Grid Bootstrap Inference for Threshold Quantile Regression

We develop bootstrap-based inference for the threshold parameter in a threshold quantile regression (TQR) model. Replacing the least-squares loss with the quantile loss, changes the bootstrap empirical process significantly since it removes the maximal inequality’s dependence on the magnitude of the regression error, and creates a bootstrap nuisance-scale problem with no least-squares counterpart. Building on the grid-bootstrap confidence interval of Hansen (1999) and the least-squares validity theory of Hidalgo, Lee and Seo (2019), we prove that the correctly rescaled bootstrap statistic converges to the correct limiting distribution under the null. The key theoretical ingredient is a conditional maximal inequality for the quantile bootstrap score process. Since the bootstrap error enters the quantile score only through its sign, the score is, conditional on the data, an i.i.d., hence a martingale argument delivers the required tightness directly, without importing the magnitude of the regression error into the bootstrap calculation. This reduces the required moment condition from the eighth-order requirement of Hidalgo et al. (2019), with no additional moment constraints contributed by the bootstrap step itself. We further show that the bootstrap has its own nuisance scale factor, generally distinct from the one governing the original-sample likelihood-ratio statistic of Kuan et al. (2017) and derive a closed-form studentizer for it directly from the known bootstrap weight density, avoiding a separate nonparametric density estimation step inside the bootstrap calculation. An extensive Monte Carlo study com-pares four nonparametric estimators of the original-sample nuisance parameter, against the asymptotic approximation across two data-generating processes, three error distributions, three sample sizes, three threshold magnitudes, and five quantile levels. The results show that under weak threshold identification, both procedures are oversized and stay that way even at larger samples. Under adequate identification the asymptotic interval is preferred at small samples, but it is conservative and does not improve with more data. The grid bootstrap’s size falls smoothly toward nominal with increasing sample size, especially at central and tail quantiles. Our implementation pairs the quantile wild bootstrap of Feng, He and Hu (2011) with a smoothed, floored conditional-scale construction that keeps both the bootstrap error density and its closed-form studentizer well behaved. An application to real data concludes.