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ABSTRACT
This article investigates alternative generalized method of moments (GMM) estimation procedures of a stochastic volatility model with realized volatility measures. The extended model can accommodate a more general correlation structure. General closed form moment conditions are derived to examine the model properties and to evaluate the performance of various GMM estimation procedures under Monte Carlo environment, including standard GMM, principal component GMM, robust GMM and regularized GMM. An application to five company stocks and one stock index is also provided for an empirical demonstration.
Acknowledgements
We would like to thank an anonymous referee and the Associate Editor for their constructive comments and suggestions. We also wish to thank for the comments from the participants in the Econometrics Workshops at the University of Western Ontario, Carleton University and Brock University, the Econometric Society Australasian Meeting at University of Adelaide and the Canadian Econometrics Study Group (CESG) Conference at Toronto.
Notes
In our working paper version, we also allow for the correlation between ut and ηt for additional statistical flexibility. Please refer to our unpublished version for more generalized theoretical results. Thanks for one referee pointing out that this correlation may not be identified. We will leave this for future investigation. In this article, we simply restrict the correlation between (6) and (7) to be zero.
We use the convention that for b < a, where fj is the functional form indexed by j.
See Chaussé (Citation2011) for a Monte Carlo study based on nonlinear moment conditions.
We compare unbiased and biased starting values to verify the robustness of the estimation procedures. The results show that the convergence is stable.
There are no typical values for β1 and β2 (depending on the quality of the realized volatility measures) in practice. In our case, we intentionally set β1 and β2 to be 0.10 and 0.90 to create some bias and scale effects between the true volatility and RV proxy in the simulation. Theoretically, if the realized measure is a good approximation for the true volatility, β1 is normally close to 0, and β2 is expected to be close to 1. Furthermore, we have also experimented with many alternative sets of the parameters’ values, including most of the scenarios in the empirical section and some other cases. Those results are available upon request.
We have also experimented with some other larger sets of the moment conditions, such as extending the lags up to 10 and increasing the power to higher orders. We found that the results are very similar as those presented in this article.
We have done simulations by changing other parameter values. Furthermore, following one anonymous referee'ssuggestion, we have also investigated the model performance under a heavy-tail distribution, such as tri-variate student-t. To save space, we do not report those results in this paper. The summary is provided in Section 4.2. However, the detailed results are available upon request.
For the detailed numerical analysis on J-test and convergence, refer to the working paper version.
To save space, we only report the simulation results based on the full set of 36 moments. The results for other combinations of the moment conditions are available upon request.
The numerical results were presented in the working paper version. As a note, we have also tested the model performance under a multivariate student-\textitt distribution. In general, we find that both the bias and RMSE measures increase as the tail becomes heavier. For example, the true data-generating process (DGP) following a multivariate student-\textitt distribution with degrees of freedom of 6. Compared to the benchmark case (2a), the bias and RMSE of β2 are observed to increase about 40% and 12%. All these results are available upon request.
We only report what we consider to be representative results. All other tables are available upon request.
As a note, only a subset of representative moments are reported in Table .
We would like to thank one anonymous referee and the associate editor pointing out this measurement.
In this article, we adopt the open-to-close return definition to capture the market open activity (Hansen et al., Citation2012, see][). In addition, we have also used the close-to-close return in the empirical estimation. These results are available upon request.
As a note, we should be careful with these results when the J-test is rejected. As shown by Hall and Inoue (Citation2003, the asymptotic distribution of the GMM estimators under misspecified models can be very different from the one under correctly specified models. Therefore, the size of all tests from models in which the J-test is rejected are biased.
Noticing that the empirical estimates across PCGMM, RGMM, and RLGMM are similar, we only construct the empirical moments based on PCGMM for demonstration.
The realized LGARCH model is estimated by using the Quasi maximum likelihood (QML) method proposed in Hansen et al. (Citation2012).
The theoretical moments would be very different and complicated if one distribution is not Gaussian in the tri-variate structure. One possible solution would be to use Copula-based method to accommodate general dependence with specified marginals. We will leave this for future research. We would like to thank one anonymous referee to point out this extension as well.
To save space, the result on SPY is only reported in the article. Other empirical results are available upon request.
It is worth mentioning that, in our working paper version, we have followed the Associate Editor's suggestion to implement the EMM estimation procedure to the proposed model. Please refer to our working paper for the EMM estimation details.