COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1781 Folklore Theorems, Implicit Maps and New Unit Root Limit Theory Peter C. B. Phillips January 2011 The delta method and continuous mapping theorem are among the most
extensively used tools in asymptotic derivations in econometrics. Extensions of these
methods are provided for sequences of functions, which are commonly encountered in
applications, and where the usual methods sometimes fail. Important examples of failure
arise in the use of simulation based estimation methods such as indirect inference. The
paper explores the application of these methods to the indirect inference estimator (IIE)
in first order autoregressive estimation. The IIE uses a binding function that is sample
size dependent. Its limit theory relies on a sequence-based delta method in the stationary
case and a sequence-based implicit continuous mapping theorem in unit root and local to
unity cases. The new limit theory shows that the IIE achieves much more than bias
correction. It changes the limit theory of the maximum likelihood estimator (MLE) when the
autoregressive coefficient is in the locality of unity, reducing the bias and the variance
of the MLE without affecting the limit theory of the MLE in the stationary case. Thus, in
spite of the fact that the IIE is a continuously differentiable function of the MLE, the
limit distribution of the IIE is not simply a scale multiple of the MLE but depends
implicitly on the full binding function mapping. The unit root case therefore represents
an important example of the failure of the delta method and shows the need for an implicit
mapping extension of the continuous mapping theorem. |
