Economics Letters 71 2001 155–163
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On the identifiability of Euler equation estimates under
q
saddlepath stability
Bernd Lucke , Christian Gaggermeier
¨ ¨
Universitat Hamburg , Fachbereich Wirtschaftswissenschaften, Institut f ur Wachstum und Konjunktur, Von-Melle-Park 5,
D- 20146 Hamburg, Germany
Received 27 June 2000; accepted 12 October 2000
Abstract
In all but the most trivial settings Euler equation estimation in saddlepath stable systems is faced with a fundamental identification problem: the Euler equation allows for an unstable root of its characteristic
equation, while the data used to estimate the Euler equation also obey the transversality condition, which rules out unstable roots. Thus, even if the model is true, then the data are completely uninformative with respect to the
unstable root of the Euler equation. But ignorance of the unstable root implies that the parameters are not identified if the relationship between parameters and roots is one to one. We illustrate the issue using a linearized
Euler equation and present an application with OECD consumption data.
2001 Elsevier Science B.V. All
rights reserved.
Keywords : Euler equation estimates; Transversality condition; Real business cycle models
JEL classification : C22; E21
1. Introduction
Today’s dynamic macroeconomic theory relies heavily on models of intertemporally optimizing agents, whose behavior is characterized by the necessary conditions of suitably formulated maxi-
mization problems. Often, estimating the structural parameters is feasible only for estimating the first order conditions, cf. Wickens 1995. Among these, Euler equation estimation is widely used to
retrieve key parameter values.
q
Former versions of this paper were distributed under the more exciting but less precise title ‘Don’t estimate Euler equations’.
Corresponding author. Tel.: 149-40-2838-2080; fax: 149-40-2838-6314. E-mail address
: luckehermes1.econ.uni-hamburg.de B. Lucke. 0165-1765 01 – see front matter
2001 Elsevier Science B.V. All rights reserved.
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. Lucke, C. Gaggermeier Economics Letters 71 2001 155 –163
In this paper, we argue that a fundamental problem of Euler equation estimation lies in the fact that if the model is true, then the data used to estimate the Euler equation are the solution to the complete
set of first order conditions including, in particular, the transversality condition. The Euler equation to be estimated represents only a subset of the restrictions the optimal trajectory is required to meet.
Specifically, Euler equations in saddlepath stable systems contain an explosive root, while the observed data which fulfil the transversality condition do not. Hence the explosive root cannot be
estimated from the data and the identifiability of parameters related to this root is not ensured.
We also show that the problem is absent in the textbook time separable case, where the change in consumption is unrelated to lagged consumption. This is Hall’s 1978 random walk scenario. While
the unit root is unstable, it is not explosive and does hence not violate the transversality condition. Empirically, however, the change in consumption is often correlated to lagged consumption, so that
momentary utility cannot be viewed as time separable, possibly due to habit persistence, durable goods, or liquidity constraints. In this case, Euler equation estimates of some of the deep parameters
will be nonsensical. As an illustration we show that estimates of the representative agents’s discount rate which should be between 0 and 1 will tend to infinity due to this problem.
The body of this paper is organized as follows: in Section 2 we propose a very general representative consumer model and derive log linear approximations to selected first order conditions.
