A bootstrap approach for generalized Autocontour testing Implications for VIX forecast densities

Abstract

We propose an extension of the Generalized Autocontour tests for dynamic specification (evaluation) of in-sample (out-of-sample) conditional densities. The new tests are based on probability integral transforms computed from bootstrap conditional densities that incorporate parameter uncertainty without relying on parametric assumptions of the error distribution. Their finite sample distributions are well approximated using standard asymptotic distributions while they are easy to implement and provide information about potential sources of misspecification. We apply the new tests to the Heterogeneous Autoregressive and the Multiplicative Error models of the VIX index and find strong evidence against the parametric assumptions of the conditional densities.

Keywords:

• HAR model
• model evaluation
• Multiplicative Error Model
• PIT

JEL classification:

• C22
• C52
• C53
• C58

Acknowledgments

We are grateful to the participants at the New Developments in Econometrics and Time Series Workshop, Madrid, October 2016, and at the IMF/IIF Workshop on Forecasting Issues on Developing Economies, Washington DC, April 2017, and to seminar participants at the Management School of the University of Liverpool, for their very useful comments. We are also thankful to J. Mencía and E. Sentana for their help with the codes to estimate the MEM model.

Notes

1 Following the suggestion of González-Rivera and Sun (2015), the variance of ${\widehat{\alpha }}_{k,i}$ is approximated using a bootstrap procedure that takes into account parameter uncertainty. $B^{(2)}$ bootstrap replicates, ${\left\{y_t^{*(b)}\right\}}_{t=1}^T$ are generated as in (6(6) $$\begin{array}{c}{a}_t^{\ast (b)}={\epsilon }_t^{\ast (b)}{\sigma }_t^{\ast (b)},\\ y_t^{\ast (b)}={\widehat{\varphi }}_0+{\widehat{\varphi }}_1{y}_{t-1}^{\ast (b)}+a_t^{\ast (b)},\end{array}$$(6) ) and ${\widehat{\alpha }}_{k,i}^{*(b)}$ is obtained using the bootstrap series as if they were the original series.

2 Results for the AR(1) model with ${\varphi }_1=0.5$ and Gaussian errors are reported in Tables A and B of the supplementary material.

3 Note that this result is expected as inference in nonlinear GARCH models requires large samples.

4 Results on the power when the DGP is the AR(2) model in (11(11) $$y_t=0.3y_{t-1}+0.6y_{t-2}+{\epsilon }_t,$$(11) ) are reported in Tables C and D of the supplementary material. The proposed tests are very powerful even for small sample sizes.

5 The effect of the forecasting scheme is an interesting question to be developed in further research. In the fully parametric autocontour context, González-Rivera and Sun (2017) provide an analysis of the effects of the forecasting schemes (fixed, rolling, and recursive) on the size and power of autocontour-based tests.

6 Results based on the asymptotic expression of the variances and covariances are very similar when H = 50 and T = 1000 ($H/T=0.05$) or T = 5000 ($H/T=0.01$). When H = 500, the results are similar if T = 5000 ($H/T=0.1$). As mentioned above, in these cases, the parameter uncertainty is irrelevant. These results are available upon request.

7 Results for the t-tests are reported in Table E of the supplementary material. For small estimation sizes, the test tends to be oversized for the middle autocontours. When T is relatively large and H/T is small, the empirical size is about 5%.

8 Fernandes et al. (2014) include explanatory variables in Eq. (15). However, we stick to a univariate model to simplify the implementation of the proposed testing procedure.

9 Estimated parameters and residual diagnostics are reported in the supplementary material.

10 Note that these authors conclude that the error distribution is better represented by a normal inverse Gaussian (NIG) or a normal-mixture distributions.

11 We use the values of the parameters estimated in Mencia and Sentana (2018) as initial conditions for our estimation. Estimation results are reported in the supplementary material.

Additional information

Funding

Financial support from the Spanish Ministry of Education and Science, research project ECO2015-70331-C2-2-R (MINECO/FEDER) is acknowledged by the four authors. The first author and fourth authors also acknowledge financial support from Coordenação de Aperfeiçoamento de Pessoal de Nivel Superior (CAPES) grant 88882.305837/2018-01 and research projects PGC2018-096977-B-100 and FCT grant UID/GES/00315/2019, respectively. Gloria González-Rivera wishes to thank the Department of Statistics at UC3M for their hospitality and the financial support of the 2015 Chair of Excellence UC3M/Banco de Santander, and the UCR Academic Senate grant.