4.1. Estimation Results
To estimate alternative RSV models using the QML method based on the Kalman filter, we use a daily return and realized volatility measure for the S&P 500 index. For the realized volatility measure, we selected the RK estimator in
Barndorff-Nielsen et al. (
2008) as it is robust to microstructure noise and jumps, as explained above. As the realized volatility is calculated using intraday data, the open-to-close return is used for
, as in
Hansen et al. (
2012). The data are obtained from the Oxford Man Institute of Quantitative Finance, and the sample period is from 22 December 2005 and to 4 December 2017, giving 3000 observations. The first
observations are used for estimating parameters, and the remaining
are reserved for forecasting. The descriptive statistics for the whole sample are presented in
Table 3. The standardized variable was calculated using
, as
is the log of RK. The return and RK have heavy tails, whereas the kurtoses of
and the standardized variable are close to three. Compared with the return, the standardized variable is close to the Gaussian distribution.
This section compares six models: SV, RSV, RSV-A, RSV
t, RSV
t-A, and two-factor RSV
t-A (2fRSV
t-A). Among these, the last model is defined by Equations (
1) and (3), with
As discussed in the previous section, the model comparison is based on the QLR test.
The QML estimates for the six models are reported in
Table 4, indicating that all parameters are significant at the five percent level. The estimates of (
) for the SV model are typical values in empirical analysis. For the RSV model, the estimates of (
) are similar to those of the SV model. As discussed in
Section 3.2, the contribution of the
equation is negligible for estimating (
) in the RSV model. In other words, the finite sample bias of the QML estimator for (
) in the SV model is corrected in the RSV model owing to the contribution of the realized volatility equation to construct the RSV model. As the estimate of
decreases, the value of
increases, keeping the variance of
,
at a similar level. The estimate of
is negative and significant, which may be caused by finite sample bias in the RK estimates. Note that it is inappropriate to compare the quasi-log-likelihood of the SV and RSV models, since the former excludes the information of
. The estimates for the RSV-A model are close to those of the RSV model. The estimate of
is negative and significant, implying the existence of the leverage effect. The QLR test rejects the null hypothesis
. For the RSV
t model, the estimate of
is 11.2. In contrast, the QLR test failed to reject the null hypothesis of the Gaussian distribution,
. As implied by the Monte Carlo results in
Table 2, there is an inefficiency on estimating
, which may derive an ambiguous result for the inference on
. Note that the descriptive statistics for the standardized variable in
Table 3 support the results of the QLR tests. The QLR tests in
Table 4 indicate that the RSV
t-A is preferred to the RSV and RSV
t model. For the 2fRSV
t-A model, the estimates of
and
in the first factor are larger than the corresponding values in the second factor. The QLR test rejects the null hypothesis of the one factor model; the tests selected the 2fRSV
t-A model among the six models.
4.2. Forecasting Performance
We compare out-of-sample forecasts of the SV and the five RSV models. For these six models, the Kalman filter prediction for the state-space form in (
5) and (
6) provides the one-step-ahead forecast,
, and
is a forecast of the quadratic variation for day
. By updating the parameter estimates, we calculate the forecasts
using the rolling window of the model for recent
observations. An alternative forecast can be considered as follows. Under the Gaussianity of
, the distribution of
conditional on the past observations is
, where
and
are defined in
Appendix A. Then, the conditional distribution of
follows the log-normal distribution, which gives the conditional mean
. Define
where
. Then,
can be a forecast as an alternative to
.
For comparison, we obtain forecasts based on the RSV, RSV-A, and RSV
t models using the 2SML method, as explained in
Koopman and Scharth (
2013) For the RSV-A model, we first compute the Kalman filter prediction of
and subsequently add the leverage effect
by calculating
. Since the RSV and RSV
t have no leverage effect, they are free from the latter part.
For comparing the out-of-sample forecasts,
Patton (
2011) suggested an approach using imperfect volatility proxies.
Patton (
2011) examined the functional form of the loss function for comparing volatility forecasts, such that the forecasts are robust to the presence of noise in the proxies. According to the definition in
Patton (
2011), a loss function is “robust” if the ranking of any two forecasts of the co-volatility matrix,
and
, by expected loss is the same whether the ranking is performed using the true covariance matrix or an unbiased volatility proxy,
.
Patton (
2011) demonstrated that squared forecast error and quasi-likelihood type loss functions, defined by
are robust to the forecast error
and the standardized forecast error
, respectively.
For these loss functions in (
10) and (11), the MCS procedure in
Hansen et al. (
2011) enables us to determine the set of models,
, that consist of the best model(s) from a collection of models,
. Define
and
as the difference in the loss functions of two competitive models and its sample mean, respectively. Under the null hypothesis
,
Hansen et al. (
2011) considered two kinds of test statistics:
where
is a bootstrap estimate of the variance of
, and the
p-values of the test statistics are determined using a bootstrap approach. If the null hypothesis is rejected at a given confidence level, the worst performing model is excluded (rejection is determined on the basis of bootstrap
p-values under the null hypothesis). Such a model is identified as follows:
where the variance is computed using a bootstrap method.
The means of the MSFE and QLIKE, defined by (
10) and (11), for the six models for the two volatility forecasts, that is,
and
, based on the QML estimation are presented in
Table 5, which reports three models for the 2SML method. Generally, the Kalman filtering prediction,
, has a smaller MSFE, while adjusted forecast,
, has the smaller QLIKE. MSFE selects the 2fRSV-
t (QML) based on the Kalman filtering prediction, while QLIKE chooses the simple RSV (QML) using the adjusted value. The
p-values of the MCS for the best model based on the
statistic are presented in brackets in
Table 5. We omitted the results for
as these are similar. The differences in model performance are explained using the
p-values. The forecast made by the SV model is significantly different from those of alternative RSV models for both loss functions. Among the RSV models, the differences may be negligible for the datasets. In general, the QML method produces better forecasts than the 2SML estimations, but the differences are statistically insignificant.
The out-of-sample forecast performance indicates that the data prefer the RSV models to the SV model. For the datasets, there are no statistical differences among the two forecasts of the six RSV models. The data contain the period of the global financial crisis caused by the collapse of the Lehman Brothers, starting from 15 September 2008. The statistical indifference among the RSV models may be caused by the effects of turbulence in the data.