October 12, 2006

The derivatives of complex characteristic roots in the econometric modelling textbook of Kuh et al.

Characteristic roots, or eigenvalues, are a useful tool in the study of linear dynamic economic models. The relation between the roots and the model coefficients are expressed in the derivatives of the former with respect to the latter.

In general, the roots are complex numbers. This note discusses the derivatives of both the modulus and the cycle time of the complex roots.

An important reference on this subject is Kuh et al. (1985) [hereafter: KNH]. Below it is shown that KNH discusses these derivatives unsatisfactory. In the next two sections the model and the derivatives of the roots are given. Section 4 presents the formulas for the derivatives of the modulus and the cycle time of complex roots. The substance of the paper is section 5, with a critical comment on the discussion of these derivatives in KNH. The last section gives a short conclusion.



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