concerning
, and can be ob-
tained by the Wald method.
) model pointed out that it is
s undeconditions on the pa-
rameters. Therefore, by Tayr expansion, we ha


For the DAR(1, Ling [5]
an ergodic procesr suitable
lo ve

1
ˆˆ
ˆˆ
2

 
 .
Hence

 
ˆˆ ˆˆ
22j

1
 

 


is asymptotically distributed as a 2
distribution with
p degrees of freedom. Thus, when

dim 1p
, the
signed log conditional likelihood ratio statistic for
and for
of the DAR(1) model is asymptotically dis-
tributed as
0,1N with rate of convergence
12.
We performed some simulation studies for testing the
no the log conditional likelihood ratio statistics
for thodel. The Kolmogorov-Smirov test is
employed. We considered a medium sample size of
50n
On
rmality of
e DAR(1) m
and a large sample size of 200n. For each
ple
ameter values.
o
sample size, we generate 10,000 sams from the D AR (1 )
model for each combination of par For
each generated sample, the signed log conditional likeli-
hood rati statistic for
and for
are calculated.
The simulatults are presented in Tables :
Ta ion res 1 and 2
aluesea sta
ble 1 records the p-values of the Kolmogorov-Smirov
test when the parameters values are on the boundary of
the weak stationary region; Table 2 records the p-values
of the Kolmogorov-Smirov test when the parameters
v are the interior points of the wktionary re-
gion. From the tables, the p-values of the Kolmogorov-
Smirov test of the signed log conditional likelihood ratio
statistic for
are large regardless of the sample sizes.
Hence no evidence that th signed log onditional likeli-
hood ratio statistic for e c
is not distributed as
0,1N.
On the other hand, the p-values of the Kolmogorov-
Smirov test of the signed log conditional likelihood ratio
statistic for
are large only for 200n. Hence for
n is sufficiently large, there is no evidence that the
signed log conditional likelihood ratio statistic for
is
not distri buted as
0,1N.
Note that when
is unknown, Ling & Li [12] sug-
 




3
211
122
2
22
1
21
2
34
11 1
1
222
22
22 1
11
.
21
2
nn
ttt
t
tt
tt
nn
tt tt t
t
tt t
tt
yy y
y
yy
yy yy y
y
y
yy
 

 

 


 






 

j









F. CHANG ET AL.
OJS
145
pe SL
Table 1. The -values of the Kolmogorov-Smirnov test for thR when ω = 1,
and α are on the boundary of the weak
stationary region.
(,
) (–0.95, 0.0975) (–0..755, 0) (0, 1) (0.5, 0.75) (0.95, 0.0975)

50n

 0.2093 0.3089 0.0923 0.2519 0.1676

200n

 0.7294 0.2544 0.2219 0.8769 0.3595
0.2162 0.6260 0.5481 0.3084 0.7428

50n


0.0054 0.0184 0.0102 0.0155 0.0217
=200n

Table 2. The p-values of the Kolmogirnov test for the SLR when,
andinterior points of the weak
staionary region.
(,
orov-Sm ω = 1 α are
) (–0.95,.0975) (–0.5, 0.5, 0.25) ((0.1(0.5(0.5, 0.5) , 0.09) 05) (–0.0, 0.5) , 0.4) , 0.25) (0.95
0.0647 0.0525 0.6512 0.4067 0.5246
0.3877 0614 0.0.9448
(n = 50)
0.1418 0.2234 0.2626 0.3933 0.9673 0.7956 0.6339 0.0996
(n = 200)
n = 50)
0.0703 0.0021 0.0628 0.0001 0.1980 0.0000 0.0011 0.0013
(
2778 0.1640 0.0659 0.1865 0.1779 0.5938 0.0660 0.1802
n = 200)
0.
(
gested a method ate to estim
and the ann Ling
& Li [12] tate alysis i
reated the estim
as
the known
. For
the rest ofon, we followine approach of Ling
& Li [12] an the msigncondi
tional likratio statistic for
the secti
d dg th
erivedodifieded log -
elihood
and
respec-
vely. er
ti When the parameter of intest is


, the nui-
sance parameter is
, anhence the consained d tr
LE oMf
is ˆ


ˆ
,

whibe obby
solving
ch can tained
ˆ
0

.
Hence,
r
can be calculated from (3). Moreover,








0
0
ˆ
,12
ˆ
,2
y
y



12
22
01
n
y
yy
yy
V
 



 
 
