Stable limits of martingale transforms with application to the estimation of GARCH parameters

A martingale-transform goodness-of-fit test for the form of the conditional variance

In the common nonparametric regression model the problem of testing for a specific parametric form of the variance function is considered. Recently Dette and Hetzler (2008) proposed a test statistic, which is based on an empirical process of pseudo residuals. The process converges weakly to a Gaussian process with a complicated covariance kernel depending on the data generating process. In the present paper we consider a standardized version of this process and propose a martingale transform to obtain asymptotically distribution free tests for the corresponding Kolmogorov-Smirnov and Cramér-vo…

A data-driven P-spline smoother and the P-Spline-GARCH-models

Penalized spline smoothing of time series and its asymptotic properties are studied. A data-driven algorithm for selecting the smoothing parameter is developed. The proposal is applied to define a semiparametric extension of the well-known Spline-GARCH, called a P-Spline-GARCH, based on the log-data transformation of the squared returns. It is shown that now the errors process is exponentially strong mixing with finite moments of all orders. Asymptotic normality of the P-spline smoother in this context is proved. Practical relevance of the proposal is illustrated by data examples and simulatio…

Hybrid quantile estimation for asymmetric power GARCH models

Asymmetric power GARCH models have been widely used to study the higher order moments of financial returns, while their quantile estimation has been rarely investigated. This paper introduces a simple monotonic transformation on its conditional quantile function to make the quantile regression tractable. The asymptotic normality of the resulting quantile estimators is established under either stationarity or non-stationarity. Moreover, based on the estimation procedure, new tests for strict stationarity and asymmetry are also constructed. This is the first try of the quantile estimation for no…

Robust estimation and inference for heavy tailed GARCH

We develop two new estimators for a general class of stationary GARCH models with possibly heavy tailed asymmetrically distributed errors, covering processes with symmetric and asymmetric feedback like GARCH, Asymmetric GARCH, VGARCH and Quadratic GARCH. The first estimator arises from negligibly trimming QML criterion equations according to error extremes. The second imbeds negligibly transformed errors into QML score equations for a Method of Moments estimator. In this case, we exploit a sub-class of redescending transforms that includes tail-trimming and functions popular in the robust esti…

Hybrid Quantile Regression Estimation for Time Series Models with Conditional Heteroscedasticity

Estimating conditional quantiles of financial time series is essential for risk management and many other applications in finance. It is well-known that financial time series display conditional heteroscedasticity. Among the large number of conditional heteroscedastic models, the generalized autoregressive conditional heteroscedastic (GARCH) process is the most popular and influential one. So far, feasible quantile regression methods for this task have been confined to a variant of the GARCH model, the linear GARCH model, owing to its tractable conditional quantile structure. This paper consid…

鞅驱动模型独立同分布路径的非参数估计及其在非自治金融模型中的应用

Nonparametric Estimation for I.I.D. Paths of a Martingale Driven Model with Application to Non-Autonomous Financial Models

本文针对鞅驱动模型构造$J_0$的投影最小二乘估计量并建立风险界，通过模型变换将结果推广至含分数布朗运动的非自治金融模型，实现漂移项等参数的非参数估计。

基于投影经验过程和p值组合的超高维参数回归的渐近分布自由检验

Asymptotic Distribution-Free Tests for Ultra-high Dimensional Parametric Regressions via Projected Empirical Processes and $p$-value Combination

本文提出了一种基于投影经验过程和p值组合的混合检验方法，用于超高维参数回归模型的拟合优度检验，通过结合低频和高频信号检测提升检验效能。

Testing for Conditional Mean Independence with Covariates through Martingale Difference Divergence

As a crucial problem in statistics is to decide whether additional variables are needed in a regression model. We propose a new multivariate test to investigate the conditional mean independence of Y given X conditioning on some known effect Z, i.e., E(Y|X, Z) = E(Y|Z). Assuming that E(Y|Z) and Z are linearly related, we reformulate an equivalent notion of conditional mean independence through transformation, which is approximated in practice. We apply the martingale difference divergence (Shao and Zhang, 2014) to measure conditional mean dependence, and show that the estimation error from app…

Transform martingale estimating functions

An estimation method is proposed for a wide variety of discrete time stochastic processes that have an intractable likelihood function but are otherwise conveniently specified by an integral transform such as the characteristic function, the Laplace transform or the probability generating function. This method involves the construction of classes of transform-based martingale estimating functions that fit into the general framework of quasi-likelihood. In the parametric setting of a discrete time stochastic process, we obtain transform quasi-score functions by projecting the unavailable score …

On the Esscher transforms and other equivalent martingale measures for Barndorff-Nielsen and Shephard stochastic volatility models with jumps

We compute and discuss the Esscher martingale transform for exponential processes, the Esscher martingale transform for linear processes, the minimal martingale measure, the class of structure preserving martingale measures, and the minimum entropy martingale measure for stochastic volatility models of Ornstein-Uhlenbeck type as introduced by Barndorff-Nielsen and Shephard. We show, that in the model with leverage, with jumps both in the volatility and in the returns, all those measures are different, whereas in the model without leverage, with jumps in the volatility only and a continuous ret…

Joint parametric specification checking of conditional mean and volatility in time series models with martingale difference innovations

Using cumulative residual processes, we propose joint goodness-of-fit tests for conditional means and variances functions in the context of nonlinear time series with martingale difference innovations. The main challenge comes from the fact the cumulative residual process no longer admits, under the null hypothesis, a distribution-free limit. To obtain a practical solution one either transforms the process in order to achieve a distribution-free limit or approximates the non-distribution free limit using a numerical or a re-sampling technique. Here the three solutions will be considered.It is …

鞅空间上的离散时间稀疏控制加权估计

Weighted Estimation by Discrete-time Sparse Domination on Martingale Spaces

本研究针对离散时间鞅变换建立稀疏控制，引入条件稀疏性概念，推导了优于现有结果的精确加权估计与混合范数估计，还将稀疏控制应用于Doob极大算子及定量双权估计。