We study the econometric properties of dynamic risk parity, which volatility scales to equalise risk through time using the precision process, the inverse of the time-varying volatility. A particular focus is on the impact of the Sharpe ratio. We give necessary and suffcient conditions that volatility scaling improves the Sharpe ratio of an investment. We approximate the Sharpe improvement using the sum of two terms: one determined by the convexity of the precision and the other the covariance of the precision and conditional mean. We show that empirically this approximation is very accurate and we document the
relative importance of the two terms.

Estimation of time-varying covariances is a key input in risk management and asset allocation. ARCH-type multivariate models are used widely for this purpose. Estimation of such models is computationally costly and parameter estimates are meaningfully biased when applied to a moderately large number of assets. Here we propose a novel estimation approach that suffers from neither of these issues, even when the number of assets is in the hundreds. The theory of this new method is developed in some detail. The performance of the proposed method is investigated using extensive simulation studies and empirical examples.

Much research and policy attention has been on socio economic gaps in participation at university, but little attention has been paid to gaps in earnings. This paper addresses this shortfall using tax and student loan administrative data to investigate the earnings of English graduates up to their mid thirties by socio economic background. We find that  graduates from higher income families (from the top fth of the income distribution of those enrolled in university) have average earnings which are 20% higher than those from lower income families. Once we condition on institution and subject choices, this premium roughly halves, to around 10%. The premium grows with age and is larger for men, in particular for men at the most selective universities. We follow Chetty et al. (2017) and estimate English mobility scorecards by university and subject, highlighting the good performance of medicine, economics, law, business, engineering, technology, math, computer science and architecture courses as well as the prominent London-based universities.

We define causal estimands for experiments on single time series, extending the potential outcome framework to dealing with temporal data.  Our approach allows the estimation of a broad class of these estimands and exact randomization based p-values for testing causal effects, without imposing stringent assumptions.  We further derive a general central limit theorem that can be used to conduct conservative tests and build confidence intervals for causal effects.  Finally, we provide three methods for generalizing our approach to multiple units that are receiving the same class of treatment, over time.  We test our methodology on simulated "potential autoregressions,"which have a causal interpretation.  Our methodology is partially inspired by data from a large number of experiments carried out by a financial company who compared the impact of two different ways of trading equity futures contracts.  We use our methodology to make causal statements about their trading methods.

We propose a novel Dirichlet-based Pólya tree (D-P tree) prior on the copula and based on the D-P tree prior, a nonparametric Bayesian inference procedure. Through theoretical analysis and simulations, we are able to show that the flexibility of the D-P tree prior ensures its consistency in copula estimation, thus able to detect more subtle and complex copula structures than earlier nonparametric Bayesian models, such as a Gaussian copula mixture. Furthermore, the continuity of the imposed D-P tree prior leads to a more favourable smoothing effect in copula estimation over classic frequentist methods, especially with small sets of observations. We also apply our method to the copula prediction between the S&P 500 index and the IBM stock prices during the 2007–08 financial crisis, finding that D-P tree-based methods enjoy strong robustness and flexibility over classic methods under such irregular market behaviours.