Economics Letters 71 2001 181–189
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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 for
r 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 of r 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
: shindwmm.ewha.ac.kr 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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. 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
