This section analyzes the performance of the previously presented estimators under different scenarios. First, we introduce the Monte Carlo setup with different kind of DGPs and parameter settings for the estimation methods. Afterwards, we present the simulation results by focusing on the bias and root mean squared error (RMSE) of the estimators.
3.1. Monte Carlo Setup
We compare the performance of the estimation methods under nine different DGPs, which are given in
Table 1. The DGP types are obtained by adapting parts of the Monte Carlo structure provided by
Qu (
2011),
Frederiksen et al. (
2012) and
Hou and Perron (
2014). The process
is modeled by an ARFIMA (1,d,0) with
, where
is a zero mean white noise process if not otherwise defined. We standardize the ARFIMA process by its own standard deviation and consider the following values of the autoregressive and persistence coefficient with
0.0, 0.2, 0.4, 0.6 and
0.0, 0.3, 0.6. By choosing
, we violate against the stationarity assumption for a variety of estimators to check the robustness in the case of non-stationarity. The sample size is set to
256, 512, 1024, 4096 and the results are based on 1000 Monte Carlo replications.
In the following, we provide further information on the different types of DGPs given in
Table 1. Unreported simulation results of a linear trend as contamination show almost no difference to the performance of a sinus trend and are therefore omitted. The components
and
of the random level shift (RLS) processes are mutually independent and the shift probability equals
. For the deterministic level shift process, we consider five shifts at fixed user-chosen dates with
for
and
with
. DGPs 7 and 8 are especially designed for the perturbed fractional estimation methods. The ARMA(0,0) process is basically an additive white noise process with a large variance. Unreported simulation results of an ARMA(1,0) process with
are showing no clear pattern and recommendations are hard to formulate. Therefore, we omit the results and instead consider the ARMA(0,1) process. The last DGP combines the two groups of contaminations with a low frequency component, in the form of a stationary RLS, and an additive noise term as a perturbation contamination.
Following
Frederiksen et al. (
2012), we set the noise-to-signal ratio equal to five and by rearranging the noise-to-signal ratio the variance of the ARFIMA (1,d,0) process is obtained by
We conduct the simulation study for all three forms of the LPWN estimator of
Frederiksen et al. (
2012). The bias and RMSE results of the three different types are very similar, especially the LPWN(0,1) and LPWN(1,0) show no significant difference. For the Monte Carlo study and the empirical example, we present the results of the LPWN(1,1) since we are varying the autoregressive coefficient for every DGP and also analyze several noise structures as perturbation. In general, we follow the commonly used parameter settings in the literature to reflect the empirical findings in the financial volatility. For all estimation methods, we choose the following values of the bandwidth
with
. The additional parameter of
Smith (
2005) is set to
. To satisfy the trimming and bandwidth condition of
Iacone (
2010), we choose
and
. We consider the adaptive procedure of
McCloskey and Perron (
2013), since this extension exhibits a smaller variance. The first step of the adaptive method,
, applies the setting of the trimmed GPH estimator with
. For the next iterations,
, the trimming changes to
with
as a consistent estimator of
d. Two different convergence criteria are used, either the procedure stops as
or when
with the final value of
. We set
and
for the estimator of
McCloskey and Perron (
2013) and the GPH estimator is used as starting value. The numerical optimization of the Monte Carlo study is based on R using the L-BFGS-B algorithm developed by
Byrd et al. (
1995).
3.2. Monte Carlo Results
In the following, the DGPs are numbered consecutively according to the order in
Table 1. The bias and root mean squared error results of all estimation methods and for the corresponding DGPs are given in the
Supplementary Materials. Before presenting some detailed results, we are summarizing the major findings of the simulation study.
The two standard estimation methods are very robust compared to the modified methods in two cases. Considering the first DGP without contamination and the ARFIMA (1,d,0) plus GARCH (1,1), the log-periodogram and Gaussian estimator are both characterized by a smaller bias and RMSE especially for
. In the case of a larger bandwidth and
, the modified estimators of
Andrews and Sun (
2004) and
Hou and Perron (
2014) perform slightly better or equivalent in terms of the RMSE than the local Whittle estimator. The same results hold for the modified estimator of
Smith (
2005) compared to the GPH estimator.
For a sinus trend or deterministic level shift as contamination type, the modified estimator of
Hou and Perron (
2014) is outperforming the local Whittle estimator in terms of bias and also for the majority cases of RMSE. Additionally, for the two variations of the random level shift process,
performs well, especially for lower values of the memory parameter (d ≤ 0.2).
