Average egg price: Mean reversion and persistence in a time series approach

Aitor Sarasola-Cullen; Luis Alberiko Gil-Alana

doi:10.1371/journal.pone.0337054

Peer Review History

Original SubmissionMay 27, 2025
2 Jul 2025 Decision Letter - Muhammad Umer Arshad, Editor Dear Dr. Gil-Alana, Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process. Please submit your revised manuscript by Aug 16 2025 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org . When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file. A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'. A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'. An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'. If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter. If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: https://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols . Additionally, PLOS ONE offers an option for publishing peer-reviewed Lab Protocol articles, which describe protocols hosted on protocols.io. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols . We look forward to receiving your revised manuscript. Kind regards, Muhammad Umer Arshad Academic Editor PLOS ONE Journal requirements: When submitting your revision, we need you to address these additional requirements. 1. Please ensure that your manuscript meets PLOS ONE's style requirements, including those for file naming. The PLOS ONE style templates can be found at https://journals.plos.org/plosone/s/file?id=wjVg/PLOSOne_formatting_sample_main_body.pdf and https://journals.plos.org/plosone/s/file?id=ba62/PLOSOne_formatting_sample_title_authors_affiliations.pdf 2. Thank you for stating the following financial disclosure: [Prof. Luis A. Gil-Alana gratefully acknowledges financial support from the PID2023-149516NB-I00/ AEI/10.13039/501100011033/ FEDER, UE project from ‘Ministerio de Economía, Industria y Competitividad’ (MINEIC), ‘Agencia Estatal de Investigación’ (AEI) Spain and ‘Fondo Europeo de Desarrollo Regional’ (FEDER), and also from Internal Projects of the Universidad Francisco de Vitoria.]. Please state what role the funders took in the study. If the funders had no role, please state: "The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript." If this statement is not correct you must amend it as needed. Please include this amended Role of Funder statement in your cover letter; we will change the online submission form on your behalf. 3. In the online submission form, you indicated that [The data are available from the authors upon request.]. All PLOS journals now require all data underlying the findings described in their manuscript to be freely available to other researchers, either 1. In a public repository, 2. Within the manuscript itself, or 3. Uploaded as supplementary information. This policy applies to all data except where public deposition would breach compliance with the protocol approved by your research ethics board. If your data cannot be made publicly available for ethical or legal reasons (e.g., public availability would compromise patient privacy), please explain your reasons on resubmission and your exemption request will be escalated for approval. Additional Editor Comments (if provided): [Note: HTML markup is below. Please do not edit.] Reviewers' comments: Reviewer's Responses to Questions Comments to the Author 1. Is the manuscript technically sound, and do the data support the conclusions? Reviewer #1: Partly Reviewer #2: Yes ******** 2. Has the statistical analysis been performed appropriately and rigorously? -->?> Reviewer #1: No Reviewer #2: Yes ****** 3. Have the authors made all data underlying the findings in their manuscript fully available??> The PLOS Data policy Reviewer #1: Yes Reviewer #2: Yes ****** 4. Is the manuscript presented in an intelligible fashion and written in standard English??> Reviewer #1: No Reviewer #2: Yes ****** Reviewer #1: The manuscript presents an important and timely economic study. However, important technical and interpretation issues must be addressed before it can be recommended for publication. Revisions are needed to correct mathematical inaccuracies, strengthen methodological arguments, improve clarity, and align conclusions with the evidence. \textbf{Recommendation:} Major Revision. Additional review comments are provided in the attached annotated PDF. Reviewer #2: This paper offers a methodologically sound and policy-relevant contribution to price dynamics in the agricultural sector. With minor revisions to include more economic interpretation, visual data summaries, and connections to recent literature, it could serve as a valuable reference for both academic and applied economists. ****** what does this mean? ). If published, this will include your full peer review and any attached files. If you choose “no”, your identity will remain anonymous but your review may still be made public. Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy Reviewer #1: Yes: Kazeem Babatunde Akande Reviewer #2: No ******** [NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.] While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/ . PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org . Please note that Supporting Information files do not need this step. Attachments Attachment Submitted filename: PONE-D-25-28658 (reviewed).pdf Attachment Submitted filename: PLOSONE.pdf Attachment Submitted filename: Report new.docx https://doi.org/10.1371/journal.pone.0337054.r001
Revision 1
