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Is the diurnal pattern sufficient to explain intraday variation in volatility? A nonparametric assessment

Kim Christensen, Ulrich Hounyo, Mark Podolskij

TL;DR

The paper tackles whether a deterministic diurnal pattern suffices to explain intraday volatility variation in high-frequency asset prices modeled as noisy Itô semimartingales with jumps. It develops a nonparametric, diurnal-corrected test based on jump-robust, pre-averaged bipower variation, proving a CLT and constructing a feasible, bootstrap-based inference procedure. A nonparametric estimator of the diurnal variance is introduced to deflate returns without imposing parametric diurnal structure, enabling practical testing in the presence of microstructure noise. Monte Carlo simulations show bootstrap-based tests achieve near-correct size with reasonable power, and an empirical application across a broad panel of US equities reveals the diurnal component accounts for a large fraction of intraday volatility but does not eliminate all heteroskedasticity. Overall, the work provides a robust framework to separate deterministic diurnal effects from stochastic volatility in high-frequency data, with direct implications for volatility modeling and intraday risk assessment.

Abstract

In this paper, we propose a nonparametric way to test the hypothesis that time-variation in intraday volatility is caused solely by a deterministic and recurrent diurnal pattern. We assume that noisy high-frequency data from a discretely sampled jump-diffusion process are available. The test is then based on asset returns, which are deflated by the seasonal component and therefore homoskedastic under the null. To construct our test statistic, we extend the concept of pre-averaged bipower variation to a general Itô semimartingale setting via a truncation device. We prove a central limit theorem for this statistic and construct a positive semi-definite estimator of the asymptotic covariance matrix. The $t$-statistic (after pre-averaging and jump-truncation) diverges in the presence of stochastic volatility and has a standard normal distribution otherwise. We show that replacing the true diurnal factor with a model-free jump- and noise-robust estimator does not affect the asymptotic theory. A Monte Carlo simulation also shows this substitution has no discernable impact in finite samples. The test is, however, distorted by small infinite-activity price jumps. To improve inference, we propose a new bootstrap approach, which leads to almost correctly sized tests of the null hypothesis. We apply the developed framework to a large cross-section of equity high-frequency data and find that the diurnal pattern accounts for a rather significant fraction of intraday variation in volatility, but important sources of heteroskedasticity remain present in the data.

Is the diurnal pattern sufficient to explain intraday variation in volatility? A nonparametric assessment

TL;DR

The paper tackles whether a deterministic diurnal pattern suffices to explain intraday volatility variation in high-frequency asset prices modeled as noisy Itô semimartingales with jumps. It develops a nonparametric, diurnal-corrected test based on jump-robust, pre-averaged bipower variation, proving a CLT and constructing a feasible, bootstrap-based inference procedure. A nonparametric estimator of the diurnal variance is introduced to deflate returns without imposing parametric diurnal structure, enabling practical testing in the presence of microstructure noise. Monte Carlo simulations show bootstrap-based tests achieve near-correct size with reasonable power, and an empirical application across a broad panel of US equities reveals the diurnal component accounts for a large fraction of intraday volatility but does not eliminate all heteroskedasticity. Overall, the work provides a robust framework to separate deterministic diurnal effects from stochastic volatility in high-frequency data, with direct implications for volatility modeling and intraday risk assessment.

Abstract

In this paper, we propose a nonparametric way to test the hypothesis that time-variation in intraday volatility is caused solely by a deterministic and recurrent diurnal pattern. We assume that noisy high-frequency data from a discretely sampled jump-diffusion process are available. The test is then based on asset returns, which are deflated by the seasonal component and therefore homoskedastic under the null. To construct our test statistic, we extend the concept of pre-averaged bipower variation to a general Itô semimartingale setting via a truncation device. We prove a central limit theorem for this statistic and construct a positive semi-definite estimator of the asymptotic covariance matrix. The -statistic (after pre-averaging and jump-truncation) diverges in the presence of stochastic volatility and has a standard normal distribution otherwise. We show that replacing the true diurnal factor with a model-free jump- and noise-robust estimator does not affect the asymptotic theory. A Monte Carlo simulation also shows this substitution has no discernable impact in finite samples. The test is, however, distorted by small infinite-activity price jumps. To improve inference, we propose a new bootstrap approach, which leads to almost correctly sized tests of the null hypothesis. We apply the developed framework to a large cross-section of equity high-frequency data and find that the diurnal pattern accounts for a rather significant fraction of intraday variation in volatility, but important sources of heteroskedasticity remain present in the data.
Paper Structure (12 sections, 8 theorems, 94 equations, 7 figures)

This paper contains 12 sections, 8 theorems, 94 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Theoretical setup
  3. Microstructure noise
  4. Test of heteroskedasticity
  5. The pre-averaging approach
  6. The pre-averaged bipower variation
  7. A truncated pre-averaged bipower variation
  8. A local estimator of diurnal variance
  9. The bootstrap
  10. Monte Carlo analysis
  11. Empirical application
  12. Conclusion

Key Result

Theorem 2.1

Let $l_{1},r_{1},l_{2}$ and $r_{2}$ be four positive real numbers and $X$ be given by equation:one. Suppose that Assumption (J) holds for some $\beta \in \lbrack 0,\min \{1,l_{1},r_{1},l_{2},r_{2}\})$ and that $(\frac{l_{1}+r_{1}-1}{2(l_{1}+r_{1}-\beta )}\vee \frac{l_{2}+r_{2}-1}{2(l_{2}+r_{2}-\beta

Figures (7)

  • Figure 1: Proportion of microstructure noise in $\check{BV}(1,1)^{n}$.
  • Figure 2: Illustration of a simulation.
  • Figure 3: Estimation of the diurnal component.
  • Figure 4: Simulation properties of $t$-statistic and H-index.
  • Figure 5: Properties of equity high-frequency data.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Theorem 2.1
  • Proposition 3.1
  • Remark 1
  • Lemma 4.1
  • Remark 2
  • Theorem 4.1
  • Corollary 4.1
  • Remark 3
  • Corollary 4.2
  • Theorem 4.2
  • ...and 2 more