IDEAS home Printed from https://ideas.repec.org/a/eee/csdana/v173y2022ics0167947322000809.html
   My bibliography  Save this article

Marginal M-quantile regression for multivariate dependent data

Author

Listed:
  • Merlo, Luca
  • Petrella, Lea
  • Salvati, Nicola
  • Tzavidis, Nikos

Abstract

An M-quantile regression model is developed for the analysis of multiple dependent outcomes by introducing the notion of directional M-quantiles for multivariate responses. In order to incorporate the correlation structure of the data into the estimation framework, a robust marginal M-quantile model is proposed extending the well-known generalized estimating equations approach to the case of regression M-quantiles with Huber's loss function. The estimation of the model and the asymptotic properties of estimators are discussed. In addition, the idea of M-quantile contours is introduced to describe the dependence between the response variables and to investigate the effect of covariates on the location, spread and shape of the distribution of the responses. To examine their variability, confidence envelopes via nonparametric bootstrap are constructed. The validity of the proposed methodology is explored both by means of simulation studies and through an application to educational data.

Suggested Citation

  • Merlo, Luca & Petrella, Lea & Salvati, Nicola & Tzavidis, Nikos, 2022. "Marginal M-quantile regression for multivariate dependent data," Computational Statistics & Data Analysis, Elsevier, vol. 173(C).
    • Handle: RePEc:eee:csdana:v:173:y:2022:i:c:s0167947322000809
    DOI: 10.1016/j.csda.2022.107500
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167947322000809
    Download Restriction: Full text for ScienceDirect subscribers only.
    File URL: https://libkey.io/10.1016/j.csda.2022.107500?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Torres Díaz, Raúl Andrés & Lillo Rodríguez, Rosa Elvira & Laniado Rodas, Henry, 2015. "A Directional Multivariate Value at Risk," DES - Working Papers. Statistics and Econometrics. WS ws1501, Universidad Carlos III de Madrid. Departamento de Estadística.
    2. Paindaveine, Davy & Siman, Miroslav, 2011. "On directional multiple-output quantile regression," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 193-212, February.
    3. Annamaria Bianchi & Enrico Fabrizi & Nicola Salvati & Nikos Tzavidis, 2018. "Estimation and Testing in M‐quantile Regression with Applications to Small Area Estimation," International Statistical Review, International Statistical Institute, vol. 86(3), pages 541-570, December.
    4. Cascos, Ignacio & Ochoa, Maicol, 2021. "Expectile depth: Theory and computation for bivariate datasets," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    5. Giorgia Giovannetti & Marco Sanfilippo & Margherita Velucchi, 2018. "Diverse twins: analysing China’s impact on Italian and German exports using a multilevel quantile regressions approach," Applied Economics, Taylor & Francis Journals, vol. 50(28), pages 3051-3065, June.
    6. Marco Alfò & Maria Francesca Marino & Maria Giovanna Ranalli & Nicola Salvati & Nikos Tzavidis, 2021. "M‐quantile regression for multivariate longitudinal data with an application to the Millennium Cohort Study," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 70(1), pages 122-146, January.
    7. Wei Pan, 2001. "Akaike's Information Criterion in Generalized Estimating Equations," Biometrics, The International Biometric Society, vol. 57(1), pages 120-125, March.
    8. ChunJing Li & Yun Li & Xue Ding & XiaoGang Dong, 2020. "DGQR estimation for interval censored quantile regression with varying-coefficient models," PLOS ONE, Public Library of Science, vol. 15(11), pages 1-17, November.
    9. Serfling, Robert, 2002. "Generalized Quantile Processes Based on Multivariate Depth Functions, with Applications in Nonparametric Multivariate Analysis," Journal of Multivariate Analysis, Elsevier, vol. 83(1), pages 232-247, October.
    10. Nikos Tzavidis & Nicola Salvati & Monica Pratesi & Ray Chambers, 2008. "M-quantile models with application to poverty mapping," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 17(3), pages 393-411, July.
    11. Luca Merlo & Lea Petrella & Valentina Raponi, 2021. "Forecasting VaR and ES using a joint quantile regression and implications in portfolio allocation," Papers 2106.06518, arXiv.org.
    12. You-Gan Wang, 2003. "Working correlation structure misspecification, estimation and covariate design: Implications for generalised estimating equations performance," Biometrika, Biometrika Trust, vol. 90(1), pages 29-41, March.
