Quantile Regression with Generated Regressors

This paper studies estimation and inference for linear quantile regression models with generated regressors. We suggest a practical two-step estimation procedure, where the generated regressors are computed in the first step. The asymptotic properties of the two-step estimator, namely, consistency and asymptotic normality are established. We show that the asymptotic variance-covariance matrix needs to be adjusted to account for the first-step estimation error. We propose a general estimator for the asymptotic variance-covariance, establish its consistency, and develop testing procedures for linear hypotheses in these models. Monte Carlo simulations to evaluate the finite-sample performance of the estimation and inference procedures are provided. Finally, we apply the proposed methods to study Engel curves for various commodities using data from the UK Family Expenditure Survey. We document strong heterogeneity in the estimated Engel curves along the conditional distribution of the budget share of each commodity. The empirical application also emphasizes that correctly estimating confidence intervals for the estimated Engel curves by the proposed estimator is of importance for inference.

LIKELIHOOD INFERENCE ON SEMIPARAMETRIC MODELS WITH GENERATED REGRESSORS

Hahn and Ridder (2013, Econometrica 81, 315–340) formulated influence functions of semiparametric three-step estimators where generated regressors are computed in the first step. This class of estimators covers several important examples for empirical analysis, such as production function estimators by Olley and Pakes (1996, Econometrica 64, 1263–1297) and propensity score matching estimators for treatment effects by Heckman, Ichimura, and Todd (1998, Review of Economic Studies 65, 261–294). The present article studies a nonparametric likelihood-based inference method for the parameters in such three-step estimation problems. In particular, we apply the general empirical likelihood theory of Bravo, Escanciano, and van Keilegom (2018, Annals of Statistics, forthcoming) to modify semiparametric moment functions to account for influences from plug-in estimates into the above important setup, and show that the resulting likelihood ratio statistic becomes asymptotically pivotal without undersmoothing in the first and second step nonparametric estimates.

An Econometric Analysis of ETF and ETF Futures in Financial and Energy Markets Using Generated Regressors

It is well known that that there is an intrinsic link between the financial and energy sectors, which can be analyzed through their spillover effects, which are measures of how the shocks to returns in different assets affect each other’s subsequent volatility in both spot and futures markets. Financial derivatives, which are not only highly representative of the underlying indices, but can also be traded on both the spot and futures markets, include Exchange Traded Funds (ETFs), a tradable spot index whose aim is to replicate the return of an underlying benchmark index. When ETF futures are not available to examine spillover effects, “generated regressors” are useful for constructing both Financial ETF futures and Energy ETF futures. The purpose of the paper is to investigate the co-volatility spillovers within and across the US energy and financial sectors in both spot and futures markets, by using “generated regressors” and a multivariate conditional volatility model, namely Diagonal BEKK. The daily data used are from 1998/12/23 to 2016/4/22. The data set is analyzed in its entirety, and are also subdivided into three distinct subsets. The empirical results show there is a significant relationship between the Financial ETF and Energy ETF in the spot and futures markets. Therefore, financial and energy ETFs are suitable for constructing a financial portfolio from an optimal risk management perspective, and also for dynamic hedging purposes.