On the probability distribution of economic growth

Three important and significantly heteroscedastic gross domestic product series are studied. Omnipresent heteroscedasticity is removed and the distributions of the series are then compared to normal, normal mixture and normal–asymmetric Laplace (NAL) distributions. NAL represents a skewed and leptokurtic distribution, which is in line with the Aghion and Howitt [1 Aghion, P. and Howitt, P. 1992. A model of growth through creative destruction. Econometrica, 60: 323–351. [Crossref], [Web of Science ®] [Google Scholar]] model for economic growth, based on Schumpeter's idea of creative destruction. Statistical properties of the NAL distributions are provided and it is shown that NAL fits the data better than the alternatives.     

A note on testing the regression functions via nonparametric smoothing

The authors present a consistent lack‐of‐fit test in nonlinear regression models. The proposed procedure possesses some nice properties of Zheng's test such as the consistency, the ability to detect any local alternatives approaching the null at rates slower than the parametric rate. What's more, for a predetermined kernel function, the proposed test is more powerful than Zheng's test and the validity of these findings is confirmed by the simulation studies and a real data example. In addition, the authors find out a close connection between the choices of normal kernel functions and the bandwidths. The Canadian Journal of Statistics 39: 108–125; 2011 © 2011 Statistical Society of Canada     

Efficient Estimation of an Additive Quantile Regression Model

Abstract. In this paper, two non‐parametric estimators are proposed for estimating the components of an additive quantile regression model. The first estimator is a computationally convenient approach which can be viewed as a more viable alternative to existing kernel‐based approaches. The second estimator involves sequential fitting by univariate local polynomial quantile regressions for each additive component with the other additive components replaced by the corresponding estimates from the first estimator. The purpose of the extra local averaging is to reduce the variance of the first estimator. We show that the second estimator achieves oracle efficiency in the sense that each estimated additive component has the same variance as in the case when all other additive components were known. Asymptotic properties are derived for both estimators under dependent processes that are strictly stationary and absolutely regular. We also provide a demonstrative empirical application of additive quantile models to ambulance travel times.     

Non‐parametric Estimation of Extreme Risk Measures from Conditional Heavy‐tailed Distributions

In this paper, we introduce a new risk measure, the so‐called conditional tail moment. It is defined as the moment of order a ≥ 0 of the loss distribution above the upper α‐quantile where α ∈ (0,1). Estimating the conditional tail moment permits us to estimate all risk measures based on conditional moments such as conditional tail expectation, conditional value at risk or conditional tail variance. Here, we focus on the estimation of these risk measures in case of extreme losses (where α ↓0 is no longer fixed). It is moreover assumed that the loss distribution is heavy tailed and depends on a covariate. The estimation method thus combines non‐parametric kernel methods with extreme‐value statistics. The asymptotic distribution of the estimators is established, and their finite‐sample behaviour is illustrated both on simulated data and on a real data set of daily rainfalls.     

Estimation of scale functions to model heteroscedasticity by regularised kernel-based quantile methods

A main goal of regression is to derive statistical conclusions on the conditional distribution of the output variable Y given the input values x. Two of the most important characteristics of a single distribution are location and scale. Regularised kernel methods (RKMs) – also called support vector machines in a wide sense – are well established to estimate location functions like the conditional median or the conditional mean. We investigate the estimation of scale functions by RKMs when the conditional median is unknown, too. Estimation of scale functions is important, e.g. to estimate the volatility in finance. We consider the median absolute deviation (MAD) and the interquantile range as measures of scale. Our main result shows the consistency of MAD-type RKMs.     

Consistency of the recursive nonparametric regression estimation for dependent functional data

We consider the recursive estimation of a regression functional where the explanatory variables take values in some functional space. We prove the almost sure convergence of such estimates for dependent functional data. Also we derive the mean quadratic error of the considered class of estimators. Our results are established with rates and asymptotic appear bounds, under strong mixing condition. Finally, the feasibility of the proposed estimator is illustrated throughout an empirical study.     

Structural Nonparametric Cointegrating Regression

Nonparametric estimation of a structural cointegrating regression model is studied. As in the standard linear cointegrating regression model, the regressor and the dependent variable are jointly dependent and contemporaneously correlated. In nonparametric estimation problems, joint dependence is known to be a major complication that affects identification, induces bias in conventional kernel estimates, and frequently leads to ill‐posed inverse problems. In functional cointegrating regressions where the regressor is an integrated or near‐integrated time series, it is shown here that inverse and ill‐posed inverse problems do not arise. Instead, simple nonparametric kernel estimation of a structural nonparametric cointegrating regression is consistent and the limit distribution theory is mixed normal, giving straightforward asymptotics that are useable in practical work. It is further shown that use of augmented regression, as is common in linear cointegration modeling to address endogeneity, does not lead to bias reduction in nonparametric regression, but there is an asymptotic gain in variance reduction. The results provide a convenient basis for inference in structural nonparametric regression with nonstationary time series when there is a single integrated or near‐integrated regressor. The methods may be applied to a range of empirical models where functional estimation of cointegrating relations is required.     

Nonparametric Instrumental Regression

The focus of this paper is the nonparametric estimation of an instrumental regression function ϕ defined by conditional moment restrictions that stem from a structural econometric model E[Y−ϕ(Z)|W]=0, and involve endogenous variables Y and Z and instruments W. The function ϕ is the solution of an ill‐posed inverse problem and we propose an estimation procedure based on Tikhonov regularization. The paper analyzes identification and overidentification of this model, and presents asymptotic properties of the estimated nonparametric instrumental regression function.     

信用风险中回收率分布的双Beta模型

本文研究了随机回收率的分布,建立了回收率的双Beta分布密度模型,它具有双峰分布的特点,这与Moody公司的最新研究相吻合,弥补了现有回收率分布模型均为单峰的不足。利用基于数论网格的序贯优化算法对所建模型的参数做出了估计,借助于核密度估计的工具,进行了实证分析,结果表明双Beta模型的拟合误差很小,远小于Beta模型的误差,它是表示回收率理想的模型。最后给出了抽取双Beta分布随机数的方法。     

VaR与CVaR的敏感性凸性及其核估计

风险价值（VaR）和条件风险价值（CVaR）是目前两大主流风险度量工具,如何准确地对它们进行估计是风险管理实践中首要而核心的问题。近年来非参数核估计方法因模型设定灵活、方便处理变量相依结构等优点备受关注。在本文,我们在核估计的框架内讨论VaR和CVaR估计量的性质;给出投资组合VaR和CVaR对组合头寸的一阶导数向量和二阶导数矩阵的核估计公式,并用它们来讨论组合VaR和CVaR对组合头寸的敏感性和凸性。最后,我们利用中国外汇市场的实际数据做实证分析。