1 Sep 2025 Author Response See the atttached file. Attachments Attachment Submitted filename: LETTER ADDRESSING THE CHANGES.docx https://doi.org/10.1371/journal.pone.0337054.r002
20 Oct 2025 Decision Letter - Muhammad Umer Arshad, Editor Dear Dr. Gil-Alana, Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process. Please submit your revised manuscript by Dec 04 2025 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org . When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file. A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'. A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'. An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'. If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: https://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols . Additionally, PLOS ONE offers an option for publishing peer-reviewed Lab Protocol articles, which describe protocols hosted on protocols.io. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols . We look forward to receiving your revised manuscript. Kind regards, Muhammad Umer Arshad Academic Editor PLOS ONE Journal Requirements: If the reviewer comments include a recommendation to cite specific previously published works, please review and evaluate these publications to determine whether they are relevant and should be cited. There is no requirement to cite these works unless the editor has indicated otherwise. Please review your reference list to ensure that it is complete and correct. If you have cited papers that have been retracted, please include the rationale for doing so in the manuscript text, or remove these references and replace them with relevant current references. Any changes to the reference list should be mentioned in the rebuttal letter that accompanies your revised manuscript. If you need to cite a retracted article, indicate the article’s retracted status in the References list and also include a citation and full reference for the retraction notice. Additional Editor Comments: The title refers to “structural breaks,” but no formal test for structural breaks has been performed; therefore, I suggest adjusting the title accordingly. Create a clearer connection between the methodological equations and the research objectives, and include a brief justification for choosing the FCVAR model over other common approaches. Remove repetition, shorten long sentences, and correct minor typographical errors (for example, Larg (Cost per Doze) should be Large (Cost per Dozen)). Also, check the notation in Equation (5) and ensure it uses a standard equation font. [Note: HTML markup is below. Please do not edit.] Reviewers' comments: Reviewer's Responses to Questions Comments to the Author Reviewer #3: All comments have been addressed ******** 2. Is the manuscript technically sound, and do the data support the conclusions??> Reviewer #3: No ****** 3. Has the statistical analysis been performed appropriately and rigorously? -->?> Reviewer #3: No ****** 4. Have the authors made all data underlying the findings in their manuscript fully available??> The PLOS Data policy Reviewer #3: No ****** 5. Is the manuscript presented in an intelligible fashion and written in standard English??> Reviewer #3: Yes ****** Reviewer #3: 1. Significance The paper examines an important and policy-relevant issue — the persistence and mean-reverting behavior of average egg prices in the United States and their co-movement with the Producer Price Index (PPI). Food price stability is a crucial macroeconomic and welfare concern. The topic is both timely and valuable, especially given the high volatility of food prices during recent supply shocks. The application of fractional integration and FCVAR models is appropriate and potentially insightful. 2. Originality The study’s originality lies in applying fractional integration and FCVAR approaches to egg price dynamics - an area that has received limited econometric attention. However, the contribution would be stronger if the authors explicitly position their study against existing literature on food price persistence that uses ARIMA/SARIMA or volatility models. The paper should better articulate how its methodological approach adds value beyond conventional time-series methods. 3. Methodology Strengths • The use of fractional integration and FCVAR is methodologically consistent with the study’s focus on persistence and long-memory behavior. • Frequency-domain causality testing (Breitung and Candelon, 2006) is a sound preliminary step before fractional cointegration analysis. Major Technical Issues 1. Estimation Details Missing: The paper does not specify how the differencing parameter d was estimated (Whittle, MLE, semiparametric, etc.), nor the software used. Estimation procedures, confidence intervals, and replication details must be clearly stated. 2. Structural Breaks Not Tested: Although the title mentions Structural Breaks, no formal test (Bai-Perron, Zivot-Andrews, or breaks in d) is presented. This omission undermines the title and conclusions. The authors should either include such tests or revise the title and discussion accordingly. 3. Seasonality Handling: Monthly data are likely to contain seasonal components (e.g., holiday demand). While the paper mentions a seasonal AR(1) model, it does not clarify how seasonality was treated. The authors should discuss the choice of specification and its effect on the persistence parameter. 4. Residual Diagnostics: The results depend on three different noise specifications (white noise, Bloomfield, seasonal AR(1)), but no diagnostic tests (Ljung–Box, ARCH tests, residual plots) are reported. Residual diagnostics are required to justify the final model choice. 