Directory UMM :Data Elmu:jurnal:E:Economics Letters:Vol71.Issue2.May2001:

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www.elsevier.com / locate / econbase

Confidence intervals for the largest root of autoregressive

models based on instrumental variable estimators

*

Dong Wan Shin , Beong Soo So

Ewha University, Department of Statistics, Seoul, 120-750, South Korea Received 13 June 2000; accepted 21 December 2000

Abstract

For estimating the largest root of autoregressive (AR) models, we propose an instrumental variable scheme which discounts a large value of regressors corresponding to the largest roots. The pivotal value of the estimator of the largest root is asymptotically normal for any value of the largest root. This fact allows us to construct a

simple confidence interval based on 6standard error, say, with good coverage probability and shorter average

length than those of [J. Monetary Economics, 28, 1991, 435–459] and [Econometrica, 61, 1993, 139–165].

 2001 Published by Elsevier Science B.V.

Keywords: Confidence interval; Instrumental variable estimation; M-estimation; Recursive mean adjustment; Unit root

JEL classification: C22

1. Introduction

The issue of unit root tests has attracted much attention from many researchers. Since Dickey and Fuller (1979), the main interest lies on the largest autoregressive root (r) of a time series and tests whether r is one. However, as Stock (1991) pointed out, reporting only unit root tests and point estimates of the largest root is unsatisfactory as a description of the data, failing to convey information about the range of models that are consistent with the observed data. This observation suggests confidence intervals forr as a more useful summary measure of persistence than unit root tests alone. There are several attempts to construct confidence intervals. By inverting percentiles of the limiting distribution of the Dickey–Fuller test statistics under the local value ofr near the unity, Stock (1991) constructed confidence intervals of r. Andrews (1993) constructed confidence intervals by inverting empirical distribution of the ordinary least squares estimator (OLSE). (Fuller, 1996, pp. 578–583)

*Corresponding author. Fax: 182-2-3277-3607.

E-mail address: [email protected] (D. Wan Shin).

0165-1765 / 01 / $ – see front matter  2001 Published by Elsevier Science B.V. P I I : S 0 1 6 5 - 1 7 6 5 ( 0 1 ) 0 0 3 8 0 - 9

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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 ofr. 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 riv such that the limiting distribution of the pivotal valuetiv(r)5(riv2r) / se(riv) is standard normal for any real r. Using the normality, we can construct a simple confidence interval

¯ ¯

riv6za/ 2se(riv), where za/ 2 is the a/ 2-percentile of the standard normal distribution. The proposed 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 AR(1) model

y_t5ry_t₂₁1e ._t (1)

Consider an instrumental variable estimator

n n

r_iv5

O

y h ( yt c t21)

YO

yt21h ( yc t21)

t52 t52

where

h (x)c 5sign(x) ifuxu$c,

5x /c ifuxu,c,

c.0 and 0 / 0 is understood to be 0 if c50. The function h is the well-known quasi-score functionc

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O

h ( yc t2ryt21)yt2150.

On the other hand, our estimator r_iv is the solution to an equation

O

( yt2ryt21)h ( yc t21)50

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_t21. We may call the discounted regressor h ( yc t21) ‘Huber-type’ instrument.

Noting

n 1 / 2 n

2 2

ˆ ˆ

se(r_iv)5

H

O

h ( yc t21)

J

Y O

yt21h ( yc t21),

t52 t52

we define the pivotal value

n 21 / 2 n

2 2

ˆ ˆ ˆ ˆ

t_iv(r)5(riv2r) /se(riv)5

H

O

h ( yc t21)

J

O

e h ( yt c t21),

t52 t52

2 2 2

where s is a consistent estimator of s . A simple consistent estimator of s is the OLSE. We

establish limiting normality of tiv(r) for all realr. Proofs of theorems are provided in Appendix A.

Theorem 1. Consider model (1). The limiting distribution of tiv(r) is standard normal for any c$0

and uru#1.

From the limiting normality of t_iv(r) for alluru#1, we can construct a (12a) confidence interval

ˆ ˆ ˆ ˆ

CI_iv5[riv2za/ 2se(riv),riv1za/ 2se(riv)].

