Conditional variance forecasts for long-term stock returns

In this paper, we apply machine learning to forecast the conditional variance of long-term stock returns measured in excess of different benchmarks, considering the short- and long-term interest rate, the earnings-by-price ratio, and the inflation rate. In particular, we apply in a two-step procedure a fully nonparametric local-linear smoother and choose the set of covariates as well as the smoothing parameters via cross-validation. We find that volatility forecastability is much less important at longer horizons regardless of the chosen model and that the homoscedastic historical average of the squared return prediction errors gives an adequate approximation of the unobserved realised conditional variance for both the one-year and five-year horizon

Smoothing and mean-covariance estimation of functional data with a Bayesian hierarchical model

Functional data, with basic observational units being functions (e.g., curves, surfaces) varying over a continuum, are frequently encountered in various applications. While many statistical tools have been developed for functional data analysis, the issue of smoothing all functional observations simultaneously is less studied. Existing methods often focus on smoothing each individual function separately, at the risk of removing important systematic patterns common across functions. We propose a nonparametric Bayesian approach to smooth all functional observations simultaneously and nonparametrically. In the proposed approach, we assume that the functional observations are independent Gaussian processes subject to a common level of measurement errors, enabling the borrowing of strength across all observations. Unlike most Gaussian process regression models that rely on pre-specified structures for the covariance kernel, we adopt a hierarchical framework by assuming a Gaussian process prior for the mean function and an Inverse-Wishart process prior for the covariance function. These prior assumptions induce an automatic mean-covariance estimation in the posterior inference in addition to the simultaneous smoothing of all observations. Such a hierarchical framework is flexible enough to incorporate functional data with different characteristics, including data measured on either common or uncommon grids, and data with either stationary or nonstationary covariance structures. Simulations and real data analysis demonstrate that, in comparison with alternative methods, the proposed Bayesian approach achieves better smoothing accuracy and comparable mean-covariance estimation results. Furthermore, it can successfully retain the systematic patterns in the functional observations that are usually neglected by the existing functional data analyses based on individual-curve smoothing.Comment: Submitted to Bayesian Analysi

Approximate Bayesian inference in semiparametric copula models

We describe a simple method for making inference on a functional of a multivariate distribution. The method is based on a copula representation of the multivariate distribution and it is based on the properties of an Approximate Bayesian Monte Carlo algorithm, where the proposed values of the functional of interest are weighed in terms of their empirical likelihood. This method is particularly useful when the "true" likelihood function associated with the working model is too costly to evaluate or when the working model is only partially specified.Comment: 27 pages, 18 figure

Wavelet Estimators in Nonparametric Regression: A Comparative Simulation Study

Wavelet analysis has been found to be a powerful tool for the nonparametric estimation of spatially-variable objects. We discuss in detail wavelet methods in nonparametric regression, where the data are modelled as observations of a signal contaminated with additive Gaussian noise, and provide an extensive review of the vast literature of wavelet shrinkage and wavelet thresholding estimators developed to denoise such data. These estimators arise from a wide range of classical and empirical Bayes methods treating either individual or blocks of wavelet coefficients. We compare various estimators in an extensive simulation study on a variety of sample sizes, test functions, signal-to-noise ratios and wavelet filters. Because there is no single criterion that can adequately summarise the behaviour of an estimator, we use various criteria to measure performance in finite sample situations. Insight into the performance of these estimators is obtained from graphical outputs and numerical tables. In order to provide some hints of how these estimators should be used to analyse real data sets, a detailed practical step-by-step illustration of a wavelet denoising analysis on electrical consumption is provided. Matlab codes are provided so that all figures and tables in this paper can be reproduced

Semi-nonparametric IV estimation of shape-invariant Engel curves

This paper studies a shape-invariant Engel curve system with endogenous total expenditure, in which the shape-invariant specification involves a common shift parameter for each demographic group in a pooled system of nonparametric Engel curves. We focus on the identification and estimation of both the nonparametric shapes of the Engel curves and the parametric specification of the demographic scaling parameters. The identification condition relates to the bounded completeness and the estimation procedure applies the sieve minimum distance estimation of conditional moment restrictions, allowing for endogeneity. We establish a new root mean squared convergence rate for the nonparametric instrumental variable regression when the endogenous regressor could have unbounded support. Root-n asymptotic normality and semiparametric efficiency of the parametric components are also given under a set of "low-level" sufficient conditions. Our empirical application using the U.K. Family Expenditure Survey shows the importance of adjusting for endogeneity in terms of both the nonparametric curvatures and the demographic parameters of systems of Engel curves

Exploring the Economic Convergence in the EU New Member States by Using Nonparametric Models

This paper analyzes the process of real economic convergence in the New Member States (NMS) bein g formerly centrally planned economies, using nonparametric methods instead of conventional parametric measurement tools like beta and sigma models. This methodological framework allows the examining of the relative income distribution in different periods of time, the number of modes of the density distribution, the existence of “convergence clubs” in the distribution and the hypothesis of convergence at a single point in time. The modality tests (e.g. the ASH-WARPing procedure) and stochastic kernel are nonparametric techniques used in the empirical part of the study to examine the income distribution in the NMS area. Additionally, random effects panel regressions are used, but only for comparison reasons. The main findings of the paper are the bimodality of the income density distribution over time and across countries, and the presence of convergence clubs in the income distribution from 1995 to 2008. The findings suggest a lack of absolute convergence in the long term (1995-2008) and also when looking only from 2003 onwards. The paper concludes that, in comparison with the parametrical approach, the nonparametric one gives a deeper, real and richer perspective on the process of real convergence in the NMS area.real convergence, nonparametric models, stochastic kernel, modality