Threshold estimation under strong dependence

Abstract We consider a threshold regression model in a long-memory setting. A method for estimating the threshold parameter is proposed. Asymptotic results are derived. The asymptotic rate of convergence turns out to be slower than under weak dependence, but approaches the usual fast rate of $$O_{p}(n^{-1})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> when the long-memory parameter of the residuals converges to zero. Furthermore, asymptotic inference for the difference of conditional means below and above an estimated threshold is considered. A statistic is defined to construct confidence intervals. Surprisingly, the rate of convergence of the statistic improves when nuisance parameters are estimated. An algorithm for constructing data driven confidence intervals is proposed. The results are illustrated by a small simulation study and an application to CBOE volumes and volatilities for S&amp;P 500 index options.