5. Inconsistent Interpretation of d: At some points, the paper claims “no evidence of short memory (I(0)),” yet elsewhere describes mean reversion (0.5 ≤ d < 1). The authors should unify their terminology and interpret d consistently throughout. 6. Data Availability and Transparency: Although the data are said to be public, there is no DOI or repository link. Please provide direct data sources (e.g., FRED codes) and, if possible, an open-access CSV and code appendix for reproducibility. 7. Robustness Checks: Consider re-estimating d using alternative semiparametric estimators (e.g., GPH, local Whittle) and provide bootstrap confidence intervals to ensure robustness. 4. Presentation and Structure Language: The manuscript is generally clear but would benefit from professional English editing to correct minor grammatical and typographical issues (e.g., inconsistent capitalization and minor spelling errors). Tables: Tables 2–5 should include the number of observations, AIC/BIC values, and explanatory notes for each column. Figures: Figures 2–3 need clearer axes labels, time scales, and detailed captions including data sources. 5. References The references are relevant but could be expanded to include recent applications of fractional cointegration and structural break tests (e.g., Bai & Perron, 2003; Robinson, 1995; Velasco, 1999). Updating these will enhance the paper’s scholarly depth. 6. Major and Minor Revisions Major Revisions Required 1. Include formal structural break tests (Bai-Perron or Zivot-Andrews) and discuss how breaks affect d and cointegration results. 2. Provide full estimation details — algorithms, software, sample size, convergence criteria, and replication code if possible. 3. Conduct robustness checks using alternative fractional estimators (e.g., GPH, local Whittle) and bootstrap confidence intervals. 4. Re-examine seasonality treatment and justify the chosen specification. 5. Add residual diagnostics and present key test statistics. 6. Provide data and code availability via a repository or appendix. Minor Revisions Improve figure and table clarity. Conduct minor language editing. Clarify the policy implications of findings (what does mean reversion imply for policymakers?). ****** what does this mean? ). If published, this will include your full peer review and any attached files. If you choose “no”, your identity will remain anonymous but your review may still be made public. Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy Reviewer #3: Yes: Hassan Tawakol Ahmed Fadol ******** [NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.] To ensure your figures meet our technical requirements, please review our figure guidelines: https://journals.plos.org/plosone/s/figures You may also use PLOS’s free figure tool, NAAS, to help you prepare publication quality figures: https://journals.plos.org/plosone/s/figures#loc-tools-for-figure-preparation. NAAS will assess whether your figures meet our technical requirements by comparing each figure against our figure specifications. Attachments Attachment Submitted filename: PLOS ONE Review Report Hassan Tawakol A. Fadol.docx https://doi.org/10.1371/journal.pone.0337054.r003
Revision 2
24 Oct 2025 Author Response Reviewers' comments: Reviewer's Responses to Questions Comments to the Author 1.) If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation. Reviewer #3: All comments have been addressed. ________________________________________ 2.) Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented. Reviewer #3: No. See comments below. ________________________________________ 3.) Has the statistical analysis been performed appropriately and rigorously? Reviewer #3: No. See comments below. ________________________________________ 4.) Have the authors made all data underlying the findings in their manuscript fully available? The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified. Reviewer #3: No. See comments below. ________________________________________ 5.) Is the manuscript presented in an intelligible fashion and written in standard English? PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here. Reviewer #3: Yes . ________________________________________ 6.) Review Comments to the Author Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters) Reviewer #3: 1. Significance The paper examines an important and policy-relevant issue — the persistence and mean-reverting behavior of average egg prices in the United States and their co-movement with the Producer Price Index (PPI). Food price stability is a crucial macroeconomic and welfare concern. The topic is both timely and valuable, especially given the high volatility of food prices during recent supply shocks. The application of fractional integration and FCVAR models is appropriate and potentially insightful. Thank you for this positive comment. 