Round 1

Reviewer 1 Report

Dear Colleagues, in my opinion, the main problem of this work is that the topic of estimation of INAR time series is already very well developed and I cannot see any theoretical contribution to the topic in this work.

The authors have to improve the history of the problem by inserting a better description of previous results. 

Author Response

Authors’ Responses to Reviewer 1

 

• In my opinion, the main problem of this work is that the topic of estimation of INAR time series is already very well developed, and I cannot see any theoretical contribution to the topic in this work. The authors have to improve the history of the problem by inserting a better description of previous results.

Authors’ response: Thank you very much for your comment. First of all, we must point out that the subject of our work is not the estimation of parameters (only) for the INAR processes. As we pointed out in the Abstract, our motivation was to describe the PGF estimation procedure "for the very general, stationary class of the NNIV time series..." After that, we considered the construction of PGF estimates for IID and INAR time series, as two important classes of stationary NNIV series. As we pointed out in the Conclusion, "the application of the PGF estimators presented here can be a starting point for some future research related to these estimates..." In any case, in the Introduction part of this revised version, we have described in more detail the previously known parameter estimation procedures that, above all, refer to INAR processes. We have also supplemented the list of references and highlighted in more detail some reasons why we believe the PGF-based estimation method deserves attention. (Please, see the parts of the revised manuscript in red.)

Author Response File: Author Response.pdf

Reviewer 2 Report

Actually, the paper is interesting, but i have some comments to make the paper better.

1- Make a single paragraph about the main contribution and the novelty of the paper.

2- Why did the authors choose those distributions in the comparison in Table 3?

3-The Abstract should be extended to cover all aspects of the paper.

After all, I recommend publication after making these comments

 

Author Response

Responses to Reviewer 2

 

• Actually, the paper is interesting, but I have some comments to make the paper better.
• Make a single paragraph about the main contribution and the novelty of the paper.

Authors’ response: Thank you very much for your comment. In this revised version, we have added some of our thoughts on the contributions, novelties, and potential benefits of PGF estimates. (Please, see the second paragraph of the Introduction and the sentences marked in red.)

• Why did the authors choose those distributions in the comparison in Table 3?

Authors’ response: Thank you very much for your comment. Poisson and geometric distributions are taken as an illustration of the application of PGF methods, primarily in describing the dynamics of real data in Sect. 5. In this version, the reason for considering these two distributions is further emphasized. (Please, see the Lines 135-137 marked in red.) We also considered another actual series, named Series A, which relates to the distribution of deaths before the onset of the COVID-19 disease in Serbia. For this time series is shown to have very weak autocorrelation and the equal-dispersion property. Therefore, it was modeled by the Poisson distribution, while the INAR(1) process with geometrically distributed innovations was used for the second one. (Please, see the parts of Sect. 5 marked in red.)

• The Abstract should be extended to cover all aspects of the paper.

Authors’ response: Thank you very much for your comment. The Abstract has now been expanded and hopefully shows all aspects of our work in more detail. (Please, see the parts marked in red.)

After all, I recommend publication after making these comments.

The authors greatly appreciate and thank Reviewer for his/her positive assessment of our work, valuable comments, and useful suggestions.

Author Response File: Author Response.pdf

Reviewer 3 Report

Please see attached file.

Comments for author File: Comments.pdf

Author Response

Authors’ Responses to Reviewer 3

• The paper focuses on a very relevant topic, it investigates clearly interesting aspects of modelling nonnegative integer-valued time series, coherently with the most recent literature. One of the most interesting things is the way the investigation is tackled. The overall assessment of the paper is therefore positive.
• I believe the paper has a very good potential, can be of much interest to the readers of Axioms and, more in general, to the audience with expertise in time series analysis of any kind.

Authors’ response: We sincerely appreciate and thank the Reviewer for his/her comments and constructive suggestions for improving our manuscript. In this revised version, all suggestions have been considered and all questions have been answered point by point. (Please see the parts of the manuscript highlighted in red.)

There are some limitations and points to be improved. In particular, the paper can improve a) in the way results are synthetized and communicated; b) in describing the potential and the possible expansion of the range of applications; c) the statistical robustness and empirical analysis.

1. For a), findings are not clear. The abstract is somehow too generic, specifically in the last part. For a), the authors should try to synthetize in a better way their main findings, discoveries, methodological and empirical contributions. This can be improved in the abstract, introduction and in the conclusions of a paper which is, on the contrary, nice to read, to follow and understandable in its main body. They should think about some sentences (in the case of the abstract) and 1 or 2 paragraphs (in the case of the introduction and conclusion sections) which can better represent the value of the paper. This is true in particular for the method’s potential to develop new empirical contributions.

Authors’ response: Thank you very much for these comments. In this revised version, we have rearranged and added some new sentences in the Abstract, Introduction, and Conclusion sections. All of them have now been expanded, so we hope they are better and, in more detail, reflect the aspects of our work that you pointed out. (Please see the parts of the manuscript marked in red.)

2. For b), the power of the method, after having been clarified, should be highlighted, as it is a prominent method which can pave the way in the analysis of different types of data. The paper should mention other relevant studies in the related literature for which the suggested methodology might convey insights... The expansion of the discussion on other types of data (e.g. natural disaster, financial data) towards which their method can improve the understanding, so they can better contextualize the contributions of the authors with respect to the extant literature in other time series fields.

Authors’ response: Thank you very much for your comments. In this revised version, we have added another set of data related to the distribution of deaths before the onset of the COVID-19 disease in Serbia. It was shown that this time series has a very weak autocorrelation, so it can be an example of the IID series, which we also analyzed from a theoretical point of view in this manuscript. Our idea was also to compare the stochastic structure of two data sets, denoted Series A and Series B, respectively, which refer to the time before and after the emergence of the SARS-CoV2 virus. In this way, we once again applied and checked the quality of the PGF estimation method. (Please see the parts of Sect. 4 highlighted in red.)

3. For c), I think the authors should complement the empirical analysis with two interrelated things. On the one hand, I suggest the authors to perform an analysis of the predictive performance of the model in an out-of-sample exercise, so to confirm the predictive ability of the model, even moving from an in-sample perspective. On the other hand, I also suggest the authors to compare their proposed method with some existing competing alternative (some that come to my mind: neg bin / Poisson autoregression, in-garch, self-exciting processes; but you could select others) to show the predictive performance of the model compared to some relevant extant alternatives.

Authors’ response: Thank you very much for your comments. In this revised version, the forecast accuracy was checked by the Diebold-Mariano test. The time period immediately before the onset of the COVID-19 disease (for series A), i.e. the period from the beginning of December to January 9 of this year (for series B), was taken as time horizons. On the other hand, we believe that a comparison of the PGF estimator with the models proposed by the Reviewer would be possible in some future research, as we pointed out in the Conclusion. (Please, see the parts of  Sect. 4 and Sect. 5 highlighted in red.)

Round 2

Reviewer 1 Report

Dear Colleagues, now you have enough references.

Reviewer 3 Report

The authors have addressed all comments satisfactorily.