We next consider the mean model

yt2m5r( yt212m)1e ,t t51, . . . , n, (2)

Note that, if mean m is adjusted by the usual sample means as

¯ ¯ ˆ

y_t2y_{( 0 )}5r( y_t212y(21 ))1e ,t

then the distribution of the resulting instrumental variable estimator has skewness forr close to one.

The reason is that the regressor ( y_t₂₁2y_{(21 )}) is correlated with the error term e arising from_t

¯ ¯

correlation of y(21 ) and e . Ast r increases to one, correlation between y(21 ) and e gets strongert

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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_t2y_t₂₁5r( y_t₂₁2y_t₂₁)1e ,_t

21 t

where y_t5t o_i₅₁y . The resulting estimator and its standard error are_i

n n

¯ ¯ ¯ ¯ ¯

r_iv5

O

( yt2yt21)h ( yc t212yt21)

YO

( yt212yt21)h ( yc t212yt21)

t52 t52

and

n 1 / 2 n

2 2

ˆ ˆ ¯ ¯ ¯

se(riv)5

H

O

h ( yc t212yt21)

J

YO

( yt212yt21)h ( yc t212yt21).

t52 t52

Note that the regressor ( y_t₂₁2y_t₂₁) is independent of the error term e . This together with the_t

¯ ¯ ¯

IV-estimation gives limiting normality of the pivotal statistic t_iv(r)5(riv2r) /se(riv).

Theorem 2. Consider model (2) with (uru51, m50) or uru,1. Assume that the density function of

e is bounded. For any c_t $0, the limiting distribution of t_iv(r) is standard normal.

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_t5ry_t₂₁1u_{1 t21}z 1 ? ? ? 1u_p_{21 t2}z _p₁₁1e ,_t (3)

˜ ˜ ˜

where z_t5y_t2y_t21 andr,u5(u1, . . . ,up)9are unknown parameters. Let r,u5(u1, . . . , up)9be the OLSE. The largest root r is estimated by applying the IV-estimation to the model

˜ ˜ _˜

y_t 2 u_{1 t21}z 2 ? ? ? 2u_p_{21 t2}z _p₁₁5ry_t₂₁1e ._t

The resulting estimator is given by

n n

˜ ˜

r_iv5

O

( yt 2 u1 t21z 2 ? ? ? 2up21 t2z p11)h ( yc t21)

Y O

yt21h ( yc t21).

t5p11 t5p11

The standard error is

n n

2 2 1 / 2

ˆ ˆ

se(r_iv)5hs

O

h ( yc t21)j

Y O

yt21h ( yc t21)

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¯ ¯

for all uru#1, which produces a confidence interval of the form riv6za/ 2se(riv).

ˆ ˆ ˆ

In Theorem 3, we establish limiting normality of t_iv(r)5(riv2r) / se(riv).

Theorem 3. Consider model (3). Assume that all the characteristic roots of u(B )512u₁B

p21

2 ? ? ? 2up21B lie outside the unit circle. For any uru#1, the limiting distribution of tiv(r) is

standard normal.

3. Numerical studies

We first compare empirical coverages and the average lengths of our confidence interval CI_iv and the intervals CI of Andrews (1993) and CI of Stock (1991). The standard normal errors e are_a _s _t generated by RNNOA, a FORTRAN subroutine of IMSL (1989). In order to simulate stationarity for

y , t_t 51, . . . , n in case ofuru,1, the model is generated for t5 219, . . . , n. Data y are generated_t by (2) with (m50; r50, 0.3, 0.6, 0.9, 0.95, 0.99, 0.995, 1; n550, 100; 10 000 replications). For

CI , we consider c_iv 5ks, k50, 1, 2, 3, 4. For CI , we only considers r50.9, 0.95, 0.99, 0.995, 1 because Stock provided tables for CI only for_s r near unity. Nominal coverage is set to 90% because Andrews prepared tables for only this coverage. All the estimators are adjusted for mean. We restrictr

to [21, 1] as Andrews (1993) did in constructing his confidence intervals. The upper limits of CI_iv

and CI are replaced by one if their right end-points are greater than one. The column under ‘NA’_s represents cases in which Stock’s table (Table 1) fails to provide a confidence interval. These are the cases where the Dickey–Fuller’s tau statistic is out of range [236.79, 1.67]. The coverage probability and average length of CI are computed excluding these cases. In Table 1, empirical coverages (%)_s and the average lengths of the confidence intervals are reported. The coverage of CI_iv is slightly smaller than the nominal coverage. However, the coverages are all close to the nominal coverage for all n and k considered here. Whenr is close to one, the average length of CI is smaller than those of_iv

advantage of CI_iv over CI and CI seems uniform for all ka s 51, 2, 3, 4. Hence, we may recommend any of 1#k#4 for a good confidence interval because the performance of CI_iv is similar for this wide range of k.

