• Abstract: We consider continuous-time models with a large panel of moment conditions, where the structural parameter depends on a set of characteristics, whose effect is of interest. The leading example is the linear factor model in financial economics where factor betas depend on observed characteristics such as firm specific instruments and macroeconomic variables, and their effect picks up long-run time-varying beta fluctuations. We specify the factor betas as the sum of characteristic effects and an orthogonal idiosyncratic parameter that captures high-frequency movements. It is often the case that researchers do not know whether or not the latter exists, or its strengths, and thus the inference about the characteristic effect should be valid uniformly over a broad class of data generating processes for idiosyncratic parameters. We construct our estimation and inference in a two-step continuous-time GMM framework. It is found that the limiting distribution of the estimated characteristic effect has a discontinuity when the variance of the idiosyncratic parameter is near the boundary (zero), which makes the usual ``plug-in" method using the estimated asymptotic variance only valid pointwise and may produce either over- or under- coveraging probabilities. We show that the uniformity can be achieved by cross-sectional bootstrap. Our procedure allows both known and estimated factors, and also features a bias correction for the effect of estimating unknown factors.

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