182 D
. Wan Shin, B. Soo So Economics Letters 71 2001 181 –189
developed confidence intervals by adjusting skewness of the empirical distribution of the ‘weighted symmetric estimator’, a version of the weighted least squares estimator. These confidence intervals are
based on the ordinary least squares estimator or variants of it. Since the distributions of these estimators are heavily skewed, the confidence intervals based on
them would have larger average lengths compared with those based on symmetrically distributed estimators. So and Shin 1999a proposed a confidence interval based on an instrumental variable
estimator, in which the sign of the regressor is used as an instrumental variable. The instrumental variable estimators are symmetrically distributed in the vicinity of one and provide us with confidence
intervals of smaller average lengths than those of Stock 1991, Andrews 1993 and Fuller 1996 for r close to one. However, due to lack of efficiency of the sign of the regressor, the average length of
the confidence interval of So and Shin 1999a would be larger than that based on an efficient and symmetrically distributed estimator for
r not close to one. In this paper, we develop a new estimator which has both symmetric distribution and high
efficiency for all values of r. The estimators are based on recursive detrending of So and Shin 1999b
and IV-estimation with a ‘Huber-type’ regressor as an instrumental variable in which a large regressor is discounted to a constant. Our instrumental variable method allows us to construct an instrumental
¯ ¯
¯ ¯
variable estimator r such that the limiting distribution of the pivotal value t r 5 r 2 r ser
iv iv
iv iv
is standard normal for any real r. Using the normality, we can construct a simple confidence interval
¯ ¯
r 6z se
r , where z is the
a 2-percentile of the standard normal distribution. The proposed
iv a 2
iv a 2
confidence interval has good coverage probability and shorter average length than those of Stock 1991 and Andrews 1993 based on percentiles of the distribution of the OLSE if
r is close to one. Section 2 introduces the new IV-estimators and establishes limiting normality of their pivotal
values. Section 3 compares average length and empirical coverage probability of the proposed confidence interval with those of Stock 1991 and Andrews 1993. Appendix A contains proofs for
theoretical results.
2. Instrumental variable estimator
We first consider the no-mean AR1 model y 5
ry 1 e .
1
t t 21
t
Consider an instrumental variable estimator
n n
ˆ r 5
O
y h y
YO
y h y
iv t
c t 21
t 21 c
t 21 t 52
t 52
where h x 5 signx if
uxu c,
c
5 x c if uxu , c,
c . 0 and 0 0 is understood to be 0 if c 5 0. The function h is the well-known quasi-score function
c
of Huber 1981, p. 71’s M-estimator which is the solution to the M-estimating equation
D . Wan Shin, B. Soo So Economics Letters 71 2001 181 –189
183
O
h y 2 ry
y 5 0.
c t
t 21 t 21
ˆ On the other hand, our estimator
r is the solution to an equation
iv
O
y 2 ry
h y 5 0
t t 21
c t 21
and is different from usual M-estimators. In the usual M-estimating equation, the quasi-score function h discounts large values of innovations to a constant. However, in our estimating equation, h reduces
c c
the influence of large values of y . We may call the discounted regressor h y
‘Huber-type’
t 21 c
t 21
instrument. Noting
n n
1 2 2
2
ˆ ˆ
se r 5 s
O
h y
Y O
y h y
,
H J
iv c
t 21 t 21
c t 21
t 52 t 52
we define the pivotal value
n n
21 2 2
2
ˆ ˆ
ˆ ˆ
t r 5 r 2 r ser 5 s
O
h y
O
e h y ,
H J
iv iv
iv c
t 21 t
c t 21
t 52 t 52
2 2
2
ˆ where
s is a consistent estimator of
s . A simple consistent estimator of s is the OLSE. We
ˆ establish limiting normality of
t r for all real r. Proofs of theorems are provided in Appendix A.
iv
¯ Theorem 1. Consider model 1. The limiting distribution of
t r is standard normal for any c 0
iv
and uru 1.