4. Concluding remarks

Based on the ‘Huber-type’ instrumental variable estimator, we have developed confidence intervals for the largest root of autoregressive models which are valid regardless of stationarity and nonstationarity. The proposed confidence intervals have several advantages over the existing confidence intervals of Stock (1991) and Andrews (1993). First, the new intervals are much simpler than the other two intervals. Second, the new intervals have smaller average lengths than the other two intervals while having reasonable coverage probabilities. Third, the new intervals are based on asymptotic normal theory and thus require no probability tables while the other two intervals require large sets of tables.

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Table 1

The empirical coverage (%) and the average lengths of confidence intervals for model y_t2m 5 r( y_t₂₁2m)1e . Note:_t ˆ

k5c /s; nominal coverage590%; number of replications510 000

n r CI_iv CIa CIs NA

k50 k51 k52 k53 k54

Coverage (%)

50 0.000 88.3 89.2 89.0 89.1 89.1 89.9

50 0.300 88.4 89.2 89.2 89.4 89.4 90.1

50 0.600 88.7 88.8 89.0 89.2 89.3 90.1

50 0.900 87.8 88.2 88.1 88.2 88.6 89.9 87.9 207

50 0.950 88.9 88.9 88.6 88.7 88.9 90.6 87.3 115

50 0.990 89.7 89.6 88.8 88.5 88.4 91.0 88.3 97

50 0.995 89.8 89.5 89.0 88.1 87.8 91.7 89.0 88

50 1.000 89.4 88.8 88.5 87.8 87.3 92.2 89.5 74

100 0.000 89.6 89.4 89.5 89.6 89.6 90.0

100 0.300 89.3 89.2 89.3 89.4 89.4 90.0

100 0.600 89.1 88.9 89.3 89.6 89.7 90.3

100 0.900 88.2 88.4 88.2 88.6 88.8 90.2 93.7 578

100 0.950 89.2 88.6 88.5 88.6 88.9 89.7 88.4 157

100 0.990 91.3 90.9 90.7 90.2 89.9 91.0 89.4 74

100 0.995 91.1 90.8 90.4 89.8 89.3 91.4 88.9 67

100 1.000 90.3 89.8 89.3 88.7 88.2 92.4 90.2 66

Average length

50 0.000 0.63 0.52 0.50 0.50 0.50 0.47

50 0.300 0.61 0.50 0.48 0.48 0.48 0.47

50 0.600 0.51 0.44 0.42 0.42 0.42 0.43

50 0.900 0.27 0.25 0.24 0.24 0.24 0.26 0.30

50 0.950 0.21 0.19 0.19 0.19 0.19 0.21 0.25

50 0.990 0.17 0.15 0.15 0.15 0.15 0.17 0.21

50 0.995 0.16 0.15 0.14 0.14 0.14 0.17 0.21

50 1.000 0.15 0.14 0.13 0.13 0.14 0.16 0.20

100 0.000 0.43 0.36 0.34 0.34 0.34 0.34

100 0.300 0.41 0.35 0.33 0.33 0.33 0.33

100 0.600 0.35 0.30 0.28 0.28 0.28 0.28

100 0.900 0.19 0.17 0.17 0.17 0.17 0.18 0.19

100 0.950 0.14 0.13 0.12 0.12 0.12 0.13 0.15

100 0.990 0.09 0.09 0.08 0.08 0.08 0.09 0.11

100 0.995 0.09 0.08 0.08 0.08 0.08 0.09 0.11

100 1.000 0.08 0.07 0.07 0.07 0.07 0.08 0.10

Acknowledgements

This research was supported by a grant for BK-21 Korea.

