The Algebraic Equivalence of the Oberhofer-Kmenta and Theil-Boot Formulae for the Asymptotic Variance of a Characteristic Root of a Dynamic Econometric Model

Abstract

IN AN important paper, Theil and Boot [4] developed a test for the stability of a dynamic econometric model. The test statistic is the (estimated) absolute value of the largest root of the model's characteristic equation. Theil and Boot made the test operational by providing a formula for the asymptotic variance of the test statistic. This formula involved the asymptotic covariance matrix of the estimated reduced form parameters.1 More recently, Oberhofer and Kmenta [3] provided a formula for the asymptotic variance of the same test statistic, which involved the asymptotic covariance matrix of the estimated structural parameters. This is, of course, more readily available than the covariance matrix of the reduced form estimates. This note shows that the two approaches are algebraically equivalent. That is, starting from a set of structural estimates, it makes no difference whether one applies the Oberhofer-Kmenta formula, or whether one calculates the derived reduced form and then uses the Theil-Boot formula; the results are exactly the same either way. 2. THE OBERHOFER-KMENTA FORMULA Let p represent the absolute value of the largest characteristic root, and p its estimate. Let 0 be the vector of structural coefficients, say of dimension N. Suppose that one has a consistent estimator 0 (say, the three-stage least squares estimate) and its (estimated) asymptotic covariance matrix. Then the Oberhofer-Kmenta result is just that (1) i=a.v. (( a.c. i=, 0j), where a.v. (p) represents the (estimated) asymptotic variance of p and a.c. (0i, 0) represents the