Testing for Jump Spillovers Without Testing for Jumps

Abstract

This article develops statistical tools for testing conditional independence among the jump components of the daily quadratic variation, which we estimate using intraday data. To avoid sequential bias distortion, we do not pretest for the presence of jumps. If the null is true, our test statistic based on daily integrated jumps weakly converges to a Gaussian random variable if both assets have jumps. If instead at least one asset has no jumps, then the statistic approaches zero in probability. We show how to compute asymptotically valid bootstrap-based critical values that result in a consistent test with asymptotic size equal to or smaller than the nominal size. Empirically, we study jump linkages between US futures and equity index markets. We find not only strong evidence of jump cross-excitation between the SPDR exchange-traded fund and E-mini futures on the S&P 500 index, but also that integrated jumps in the E-mini futures during the overnight period carry relevant information. Supplementary materials for this article are available as an online supplement.

Keywords:

• Conditional independence
• Jump intensity
• Kernel smoothing
• Quadratic variation
• Realized measure

Acknowledgments

The authors thank the associate editor and anonymous referee as well as Yacine Aït-Sahalia, Dante Amengual, Jesus Gonzalo, Yongmiao Hong, Nour Meddahi, Aureo de Paula, Bradley Paye, Enrique Sentana, and Stefan Sperlich for valuable comments. They are indebted to seminar and conference participants at Carlos III, CEMFI, Essex Business School, ETH Zurich, Toulouse School of Economics, Frontiers of Finance (Warwick Business School, 2012), SKBI Annual Conference on Financial Econometrics (Singapore, 2012), Symposium of Econometric Theory and Applications (Shanghai, 2012), Fifth Italian Congress of Econometrics and Empirical Economics (Genova, 2013), International Conference on Systemic Risk, Contagion and Jumps (Cass Business School, 2013), Meetings of the Brazilian Finance Society (Recife, 2014), New Methods for the Empirical Analysis of Financial Markets (Santander Financial Institute, 2017), Southampton Finance and Econometrics Workshop (University of Southampton, 2017), Third International Workshop on Financial Econometrics (Arraial d’Ajuda, 2017), and SoFiE Annual Conference (Lugano, 2018).

Notes

1 See the recent works by Li, Todorov, and Tauchen (2017a, 2017b) and Li et al. (2017) for a more formal framework to handle jump regressions.

2 Stephan and Whaley (1990), Easley, O’Hara, and Srinivas (1998), and Chakravarty, Gulen, and Mayhew (2004) offer some counterexamples in stock option markets and Yang (2009) for currency markets, though.

3 We restrict attention to the past components of the quadratic variation of the price of asset A in the smaller information set, but one could certainly think of other conditioning variables. In addition, we control only for the last realization of the quadratic variation components to mitigate dimensionality issues. This is enough to accommodate both Bates’s (2000) geometric jump-diffusion process with stochastic volatility and Duffie, Pan, and Singleton’s (2000) affine jump-diffusion process, for instance.

4 See, for instance, Aït-Sahalia and Jacod (2009) for a formal definition.

5 Boudt and Zhang (2015) applied a thresholding to robustify the two-scale realized measure, but the resulting estimator yields a not-fast-enough convergence rate of $a_M=M^{-1/3}$.

6 One obvious alternative is to employ subsampling, which also boils down to resampling m out of n observations, though without replacement. See Politis, Romano, and Wolf (1999) for more details.

7 Note that the above condition for δ constrains the dimension of the larger information set to at most 3 conditioning variables. In a previous version of this article, we had a less restrictive condition for at most two conditioning variables (q = 2): namely, $27/40<\delta <1$.

8 To avoid too much undersmoothing in the higher dimensional case in which we also control for the past realized variance, we multiply these adjusted rule-of-thumb bandwidths by 10 similarly to Corradi, Distaso, and Fernandes (2012).

9 The qualitative results do not change if we do not control for market microstructure noise and estimate the realized jump component by the difference between the realized variance and the bipower variation.

10 Our choice for subsamples is arbitrary, essentially resting on visual inspection of the realized measures in Figure 1.

11 Further analysis nonetheless rejects at the 1% significance level the null hypothesis that today’s jump component for ES during NYSE trading hours does not depend on yesterday’s given their overnight counterpart.

12 We deliberately avoid saying co-jumps because it may give the impression that jumps are necessarily at the same time, rather than just over the same trading day.

Additional information

Funding

This work was partially supported by the ESRC (RES-062-23-0311), FAPESP (2013/22930-0), and CNPq (302272/2014-3).