Appendix A. Proofs

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21 / 2 2

O

e sign( y_t _t₂₁)⇒N(0,s ).

t52

Thus, limiting normality of t_iv(r) for uru51 follows from the fact that

n n

21 / 2 21 / 2

ifuru51, n

O

e h ( y_t _c _t₂₁)5n

O

e sign( y_t _t₂₁)1o (1),_p

t52 t52

21 2

ifuru51, n

O

h ( y_c _t₂₁)511o (1)_p

t52

which are established in Lemma 1 of Shin and So (1999b). Also, limiting normality of t_iv(r) for

uru,1 follows from the fact that

21 / 2 2 2

ifuru,1, n

O

e h ( y_t _c _t21)⇒N[0,s Ehh ( y )c 1 j],

t52

21 2 2

ifuru,1, n

O

h ( y_c _t21)→E[h ( y )],c 1 in probability,

t52

which are direct consequences of standard ergodic theory for stationary time series.

Proof of Theorem 2. Proof for (uru51, m50) is the same as that for (uru51) for no mean model in Theorem 1. In the sequel of this proof, we assume uru,1. We have

n n

21 / 2 _¯ 21 2 _¯ 2 1 / 2

¯ ¯ ˆ

t_iv(r)5n

O

h(12r)(m2 y(t21 ))1etjhc,t21/hn s

O

(hc,t21) j ,

t52 t52

¯ _¯

where hc,t215h ( yc t212y(t21 )). Observe that

t21

¯ ¯

y_t₂₁2y_{(t21 )}5m1u_t₂₁2

O

(m1u ) /(t_i 21)5u_t₂₁2u_t₂₁,

t51

where u_t5y_t2m and

t21 21 ¯ut215(t21)

O

u .i

i51

Note that

¯ ¯ ¯

h (u_c _t2u )_t 2h (u )_c _t 50 if (u_t2u_t# 2c, u_t# 2c) or (u_t2u_t$c, u_t$c)

¯ ¯

5 2u if_t 2c,u_t2u_t,c and 2c,u_t,c, ¯

5r if_t 2c or c is between u_t2u and u ,_t _t ¯

where r are determined from u , c, and u and satisfies_t _t _t ur_tu#3c. Let S denote the event [_t 2c,u_t2

¯ ¯

ut,c and 2c,ut,c] and let R denote the event that [t 2c or c is between ut2u and u ]. At thet t

21 / 2

end of this proof, we show that P(R )_t 5O(t ). We now have

¯ ¯

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where I(S ) and I(R ) are the indicator functions of the corresponding events. Observe that thet t

numerator of t_iv(r) is

21 / 2

¯ ¯

O

h2(1 2 r)u_t₂₁)1e_tjhh (u_c _t₂₁)1h (u_c _t₂₁2u_t₂₁)2h (u_c _t₂₁)j. (A.1)

t52

Now, from the proof of Theorem 1,

21 / 2 2 2

O

e h (u_t _c _t₂₁)⇒N[0,s Eh (u )]._c ₁

t52

Also,

21 / 2 21 / 4

An5n

O

ethh (uc t212ut21)2h (uc t21)j5O (np ) (A.2)

t52

because A is a martingale with variancen n

2 21 2

var(A )n 5s n

O

Eh2ut21I(St21)1rt21I(Rt21)j

t52

2 21 2 2 21 21 / 2 21 / 2

#s n

O

hE(u_t21)16cE(uu1u)P(Rt21)19c P(Rt21)]5O(n

O

t )5O(n ).

t52

Similarly,

21 / 2 21 / 4

¯ ¯

O

u_t₂₁hh (u_c _t2u )_t 2h (u )_c _t j5O (n_p ).

t52

2 2

Therefore, the numerator of t_iv(r) in (A.1) converges in distribution to N[0,s Eh (u )]. By similarc 1

arguments leading to (A.2),

n n

21_ˆ2 _¯ 2 21_ˆ 2 2

n s

O

(h_c,t₂₁) 5n s

O

h (u_c _t₂₁)1o (1)_p

t52 t52

and we get the result.

21 / 2

¯ ¯

It remains to show P(R )_t 5O(t ). Observe that if c is between u_t2u and u then_t _t uc2u_tu#uu_tu.

¯ ¯

Therefore, R implies that_t uc2u_tu#uu_tu or uc1u_tu#uu_tu. We thus have

¯ ¯

P(R )_t #P[uc2u_tu#uu_tu]1P[uc1u_tu#uu_tu]

¯ ¯

5P[uc2u₁u#uu_tu]1P[uc1u₁u#uu_tu].

