Smoothed gradient least squares estimator for linear threshold models

Abstract

In the presence of fixed threshold effects, the least squares (LS) estimator of the threshold parameter poses challenges for statistical inference due to its nonstandard limiting distribution, which also presents challenges for bootstrap methods. To address this issue, we propose a novel estimator: a two-step smoothed gradient least squares (SGLS) estimator. Our proposed method achieves a normal limiting distribution for the threshold parameter with minimal efficiency loss compared to the LS estimator. Furthermore, our modified bootstrap method significantly enhances computational efficiency, leading to improved bootstrap confidence intervals (CIs) for the threshold parameter compared to asymptotic CIs. Our method is validated through a small Monte Carlo study and demonstrated with an empirical application.

Keywords:

• Bootstrap
• confidence interval
• estimation
• linear threshold model
• nascent delta function
• smoothed gradient

JEL Classification::

• C13
• C21
• C51
• G30

Notes

1. Yu and Phillips (2018) introduce a nonparametric integrated difference kernel (IDK) estimator for γ₀, which is also n-consistency and has a limiting distribution depending on a compound Poisson process.

2. Let $\overrightarrow{\gamma }$ be the GMM estimator of γ₀. Then, $\overrightarrow{\gamma }-{\gamma }_0=O_p\left(n^{-1/2+\nu }\right)$; see, Seo and Shin (2016) and Konstantinidi et al. (2023).

3. With ${\lambda }_{2,0}\in \{.25,.5,1,2\}$, our Monte Carlo results support our discussion. Of course, the estimation results of the slope parameters do vary with the value of λ_{2, 0}. The results are omitted to save space and can be obtained from the author upon request.

4. Z-score = 1.2 (working capital/total assets) + 1.4 (retained earnings/total assets) + 3.3 (earnings before interest and taxes/total assets) + 0.6 (market value of equity/book value of total liabilities) + 0.999 (sales/total assets), where book value of total liabilities, earnings before interest and taxes, market value of equity, retained earnings, sales, total assets, and working capital are Compustat item LT, EBIT, PRCC_F×CSHO, RE, SALE, AT, and WCAP, respectively.

5. To save space, we did not report the bootstrap percentile-t confidence intervals, which are very close to the bootstrap percentile confidence intervals with slightly narrower interval width in general.