On Bivariate Time-Varying Price Staleness

Abstract

Price staleness refers to the extent of zero returns in price dynamics. Bandi, Pirino, and Reno introduce two types of staleness: systematic and idiosyncratic staleness. In this study, we allow price staleness to be time-varying and study the statistical inference for idiosyncratic and common price staleness between two assets. We propose consistent estimators for both time-varying idiosyncratic and systematic price staleness and derive their asymptotic theory. Moreover, we develop a feasible nonparametric test for the simultaneous constancy of idiosyncratic and common price staleness. Our inference is based on infill asymptotics. Finally, we conduct simulation studies under various scenarios to assess the finite sample performance of the proposed approaches and provide an empirical application of the proposed theory.

Keywords:

• Joint price staleness
• Market liquidity
• Price staleness
• Stable convergence

Supplementary Materials

The supplementary file provides all of the proof of the theorems in the main text.

Acknowledgments

The authors would like to thank the Editor Professor Christian Hansen, an associate editor, and two anonymous referees for their extensive and constructive suggestions that helped to improve this article considerably. We are grateful to Aleksey Kolokolov for the valuable comments. All errors are ours.

Disclosure Statement

The authors report there are no competing interests to declare.

Notes

1 The explicit form of $p_t^{(\mathcal{l})}$ can be derived from the It $\widehat{\mathrm{o}}$ formula. Precisely, for any smooth function $\Phi (·):\mathbb{R}\mapsto (0,1)$, we have

$\begin{array}{c}d{p}_t^{(\mathcal{l})}=((S_0^{(\mathcal{l})}-S_t^{(\mathcal{l})})\frac{\partial \Phi }{\partial x}{|}_{x=S_t^{(\mathcal{l})}}+\frac{{({\sigma }^S)}^2}{2}\frac{{\partial }^2\Phi }{\partial x^2}{|}_{x=S_t^{(\mathcal{l})}})dt\\ \hspace{1em}+{\sigma }^S\left(\frac{\partial \Phi }{\partial x}{|}_{x=S_t^{(\mathcal{l})}}\right)d{W}_t^{(\mathcal{l})}.\end{array}$

The other well-known candidate of $\Phi$ is the sigmoid function, $\Phi (S_t^{(\mathcal{l})})=\frac{1}{1+\hspace{0.17em}\exp \hspace{0.17em}\{-S_t^{(\mathcal{l})}\}}$.

Additional information

Funding

Zhi Liu’s research is supported by NSFC (No. 11971507), MYRG2018-00107-FST from the University of Macau, and The Science and Technology Development Fund (FDCT) of Macau (No. 0041/2021/ITP).