Paul Sangrey

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.

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.