2. Originality The study’s originality lies in applying fractional integration and FCVAR approaches to egg price dynamics - an area that has received limited econometric attention. However, the contribution would be stronger if the authors explicitly position their study against existing literature on food price persistence that uses ARIMA/SARIMA or volatility models. The paper should better articulate how its methodological approach adds value beyond conventional time-series methods. Thanks for this relevant comment. We have added the following paragraph in the literature review section: “ … There are numerous studies that have focused the attention in the modelization of food prices throughout different techniques. Thus, for example, Mao et al. (2020) investigated China’s vegetable prices by using ARIMA models in order to determine the factors that influence vegetable price fluctuations. Jayatri (2025) used a more extended ARIMA-GARCH model to examine the volatility in various essential food prices in Indonesia. In a similar vein, Magnogna et al. (2025) employed a hybrid long short-term memory model for the analysis of agricultural price volatility in India. Other studies have performed empirical studies on price forecasting of individual agricultural prices such as soybean (Yin and Wang, 2021), pork (Wang and Sun, 2021; Ye et al., 2021), corn (Xu and Zhang, 2021) and beef (Zeng et al., 2019). Nevertheless, in the economic literature, the statistical properties of the average egg price have received limited attention, …”. References: Manogna, R.L., Dharmaji, V. & Sarang, S. (2025), A novel hybrid neural network-based volatility forecasting of agricultural commodity prices: empirical evidence from India. Journal of Big Data 12, 85. https://doi.org/10.1186/s40537-025-01131-8 Mao, L, Y. Huang, X. Zhang, S. LI and X. Huang (2020), ARIMA model forecasting analysis of the prices of multiple vegetables under the impact of the COVID-19, PLoS ONE 17(7): e0271594. https://doi.org/10.1371/journal.pone.0271594 Xu, X.J. and Y. Zhang (2021), Corn cash price forecasting with neural networks. Computers and Electronics in Agriculture,184, 106120 Ye K., Piao, Y.H.R., Zhao, K. and X.H. Cui (2021), A Heterogeneous Graph Enhanced LSTM Network for Hog Price Prediction Using Online Discussion. Agriculture,11(4) 359; https://doi.org/10.3390/agriculture11040359. Yin, T and Y.M. Wang (2021), Market Efficiency and Nonlinear Analysis of Soybean Futures. SustainabilitY13(2), 518, https://doi.org/10.3390/su13020518. Wang, C.S. and Z.H. Sun (2021),Monthly pork price forecasting method based on Census X12-GM(1,1) combination model.[J]. Plos one,16(5). https://doi.org/10.1371/journal.pone.0251436 Zeng, B., S.L. Li, W. Meng and D.H. Zhang (2019), An improved gray prediction model for China’s beef consumption forecasting. Plos one,14(9), pmid:31490952 3. Methodology Strengths • The use of fractional integration and FCVAR is methodologically consistent with the study’s focus on persistence and long-memory behavior. • Frequency-domain causality testing (Breitung and Candelon, 2006) is a sound preliminary step before fractional cointegration analysis. Thank you! Major Technical Issues 1. Estimation Details Missing: The paper does not specify how the differencing parameter d was estimated (Whittle, MLE, semiparametric, etc.), nor the software used. Estimation procedures, confidence intervals, and replication details must be clearly stated. Thanks for the comment. We have added the following in the methodological section: “ … The estimation relies on the parametric approach of Robinson (1994), which is a testing procedure based on the Lagrange Multiplier (LM) principle and that uses the likelihood function in the frequency domain. We use a very simple approach of this method that has been widely used in the empirical literature (see Gil-Alana and Robinson, 1997 for the specific functional form of the tests used in this application, whose codes can be found in Gil-Alana, 1998). The main features of this procedure are …”. Also, “ …“ … Data and codes used in this application are available in the following repository 10.5281/zenodo.17415399 at https://zenodo.org/uploads/17415399. …”. References Gil-Alana, L.A. (1998), Testing fractional integration in macroeconomic time series. PhD thesis, London School of Economics and Political Science, London, UK. Gil-Alana, L.A. and P.M. Robinson, (1997). Testing of unit root and other nonstationary hypotheses in macroeconomic time series. Journal of Econometrics, 80(2), 241-268. doi:10.1016/s0304-4076(97)00038-9 Robinson, P. M. (1994). Efficient Tests of Nonstationary Hypotheses. Journal of the American Statistical Association, 89(428), 1420–1437. https://doi.org/10.2307/2291004 2. Structural Breaks Not Tested: Although the title mentions Structural Breaks, no formal test (Bai-Perron, Zivot-Andrews, or breaks in d) is presented. This omission undermines the title and conclusions. The authors should either include such tests or revise the title and discussion accordingly. Thanks for letting us know this point. We have proceeded accordingly, removing the presence of “breaks” in the title, and extending a comment in the conclusions about potential lines of future research. The new title is now as follows: “Average Egg Price: Mean reversion and Persistence in a Time Series Approach”. We have also added the following comment in the conclusions: “ … The paper contains some limitations. First, the possibility of structural breaks is an issue that has not been investigated in the present work. This is relevant since long memory and fractional integration are closely related to the issue of breaks (see, Granger and Hyung, 2008; Barassi et al., 2018; etc.). In this context, the methodology proposed in Gil-Alana (2008), which is basically an extension of Bai and Perron’s (2003) approach to the fractional case can be implemented to the same data used in this application. …”. References Bai, J. and P. Perron (2003), Computation and analysis of multiple structural change models, Journal of Applied Econometrics 18, 1, 1-22. Barassi, M.R., Spagnolo, N. & Zhao, Y. (2018), Fractional Integration Versus Structural Change: Testing the Convergence of Emissions. Environ Resource Econ 71, 923–968. https://doi.org/10.1007/s10640-017-0190-z Gil-Alana, L.A. (2008), Fractional integration and structural breaks at unknown periods of time, Journal of Time Series Analysis 29, 1, 163-185. Granger, C.W.J. and N. Hyung (2004), Occasional structural breaks and long memory with an application to the S&P 500 absolute stock returns, Journal of Empirical Finance 11, 3, 399-421. 