COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1613 Exact Distribution Theory in Structural Estimation with an Identity Peter C.B. Phillips June 2007 Some exact distribution theory is developed for structural equation models with and
without identities. The theory includes LIML, IV and OLS. We relate the new results to
earlier studies in the literature, including the pioneering work of Bergstrom (1962).
General IV exact distribution formulae for a structural equation model without an identity
are shown to apply also to models with an identity by specializing along a certain
asymptotic parameter sequence. Some of the new exact results are obtained by means of a
uniform asymptotic expansion. An interesting consequence of the new theory is that the
uniform asymptotic approximation provides the exact distribution of the OLS estimator in
the model considered by Bergstrom (1962). This example appears to be the first instance in
the statistical literature of a uniform approximation delivering an exact expression for a
probability density. |
