Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Time Series Econometrics

Editor-in-Chief: Hidalgo, Javier

2 Issues per year

CiteScore 2017: 0.25

SCImago Journal Rank (SJR) 2017: 0.236
Source Normalized Impact per Paper (SNIP) 2017: 0.682

See all formats and pricing
More options …

Methods for Computing Numerical Standard Errors: Review and Application to Value-at-Risk Estimation

David Ardia
  • Corresponding author
  • Institute of Financial Analysis, University of Neuchâtel, Neuchâtel, Switzerland; Department of Finance, Insurance and Real Estate, Laval University, Québec City, Canada; University of Neuchâtel, Rue A.-L. Breguet 2, CH-2000 Neuchâtel, Switzerland
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Keven Bluteau
  • Vrije Universiteit Brussel, Solvay Business School, Brussel, Belgium
  • Institute of Financial Analysis, University of Neuchâtel, Neuchâtel, Switzerland
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Lennart F. Hoogerheide
Published Online: 2018-07-21 | DOI: https://doi.org/10.1515/jtse-2017-0011


Numerical standard error (NSE) is an estimate of the standard deviation of a simulation result if the simulation experiment were to be repeated many times. We review standard methods for computing NSE and perform a Monte Carlo experiments to compare their performance in the case of high/extreme autocorrelation. In particular, we propose an application to risk management where we assess the precision of the value-at-risk measure when the underlying risk model is estimated by simulation-based methods. Overall, heteroscedasticity and autocorrelation estimators with prewhitening perform best in the presence of large/extreme autocorrelation.

Keywords: bootstrap; GARCH; HAC kernel; numerical standard error (NSE); Monte Carlo; Markov chain Monte Carlo (MCMC); spectral density; value-at-risk; Welch


  • Akaike, H. 1974. “A New Look at the Statistical Model Identification.” IEEE Transactions on Automatic Control 19: 716–23. doi:.CrossrefGoogle Scholar

  • Andrews, D. W. K. 1991. “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation.” Econometrica 59: 817–58. doi:.CrossrefGoogle Scholar

  • Andrews, D. W. K., and J. C. Monahan. 1992. “An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator.” Econometrica 60: 953–66. doi:.CrossrefGoogle Scholar

  • Ardia, D. 2008. Financial Risk Management with Bayesian Estimation of GARCH Models. Berlin/Heidelberg: Springer. doi:.CrossrefGoogle Scholar

  • Ardia, D., and K. Bluteau. 2017. “nse: Computation of Numerical Standard Errors in R.” Journal of Open Source Software 10. doi:.Crossref

  • Ardia, D., K. Bluteau, K. Boudt, and D.-A. Trottier. Markov-Switching GARCH Models in R: The MSGARCH package. Working paper 2016a https://ssrn.com/abstract=2845809. Forthcoming in Journal of Statistical Software.

  • Ardia, D., K. Bluteau, K. Bout, B. Peterson, and D.-A. Trottier. 2016b. “MSGARCH: Markov Switching GARCH Models in R.” https://cran.r-project.org/package=MSGARCH.Google Scholar

  • Flegal, J. M., and G. L. Jones. 2010. “Batch Means and Spectral Variance Estimators in Markov Chain Monte Carlo.” Annals of Statistics 38: 1034–70. doi:.CrossrefWeb of ScienceGoogle Scholar

  • Geyer, C. J. 1992. “Practical Markov Chain Monte Carlo.” Statistical Science 7: 473–83. doi:.CrossrefGoogle Scholar

  • Haas, M., S. Mittnik, and M. Paollela. 2004. “A New Approach to Markov-Switching GARCH Models.” Journal of Financial Econometrics 2: 493–530. doi:.CrossrefGoogle Scholar

  • Heidelberger, P., and P. D. Welch. 1981. “A Spectral Method for Confidence Interval Generation and Run Length Control in Simulations.” Communications of the ACM 24: 233–45. doi:.CrossrefGoogle Scholar

