# Paul Sangrey

#### Jumps, Realized Densities, and News Premia

Announcements and other news continuously barrage financial markets, causing asset prices to jump hundreds of times each day. If price paths are continuous, then diffusive volatility nonparametrically summarizes the return distributions' dynamics, and risk premia are instantaneous covariances. But this is not true in the empirically-relevant case involving price jumps. To address this impasse, I derive both a tractable nonparametric continuous-time representation for the price jumps and an implied sufficient statistic for their dynamics. This statistic — *jump volatility* — is the instantaneous variance of the jump part and measures news risk. The realized density then depends (exclusively) on the diffusive volatility and the jump volatility. I develop estimators for both and show how to use them to nonparametrically identify continuous-time jump dynamics and associated risk premia. I provide a detailed empirical application to the S&P 500 and show that the jump volatility premium is less than the diffusive volatility premium.

#### Bypassing the Curse of Dimensionality: Feasible Multivariate Density Estimation (with Minsu Chang)

Most economic data are multivariate and so estimating multivariate densities is a classic problem in the literature. However, given vector-valued data, the *curse of dimensionality* makes nonparametrically estimating the data’s density infeasible if the number of series D is large. Hence, we do not seek to provide estimators that perform well all of the time (it is impossible), but rather seek to provide estimators that perform well most of the time. We adapt the ideas in the Bayesian compression literature to this environment by randomly binning the data. The binning randomly determines both the number of bins and which observation is placed in which bin. This novel procedure induces a simple mixture representation for the data’s density. For any finite number of periods T, the number of mixture components used is random. We construct a bound for this variable as a function of T that holds with high probability. We adopt the nonparametric Bayesian framework and construct a computationally efficient density estimator using Dirichlet processes. Since the number of mixture components is the key determinant of our model’s complexity, our estimator’s convergence rates — log(T)/√/T in the unconditional case and log(T)/√T in the conditional case — depend on D only through the constant term. We then analyze our estimators' performance in a monthly macroeconomic panel and a daily financial panel. Our procedure performs well in capturing the data’s stylized features such as time-varying volatility and fat-tails.

#### Identification Robust Inference for Risk Prices in Structural Stochastic Volatility Models (with Xu Cheng and Eric Renault)

In structural stochastic volatility asset pricing models, changes in volatility affect risk premia through two channels: (1) the investor's willingness to bear high volatility in order to get high expected returns as measured by the market risk price, and (2) the investor’s direct aversion to changes in future volatility as measured by the volatility risk price. Disentangling these channels is difficult and poses a subtle identification problem that invalidates standard inference. We adopt the discrete-time exponentially affine model of Han, Khrapov, and Renault (2018), which links the identification of volatility risk price to the leverage effect. In particular, we develop a minimum distance criterion that links the market risk price, the volatility risk price, and the leverage effect to the well-behaved reduced-form parameters governing the return and volatility's joint distribution. The link functions are almost flat if the leverage effect is close to zero, making estimating the volatility risk price difficult. We adapt the conditional quasi-likelihood ratio test Andrews and Mickusheva (2016) develop in a nonlinear GMM framework to a minimum distance framework. The resulting conditional quasi-likelihood ratio test is uniformly valid. We invert this test to derive robust confidence sets that provide correct coverage for the prices regardless of the leverage effect's magnitude.

Econometrics, Financial Economics

Department of Economics, University of Pennsylvania, PCSPE 628, 133 South 36th Street, Philadelphia, PA 19104

Francis X. Diebold

Frank Schorfheide

Francis X. Diebold

Department of Economics, PCSPE 607, 133 South 36th Street, Philadelphia, PA 19104

fdiebold@upenn.edu

Frank Schorfheide

Department of Economics, PCSPE 621, 133 South 36th Street, Philadelphia, PA 19104

schorf@ssc.upenn.edu

Xu Cheng

Department of Economics, PCPSE 620, 133 South 36th Street, Philadelphia, PA 19104

xucheng@upenn.edu

Amir Yaron

Department of Finance, The Wharton School, University of Pennsylvania, SH-DH 2325, 3620 Locust Walk, Philadelphia, PA 19104

yaron@wharton.upenn.edu