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A semiparametric nonlinear quantile regression model for financial returns

  • Krenar Avdulaj EMAIL logo and Jozef Barunik
Published/Copyright: June 3, 2016
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Studies in Nonlinear Dynamics & Econometrics
From the journal Studies in Nonlinear Dynamics & Econometrics Volume 21 Issue 1

Abstract

Accurately measuring and forecasting value-at-risk (VaR) remains a challenging task at the heart of financial economic theory. Recently, quantile regression models have been used successfully to capture the conditional quantiles of returns and to forecast VaR accurately. In this paper, we further explore nonlinearities in data and propose to couple realized measures with the nonlinear quantile regression framework to explain and forecast the conditional quantiles of financial returns. The nonlinear quantile regression models are implied by the copula specifications and allow us to capture possible nonlinearities, tail dependence, and asymmetries in the conditional quantiles of financial returns. Using high frequency data that covers most liquid US stocks in seven sectors, we provide ample evidence of asymmetric conditional dependence with different levels of dependence, which are characteristic for each industry. The backtesting results of estimated VaR favour our approach.

Keywords: copula quantile regression; realized volatility; value-at-risk
JEL Classification: C14; C32; C58; F37; G32

Acknowledgments

We are grateful to the editor, Fredj Jawadi, the two anonymous referees, and the participants at the 2nd International Workshop on “Financial Markets and Nonlinear Dynamics” for many useful comments and suggestions. Support from the Czech Science Foundation under the P402/12/G097 DYME Dynamic Models in Economics project is gratefully acknowledged. Avdulaj gratefully acknowledges financial support from the Grant Agency of the Charles University (GA UK) under the project 162815.

Appendix A Proofs

Probability distribution of rt+1 conditional on ϑt

τ(rt+1|ϑt)=∂C(ut, vt+1)∂ut

Proof. from

ℙ(ℛ<rt+1, V=ϑt)=∂FV,ℛ(ϑt, rt+1)∂ϑt

it follows that

ℙ(R<rt+1, V=ϑt)=limϵ→0ℙ(ℛ≤rt+1|ϑt≤V≤ϑt+ϵ)=limϵ→0F(ϑt+ϵ, rt+1)−F(ϑt, rt+1)FV(ϑt+ϵ)−FV(ϑt)≈(∂F(ϑt, rt+1)/∂ϑt)⋅ϵfV(ϑt)⋅ϵ=1fV(ϑt)∂C(FV(ϑt), Fℛ(rt+1))∂ϑt=1fV(ϑt)⎵*∂C(ut, vt+1)∂ut∂FV(ϑt)∂ϑt⎵*=∂C(ut, vt+1)/∂ut

where ut=F𝒱 (ϑt), νt+1=Fℛ (rt+1) and * terms cancel out.□

Following the same path it is easy to show that

τ(ϑt|rt+1)=∂C(ut, vt+1)∂vt+1

Appendix B Tables and Figures

Table 4:

Descriptive statistics for daily returns and realized volatility over the sample period extending from August 2004 to December 2011.

Returns
Information technologyConsumer discretionaryConsumer staplesTelecommunication services
AAPLINTCMSFTAMZNDISMCDKOPGWMTCMCSATVZ
Mean–0.0003–0.0001–0.00010.00150.00090.00040.00010.0006–0.00010.0002–0.0001–0.0004
Std dev0.02010.01640.01400.02240.01560.01210.01060.00990.01080.01870.01320.0127
Skewness–0.30970.06410.14830.30040.46820.29670.0474–0.05800.44040.62740.58770.5984
Kurtosis3.29143.34025.81214.41356.97696.06838.12746.65946.597918.23229.59648.3887
Minimum–0.1223–0.0907–0.0755–0.1313–0.0909–0.0799–0.0717–0.0660–0.0653–0.1416–0.0629–0.0760
Maximum0.11230.08800.11020.13880.11850.10350.07950.07760.07620.23250.12420.1118
FinancialsEnergyHealth care
BACCWFCCVXSLBXOMJNJMRKPFE
Mean–0.0023–0.0042–0.00020.0001–0.00020.00050.00010.0000–0.0006
Std dev0.03270.03410.02720.01540.02150.01470.00920.01520.0133
Skewness–0.4071–1.78890.24580.0847–0.4012–0.01080.0305–0.17100.1302
Kurtosis13.628720.858813.052611.96755.427810.39409.65996.89053.2493
Minimum–0.2509–0.3468–0.2081–0.1296–0.1552–0.1261–0.0803–0.1092–0.0696
Maximum0.20140.19920.19330.14600.12530.11890.07280.09190.0714
Realized volatility
Information technologyConsumer discretionaryConsumer staplesTelecommunication services
AAPLINTCMSFTAMZNDISMCDKOPGWMTCMCSATVZ
Mean0.00040.00030.00020.00060.00030.00020.00010.00010.00020.00040.00020.0002
Std dev0.00080.00050.00040.00090.00050.00040.00030.00040.00040.00070.00050.0005
Skewness11.976410.77007.69197.67079.877525.379910.489526.074520.718912.790412.054115.4769
Kurtosis209.4541191.673090.607180.7537151.3225868.6006175.7496895.9773625.6571243.4969242.6791382.1494
Minimum0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
Maximum0.01920.01210.00700.01440.01170.01570.00630.01430.01260.01690.01420.0148
FinancialsEnergyHealth care
BACCWFCCVXSLBXOMJNJMRKPFE
Mean0.00090.00110.00070.00030.00050.00020.00010.00030.0002
Std dev0.00260.00400.00170.00070.00090.00070.00030.00070.0004
Skewness7.960710.84885.796017.12728.325318.396118.560413.50838.8700
Kurtosis95.3976166.798545.1057435.1365111.8150490.6232486.2773263.8237125.7655
Minimum0.00000.00000.00000.00000.00000.00000.00000.00000.0000
Maximum0.04890.08660.02310.02070.01780.02050.00900.01650.0079

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Supplemental Material:

The online version of this article (DOI: 10.1515/snde-2016-0044) offers supplementary material, available to authorized users.

Published Online: 2016-6-3
Published in Print: 2017-2-1

©2017 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. Introduction: recent developments of switching models for financial data
  4. Research Articles
  5. On the estimation of regime-switching Lévy models
  6. RALS-LM unit root test with trend breaks and non-normal errors: application to the Prebisch-Singer hypothesis
  7. Modeling threshold effects in stock price co-movements: a vector nonlinear cointegration approach
  8. Specification analysis in regime-switching continuous-time diffusion models for market volatility
  9. A semiparametric nonlinear quantile regression model for financial returns
  10. A model of the euro-area yield curve with discrete policy rates
https://doi.org/10.1515/snde-2016-0044
Keywords for this article
copula quantile regression; realized volatility; value-at-risk

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. Introduction: recent developments of switching models for financial data
  4. Research Articles
  5. On the estimation of regime-switching Lévy models
  6. RALS-LM unit root test with trend breaks and non-normal errors: application to the Prebisch-Singer hypothesis
  7. Modeling threshold effects in stock price co-movements: a vector nonlinear cointegration approach
  8. Specification analysis in regime-switching continuous-time diffusion models for market volatility
  9. A semiparametric nonlinear quantile regression model for financial returns
  10. A model of the euro-area yield curve with discrete policy rates
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