Parametric and nonparametric regressions on spot volatility

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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.