3. Seasonality Handing: Monthly data are likely to contain seasonal components (e.g., holiday demand). While the paper mentions a seasonal AR(1) model, it does not clarify how seasonality was treated. The authors should discuss the choice of specification and its effect on the persistence parameter. This is another relevant point that was not clearly explained in the old manuscript. We have added the following: “ … Note that seasonality does not appear as a relevant issue in any of the two series investigated since the estimates of d are practically identical under the seasonal AR specification and the white noise form for the error term. In fact, though unreported, the seasonal AR coefficients were statistically insignificant in the two cases. …”. 4. Residual Diagnostics: The results depend on three different noise specifications (white noise, Bloomfield, seasonal AR(1)), but no diagnostic tests (Ljung–Box, ARCH tests, residual plots) are reported. Residual diagnostics are required to justify the final model choice. Thanks for the comment. Note that this cannot be performed in the context of the model of Bloomfield (1973) since this is a non-parametric approach with no explicit functional form. However, performing tests of no autocorrelation in the other two specifications, the results supported the view of no additional time dependence. “ … It should finally be noted that several Ljung-Box Q-test statistics were conducted on the residuals from the white noise and seasonal AR(1) specifications and the results suggested that they were no additional serial correlation in the data. …”. References Bloomfield, P. (1973). An exponential model in the spectrum of a scalar time series, Biometrika 60, 217-226. 5. Inconsistent Interpretation of d: At some points, the paper claims “no evidence of short memory (I(0)),” yet elsewhere describes mean reversion (0.5 ≤ d < 1). The authors should unify their terminology and interpret d consistently throughout. Thanks for the comment. We refer to short memory or I(0) behaviour if the hypothesis of d = 0 is rejected in the data. This happens if that value does not belong to the 95% confidence intervals reported in the tables. Mean reversion takes place if d is smaller than 1. We have clarified these points all over the manuscript. In this context, we have changed two sentences in the Abstract, that reads now as follows: “This study investigates the long-term behavior of average egg prices in the U.S., using fractional integration techniques. The analysis also explores the relationship between egg prices and the Producer Price Index (PPI) through cointegration methods. The findings suggest long memory in both series and mean reversion for PPI. Additionally, the FCVAR model indicates a potential long-run equilibrium relationship between average egg prices and the PPI.”. Also, in the conclusions we have made this point much more explicit: “ … The primary discovery of this paper supports the notion that disturbances in average egg prices will have enduring effects in the U.S., as none of the series displays short memory traits (i.e., I(0)). Thus, long memory takes place in all cases with positive values of d across all confidence intervals. This suggests that abrupt changes or fluctuations in prices are likely to have a more prolonged impact. Moreover, the fact that the intervals include the value of d = 1 indicates that we cannot reject the hypothesis of a unit root in the series and therefore we may reject the hypothesis of reversion to the mean. On the other hand, regarding the Producer Price Index, despite the absence of short memory traits, the parameter d falls within the range 0.5 ≤ d < 1. This indicates that the time series is nonstationary with mean reversion, 6. Data Availability and Transparency: Although the data are said to be public, there is no DOI or repository link. Please provide direct data sources (e.g., FRED codes) and, if possible, an open-access CSV and code appendix for reproducibility. We have added the following comment: “ … We use a very simple approach of this method that has been widely used in the empirical literature (see Gil-Alana and Robinson, 1997 for the specific functional form of the tests used in this application, whose codes can be found in Gil-Alana, 1998). …”. Also, in the data section: “ … Data and codes used in this application are available in the following repository 10.5281/zenodo.17415399 at https://zenodo.org/uploads/17415399. …”. References Gil-Alana, L.A. (1998), Testing fractional integration in macroeconomic time series. PhD thesis, London School of Economics and Political Science, London, UK. Robinson, P. M. (1994). Efficient Tests of Nonstationary Hypotheses. Journal of the American Statistical Association, 89(428), 1420–1437. https://doi.org/10.2307/2291004 7. Robustness Checks: Consider re-estimating d using alternative semiparametric estimators (e.g., GPH, local Whittle) and provide bootstrap confidence intervals to ensure robustness. Thanks for