The two trimmed versions of the local Whittle and GPH estimator are both exhibiting a very similar pattern over the DGPs. Similar to the estimator of
Hou and Perron (
2014), the estimators of
Iacone (
2010) and
McCloskey and Perron (
2013) also have their strength in the presence of level shifts, irrespective of whether they occur randomly (non-)stationary or deterministic. The good performance in the aforementioned situations is restricted to smaller values of
d and to less short-run effects.
While the estimators of
Hou and Perron (
2014),
Iacone (
2010) and
McCloskey and Perron (
2013) often improve the standard estimators in the short-memory case or without short-run dynamics, the methods of
Smith (
2005),
Andrews and Sun (
2004),
Hurvich et al. (
2005) and
Frederiksen et al. (
2012) outperform their respective standard methods for larger values of the persistence parameter (d ≥ 0.4) as well as for a higher short-run parameter (
). This finding is not very surprising for the three estimation methods designed for perturbation, however the good performance of
in these situations is rather unexpected. In particular, the DGPs especially constructed for the perturbation setup are dominated by the perturbation estimators and by
Smith (
2005). Moreover, unreported simulation studies stress the robustness of the estimation method of
Smith (
2005) to short-run effects with values larger than
.
After a broad overview of the simulation results, we focus on three different contamination types in more detail: a sinus trend, a stationary random level shift and an additive noise term. Since the parameter intervals of the Monte Carlo study are very comprehensive, we concentrate on a sample size of for the following three tables and distinguish between two bandwidths. Further, we omit the persistence value of for the first two detailed DGPs and for the third contamination type we add the memory value but exclude the short-memory case.
The results of a sinus trend as contamination are presented in
Table 2. The best performance in terms of lowest bias and RMSE is obtained by the GPH and modified GPH estimator of
McCloskey and Perron (
2013). Further, the trimmed local Whittle estimator of
Iacone (
2010) exhibits a comparable small bias and RMSE for
. In the next step, we compare the modified methods to the respective standard estimators. The modified estimator of
Hou and Perron (
2014) outperforms the local Whittle estimator for all combinations in terms of bias and RMSE. Generally, the estimator improves the performance with a larger bandwidth, except for situations with a larger autoregressive parameter of the ARFIMA model. The performance of the estimator of
Iacone (
2010) is very similar with an even smaller bias and RMSE for
and for
combined with a smaller bandwidth. Although the trimmed GPH estimator of
McCloskey and Perron (
2013) performs well in terms of the bias for
, the inflated variance increases the RMSE compared to the GPH estimator in most cases. The estimator of
Smith (
2005) performs better with an increasing autoregressive parameter and for a larger bandwidth. However, the three estimation methods developed for perturbed fractional processes are not an alternative to the standard methods. For the trimmed local Whittle estimator, the reduction of the RMSE is in the range of 60–80% for
and remains within 30–75% for
. In contrast to the aforementioned estimator, the estimator of
Hou and Perron (
2014) has a smaller reduction of the RMSE for
with 20–65% due to the higher bias. For
, the reduction of the RMSE of
is very similar to the estimator of
Iacone (
2010) with 35–80% and remains within 20–75% for
, except for
. The estimator of
Smith (
2005) provides a smaller reduction of the RMSE compared to the estimator of
Hou and Perron (
2014) for the highest value of the autoregressive parameter with 5–50%.
Table 3 presents the results for a stationary random level shift as contamination. In general, the results of the non-stationary RLS in the
Supplementary Materials have a similar pattern but with a larger bias due to the non-stationarity and, thus, an increased RMSE. Three out of the four modified estimators developed especially for low frequency contaminations, such as a random level shifts, improve the standard methods, especially for smaller persistence values. The highest reduction in the RMSE for the estimation methods of
Hou and Perron (
2014),
Iacone (
2010) and
McCloskey and Perron (
2013) is obtained for
with 50–55%, 35–50% and 1–35%, respectively, and with less improvement for an increased short-run effect. The performance of
is rather unexpected and shows no improvement against the GPH estimator. The poor results of
Smith (
2005) are stressed by the fact that the estimator of
Andrews and Sun (
2004) outperforms
, even though the estimator of
Smith (
2005) has originally been developed for random level shifts, unlike
. The three estimation methods not explicitly developed for low frequency contaminations are offering no improvement. Nevertheless, compared to the sinus trend above the bias and RMSE decreased and the findings are getting closer to the other modified estimators, especially for a higher short-run dynamic.
