Different continuous-time models for interest rates coexist in the literature. We test parametric models by comparing their implied parametric density to the same density estimated nonparametrically. We do not replace the continuous-time model by discrete approximations, even though the data are recorded at discrete intervals. The principal source of rejection of existing models is the strong nonlinearity of the drift. Around its mean, where the drift is essentially zero, the spot rate behaves like a random walk. The drift then mean-reverts strongly when far away from the mean. The volatility is higher when away from the mean.

Testing Continuous-Time Models of the Spot Interest Rate

Introduction.- Histogram.- Nonparametric Density Estimation.- Nonparametric Regression.- Semiparametric and Generalized Regression Models.- Single Index Models.- Generalized Partial Linear Models.- Additive Models and Marginal Effects.- Generalized Additive Models.

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Nonparametric and Semiparametric Models

While the additive model is a popular nonparametric regression method, many of its theoretical properties are not well understood, especially when the backfitting algorithm is used for computation of the estimators. This article explores those properties when the additive model is fitted by local polynomial regression. Sufficient conditions guaranteeing the asymptotic existence of unique estimators for the bivariate additive model are given. Asymptotic approximations to the bias and the variance of a homoscedastic bivariate additive model with local polynomial terms of odd and even degree are computed. This model is shown to have the same rate of convergence as that of univariate local polynomial regression.

Fitting a Bivariate Additive Model by Local Polynomial Regression

Useful in the theoretical and empirical analysis of nonlinear time series data, semiparametric methods have received extensive attention in the economics and statistics communities over the past twenty years. Recent studies show that semiparametric methods and models may be applied to solve dimensionality reduction problems arising from using fully nonparametric models and methods. Answering the call for an up-to-date overview of the latest developments in the field, "Nonlinear Time Series: Semiparametric and Nonparametric Methods" focuses on various semiparametric methods in model estimation, specification testing, and selection of time series data.After a brief introduction, this book examines semiparametric estimation and specification methods and then applies these approaches to a class of nonlinear continuous-time models with real-world data. It also assesses some newly proposed semiparametric estimation procedures for time series data with long-range dependence. Even though this book only deals with climatological and financial data, the estimation and specifications methods discussed can be applied to models with real-world data in many disciplines. This resource covers key methods in time series analysis and provides the necessary theoretical details. The latest applied finance and financial econometrics results and applications presented in this book enable researchers and graduate students to keep abreast of developments in the field.

Nonlinear Time Series: Semiparametric and Nonparametric Methods

This survey intends to collect the developments on Goodness-of-Fit for regression models during the last 20 years, from the very first origins with the proposals based on the idea of the tests for density and distribution, until the most recent advances for complex data and models. Far from being exhaustive, the contents in this paper are focused on two main classes of tests statistics: smoothing-based tests (kernel-based) and tests based on empirical regression processes, although other tests based on Maximum Likelihood ideas will be also considered. Starting from the simplest case of testing a parametric family for the regression curves, the contributions in this field provide also testing procedures in semiparametric, nonparametric, and functional models, dealing also with more complex settings, as those ones involving dependent or incomplete data.

https://www.eco.uc3m.es/temp/Gonzalez%203.pdf

An updated review of Goodness-of-Fit tests for regression models

In the common nonparametric regression model with high dimensional predictor several tests for the hypothesis of an additive regression are investigated. The corresponding test statistics are either based on the diiferences between a fit under the assumption of additivity and a fit in the general model or based on residuals under the assumption of additivity. For all tests asymptotic normality is established under the null hypothesis of additivity and under fixed alternatives with different rates of convergence corresponding to both cases. These results are used for a comparison of the different methods. It is demonstrated that a statistic based on an empirical L1 - distance of the Nadaraya Watson and the marginal integration estimator yields the asymptotically most efficient procedure if these are compared with respect to the asymptotic behaviour under fixed alternatives.

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Testing additivity by kernel-based methods - what is a reasonable test?

. A new specification test for the parametric form of the variance function in diffusion processes is proposed, which does not require specific knowledge of the functional form of the model. The corresponding test statistic has an asymptotic normal distribution under the null hypothesis and diverges at an appropriate rate under the alternative. In contrast to recent work the approach of the present paper does not require the specification of particular time points at which the hypothesis of a parametric form is checked. As a by-product we obtain a very simple test for homoscedasticity in diffusion processes. Moreover, the new test does not use nonparametric estimation techniques for estimating the variance function and is therefore independent of the specification of a particular smoothing parameter. The results are illustrated by a small simulation study and a data example is analyzed.

On a test for a parametric form of volatility in continuous time financial models

In this article we highlight the main differences of available methods for the analysis of regression functions that are probably additive separable. We first discuss definition and interpretation of the most common estimators in practice. This is done by explaining the different ideas of modeling behind each estimator as well as what the procedures are doing to the data. Computational aspects are mentioned explicitly. The illustrated discussion concludes with a simulation study on the mean squared error for different marginal integration approaches. Next, various test statistics for checking additive separability are introduced and accomplished with asymptotic theory. Based on the asymptotic results under hypothesis as well as under the alternative of non additivity we compare the tests in a brief discussion. For the various statistics, different smoothing and bootstrap methods we perform a detailed simulation study. A main focus in the reported results is directed on the (non-) reliability of the methods when the covariates are strongly correlated among themselves. Again, a further point are the computational aspects. We found that the most striking differences lie in the different pre-smoothers that are used, but less in the different constructions of test statistics. Moreover, although some of the observed differences are strong, they surprisingly can not be revealed by asymptotic theory.

A comparison of different nonparametric methods for inference on additive models

In this article, we highlight the main differences of available methods for the analysis of regression functions that are probably additive separable. We first discuss definition and interpretation of the most common estimators in practice explaining the different ideas of modeling behind each estimator as well as what the procedures are doing to the data. Computational aspects are mentioned explicitly. The discussion concludes with a simulation study on the mean squared error for different marginal integration approaches. Next, various test statistics for checking additive separability are introduced and accomplished with asymptotic theory. For the statistics, different smoothing and bootstrap methods, we perform a detailed simulation study. A main focus in the reported results is directed on the (non)reliability of the methods when the covariates are strongly correlated among themselves. We found that the most striking differences lie in the different pre-smoothers that are used, but less in the differe...

/pdf/a-comparison-of-different-nonparametric-methods-for-oka7kh4a3n.pdf

Carsten von Lieres und Wilkau

Papers

Testing additivity by kernel-based methods - what is a reasonable test?

On a test for a parametric form of volatility in continuous time financial models

A comparison of different nonparametric methods for inference on additive models

A comparison of different nonparametric methods for inference on additive models

On a test for constant volatility in continuous time financial models