01 1
2
01 2111
22 2
01 1
01 1
ˆˆ ˆ
ˆˆ ˆ
2 2
.
n
nn n
n
n
n
y
y yyyyyy
y
yy yy y
yy y
vv v

 



 
 



For 11tn ,
yy
y
01 1n




0
1
1
22
2
1
21
tt
tt t
tt
yt
tt
t
yy
yyy
yy
y
yy
y





22


and for tn,
0
1
21
nn
nn
y
y
y
y
y


Then, the locally defined canonical parameter
12
,
is







22 2
1 1
11 1
11 11
122 2
2
1
222
1111 1
11
22
2
1
n n
tt tt
tt tttt t
tt
ntttttt
tt tt
tt
t
yy yy
yy yyyy y
yy
y
yyyyy
yy yy
y
y


  





 
 
 





 





 






11
t
tt
t
y y
F. CHANG ET AL.
146







22
1 1
11 1
11
222
2
1 1
22
1111 1
11
22
2
1
n n
tt tt
tt tt
t t
t
nttttt t
tt tt
tt
t
vy yv
yv yyyv
y
vyyvy v
yv yy
y
y


 





 
 
 11
1
2
1
.
tt t t
tt
tt
y v
yy
yv








Thus Q can be obtained from (7) with
 
  








 



   
,1,0




. Finally,
nal likelihood ratio sta-
tistic

*
r
 
 



the modified signed log conditio
can be calculated from (5) and, therefore, a
1

100% confidenterval for ce in
can be obtained
om f interest is

fr (6).
On theother hand, when the parameter o

, the nuisance pter is arame
strained MLE
. The con-
of
is

=,
ˆˆ


which can be ob-
y tainedng

bsolvi
ˆ


r
0
.
Again
can be calculated fromhe tangent (3). T
inhanged as a
om (7) wit h
 
,0,1
 
 

 



. And once
again, the modified signed log connal likelihood
ratio statistic
direction V remas uncbove and hence Q
can be obtai ned fr
ditio
*
r
can be calculated from (5) and
therefore a
1100% confidence interval for
can
be obtaine d from (6).
4. Numerical Studies
Anderson [13] considered the closing prices of the Impe-
ary, 1
calibra t
secutive data points after a log
transformation. A scatter plot for the calibrated data is
rial Chemical Industries (I.C.I.) for the period 25 August,
1972 to 19 Janu973. Instead of using the raw data,
we use the ted data, which is obtained byaking
the difference of two con
shown in Figure 2.
The DAR(1) model is employed. By applying Ling &
Li [12], we have an estimate of
being 0.0002. The
overall MLE is obtained by maximizing (8) and we have
ˆ
is

0.1553,0.1330
. Table 3 reports the 90% con-
Figure 2. Scatter plot for calibrated data.
F. CHANG ET AL. 147
Table 3. 90% central confidence intervals for
and α.
Method 90% Confidence Interval for
90% Confidence Interval for α
BN (–0.3629, 0.03 89 ) (0.0048, 0.4719)
SLR (–0.3490, 0.0352) (0.0 0 00 , 0.4072)
Ling (–0.3438, 0.0333) (0.0048, 0.3588)
fidervals for nce inte
, and the 90% confidence inter-
vals for
, obtained from the methods discussed in this
paper: is the proposed method based on Equatio n (5),
SLR is te signed log-likelihood ratio statistic (3), Ling
is the mhod discussed in Ling [5]. SLR and Ling give
similar confidence intervals which are quite different
from thlts obtained by BN. Moreover, since
BN
h
et
e resu
is
bounded by 0, both SLR and Ling have deficiency on the
left boundary.
The p-value functions are presented in Figures 3 and 4.
Additionally, the two horizontal lines indicate upper and
lower 0.05 levels respectively. The plots show that re-
sults obtained by the three methods discussed in this pa-
per are quite different.
To examine the accuracy of the methods discussed in
this paper, Monte Carlo simulation studies are performed.
For each coon of

,
mbinati
, 10,000 Monte Carlo
samples, for each sample sizes takeing values n = 50,
200 and 4 are generated from the DAR(1) model
, 0,,tn. Withou
ality,
of
00,

