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.
Research and Education Activities: As part of the project, the following papers were written: (1) "Semi-Nonparametric Estimation and Misspecification Testing of Diffusion Models," published in Journal of Econometrics, Vol. 164, pp. 382–403. (2) "Testing Conditional Factor Models," published in Journal of Financial Economics, Vol. 106, pp. 132-156. (3) "Nonparametric Detection and Estimation of Structural Change," published in Econometrics Journal, Vol. 15, pp. 420-461. (4) "Estimation of Dynamic Models with Nonparametric Simulated Maximum Likelihood," published in Journal of Econometrics, Vol. 167, pp. 76-94. (5) "Estimation of Dynamic Latent Variable Models Using Simulated Nonparametric Moments," published in Econometrics Journal, Vol. 15, pp. 490-515. (6) "Nonparametric Identification and Estimation of Transformation Models," working paper, under revision at Journal of Econometrics. The above papers develop new models and techniques for the analysis of economic and financial data, allowing applied researchers to draw inference about underlying economic structures and to carry out forecasting and policy analysis. Below, I provide a brief summary of each paper: (1) develops new specification tests of so-called diffusion models. These models are widely used in economics and finance to describe the evolution of economic variables of time. I first demonstrate that existing tests have certain shortcomings and then develop new kernel- based tests that do not suffer from these. These new tests should lead to better guidance in the search for appropriate descriptions/models of financial and economic time series. (2) revisits so-called 'rolling-window' estimators of Fama-French factor models of stock returns. These estimators are widely used in empirical finance, but their theoretical properties have not been fully analyzed. We demonstrate how this class of estimators can be described as kernel-type estimators and so import techniques from the literature on nonparametric estimators to analyze the asymptotic properties of them. This enable empirical researchers to draw correct inference based on these popular estimators. (3) develops some refinements of the basic techniques found in (2). In particular, we consider semiparametric null hypotheses and develop statistical tests of these. We also show how the methods in (2) can be used in a broader setting to test for parameter instability in time series regressions. (4) develops new methods for likelihood-based inference in dynamic models where the likelihood cannot be written up on closed-form. This situation often arises in estimation of economic and financial models. We show how simulations and nonparametric techniques can be combined to obtain a simulated maximum-likelihood estimator and analyze its asymptotic properties. As an application, we demonstrate its usefulness in estimation of continuous-time models as used in financial asset pricing. (5) is related to (4) in that the same problem is considered. However, instead of basing estimation and inference on the likelihood, we develop moment-based methods. This is useful for forecasting and for robust inference. (6) develops new identification and estimation techniques for a class of transformation models that include many important econometric models. This is the first paper to develop estimators in a fully nonparametric setting. As part of the above research, a PhD student at Columbia Uni was hired as a research assistant. Under my guidance, the RA wrote up computer code for the implementation of the methods developed as part of the project, thereby providing applied researchers easy acces to the techniques that were developed. Training and Development: The RA hired for the project received training in the econometric techniques employed and how to implement those in practice. He received his PhD in 2012 and is now working in a financial institution in South Korea. I furthermore included parts of the research in my teaching of PhD students at Columbia University. I gave a seminar for people working in the financial industry, presenting some of the results of this research proposal. Outreach Activities: I have presented parts of the research at: * PhD course at University of Copenhagen * Seminars: UC San Diego; NYU Stern; Rutgers University; University of Rochester; University of Groningen; Tilburg University; Brown Uni; National Uni of Singapore; Singapore Management Uni; Harvard-MIT; LSE; Uni Southampton; Uni Paris-Dauphine. * Conferences: NBER Summer Institute Forecasting seminar, Cambridge; Greater NY Metropolitian Area Econometrics Colloquium, NYU; 2010 CAM Workshop, University of Copenhagen; 'Latest Developments in Financial Econometrics,' Uni Libre de Bruxelles; 'Econometric and Statistical Modelling of Multivariate Time Series,' CORE; '2nd Humboldt-Copenhagen Conference,' Uni of Copenhagen; Asian Meetings of the Econometric Society, Seoul.
