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The realized empirical distribution function of stochastic variance with application to goodness-of-fit testing

Kim Christensen, Martin Thyrsgaard, Bezirgen Veliyev

TL;DR

This work addresses validating stochastic volatility models when spot volatility is unobserved by constructing a nonparametric, noise-robust realized EDF (REDF) of latent volatility from tick-by-tick data. It develops a pre-averaging-based estimator for local spot variance, debiases microstructure noise, and proves infill consistency and a joint infill-long-span functional CLT, yielding a feasible Gaussian limit with long-run variance $ obreak $. Using this, the authors define realized Kolmogorov–Smirnov and weighted $L^{2}$ goodness-of-fit statistics, which, with bootstrap, allow model validation for stochastic volatility specifications. The empirical application to SPY and sector ETFs shows inverse Gaussian volatility distributions typically fit better than gamma or log-normal models, suggesting potential benefits from more flexible families like the generalized inverse Gaussian. Overall, the paper provides a robust framework for assessing stochastic volatility models using high-frequency data and paves the way for refined distributional assumptions of volatility such as $V_t hicksim IG( ext{parameters})$ or $V_t hicksim GIG( ext{parameters})$.

Abstract

We propose a nonparametric estimator of the empirical distribution function (EDF) of the latent spot variance of the log-price of a financial asset. We show that over a fixed time span our realized EDF (or REDF) -- inferred from noisy high-frequency data -- is consistent as the mesh of the observation grid goes to zero. In a double-asymptotic framework, with time also increasing to infinity, the REDF converges to the cumulative distribution function of volatility, if it exists. We exploit these results to construct some new goodness-of-fit tests for stochastic volatility models. In a Monte Carlo study, the REDF is found to be accurate over the entire support of volatility. This leads to goodness-of-fit tests that are both correctly sized and relatively powerful against common alternatives. In an empirical application, we recover the REDF from stock market high-frequency data. We inspect the goodness-of-fit of several two-parameter marginal distributions that are inherent in standard stochastic volatility models. The inverse Gaussian offers the best overall description of random equity variation, but the fit is less than perfect. This suggests an extra parameter (as available in, e.g., the generalized inverse Gaussian) is required to model stochastic variance.

The realized empirical distribution function of stochastic variance with application to goodness-of-fit testing

TL;DR

This work addresses validating stochastic volatility models when spot volatility is unobserved by constructing a nonparametric, noise-robust realized EDF (REDF) of latent volatility from tick-by-tick data. It develops a pre-averaging-based estimator for local spot variance, debiases microstructure noise, and proves infill consistency and a joint infill-long-span functional CLT, yielding a feasible Gaussian limit with long-run variance . Using this, the authors define realized Kolmogorov–Smirnov and weighted goodness-of-fit statistics, which, with bootstrap, allow model validation for stochastic volatility specifications. The empirical application to SPY and sector ETFs shows inverse Gaussian volatility distributions typically fit better than gamma or log-normal models, suggesting potential benefits from more flexible families like the generalized inverse Gaussian. Overall, the paper provides a robust framework for assessing stochastic volatility models using high-frequency data and paves the way for refined distributional assumptions of volatility such as or .

Abstract

We propose a nonparametric estimator of the empirical distribution function (EDF) of the latent spot variance of the log-price of a financial asset. We show that over a fixed time span our realized EDF (or REDF) -- inferred from noisy high-frequency data -- is consistent as the mesh of the observation grid goes to zero. In a double-asymptotic framework, with time also increasing to infinity, the REDF converges to the cumulative distribution function of volatility, if it exists. We exploit these results to construct some new goodness-of-fit tests for stochastic volatility models. In a Monte Carlo study, the REDF is found to be accurate over the entire support of volatility. This leads to goodness-of-fit tests that are both correctly sized and relatively powerful against common alternatives. In an empirical application, we recover the REDF from stock market high-frequency data. We inspect the goodness-of-fit of several two-parameter marginal distributions that are inherent in standard stochastic volatility models. The inverse Gaussian offers the best overall description of random equity variation, but the fit is less than perfect. This suggests an extra parameter (as available in, e.g., the generalized inverse Gaussian) is required to model stochastic variance.
Paper Structure (13 sections, 13 theorems, 151 equations, 4 figures, 2 tables)

This paper contains 13 sections, 13 theorems, 151 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The setting
  3. A discrete and noisy high-frequency record of $X$
  4. The realized EDF
  5. Asymptotic properties of the REDF
  6. Infill setting
  7. Joint infill and long-span setting
  8. Goodness-of-fit of the volatility distribution
  9. Simulation study
  10. Empirical application
  11. Conclusion
  12. Appendix
  13. Simulation of critical values for goodness-of-fit testing

Key Result

Lemma 2.1

Suppose that Assumption assumption:noise holds and $r \in[0,2]$. For each $t \in [0,T]$, as $\Delta_{n} \rightarrow 0$,

Figures (4)

  • Figure 1: Estimation of the EDF.
  • Figure 2: Mean relative bias and absolute error.
  • Figure 3: Inference about marginal distribution, $T = 250$.
  • Figure 4: Properties of stochastic variance in SPY.

Theorems & Definitions (13)

  • Lemma 2.1
  • Theorem 3.1
  • Corollary 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Theorem 3.5
  • Corollary 3.6
  • Proposition 3.7
  • Corollary 3.8
  • Lemma A.1
  • ...and 3 more