Model Selection and Post Selection to Improve the Estimation of the ARCH Model

Round 1

Reviewer 1 Report

The authors suggest improved post-selection estimating strategies in this study and compare them to a benchmark estimator. They test the proposed methods on the daily closing prices of the S&P500 stock market index.

The paper is mostly a technical study that proposes and tests econometric models based on the ARCH model. The paper is interesting because of the proposed methodology, particularly in terms of deriving the asymptotic properties of the estimators and comparing their performances using risk analysis and mean square error, as well as the fact that they conducted extensive simulation studies for the chosen model and demonstrated the application of the proposed estimators in real-world problems.

However, before it can be published, the article must be improved. The article is difficult to understand, on one side because the English expression is weak, and on the other hand because the work is highly technical.

Furthermore, the study's implications must be explained in the conclusion section; who will benefit from them? What is the research's added value?

The final section (Conclusion) might be expanded to include a more in-depth discussion of issues such as comparisons to previous study results, study limitations, future research directions, and practical implications, among others.

Author Response

Reviewer #1 comments

We do appreciate and thank the reviewer for the valuable comments

The authors suggest improved post-selection estimating strategies in this study and compare them to a benchmark estimator. They test the proposed methods on the daily closing prices of the S&P500 stock market index.

The paper is mostly a technical study that proposes, and tests econometric models based on the ARCH model. The paper is interesting because of the proposed methodology, particularly in terms of deriving the asymptotic properties of the estimators and comparing their performances using risk analysis and mean square error, as well as the fact that they conducted extensive simulation studies for the chosen model and demonstrated the application of the proposed estimators in real-world problems.

However, before it can be published, the article must be improved. The article is difficult to understand, on one side because the English expression is weak, and on the other hand because the work is highly technical.

Response: Thanks for the reviewer comment:

We updated our manuscript and added anew section about the literature  review to make it much easier for the reader to understand the new procedure of estimating the parameters of the ARCH model as a new research and contribution to this model.

Furthermore, the study's implications must be explained in the conclusion section; who will benefit from them? What is the research's added value?

Response: We do thank  the reviewer for this comment:

We updated the conclusion section (it is in red) in the new version of the manuscript, we pointed out to the usefulness of using the recommended estimator of the parameter vector of the ARCH model, and the importance of having or obtaining some previous knowledge as using the AIC, BIC   selection criterion. The value gained in this case is the reduction of the Mean Square error of the proposed estimator. 

The final section (Conclusion) might be expanded to include a more in-depth discussion of issues such as comparisons to previous study results, study limitations, future research directions, and practical implications, among others.

Response: We  thank the reviewer for this  comments as well.

Due to complexity of the ARCH model, we compared the proposed estimation strategy with respect to the classical OLS method. To the best of our knowledge, no previous estimation strategies exist in the literature. That is why we limited our comparison of the proposed estimators to the classical OLS estimator.  

Reviewer 2 Report

This paper proposes a post selection estimation strategy. It investigates and develop some asymptotic properties of suggested strategies and compare with benchmark estimator. Further, it conducts Monte Carlo simulation study to reappraise the relative characteristics of the listed estimators.

The topic per se is interesting and the authors have done a reasonable job of conducting the work.

The authors must provide an extra section with literature review

What are the limitations and disadvantages of the proposed estomators?

Comments for author File: Comments.docx

Author Response

Reviewer #2 comments

We do appreciate and thank the reviewer for the valuable comments

This paper proposes a post selection estimation strategy. It investigates and develop some asymptotic properties of suggested strategies and compare with benchmark estimator. Further, it conducts Monte Carlo simulation study to reappraise the relative characteristics of the listed estimators.

The topic per se is interesting and the authors have done a reasonable job of conducting the work.

The authors must provide an extra section with literature review

Response: Thanks for the reviewer comment:

An extra section- section 2.-  of literature review has been added to the new version of our manuscript.

What are the limitations and disadvantages of the proposed estimators?

Response: Thanks for the reviewer comment:

One limitation is the dimension of the parameter vector  that we are interested to estimate, it supposed to be 3 more than the rank of the matrix given by R in our manuscript. Also, the values of the response variable   must be positive to be defined. Disadvantages is bias of all estimators, all estimators are biased except the classical one, and the restricted estimator if the restriction is correct. One may balance between the bias and having small mean squared error (MSE)  of the estimator. if we are interested in having an unbiased estimator, then OLS will our choice. Otherwise, the positive shrinkage estimator is recommended which has a smaller MSE, and asymptotically is unbiased.

Reviewer 3 Report

1. This study proposes an improved selection estimation method and compares it with benchmark estimators. The relative characteristics of the listed estimators were then reassessed through a Monte Carlo simulation method.

2. The derivation of the asymptotic characteristics of the ARCH model is quite rigorous, and the subsequent simulation methods and results are reasonable.

3. Although the ARCH model can be used to deal with the volatility in financial time series data that the ARIMA model cannot handle, the applicability of this improved ARCH estimation method needs to be reconsidered if there are asymmetric phenomena.

4. In addition, if the numerical analysis can be matched with relevant empirical research, the model suitability test and comparison can be carried out, which can improve the robustness of the new estimation method.

Author Response

Reviewer #3 comments

We do appreciate and thank the reviewer for the valuable comments

Comments and Suggestions for Authors

1. This study proposes an improved selection estimation method and compares it with benchmark estimators. The relative characteristics of the listed estimators were then reassessed through a Monte Carlo simulation method.

Response: Thanks for the reviewer comment:

Our goal was to prove the proposed estimation analytically and numerically, so we included both techniques and apply it to a real data example.

2. The derivation of the asymptotic characteristics of the ARCH model is quite rigorous, and the subsequent simulation methods and results are reasonable.

Response: Thanks for the reviewer comment:

Yes, we agree with the reviewer comment, we spent some time to prepare the required theorems that helped us to prove our expected superior performance of the proposed estimators.

3. Although the ARCH model can be used to deal with the volatility in financial time series data that the ARIMA model cannot handle, the applicability of this improved ARCH estimation method needs to be reconsidered if there are asymmetric phenomena.

Response: Thanks for the reviewer comment:

There are many studies about  the dilemma of having a financial time series with a high volatility, for example some of these studies results revealed that both micro-economic and macro-economic conditions had a direct impact on the Stock Market volatility. We pointed out in the literature review about this issue.

4. In addition, if the numerical analysis can be matched with relevant empirical research, the model suitability test and comparison can be carried out, which can improve the robustness of the new estimation method.

Response: Thanks for the reviewer comment:

The pre-test and shrinkage estimation strategies have been studied a lot in the literature via different regression models, as linear, partial linear, conditional autoregressive, special error models, and others. Numerical results of the proposed estimators behaved in a similar manner as in ARCH model, this issue appeared in the introduction and the new section added to the new version of the manuscript. Our goal was to show the superiority of the proposed shrinkage estimator when some prior information are available about the parameter vector, which can be either historical non sample information, or using the sample itself by applying some selection techniques as for example the AIC, BIC. Which had been shown in our results via the graphs or analytical results

Round 2

Reviewer 1 Report

We thank the authors for taking into account our indications and modifying their article accordingly. The article has therefore been improved and is suitable for publication in its current form. 

Author Response

Dear Respected Reviewer,

We do than you for your efforts and time you spent in reading different papers. please note that the suggested corrections might belong to another research  paper. 

thanks

Authors 

 

Author Response File: Author Response.pdf

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.