# Improved Semi-Nonparametric Estimation and Testing by Modified Likelihood

*National Science Foundation, SES-0961596*

Principal Investigator: Dennis Kristensen

__Abstract__

Economic theory often has little to say about the specific functional forms of components entering economic models. This has lead to an increased use of non- and semiparametric estimation and testing methods in economics since these in general impose weaker functional restrictions on the models of interest.

Many popular non- and semiparametric estimators involve so-called kernel-smoothing. However, these can be challenging to implement since they involve choosing appropriate bandwidths which are an integral part of the estimators: The resulting estimators are in general sensitive to the bandwidth choice. Unfortunately, theory offers few guidelines for how these should be chosen in finite samples: First of all, the bandwidth does not appear in the asymptotic distribution of the parametric estimator. Secondly, for the first-step estimation error to vanish at an optimal rate, undersmoothing is required. This rules out standard bandwidth selection methods such as plug-in and cross-validation. For a few special estimators, methods have been developed, but these can be complicated to implement and do not always perform well.

We here propose a novel class of semiparametric profile estimators that do not suffer from these problems: We develop a modified version of the standard objective (likelihood) function defining the estimator. The modification entails that it can be used to estimate both the nonparametric component and the parametric one. The advantages of this modification are three-fold: First, the modification ensures that an error term normally appearing in the expansion of the parametric estimator now vanishes. Thus we expect that the modified version will have better finite-sample properties. Second, by removing the error term, we do not have to undersmooth in order for the first-step estimation error to vanish at an optimal rate. Hence standard bandwidth selection methods can be used. Finally, the proposed modified estimator is no more difficult to implement than standard estimators and require no heavy computations.

We also demonstrate how the modified objective function can be used to improve on existing non- and semiparametric testing procedures using kernel-smoothing methods. The modified tests are shown to dominate the original ones in terms of Pitman's relative efficiency criterion and as such are more powerful. As with the kernel-based semiparametric estimation procedures, the issue of how to select bandwidths in the implementation of kernel-based testing procedures is to a large extent unresolved. We will examine the issue of bandwidth selection for the new class of test statistics developed in this project.

The novel procedures can be used to improve upon existing econometric methods developed for many semiparametric models, including partially linear models, single-index models, (semi-)varying-coefficient models, and models with time-varying parameters. These and many other models will be considered in the project.

http://www.nsf.gov/awardsearch/showAward.do?AwardNumber=0961596

__Working Papers and Publications__

Dennis Kristensen & Bernard Salanié, 2010. "Higher Order Improvements for Approximate Estimators," CAM Working Papers 2010-04, University of Copenhagen. Department of Economics. Centre for Applied Microeconometrics.

Full text: http://www.econ.ku.dk/cam/wp0910/2010-04.pdf

Dennis Kristensen, 2010. “Indirect Likelihood Inference”

Full text: http://research.barcelonagse.eu/tmp/working_papers/558.pdf

Dennis Kristensen, 2010. "Semi-Nonparametric Estimation and Misspecification Testing of Diffusion Models," Discussion Papers 10-10, University of Copenhagen. Department of Economics.

Full text: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1571059

Dennis Kristensen & Anders Rahbek, 2010. "Testing and Inference in Nonlinear Cointegrating Vector Error Correction Models," Discussion Papers 10-25, University of Copenhagen. Department of Economics.

Full text: http://www.econ.ku.dk/english/research/publications/wp/dp_2010/1025.pdf/

Dennis Kristensen & Anders Rahbek, 2007. "Likelihood-Based Inference in Nonlinear Error-Correction Models," CREATES Research Papers 2007-38, School of Economics and Management, University of Aarhus.

Full text: http://www.columbia.edu/~ao2027/Kristensen.pdf

Dennis Kristensen, 2011. “Nonparametric Detection and Estimation of Structural Change”

Full Text: http://econ.au.dk/fileadmin/site_files/filer_oekonomi/Working_Papers/CREATES/2011/rp11_13.pdf