this comment. We have included the following comment at the end of the Methodological section: “ … Nevertheless, as a robustness check, we also employed alternative approaches like the parametric method of Sowell (1992) and the semiparametric log-periodogram estimate of Geweke and Porter-Hudak (1983) and others like Robinson (1995) and Velasco (1999), all producing very similar results to those reported in this work. …”. References Geweke, J. & Porter-Hudak, S. (1983), The estimation and application of long memory time series. Journal of Time Series Analysis 4(4), 221-238. https://doi.org/10.1111/j.1467-9892.1983.tb00371.x Robinson, P. M. (1995), Gaussian semiparametric estimation of long range dependence, Annals of Statistics 23, 1630-1661. Sowell, F. (1992), Maximum likelihood estimation of stationary univariate fractionally integrated time series models, Journal of Econometrics 53(1-3), 165-188. https://doi.org/10.1016/0304-4076(92)90084-5 Velasco, C. (1999), Gaussian Semiparametric Estimation of Non-stationary Time Series, Journal of Time Series Analysis 20, 1, 87-127. 8. Presentation and Structure Language: The manuscript is generally clear but would benefit from professional English editing to correct minor grammatical and t Attachments Attachment Submitted filename: new letter addressing the changes.docx https://doi.org/10.1371/journal.pone.0337054.r004
29 Oct 2025 Decision Letter - Muhammad Umer Arshad, Editor Dear Dr. Gil-Alana, Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process. Please submit your revised manuscript by Dec 13 2025 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org . When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file. A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'. A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'. An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'. If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: https://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols . Additionally, PLOS ONE offers an option for publishing peer-reviewed Lab Protocol articles, which describe protocols hosted on protocols.io. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols . We look forward to receiving your revised manuscript. Kind regards, Muhammad Umer Arshad Academic Editor PLOS ONE Journal Requirements: If the reviewer comments include a recommendation to cite specific previously published works, please review and evaluate these publications to determine whether they are relevant and should be cited. There is no requirement to cite these works unless the editor has indicated otherwise. Please review your reference list to ensure that it is complete and correct. If you have cited papers that have been retracted, please include the rationale for doing so in the manuscript text, or remove these references and replace them with relevant current references. Any changes to the reference list should be mentioned in the rebuttal letter that accompanies your revised manuscript. If you need to cite a retracted article, indicate the article’s retracted status in the References list and also include a citation and full reference for the retraction notice. Additional Editor Comments: The abstract is too short. it doesn’t explain the importance of the study, the data used, or what the results mean. Please revise and expand to clearly say what question the paper answers, what data period was used, what methods were applied, and what the main findings mean in simple terms. Table 1 and 5 the headings for tables are too brief. they should be more descriptiv, please revise. In addition, the conclusion is too long and repeats earlier parts of the paper. it should be shorter and only focus on what was found, why it matters, and what could be done in future work. [Note: HTML markup is below. Please do not edit.] [NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.] To ensure your figures meet our technical requirements, please review our figure guidelines: https://journals.plos.org/plosone/s/figures You may also use PLOS’s free figure tool, NAAS, to help you prepare publication quality figures: https://journals.plos.org/plosone/s/figures#loc-tools-for-figure-preparation. NAAS will assess whether your figures meet our technical requirements by comparing each figure against our figure specifications. https://doi.org/10.1371/journal.pone.0337054.r005
Revision 3
30 Oct 2025 Author Response Additional Editor Comments: The abstract is too short. it doesn’t explain the importance of the study, the data used, or what the results mean. Please revise and expand to clearly say what question the paper answers, what data period was used, what methods were applied, and what the main findings mean in simple terms. Table 1 and 5 the headings for tables are too brief. they should be more descriptiv, please revise. In addition, the conclusion is too long and repeats earlier parts of the paper. it should be shorter and only focus on what was found, why it matters, and what could be done in future work. Reply: Dear Editor, Many thanks for these additional comments. Following your suggestions, we have first modified the Abstract that now reads as follows: “ This study investigates the long-term behavior of average egg prices and the Producer Price Index (PPI) in the U.S., using fractional integration and cointegration techniques. Using monthly data from 1980 to 2023, the results indicate evidence of long memory in both series and mean reversion for PPI. Thus, exogenous shocks in the PPI series will return by themselves to the original values. Additionally, there is a long-run equilibrium relationship between average egg prices and the PPI.”