Folklore Theorems, Implicit Maps, and Indirect Inference

By Peter C. B. Phillips. 2012

57 citations recorded by the source corpus.

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 that 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 (partial) 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.

Subfields: Economics and Econometrics, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c78f5757bd0bb75e0739e03b97fa5574b04b1ed4

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Citations in the corpus

Bias in parametric estimation: reduction and useful side‐effects (2013, result): “eneralized linear models with random eﬀects and Jiang and Turnbull (2004) give a comprehensive review of indirect inference from a statistical point of view. Furthermore, Gourieroux et al. (2000) and Phillips (2012) discuss bias reduction through indirect inference in econometric applications. Pfeﬀe”
Least Squares Bias in Time Series with Moderate Deviations from a Unit Root (2018, background): “(A.8) For c = 0, corresponding to |α| = 1, the algebra reduces to the one found in Phillips (2012).”
Uniformity and the Delta Method (2015, methodology): “Phillips (2012) has extended the pointwise delta-method in such a direction using an asymptotically locally relative equicontinuity condition.”
X-DIFFERENCING AND DYNAMIC PANEL MODEL ESTIMATION (2013, background): “This limit theory has only recently been obtained for the time series model (Phillips, 2012), and the panel extension is left for a subsequent contribution.”