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Robustified Gaussian quasi-BIC for volatility

Shoichi Eguchi, Hiroki Masuda

Abstract

We develop a theoretical foundation for robust model comparison in a class of non-ergodic continuous volatility regression models contaminated by finite-activity jumps. Using the density-power weighting and the Hölder(-inequality)-based normalization of the conventional Gaussian quasi-likelihood function, we propose two Schwarz-type statistics and also establish their model selection consistency with respect to the minimal true parametric volatility coefficient. Numerical experiments are conducted to illustrate our theoretical findings.

Robustified Gaussian quasi-BIC for volatility

Abstract

We develop a theoretical foundation for robust model comparison in a class of non-ergodic continuous volatility regression models contaminated by finite-activity jumps. Using the density-power weighting and the Hölder(-inequality)-based normalization of the conventional Gaussian quasi-likelihood function, we propose two Schwarz-type statistics and also establish their model selection consistency with respect to the minimal true parametric volatility coefficient. Numerical experiments are conducted to illustrate our theoretical findings.

Paper Structure

This paper contains 17 sections, 5 theorems, 73 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notation and setup
  4. Robustfied Gaussian quasi-likelihood inference
  5. Gaussian quasi-BIC
  6. Gaussian quasi-Bayesian information criterion
  7. Asymptotic probability of relative model selection
  8. Numerical experiments
  9. Time-inhomogeneous Wiener process
  10. Jump-diffusion process
  11. Proofs
  12. Proof of Theorem \ref{['se:thm_denbic']}
  13. Proof of Theorem \ref{['se:thm_consis']}
  14. Proof of (i)
  15. Proof of (ii)
  16. ...and 2 more sections

Key Result

Theorem 3.2

Suppose that Assumptions hm:A_diff.coeff--hm:A_lam and se:prior hold. Then, we have $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: One of 1000 sample paths in Section \ref{['se:simu1']} (left: $n=100$, center: $n=500$, right: $n=1000$).
  • Figure 2: Boxplots of $\hat{\theta}_{2,n}(\lambda)-\theta_{2,0}$ and $\hat{\theta}_{2,n}^{\flat}(\lambda)-\theta_{2,0}$ for each $\lambda$ when the candidate coefficient is Diff 2 in Section \ref{['se:simu1']} ($q=0.01n$, $n=500$).
  • Figure 3: Boxplots of $\hat{\theta}_{2,n}(\lambda)-\theta_{2,0}$ and $\hat{\theta}_{2,n}^{\flat}(\lambda)-\theta_{2,0}$ for each $\lambda$ when the candidate coefficient is Diff 2 in Section \ref{['se:simu1']} ($q=0.1n$, $n=500$).
  • Figure 4: One of 1000 sample paths in Section \ref{['se:simu2']} (left: $n=100$, center: $n=500$, right: $n=1000$).

Theorems & Definitions (9)

  • Remark 2.1
  • Theorem 3.2
  • Theorem 3.4
  • Remark 3.5
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • Theorem A.6