Mixed-scale jump regressions with bootstrap inference

We develop an efficient mixed-scale estimator for jump regressions using high-frequency asset returns. A fine time scale is used to accurately identify the locations of large rare jumps in the explanatory variables such as the price of the market portfolio. A coarse scale is then used in the estimation in order to attenuate the effect of trading frictions in the dependent variable such as the prices of potentially less liquid assets. The proposed estimator has a non-standard asymptotic distribution that cannot be made asymptotically pivotal via studentization. We propose a novel bootstrap procedure for feasible inference and justify its asymptotic validity. We show that the bootstrap provides an automatic higher-order asymptotic approximation by accounting for the sampling variation in estimates of nuisance quantities that are used in efficient estimation. The Monte Carlo analysis indicates good finite-sample performance of the general specification test and confidence intervals based on the bootstrap. We apply the method to a high-frequency panel of Dow stock prices together with the market index defined by the S&P 500 index futures over the period 2007–2014. We document remarkable temporal stability in the way that stocks react to market jumps. However, this relationship for many of the stocks in the sample is significantly noisier and more unstable during sector-specific jump events.