1 / 2 2 2 2 2

Now, using t u_t⇒N(0, g ),g 5s /(12r ), we have

21 1 / 2 21 1 / 2 21 1 / 2 21 1 / 2

¯ ¯

P(Rtuu )1 #P[g t uc2u1u#g t uutuuu ]1 1P[g t uc1u1u#g t uutuuu ]1 21 1 / 2 21 1 / 2

¯ ¯

(2[F(g t uc2u1u)1F(g t uc1u1u).

21 21 2 2 21 2 2

#(2p) [exp(22 t(c2u ) /1 g )1O(exp(22 t(c1u ) /1 g )] because

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` `

21 / 2 2 21 / 2 2 21 / 2 2

F(x)5(2p)

E

exp(2v / 2) dv#(2p)

E

vexp(2v / 2) dv5(2p) exp(2x / 2),

x x

x 21 / 2 2

where F(x)512F(x), F(x)5e f(v) dv, and f(x)5(2p) exp(2x / 2). Since

` `

21 2 2 21 2 2 21 / 2

E

exp(22 t(c2u) /g )f(u) du #M

E

exp(22 t(c 2 u) /g ) du5O(t ),

2` 2`

21 / 2

we have P(R )_t 5O(t ), where M5sup f(u) and f is the density function of u ._u _t

21 / 2 ˜

Proof of Theorem 3. By Chan and Wei (1988), (u2u)5O (n_p ). Therefore, the limiting

ˆ ˆ

distribution oft_iv(r) is the same as that with known u. Now, tiv(r) if u in it is replaced by the true valueu, becomes

n 21 / 2 n

2 2

O

h ( y )

O

( y 2u z 2 ? ? ? 2u z )h ( y )

H

c t21

J

t 1 t21 p21 t2p11 c t21

t5p11 t5p11

n n

2 2 21 / 2

5hs

O

h ( y_c _t₂₁)j

O

e h ( y_t _c _t₂₁),

t5p11 t5p11

which has the limiting normal distribution as in Theorem 1.

References

Andrews, D.W.K., 1993. Exactly median-unbiased estimation of first order autoregressive unit root models. Econometrica 61, 139–165.

Chan, N.H., Wei, C.Z., 1988. Limiting distributions of least squares estimates of unstable autoregressive processes. Annals of Statistics 16, 367–401.

Dickey, D.A., Fuller, W.A., 1979. Distribution of the estimators for autoregressive time series with a unit root. J. American Statistical Assocication 74, 427–431.

Fuller, W.A., 1996. Introduction To Statistical Time Series, 2nd ed. John Wiley & Sons, New York. Huber, P.J., 1981. Robust Statistics. John Wiley, New York.

IMSL, 1989. User’s Manual. IMSL, Houston, Texas.

Shin, D.W., So, B.S., 1999a. Recursive mean adjustment in time series inferences. Statistics & Probability Letters 43, 65–73. Shin, D.W., So, B.S., 1999b. Normal tests for unit roots based on instrumental variable estimators. Journal of Time Series

Analysis, forthcoming.

So, B.S., Shin, D.W., 1999a. Cauchy estimation for autoregressive process with applications to unit root tests and confidence interval. Econometric Theory 15, 165–176.

So, B.S., Shin, D.W., 1999b. Recursive mean adjustment in time series inferences. Statistic & Probability Letters 43, 65–73. Stock, J.H., 1991. Confidence intervals for the largest autoregressive root in U.S. macroeconomic time series. J. Monetary

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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_t2y_t₂₁5r( y_t₂₁2y_t₂₁)1e ,_t 21 t

where y_t5t o_i₅₁y . The resulting estimator and its standard error are_i

n n

¯ ¯ ¯ ¯ ¯

r_iv5

O

( yt2yt21)h ( yc t212yt21)

YO

( yt212yt21)h ( yc t212yt21)

t52 t52

and

n 1 / 2 n

2 2

ˆ ˆ ¯ ¯ ¯

se(riv)5

H

O

h ( yc t212yt21)

J

YO

( yt212yt21)h ( yc t212yt21). t52 t52

Note that the regressor ( y_t₂₁2y_t₂₁) is independent of the error term e . This together with the_t

¯ ¯ ¯

IV-estimation gives limiting normality of the pivotal statistic t_iv(r)5(riv2r) /se(riv).