0, 1N
where t
is set to be 1.
Tables 4-9 recorded the central coverage probability
(CCP) which is the proportion of intervals that contains
the true
, the lower error probability (L) which is the
proportion of true
that falls outside the lower bound
of the confidence interval, and upper error probability (U)
which is the proportion of true
that falls outside the
upper bound of the confidence interval. The nominal
values for the central coverage pro bability, and the lower
and upper errors probabilities are 0.90, 0.05, and 0.05
respectively. In additional to this, we also report the av-
erage bias (Avg Bias) defined by L0.05U 0.05
2
,
which has the nominal value of 0.
For both
t loss of gener-
and
, simulation studies show
that BN is remarkably accurate even when the sample
size is small. As n increases, there is significant im-
provement on the precision of both the SLR method and
the Ling’s method in terms of both cental coverage and
average bias but the results are still not as accurate as
those obtained by the proposed method.
5. Conclusion
A conditional likelihood based method is proposed to
obtain confidence intervals for and for
of the
weak stationary DAR(1) model. Theoretically, the pro-
Figure 3. p-value functions for
of I.C.I. Data.
F. CHANG ET AL.
148
Figure 4. p-value functions for α of I.C.I. Data.
Table 4. Simulation results for some boundary points of the DAR(1) model when n = 50.
Me th od L U CCP Avg BiasL U CCP A vg Bias
–0.95 0.0975 BN 0.0544 0.0451 0.9005 0.0046 0.0372 0.0573 0.9055 0.0101
SLR 0.0343 0.0760 0.8897 0.0209 0.0870 0.0314 0.8816 0.0278
Ling 0.0384 0.0924 0.8692 0.0270 0.1228 0.0020 0.8752 0.0604
–0.5 0.75 BN 0.0542 0.0494 0.8964 0.0024 0.0542 0.0494 0.8964 0.0024
SLR 0.0950 0.0272 0.8778 0.0339 0.0950 0.0272 0.8778 0.0339
Ling 0.1558 0.0041 0.8401 0.0759 0.1558 0.0041 0.8401 0.0759
0 1 BN 97 0.0524 0.8979 0.0014
SLR 0.0560 0.0571 0.8869 0.0065 0.0987 0.0275 0.8738 0.0356
Ling 0.0625 0.0645 0.8730 0.0135 0.1532 0.0044 0.8424 0.0744
0.5 0.75 BN 0.0487 0.0506 0.9007 0.0009 0.0536 0.0492 0.8972 0.0022
SLR 0.0630 0.0491 0.8879 0.0070 0.0964 0.0264 0.8772 0.0350
Ling 0.0745 0.0565 0.8690 0.0155 0.1545 0.0026 0.8429 0.0760
0.95 0.0975 BN 0.0446 0.0576 0.8978 0.0065 0.0375 0.0542 0.9083 0.0083
SLR 0.0761 0.0369 0.8870 0.0196 0.0856 0.0331 0.8813 0.0262
Ling 0.0893 0.0408 0.8699 0.0243 0.1223 0.0014 0.8763 0.0605
0.0521 0.0529 0.8950 0.0025 0.04
F. CHANG ET AL. 149
Table 5. Simulation results for some interior points of the DAR(1) model when n = 50.
Method L U CCP Avg BiasL U CCP Avg Bias
–0.95 0.09 BN 0.0574 0.0435 0.8991 0.0070 0.0322 0.0533 0.9145 0.0106
SLR 0.0363 0.0734 0.8903 0.0186 0.0838 0.0290 0.8872 0.0274
Ling 0.0399 0.0880 0.8721 0.0240 0.1161 0.0017 0.8822 0.0572
–0.5 0.5 BN 0.0536 0.0483 0.8981 0.0026 0.0497 0.0524 0.8979 0.0014
SLR 0.0508 0.0606 0.8886 0.0057 0.1078 0.0291 0.8631 0.0394
Ling 0.0569 0.0702 0.8729 0.0135 0.1724 0.0032 0.8244 0.0846
–0.5 0.25 BN 0.0507 0.0488 0.9005 0.0009 0.0113 0.0514 0.9373 0.0201
SLR 0.0467 0.0598 0.8935 0.0066 0.0997 0.0249 0.8754 0.0374
Asymptotic Inference for the Weak Stationary Double AR(1) Model