Evaluation of realized volatility predictions from models with leptokurtically and asymmetrically distributed forecast errors

Abstract

Accurate volatility forecasting is a key determinant for portfolio management, risk management and economic policy. The paper provides evidence that the sum of squared standardized forecast errors is a reliable measure for model evaluation when the predicted variable is the intra-day realized volatility. The forecasting evaluation is valid for standardized forecast errors with leptokurtic distribution as well as with leptokurtic and asymmetric distributions. Additionally, the widely applied forecasting evaluation function, the predicted mean-squared error, fails to select the adequate model in the case of models with residuals that are leptokurtically and asymmetrically distributed. Hence, the realized volatility forecasting evaluation should be based on the standardized forecast errors instead of their unstandardized version.

Keywords:

• integrated volatility
• intra-day
• predicted mean-squared error
• realized volatility
• standardized prediction error criterion
• simulating forecast errors
• ultra-high frequency
• volatility forecasting evaluation

JEL Classifications:

• G17
• G15
• C15
• C32
• C53

Acknowledgments

Authors acknowledge the support from the European Community's Seventh Framework Programme (FP7-PEOPLE-2010-RG) funded under grant agreement no. PERG08-GA-2010-276904. Also, we would like to thank the editor Robert Aykroyd and the anonymous referees for their constructive suggestions. The views expressed are those of the authors and should not be interpreted as those of their respective institutions. The authors are solely responsible for any remaining errors and deficiencies.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. The most known evaluation functions for volatility forecasts are the heteroskedasticity adjusted absolute error Andersen et al. [2] and the logarithmic error Pagan and Schwert [23].

2. The most widely applied information criterion is the Schwarz's [25] Bayesian criterion.

3. For example, Granger and Pesaran [17] linked forecast evaluation with the decisions made based on the predictions. Engle et al. [15] and Xekalaki and Degiannakis [26] developed an evaluation function that measures the profitability of trading options.

4. The values of the parameters are based on the estimation of the model for the realized volatility of the CAC40 index; see Degiannakis and Floros [12].

5. The infinite expansions of the fractional differencing operator, for $d>0$, are defined as $(1-L{)}^{-d}={\sum }_{j=0}^{\mathrm{\infty }}\left(\right(\mathrm{\Gamma }(j+d)/\mathrm{\Gamma }\left(d\right)\mathrm{\Gamma }(j+1)\left)L^j\right)=1+(1/1!)\mathrm{d}L+(1/2!)d(1+d)L^2-\dots$, see Xekalaki and Degiannakis [27, p. 113] and Baillie [3, p. 18].

6. The values of the parameters are based on the estimation of the model for the realized volatility of the CAC40 index.

7. We assume that the rolling-sample estimated parameters of the ARFIMA-GARCH model do not change across time. In example, for each point $t$ in time and ${\theta }^{\left(t\right)}\equiv (d^{\left(t\right)},d_1^{\left(t\right)},{\beta }_0^{\left(t\right)},a_0^{\left(t\right)},a_1^{\left(t\right)},b_1^{\left(t\right)},v^{\left(t\right)}{)}^{\mathrm{\prime }}$ we assume that ${\mathit{\theta }}^{\left(t\right)}={\mathit{\theta }}^{\left(t^{\mathrm{\prime }}\right)}$, for $\mathrm{\forall }t\ne t^{\mathrm{\prime }}$.

8. Descriptive statistics of the simulated variables $\{z_t{\}}_{t=1}^{10,000}$, $\{{\epsilon }_t{\}}_{t=1}^{10,000}$, $\{h_t^2{\}}_{t=1}^{10,000}$, $\{\log ({\mathrm{RV}}_t^{\left(\tau \right)}){\}}_{t=1}^{10,000}$, $\{{\mathrm{RV}}_t^{\left(\tau \right)}{\}}_{t=1}^{10,000}$ and ${\left\{\sqrt{252{\mathrm{RV}}_t^{\left(\tau \right)}}\right\}}_{t=1}^{10,000}$ from the Student t distribution, are available upon request.

9. Plots and frequency distributions of the one-step-ahead simulated forecasts $\{z_{t+1|t}{\}}_{t=1}^{10,000}$, $\{{\epsilon }_{t+1|t}{\}}_{t=1}^{10,000}$, ${\left\{\sqrt{252{\mathrm{RV}}_t^{\left(\tau \right)}}\right\}}_{t=1}^{10,000}$, as well as, descriptive statistics of the simulated draws $\{z_t{\}}_{t=1}^{10,000}$, $\{{\epsilon }_t{\}}_{t=1}^{10,000}$, $\{h_t^2{\}}_{t=1}^{10,000}$ from the GED distribution, are available upon request.

10. Descriptive statistics of the simulated variables from the skewed Student t distribution, are available upon request.

11. If we denote the realized volatility forecasts produced by models A and B as ${\mathrm{RV}}_{t+1\left|t\right(A)}^{\left(\tau \right)}$ and ${\mathrm{RV}}_{t+1\left|t\right(B)}^{\left(\tau \right)}$, respectively, then the forecasts are comparable (in terms of forecasting ability) through testing the null hypothesis that the models produce statistically equivalent predictions against the alternative hypothesis that model A produces more accurate predictions than model B (see also Xekalaki and Degiannakis [27]). The statistic ${\sum }_{t=1}^{\tilde{T}}{z}_{t+1\left|t\right(B)}^2/\phantom{{\sum }_{t=1}^{\tilde{T}}{z}_{t+1\left|t\right(B)}^2{\sum }_{t=1}^{\tilde{T}}{z}_{t+1\left|t\right(A)}^2}{\sum }_{t=1}^{\tilde{T}}{z}_{t+1\left|t\right(A)}^2$ has known distributional form, the correlated gamma ratio distribution.

12. Figure that plots the time-varying estimates of the vector of parameters, ${\mathit{\theta }}^{\left(t\right)}\equiv (d^{\left(t\right)},d_1^{\left(t\right)},{\beta }_0^{\left(t\right)},a_0^{\left(t\right)},a_1^{\left(t\right)},b_1^{\left(t\right)},v^{\left(t\right)}{)}^{\mathrm{\prime }}$, of the ARFIMA(0,d,1)-GARCH(1,1) model for GED distributed innovations is available upon request.

13. The relative figures for normally, Student t and GED distributed standardized innovations are available upon request.

14. The Mann and Whitney [21] proposed the U statistic for testing the null hypothesis that two random variables with continuous cumulative distribution functions f and g have stochastically equal distributions against the alternative hypothesis that one distribution is stochastically smaller than the other.