Theorem 2. Consider model (2) with (uru51, m50) or uru,1. Assume that the density function of

e is bounded. For any c_t $0, the limiting distribution of t_iv(r) is standard normal.

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_t5ry_t₂₁1u_{1 t}z₂₁1 ? ? ? 1u_p₂_{1 t}z₂_p₁₁1e ,_t (3)

˜ ˜ ˜

where z_t5y_t2y_t21 andr,u5(u1, . . . ,up)9are unknown parameters. Let r,u5(u1, . . . , up)9be the OLSE. The largest root r is estimated by applying the IV-estimation to the model

˜ ˜ _˜

y_t 2 u_{1 t}z₂₁2 ? ? ? 2u_p₂_{1 t}z₂_p₁₁5ry_t₂₁1e ._t

The resulting estimator is given by

n n

˜ ˜

r_iv5

O

( yt 2 u1 tz212 ? ? ? 2up21 tz2p11)h ( yc t21)

Y O

yt21h ( yc t21). t5p11 t5p11

The standard error is

n n

2 2 1 / 2

ˆ ˆ

se(r_iv)5hs

O

h ( yc t21)j

Y O

yt21h ( yc t21) t5p11 t5p11

(2)

¯ ¯

for all uru#1, which produces a confidence interval of the form riv6za/ 2se(riv).

ˆ ˆ ˆ

In Theorem 3, we establish limiting normality of t_iv(r)5(riv2r) / se(riv).

Theorem 3. Consider model (3). Assume that all the characteristic roots of u(B )512u₁B

p21

2 ? ? ? 2up21B lie outside the unit circle. For any uru#1, the limiting distribution of tiv(r) is

standard normal.

3. Numerical studies

to [21, 1] as Andrews (1993) did in constructing his confidence intervals. The upper limits of CI_iv

CI and CI . For example, when n_a _s 550 and r$0.99, the average length of CI_iv is about 9 / 10 of that of CI and 3 / 4 of that of CI . When_a _s r50, 0.3, 0.6, length of CI_iv is almost as good as CI . Thea advantage of CI_iv over CI and CI seems uniform for all ka s 51, 2, 3, 4. Hence, we may recommend any of 1#k#4 for a good confidence interval because the performance of CI_iv is similar for this wide range of k.

4. Concluding remarks

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Table 1

The empirical coverage (%) and the average lengths of confidence intervals for model y_t2m 5 r( y_t₂₁2m)1e . Note:_t ˆ

k5c /s; nominal coverage590%; number of replications510 000

n r CI_iv CIa CIs NA

k50 k51 k52 k53 k54

Coverage (%)

