Uniform Inference for Conditional Factor Models with Instrumental and Idiosyncratic Betas

It has been well known in financial economics that factor betas depend on observed instruments such as firm specific characteristics and macroeconomic variables, and a key object of interest is the effect of instruments on the factor betas. One of the key features of our model is that we specify the factor betas as functions of time-varying observed instruments that pick up long-run beta fluctuations, plus an orthogonal idiosyncratic component that captures high-frequency movements in beta. It is often the case that researchers do not know whether or not the idiosyncratic beta exists, or its strengths, and thus uniformity is essential for inferences. It is found that the limiting distribution of the estimated instrument effect has a discontinuity when the strength of the idiosyncratic beta is near zero, which makes usual inferences fail to be valid and produce misleading results. In addition, the usual "plug-in" method using the estimated asymptotic variance is only valid pointwise. The central goal is to make inference about the effect on the betas of firms' instruments, and to conduct out-of-sample forecast of integrated volatilities using estimated factors. Both procedures should be valid uniformly over a broad class of data generating processes for idiosyncratic betas with various signal strengths and degrees of time-variant. We show that a cross-sectional bootstrap procedure is essential for the uniform inference, and our procedure also features a bias correction for the effect of estimating unknown factors.