  • Hirukawa, M. 2010. A Two-Stage Plug-in Bandwidth Selection and Its Implementation for Covariance Estimation.” Econometric Theory 26: 710–43. doi:.CrossrefWeb of ScienceGoogle Scholar

  • Hoogerheide, L., and H. K. van Dijk. 2010. “Bayesian Forecasting of Value at Risk and Expected Shortfall Using Adaptive Importance Sampling.” International Journal of Forecasting 26: 231–47. doi:.CrossrefWeb of ScienceGoogle Scholar

  • Hurvich, C. M. 1985. “Data-Driven Choice of a Spectrum Estimate: Extending the Applicability of Cross-Validation Methods.” Journal of the American Statistical Association 80: 933–40. doi:.CrossrefGoogle Scholar

  • Lahiri, S. N. 1999. “Theoretical Comparisons of Block Bootstrap Methods.” Annals of Statistics 27: 386–404. doi:.CrossrefGoogle Scholar

  • Newey, W. K., and K. D. West. 1987. “A Simple, Positive Semi-definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix.” Econometrica 55: 703–8. doi:.CrossrefGoogle Scholar

  • Newey, W. K., and K. D. West. 1994. “ Automatic Lag Selection in Covariance Matrix Estimation.” Review of Economic Studies 61: 631–53. doi:.CrossrefGoogle Scholar

  • Nordman, D. J. 2009. “A Note on the Stationary Bootstrap’s Variance.” Annals of Statistics 37: 359–70. doi:.CrossrefWeb of ScienceGoogle Scholar

  • Parzen, E. 1957. “On Consistent Estimates of the Spectrum of a Stationary Time Series.” The Annals of Mathematical Statistics 28: 329–48.CrossrefGoogle Scholar

  • Patton, A., D. N. Politis, and H. White. 2009. “Correction to ‘Automatic Block-length Selection for the Dependent Bootstrap’ by D. Politis and H. White.” Econometric Reviews 28: 372–75. doi:.CrossrefGoogle Scholar

  • Percival, D., and W. Constantine. 2006. “Exact Simulation of Gaussian Time Series from Nonparametric Spectral Estimates with Application to Bootstrapping.” Statistics and Computing 16: 25–35. doi:.CrossrefGoogle Scholar

  • Politis, D. N., and J. P. Romano. 1992. “A Circular Block-Resampling Procedure for Stationary Data.” In Exploring the Limits of Bootstrap, 263–70. John Wiley & Sons.

  • Politis, D. N., and J. P. Romano. 1994. “The Stationary Bootstrap.” Journal of the American Statistical Association 89: 1303–13. doi:.CrossrefWeb of ScienceGoogle Scholar

  • Politis, D. N., and H. White. 2004. “Automatic Block-Length Selection for the Dependent Bootstrap.” Econometric Reviews 23: 53–70. doi:.CrossrefWeb of ScienceGoogle Scholar

  • Press, H., and J. Tukey. 1956. “Power Spectral Methods of Analysis and Application in Airplane Dynamics.” In Vol. IV of AGARD Flight Test Manual, 1–41. Paris, Ch. C.

  • R Core Team. 2015. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. http://www.R-project.org/

  • Vihola, M. 2012. “Robust Adaptive Metropolis Algorithm with Coerced Acceptance Rate.” Statistics and Computing 22: 997–1008. doi:.CrossrefWeb of ScienceGoogle Scholar

  • Welch, P. D. 1967. “The Use of fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short, Modified Periodograms.” IEEE Transactions on Audio and Electroacoustics 15: 70–73. doi:.CrossrefGoogle Scholar

About the article

Published Online: 2018-07-21

Citation Information: Journal of Time Series Econometrics, Volume 10, Issue 2, 20170011, ISSN (Online) 1941-1928, DOI: https://doi.org/10.1515/jtse-2017-0011.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in