50 0.000 88.3 89.2 89.0 89.1 89.1 89.9

50 0.300 88.4 89.2 89.2 89.4 89.4 90.1

50 0.600 88.7 88.8 89.0 89.2 89.3 90.1

50 0.900 87.8 88.2 88.1 88.2 88.6 89.9 87.9 207

50 0.950 88.9 88.9 88.6 88.7 88.9 90.6 87.3 115

50 0.990 89.7 89.6 88.8 88.5 88.4 91.0 88.3 97

50 0.995 89.8 89.5 89.0 88.1 87.8 91.7 89.0 88

50 1.000 89.4 88.8 88.5 87.8 87.3 92.2 89.5 74

100 0.000 89.6 89.4 89.5 89.6 89.6 90.0

100 0.300 89.3 89.2 89.3 89.4 89.4 90.0

100 0.600 89.1 88.9 89.3 89.6 89.7 90.3

100 0.900 88.2 88.4 88.2 88.6 88.8 90.2 93.7 578

100 0.950 89.2 88.6 88.5 88.6 88.9 89.7 88.4 157

100 0.990 91.3 90.9 90.7 90.2 89.9 91.0 89.4 74

100 0.995 91.1 90.8 90.4 89.8 89.3 91.4 88.9 67

100 1.000 90.3 89.8 89.3 88.7 88.2 92.4 90.2 66

Average length

50 0.000 0.63 0.52 0.50 0.50 0.50 0.47

50 0.300 0.61 0.50 0.48 0.48 0.48 0.47

50 0.600 0.51 0.44 0.42 0.42 0.42 0.43

50 0.900 0.27 0.25 0.24 0.24 0.24 0.26 0.30

50 0.950 0.21 0.19 0.19 0.19 0.19 0.21 0.25

50 0.990 0.17 0.15 0.15 0.15 0.15 0.17 0.21

50 0.995 0.16 0.15 0.14 0.14 0.14 0.17 0.21

50 1.000 0.15 0.14 0.13 0.13 0.14 0.16 0.20

100 0.000 0.43 0.36 0.34 0.34 0.34 0.34

100 0.300 0.41 0.35 0.33 0.33 0.33 0.33

100 0.600 0.35 0.30 0.28 0.28 0.28 0.28

100 0.900 0.19 0.17 0.17 0.17 0.17 0.18 0.19

100 0.950 0.14 0.13 0.12 0.12 0.12 0.13 0.15

100 0.990 0.09 0.09 0.08 0.08 0.08 0.09 0.11

100 0.995 0.09 0.08 0.08 0.08 0.08 0.09 0.11

100 1.000 0.08 0.07 0.07 0.07 0.07 0.08 0.10

Acknowledgements

This research was supported by a grant for BK-21 Korea.

Appendix A. Proofs

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21 / 2 2

O

e sign( y_t _t₂₁)⇒N(0,s ).

t52

Thus, limiting normality of t_iv(r) for uru51 follows from the fact that

n n

21 / 2 21 / 2

ifuru51, n

O

e h ( y_t _c _t₂₁)5n

O

e sign( y_t _t₂₁)1o (1),_p

t52 t52

21 2

ifuru51, n

O

h ( y_c _t₂₁)511o (1)_p

t52

which are established in Lemma 1 of Shin and So (1999b). Also, limiting normality of t_iv(r) for uru,1 follows from the fact that

21 / 2 2 2

ifuru,1, n

O

e h ( yt c t21)⇒N[0,s Ehh ( y )c 1 j],

t52

21 2 2

ifuru,1, n

O

h ( y_c _t21)→E[h ( y )],c 1 in probability,

t52

which are direct consequences of standard ergodic theory for stationary time series.

Proof of Theorem 2. Proof for (uru51, m50) is the same as that for (uru51) for no mean model in Theorem 1. In the sequel of this proof, we assume uru,1. We have

n n

21 / 2 _¯ 21 2 _¯ 2 1 / 2

¯ ¯ ˆ

t_iv(r)5n

O

h(12r)(m2 y(t21 ))1etjhc,t21/hn s

O

(hc,t21) j ,

t52 t52

¯ _¯

where hc,t215h ( yc t212y(t21 )). Observe that t21

¯ ¯

y_t₂₁2y_(t₂_{1 )}5m1u_t₂₁2

O

(m1u ) /(t_i 21)5u_t₂₁2u_t₂₁,

t51

where u_t5y_t2m and t21

¯ut215(t21)

O

u .i

i51

Note that

¯ ¯ ¯

h (u_c _t2u )_t 2h (u )_c _t 50 if (u_t2u_t# 2c, u_t# 2c) or (u_t2u_t$c, u_t$c)

¯ ¯

5 2u if_t 2c,u_t2u_t,c and 2c,u_t,c, ¯

5r if_t 2c or c is between u_t2u and u ,_t _t ¯

where r are determined from u , c, and u and satisfies_t _t _t ur_tu#3c. Let S denote the event [2_t c,u_t2

¯ ¯

ut,c and 2c,ut,c] and let R denote the event that [t 2c or c is between ut2u and u ]. At thet t

21 / 2

end of this proof, we show that P(R )_t 5O(t ). We now have

¯ ¯

(5)

where I(S ) and I(R ) are the indicator functions of the corresponding events. Observe that thet t

numerator of t_iv(r) is n

21 / 2 _¯ _¯

O

h2(1 2 r)u_t₂₁)1e_tjhh (u_c _t₂₁)1h (u_c _t₂₁2u_t₂₁)2h (u_c _t₂₁)j. (A.1)

t52

Now, from the proof of Theorem 1, n

21 / 2 2 2

O

e h (u_t _c _t₂₁)⇒N[0,s Eh (u )]._c ₁

t52

Also,

21 / 2 21 / 4

An5n

O

ethh (uc t212ut21)2h (uc t21)j5O (np ) (A.2)

t52

because A is a martingale with variancen n

2 21 2

var(A )n 5s n

O

Eh2ut21I(St21)1rt21I(Rt21)j

t52

2 21 2 2 21 21 / 2 21 / 2

#s n

O

hE(u_t21)16cE(uu1u)P(Rt21)19c P(Rt21)]5O(n

O

t )5O(n ).