. The heading in Tables 1 and 5 have been extended. Thus, Table 1: Descriptive statistics of Average Egg Prices and PPI Egg Prices Table 5: Results of cointegration between Average Egg Prices and PPI using the Fractional Cointegration VAR (FCVAR) approach Finally, the conclusions section has been reduced, removing several paragraphs that were repetitions and no much informative for the purpose of the work. Thus, the new section is only formed by three paragraphs: one explaining the issue investigated, a second one displaying the main results, and a final one with the limitation of the present work: “The objective of this paper has been to analyze the stochastic behavior of the average egg price (cost per dozen) in the U.S. by looking at the statistical properties of the series and using fractional integration. Additionally, the study aimed to understand the possible long run relationship between PPI and the average egg price. The results indicate that for the average egg price, we observe lack of reversion to the mean, implying permanency of shocks and requiring policy actions (e.g., subsidy design or inventory stabilization) in case of shocks to recover the original trends. Conversely, for the Producer Price index data, mean reversion seems to take place in all cases, making it useless for policymakers to take action when sudden peaks occur. Furthermore, both the correlation coefficient and the causality test prove the existence of some a positive relationship between the two variables. Furthermore, the results obtained using the FCVAR model indicate that the average egg price and the producer price index are cointegrated with the cointegrating errors displaying an I(0) behaviour. These findings suggest the presence of a stable long-run equilibrium relationship between the Average Egg price and the Producer Price Index in the U.S., indicating a sustained co-movement between the two variables over time. The paper contains some limitations. First, the possibility of structural breaks is an issue that has not been investigated in the present work. This is relevant since long memory and fractional integration are closely related to the issue of breaks (see, Granger and Hyung, 2008; Barassi et al., 2018; etc.). In this context, the methodology proposed in Gil-Alana (2008), which is basically an extension of Bai and Perron’s (2003) approach to the fractional case can be implemented to the same data used in this application. In addition, non-linear trends can replace the linear structure employed in this work. Other alternative long memory model can also be used.”. Hoping this time the paper will be finally accepted in PlosOne, Yours sincerely, The authors https://doi.org/10.1371/journal.pone.0337054.r006
2 Nov 2025 Decision Letter - Muhammad Umer Arshad, Editor Dear Dr. Gil-Alana, Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process. Please submit your revised manuscript by Dec 17 2025 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org . When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file. A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'. A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'. An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'. If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: https://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols . Additionally, PLOS ONE offers an option for publishing peer-reviewed Lab Protocol articles, which describe protocols hosted on protocols.io. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols . We look forward to receiving your revised manuscript. Kind regards, Muhammad Umer Arshad Academic Editor PLOS ONE Journal Requirements: If the reviewer comments include a recommendation to cite specific previously published works, please review and evaluate these publications to determine whether they are relevant and should be cited. There is no requirement to cite these works unless the editor has indicated otherwise. Please review your reference list to ensure that it is complete and correct. If you have cited papers that have been retracted, please include the rationale for doing so in the manuscript text, or remove these references and replace them with relevant current references. Any changes to the reference list should be mentioned in the rebuttal letter that accompanies your revised manuscript. If you need to cite a retracted article, indicate the article’s retracted status in the References list and also include a citation and full reference for the retraction notice. Additional Editor Comments: The author has attempted to revise the abstract, but it still needs further improvement. I recommend reviewing abstracts from similar published papers to better understand how to clearly present the study’s purpose, key findings, and significance. [Note: HTML markup is below. Please do not edit.] [NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.] To ensure your figures meet our technical requirements, please review our figure guidelines: https://journals.plos.org/plosone/s/figures You may also use PLOS’s free figure tool, NAAS, to help you prepare publication quality figures: https://journals.plos.org/plosone/s/figures#loc-tools-for-figure-preparation. NAAS will assess whether your figures meet our technical requirements by comparing each figure against our figure specifications. https://doi.org/10.1371/journal.pone.0337054.r007
