In this paper, we introduce the weighted-average quantile regression framework, R 1 0 qY |X(u)ψ(u)du = X0β, where Y is a dependent variable, X is a vector of covariates, qY |X is the quantile function of the conditional distribution of Y given X, ψ is a weighting function, and β is a vector of parameters. We argue that this framework is of interest in many applied settings and develop an estimator of the vector of parameters β. We show that our estimator is √ T-consistent and asymptotically normal with mean zero and easily estimable covariance matrix, where T is the size of available sample. We demonstrate the usefulness of our estimator by applying it in two empirical settings. In the first setting, we focus on financial data and study the factor structures of the expected shortfalls of the industry portfolios. In the second setting, we focus on wage data and study inequality and social welfare dependence on commonly used individual characteristics.
We find that three factors – cryptocurrency market, size, and momentum – capture the cross-sectional expected cryptocurrency returns. We consider a comprehensive list of price- and market-related return predictors in the stock market, and construct their cryptocurrency counterparts. Ten cryptocurrency characteristics form successful long-short strategies that generate sizable and statistically significant excess returns, and we show that all of these strategies are accounted for by the cryptocurrency three-factor model. Lastly, we examine potential underlying mechanisms of the cryptocurrency size and momentum effects.