t52

Similarly, n

21 / 2 21 / 4

¯ ¯

O

u_t₂₁hh (u_c _t2u )_t 2h (u )_c _t j5O (n_p ).

t52

2 2

Therefore, the numerator of t_iv(r) in (A.1) converges in distribution to N[0,s Eh (u )]. By similarc 1 arguments leading to (A.2),

n n

21_ˆ2 _¯ 2 21_ˆ 2 2

n s

O

(h_c,t₂₁) 5n s

O

h (u_c _t₂₁)1o (1)_p

t52 t52

and we get the result.

21 / 2

¯ ¯

It remains to show P(R )t 5O(t ). Observe that if c is between ut2u and u thent t uc2utu#uutu.

¯ ¯

Therefore, R implies thatt uc2utu#uutu or uc1utu#uutu. We thus have

¯ ¯

P(R )t #P[uc2utu#uutu]1P[uc1utu#uutu]

¯ ¯

5P[uc2u1u#uutu]1P[uc1u1u#uutu].

1 / 2 2 2 2 2

Now, using t u_t⇒N(0, g ),g 5s /(12r ), we have

21 1 / 2 21 1 / 2 21 1 / 2 21 1 / 2

¯ ¯

P(Rtuu )1 #P[g t uc2u1u#g t uutuuu ]1 1P[g t uc1u1u#g t uutuuu ]1

21 1 / 2 21 1 / 2

¯ ¯

(2[F(g t uc2u1u)1F(g t uc1u1u).

21 21 2 2 21 2 2

#(2p) [exp(22 t(c2u ) /g1 )1O(exp(22 t(c1u ) /g1 )] because

(6)

` `

21 / 2 2 21 / 2 2 21 / 2 2

F(x)5(2p)

E

exp(2v / 2) dv#(2p)

E

vexp(2v / 2) dv5(2p) exp(2x / 2),

x x

x 21 / 2 2

where F(x)512F(x), F(x)5e f(v) dv, and f(x)5(2p) exp(2x / 2). Since 2`

` `

21 2 2 21 2 2 21 / 2

E

exp(22 t(c2u) /g )f(u) du #M

E

exp(22 t(c 2 u) /g ) du5O(t ),

2` 2`

21 / 2

we have P(R )_t 5O(t ), where M5sup f(u) and f is the density function of u ._u _t

21 / 2

Proof of Theorem 3. By Chan and Wei (1988), (u2u)5O (n_p ). Therefore, the limiting

ˆ ˆ

distribution oft_iv(r) is the same as that with known u. Now, tiv(r) if u in it is replaced by the true valueu, becomes

n 21 / 2 n

2 2

O

h ( y )

O

( y 2u z 2 ? ? ? 2u z )h ( y )

H

c t21

J

t 1 t21 p21 t2p11 c t21 t5p11 t5p11

n n

2 2 21 / 2

5hs

O

h ( y_c _t₂₁)j

O

e h ( y_t _c _t₂₁), t5p11 t5p11

which has the limiting normal distribution as in Theorem 1.

References

Andrews, D.W.K., 1993. Exactly median-unbiased estimation of first order autoregressive unit root models. Econometrica 61, 139–165.

Chan, N.H., Wei, C.Z., 1988. Limiting distributions of least squares estimates of unstable autoregressive processes. Annals of Statistics 16, 367–401.

Dickey, D.A., Fuller, W.A., 1979. Distribution of the estimators for autoregressive time series with a unit root. J. American Statistical Assocication 74, 427–431.

Fuller, W.A., 1996. Introduction To Statistical Time Series, 2nd ed. John Wiley & Sons, New York. Huber, P.J., 1981. Robust Statistics. John Wiley, New York.

IMSL, 1989. User’s Manual. IMSL, Houston, Texas.

Analysis, forthcoming.

So, B.S., Shin, D.W., 1999a. Cauchy estimation for autoregressive process with applications to unit root tests and confidence interval. Econometric Theory 15, 165–176.