Revision 4
3 Nov 2025 Author Response Additional Editor Comments: The author has attempted to revise the abstract, but it still needs further improvement. I recommend reviewing abstracts from similar published papers to better understand how to clearly present the study’s purpose, key findings, and significance. Reply: Thanks for the comment. See the new Abstract: The evolution over time of the Consumer Price Index (CPI) is regarded as a key indicator of the general health and direction of any given economy. As the CPI continues to rise, the purchasing power of consumers decreases and their spending habits change significantly, making it imperative for policymakers to understand the underlying reasons that lead to such changes. A key component of the CPI basket is represented by the food and beverages items, within which eggs have undergone a significant price increase during the past years. Egg prices have a significant impact on consumers, given that eggs are a staple product, serving as the lowest cost protein alternative. This paper analyzes the long term behavior of the average egg price (cost per Dozen) in the U.S. by looking at the statistical properties of the series and using a methodology based on the concept of fractional integration. The primary goal is to determine whether the average egg price exhibits traits of long memory or mean reversion. Long memory describes the scenario where observations from a distant past have an influence on the present value of the series. Conversely, mean reversion refers to the phenomenon where data points eventually return to the long-term average after deviating from the mean for a certain period. The analysis also explores the relationship between egg prices and the Producer Price Index (PPI) through cointegration methods. Preliminary findings indicate that long memory takes place in both series and mean reversion in the PPI. Also, the two series seem to be cointegrated. This suggests the presence of a stable long-run equilibrium relationship between the Average Egg price and the Producer Price Index in the U.S., indicating a sustained co-movement between the two variables over time. https://doi.org/10.1371/journal.pone.0337054.r008
5 Nov 2025 Decision Letter - Muhammad Umer Arshad, Editor Average Egg Price: Mean reversion and Persistence in a Time Series Approach PONE-D-25-28658R4 Dear Dr. Gil-Alana, We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements. Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication. An invoice will be generated when your article is formally accepted. Please note, if your institution has a publishing partnership with PLOS and your article meets the relevant criteria, all or part of your publication costs will be covered. Please make sure your user information is up-to-date by logging into Editorial Manager at Editorial Manager® and clicking the ‘Update My Information' link at the top of the page. For questions related to billing, please contact billing support . If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they’ll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org. Kind regards, Muhammad Umer Arshad Academic Editor PLOS ONE Additional Editor Comments (optional): Reviewers' comments: https://doi.org/10.1371/journal.pone.0337054.r009
Formally Accepted
Acceptance Letter - Muhammad Umer Arshad, Editor PONE-D-25-28658R4 PLOS One Dear Dr. Gil-Alana, I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS One. Congratulations! Your manuscript is now being handed over to our production team. At this stage, our production department will prepare your paper for publication. This includes ensuring the following: * All references, tables, and figures are properly cited * All relevant supporting information is included in the manuscript submission, * There are no issues that prevent the paper from being properly typeset You will receive further instructions from the production team, including instructions on how to review your proof when it is ready. Please keep in mind that we are working through a large volume of accepted articles, so please give us a few days to review your paper and let you know the next and final steps. Lastly, if your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org. You will receive an invoice from PLOS for your publication fee after your manuscript has reached the completed accept phase. If you receive an email requesting payment before acceptance or for any other service, this may be a phishing scheme. Learn how to identify phishing emails and protect your accounts at https://explore.plos.org/phishing. If we can help with anything else, please email us at customercare@plos.org. Thank you for submitting your work to PLOS ONE and supporting open access. Kind regards, PLOS ONE Editorial Office Staff on behalf of Dr. Muhammad Umer Arshad Academic Editor PLOS One https://doi.org/10.1371/journal.pone.0337054.r010

Open letter on the publication of peer review reports

PLOS recognizes the benefits of transparency in the peer review process. Therefore, we enable the publication of all of the content of peer review and author responses alongside final, published articles. Reviewers remain anonymous, unless they choose to reveal their names.

We encourage other journals to join us in this initiative. We hope that our action inspires the community, including researchers, research funders, and research institutions, to recognize the benefits of published peer review reports for all parts of the research system.

Learn more at ASAPbio .