After two typical examples of low frequency contaminations in the form of a time trend and level shift, we consider the effect of an additive noise term with the results given in
Table 4. Overall, the estimators of
Smith (
2005),
Hurvich et al. (
2005) and
Frederiksen et al. (
2012) are the three best performing estimation methods in terms of bias and RMSE. In the following, we compare the modified methods to the respective standard estimators. As expected, the three estimation methods developed for perturbed fractional processes are dominating the standard methods in terms of a lower bias and RMSE, although a RMSE reduction is restricted to larger memory values of
. Additionally, the estimator of
Smith (
2005) performs well compared to the GPH estimator. The reduction of the RMSE for the estimators of
Smith (
2005) and
Andrews and Sun (
2004) is decreasing over the
interval and ranging from 20–30% to 15–30% and finally to 10–30%. A similar but slightly larger reduction of the RMSE is obtained for the estimation methods of
Hurvich et al. (
2005) and
Frederiksen et al. (
2012) with 20–60%, 15–60% and still 5–60% for the same aforementioned region of
. The largest reduction is gained for
and
, irrespective of the previous estimation methods. In the presence of an additive noise term, the two standard methods are more robust than the four modified estimators developed for low frequency contaminations, except for
Smith (
2005).
The last detailed DGP in
Table 5 contains two groups of contamination: a stationary random level shift and an additive noise term. The overall outcome combines the individual results of each contamination type from
Table 3 and
Table 4. Excluding the short-memory process, the estimators of
Smith (
2005) and
Andrews and Sun (
2004) perform best in terms of bias and RMSE. For
the reduction of the RMSE to the respective standard method of
Smith (
2005) and
Andrews and Sun (
2004) equals on average
, irrespectively of the short-run influence. On the contrary, the estimators of
Iacone (
2010) and
Hou and Perron (
2014) provide the best results for a short-memory process being contaminated. With an increasing influence of the short-run dynamic, the reduction of the RMSE for
decreases on average from 55% to 50% and then to 30% for the estimators of
Iacone (
2010) and
Hou and Perron (
2014). Therefore, no procedure provides stable results over the whole range of the persistence parameter in the case of two contamination types.
In the following, we give a broad user-guideline by focusing on the RMSE. Usually, an economist is not aware of specific trends, level shifts or perturbation types when empirical data are analyzed. In the following, we compare the effects of two possible scenarios. The first situation is characterized by a perturbation process and instead of using the more appropriate estimators designed for this specific case one falsely applies estimation methods developed for low frequency contamination. The Monte Carlo results indicate a moderate increase of the RMSE for low frequency contaminated estimators, apart from . The next scenario summarizes the reverse case, which means that, for a low frequency contaminated process, the estimators developed to capture perturbation are used. Depending on the specific type of low frequency contamination, the Monte Carlo study shows mixed results. Considering a sinus trend or deterministic level shift the RMSE increases substantially for the perturbation estimators, in some cases three times as much as the estimators developed for low frequency contamination. The increase of the RMSE for a (non-)stationary RLS is lower than for the two aforementioned processes but still more than twice as high as the RMSE of low frequency contaminated estimators. The random level shift process plus an additive noise term shows slightly higher root mean squared errors. Considering the three different types of random level shifts an improvement of the RMSE is obtained for when the perturbation estimators are used instead of the methods designed for low frequency contamination. Therefore, we recommend the estimators developed for low frequency contamination, since the increase in a perturbation setup is not as severe as in the reverse case, especially in the stationary region of the memory parameter.
In summary, our results show a good performance of
in situations of a high short-run dynamic and in typical perturbation situations, however, in cases of low frequency contaminations, the estimator shows some weaknesses. The estimators of
Iacone (
2010) and
Hou and Perron (
2014) perform very similar in terms of bias and RMSE, whereas
is slightly better. In many situations, the estimator of
McCloskey and Perron (
2013) cannot compete with the other modified estimators developed for low frequency contamination in terms of RMSE due to the higher variance. Finally, the estimators of
Hurvich et al. (
2005) and
Frederiksen et al. (
2012) yield almost identical findings, which are in most situations better than the results of
Andrews and Sun (
2004).