Parametric and nonparametric regressions on spot volatility

本研究项目为高频采样的连续半鞅模型开发了新的估计和推理工具。半鞅模型是最通用的资产价格模型，它排除了套利机会，因此一直是现代资产定价的主力模型。所提出的活动的主要智力优点是发展了新的非线性回归方法，以半鞅的潜在波动过程作为回归量。波动性过程测量半鞅的内在变异性。这些方法将允许研究人员调查经济变量和波动过程之间的统计关系，而无需强加强假设。建议的活动可分为三个部分。第一部分涉及涉及波动率的基线向量非线性回归模型。估计分两个步骤执行。在第一步中，以无模型的方式从高频数据中恢复潜在波动过程，在第二步中，通过广义矩量方法（GMM）估计回归模型。研究了该过程的统计性质。这些工具允许用户探索波动过程如何驱动其他经济变量，并做出统计上的正式声明。包括一个经验应用来说明该方法的使用。第二部分扩展了第一部分，允许错误指定回归模型。这个扩展揭示了在一个现实的设置估计方法的鲁棒性，其中回归模型只被认为是真实模型的近似值。对错误指定模型的分析有助于对竞争模型进行比较和评价。第三部分介绍了一种新的回归框架，该框架可用于将潜在波动过程的样本路径非线性投影到另一个波动过程的样本路径上。在金融应用中，该方法可用于探索多个资产的波动率如何相互协变。虽然在提议的活动中激励的例子是那些金融模型，在这个项目中开发的方法是有效的一般半鞅。除了经济和金融，半鞅还被用于生物、化学和电气领域，在这些领域也可以获得高频数据。人们可以希望，这里开发的一些统计方法可以在这些领域找到应用。

This research project develops new estimation and inference tools for continuous-time semimartingale models sampled at high frequency. The semimartingale model is the most general model for asset prices that precludes arbitrage opportunities and, as a result, has been the workhorse model in modern asset pricing.The primary intellectual merit of the proposed activities is the development of new nonlinear regression methods with the latent volatility process of a semimartingale as a regressor. The volatility process measures the intrinsic variability of the semimartingale. The methods will allow researchers to investigate the statistical relationship between economic variables and the volatility process without imposing strong assumptions.The proposed activity can be divided into three sections. The first section concerns a baseline vector nonlinear regression model involving the volatility. The estimation is performed in two steps. In the first step, the latent volatility process is recovered from high frequency data in a model-free fashion, and in the second step, the regression model is estimated via the generalized methods of moments (GMM). The statistical property of this procedure is studied. These tools allow the user to explore how the volatility process drives other economic variables and to make statistically formal statements. An empirical application is included for illustrating the use of the method.The second section extends the first section by allowing the regression model to be possibly misspecified. This extension sheds light on the robustness of the estimation method in a realistic setting in which the regression model is only considered as an approximation of the true model. The analysis on misspecified models facilitates the comparison and evaluation of competing models.The third section introduces a new regression framework which can be applied to perform nonlinear projection of the sample path of a latent volatility process onto that of another volatility process. In financial applications, the method can be used to explore how volatilities of multiple assets co-vary with each other. While the motivating examples in the proposed activity are those of financial models, the methods developed in this project are valid for generic semimartingales. Besides economics and finance, semimartingales have also been used in biological, chemical, and electrical applications, where high frequency data are also available. One can hope that some of the statistical methods developed here can find applications in these fields.