ˆ From the limiting normality of
t r for all uru 1, we can construct a 1 2 a confidence interval
iv
ˆ ˆ
ˆ ˆ
CI 5 [ r 2 z
se r , r 1 z
se r ].
iv iv
a 2 iv
iv a 2
iv
We next consider the mean model y 2
m 5 r y 2
m 1 e , t 5 1, . . . , n,
2
t t 21
t
Note that, if mean m is adjusted by the usual sample means as
¯ ¯
ˆ y 2 y
5 r y
2 y 1 e ,
t t 21
21 t
then the distribution of the resulting instrumental variable estimator has skewness for r close to one.
¯ The reason is that the regressor y
2 y is correlated with the error term e arising from
t 21 21
t
¯ ¯
correlation of y and e . As
r increases to one, correlation between y and e gets stronger
21 t
21 t
ˆ causing higher asymmetry in the distribution of
t r. In order to resolve this problem, we adopt the
o
184 D
. Wan Shin, B. Soo So Economics Letters 71 2001 181 –189
recursive mean adjustment of So and Shin 1999b and Shin and So 1999a. The AR parameter is estimated by applying the IV-estimation to the recursively mean adjusted model
¯ ¯
ˆ y 2 y
5 r y
2 y 1 e ,
t t 21
t 21 t 21
t 21
t
¯ where y 5 t
o y . The resulting estimator and its standard error are
t i 51
i n
n
¯ ¯
¯ ¯
¯ r 5
O
y 2 y h y
2 y
YO
y 2 y
h y 2 y
iv t
t 21 c
t 21 t 21
t 21 t 21
c t 21
t 21 t 52
t 52
and
n n
1 2 2
2
ˆ ˆ
¯ ¯
¯ se
r 5 s
O
h y 2 y
YO
y 2 y
h y 2 y
.
H J
iv c
t 21 t 21
t 21 t 21
c t 21
t 21 t 52
t 52
¯ Note that the regressor y
2 y is independent of the error term e . This together with the
t 21 t 21
t
¯ ¯
¯ IV-estimation gives limiting normality of the pivotal statistic
t r 5 r 2 r ser .
iv iv
iv
Theorem 2. Consider model 2 with
uru 5 1, m 5 0 or uru , 1. Assume that the density function of ¯
e is bounded. For any c 0, the limiting distribution of t r is standard normal.
t iv
The bounded density condition is for technical simplicity of the proof. The usual normal and many other common distributions satisfy this condition.
Our method can be extended to higher order autoregressions. Consider an AR p model in the form of the Dickey–Fuller regression
y 5 ry
1 u z
1 ? ? ? 1 u
z 1 e ,
3
t t 21
1 t 21 p 21 t 2p 11
t
˜ ˜
˜ ˜
where z 5 y 2 y and
r, u 5 u , . . . , u 9 are unknown parameters. Let r, u 5 u , . . . , u 9 be the
t t
t 21 1
p 1
p
OLSE. The largest root r is estimated by applying the IV-estimation to the model
˜ ˜
˜ y 2
u z 2 ? ? ? 2
u z
5 ry
1 e .
t 1 t 21
p 21 t 2p 11 t 21
t
The resulting estimator is given by
n n
˜ ˜
ˆ r 5
O
y 2 u z
2 ? ? ? 2 u
z h y
Y O
y h y
.
iv t
1 t 21 p 21 t 2p 11
c t 21
t 21 c
t 21 t 5p 11
t 5p 11
The standard error is
n n
2 2
1 2
ˆ ˆ
se r 5
hs
O
h y j
Y O
y h y
iv c
t 21 t 21
c t 21
t 5p 11 t 5p 11
D . Wan Shin, B. Soo So Economics Letters 71 2001 181 –189
185
¯ ¯
for all uru 1, which produces a confidence interval of the form r 6z
se r .
iv a 2
iv
ˆ ˆ
ˆ In Theorem 3, we establish limiting normality of
t r 5 r 2 r ser .
iv iv
iv
Theorem 3. Consider model 3. Assume that all the characteristic roots of uB 5 1 2 u B
1 p 21
¯ 2 ? ? ? 2
u B
lie outside the unit circle. For any uru 1, the limiting distribution of t r is
p 21 iv
standard normal.
3. Numerical studies
