Economics Letters 67 2000 261–271
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Alternative representation for asymptotic distributions of impulse responses in cointegrated VAR systems
Yoichi Arai, Taku Yamamoto
Department of Economics , Hitotsubashi University, Kunitachi, Tokyo 186-8601, Japan
Received 12 July 1999; accepted 21 October 1999
Abstract
We show an alternative representation for the asymptotic distributions of impulse responses in cointegrated VAR systems. Our representation has the advantage that the asymptotic variances are convergent at long
horizons.
2000 Elsevier Science S.A. All rights reserved.
Keywords : Cointegration; Impulse response; Reduced rank regression
JEL classification : C32
1. Introduction
Impulse responses have been important tools for analyzing the interrelationships among variables. ¨
For stationary systems, Lutkepohl 1990 provides the simple representations for the asymptotic distributions of the impulse responses estimated by an OLS regression. For cointegrated systems,
¨ Lutkepohl and Reimers 1992 derive the asymptotic distributions of the impulse responses estimated
by an OLS regression for a fixed finite lead time i. Phillips 1998, Theorem 2.3 shows that the impulse response matrices estimated by an OLS regression converge in distribution to random
matrices and are inconsistent when i →
`. Phillips 1998, Theorem 2.9 also shows that the impulse responses estimated by a reduced rank regression RRR are consistent even when i
→ `, and derived
the asymptotic distributions for cointegrated VAR systems. The asymptotic distributions have an undesirable property in that the asymptotic variances diverge as the lead time goes to infinity. This is
rather puzzling in view of the fact that the true impulse response matrices converge to a fixed finite matrix as i
→ ` and the impulse responses estimated by a reduced rank regression RRR are
consistent.
Corresponding author. Tel.: 181-42-580-8793; fax: 181-42-580-8793. E-mail address
: yamamotoecon.hit-u.ac.jp T. Yamamoto 0165-1765 00 – see front matter
2000 Elsevier Science S.A. All rights reserved.
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262 Y
. Arai, T. Yamamoto Economics Letters 67 2000 261 –271
In this paper we present alternative representations of the asymptotic distributions of the RRR estimates of impulse responses with the convergent asymptotic variances for cointegrated VAR
systems. The derivation closely follows that of Phillips 1998 except one important difference concerning the treatment of unit roots. Our derivation explicitly utilizes the fact that s 5 m 2 r unit
roots are not estimated by a RRR, where m is the dimension of VAR system and r is the cointegration rank. This difference leads to the asymptotic distributions with the convergent asymptotic variances
even if the lead time goes to infinity.
In Section 2, we will give a brief description of the model. In Section 3, we will show alternative representations for the asymptotic distributions of impulse responses estimated by a RRR. Section 4
concludes the paper.
2. The model
