Simulation-Based Exact Tests for Jump-Diffusions with Unidentified Nuisance Parameters. An Application to Commodities Spot Prices

In this paper, we propose to use the Monte-Carlo (MC) test technique to obtain valid p-values when testing for the presence of discontinuities in jump-diffusion models. Indeed, the LR statistic used to test for discontinuities has typically a complex non-standard distribution, for at least two reasons: the jump frequency parameter lies on the boundary of its domain, and unidentified nuisance parameters intervene under the null hypothesis. We show that, if no other (identified) nuisance parameters are present (e.g. the geometric Brownian motion case), the proposed p-value is finite sample exact. Otherwise, we derive nuisance-parameter free bounds on the null distribution of the LR and obtain exact bounds p-values. We illustrate our approach with four classes of jump diffusion models (geometric Brownian motion and logarithmic Ornstein-Uhlenbeck, with and without a GARCH(1,1) error structure), which we apply to weekly and monthly spot prices of non-precious metals, gold, and crude oil. We find significant jumps in all weekly time series, but only in a few monthly time series.