An Alternative Approach to Obtaining Nagar-Type Moment Approximations in Simultaneous Equation Models

Paper number: 99/05

Paper date: November 1998

Year: 1999

Paper Category: Discussion Paper


Garry D.A. Phillips


The paper examines asymptotic expansions for estimation errors expressed explicitly as functions of underlying random variables. Taylor series expansions are obtained from which first and second moment approximations are derived. While the expansions are essentially equivalent to the traditional Nagar-type, the terms are expressed in a form which enables moment approximations to be obtained in a particularly straightforward way, once the partial derivatives have been found. The approach is illustrated by considering the k-class estimators in a static simultaneous equation model where the disturbances are non-spherical.

JEL Classification Nos: C30
Keywords: Simultaneous equation model, non-spherical disturbances, Nagar expansions, bias and second moment approximations

Corresponding Author: Garry D.A. Phillips, School of Business and Economics, University of Exeter, Streatham Court, Rennes Drive, Exeter, EX4 4PU, UK, tel: (44) 1392 263341, fax (44) 1392 263242, email:


Acknowledgements: Generous assistance by Jan Magnus and Heinz Neudecker in deriving the results in the appendices is gratefully acknowledged. Helpful comments and criticisms were received from participants at seminars and conferences; in particular from Gordon Fisher, David Giles, Nanak Kakwani, Essie Maasoumi, Adrian Pagan, Tom Rothenberg, Chiang Tsiao and Aman Ullah. In addition, I much benefitted from discussions with Rob Engle.