Parametric and nonparametric regressions on spot volatility

该研究项目为高频采样的连续时间半鞅模型开发了新的估计和推断工具。半鞅模型是最普遍的资产价格模型，排除了套利机会，因此成为现代资产定价中的主力模型。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为回归变量。波动率过程衡量的是半令牌的内在变异性。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，而无需强加强烈的假设。拟议的活动可分为三个部分。第一节介绍了一个包含波动率的基线向量非线性回归模型。估算分两步进行。在第一步中，从高频数据中以无模型的方式恢复潜在波动过程，第二步，利用广义矩方法(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。第二节扩展了第一节，允许回归模型可能被错误地指定。这一扩展揭示了估计方法在现实环境中的稳健性，其中回归模型仅被视为真实模型的近似。对错误指定模型的分析有利于竞争模型的比较和评价。第三部分介绍了一个新的回归框架，该框架可以用于将一个潜在波动率过程的样本路径非线性地投影到另一个波动率过程的样本路径上。在金融应用中，该方法可以用来探索多个资产的波动性是如何彼此协变的。虽然拟议活动中的激励例子是金融模型的那些例子，但本项目中开发的方法对于一般半鞅是有效的。除了经济和金融之外，半鞅还被用于生物、化学和电气领域，这些领域也有高频数据可用。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.