    13. Xiaoming Lu & Zhaozhi Fan, 2015. "Weighted quantile regression for longitudinal data," Computational Statistics, Springer, vol. 30(2), pages 569-592, June.
    14. Marc Hallin & Davy Paindaveine & Miroslav Siman, 2008. "Multivariate quantiles and multiple-output regression quantiles: from L1 optimization to halfspace depth," Working Papers ECARES 2008_042, ULB -- Universite Libre de Bruxelles.
    15. Koenker,Roger, 2005. "Quantile Regression," Cambridge Books, Cambridge University Press, number 9780521845731.
    16. Molchanov,Ilya & Molinari,Francesca, 2018. "Random Sets in Econometrics," Cambridge Books, Cambridge University Press, number 9781107121201.
    17. Ray Chambers & Nikos Tzavidis, 2006. "M-quantile models for small area estimation," Biometrika, Biometrika Trust, vol. 93(2), pages 255-268, June.
    18. Ra'ul Torres & Rosa E. Lillo & Henry Laniado, 2015. "A Directional Multivariate Value at Risk," Papers 1502.00908, arXiv.org.
    19. Ray Chambers & Hukum Chandra & Nicola Salvati & Nikos Tzavidis, 2014. "Outlier robust small area estimation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 76(1), pages 47-69, January.
    20. Luca Merlo & Lea Petrella & Nikos Tzavidis, 2022. "Quantile mixed hidden Markov models for multivariate longitudinal data: An application to children's Strengths and Difficulties Questionnaire scores," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 71(2), pages 417-448, March.
    21. You-Gan Wang & Vincent J. Carey, 2004. "Unbiased Estimating Equations From Working Correlation Models for Irregularly Timed Repeated Measures," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 845-853, January.
    22. Reich, Brian J. & Fuentes, Montserrat & Dunson, David B., 2011. "Bayesian Spatial Quantile Regression," Journal of the American Statistical Association, American Statistical Association, vol. 106(493), pages 6-20.
    23. Yen-Chi Chen & Christopher R. Genovese & Larry Wasserman, 2017. "Density Level Sets: Asymptotics, Inference, and Visualization," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(520), pages 1684-1696, October.
    24. Torres, Raúl & Lillo, Rosa E. & Laniado, Henry, 2015. "A directional multivariate value at risk," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 111-123.
    25. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
    26. White, Halbert, 1980. "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity," Econometrica, Econometric Society, vol. 48(4), pages 817-838, May.
    27. Lin, Fangzheng & Tang, Yanlin & Zhu, Zhongyi, 2020. "Weighted quantile regression in varying-coefficient model with longitudinal data," Computational Statistics & Data Analysis, Elsevier, vol. 145(C).
    28. Nikos Tzavidis & Nicola Salvati & Timo Schmid & Eirini Flouri & Emily Midouhas, 2016. "Longitudinal analysis of the strengths and difficulties questionnaire scores of the Millennium Cohort Study children in England using M-quantile random-effects regression," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 179(2), pages 427-452, February.
    29. Weihua Zhao & Weiping Zhang & Heng Lian, 2020. "Marginal quantile regression for varying coefficient models with longitudinal data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(1), pages 213-234, February.
    30. Chi-Chuan Yang & Yi-Hau Chen & Hsing-Yi Chang, 2017. "Joint regression analysis of marginal quantile and quantile association: application to longitudinal body mass index in adolescents," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 66(5), pages 1075-1090, November.
    31. Fu, Liya & Wang, You-Gan, 2012. "Quantile regression for longitudinal data with a working correlation model," Computational Statistics & Data Analysis, Elsevier, vol. 56(8), pages 2526-2538.
    32. Cho, Hyunkeun, 2016. "The analysis of multivariate longitudinal data using multivariate marginal models," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 481-491.
    33. Maria Marino & Alessio Farcomeni, 2015. "Linear quantile regression models for longitudinal experiments: an overview," METRON, Springer;Sapienza Università di Roma, vol. 73(2), pages 229-247, August.
    34. Lindsey, J. K., 1999. "Models for Repeated Measurements," OUP Catalogue, Oxford University Press, edition 2, number 9780198505594.
    35. Jones, M. C., 1994. "Expectiles and M-quantiles are quantiles," Statistics & Probability Letters, Elsevier, vol. 20(2), pages 149-153, May.
    36. N. Salvati & E. Fabrizi & M. G. Ranalli & R. L. Chambers, 2021. "Small area estimation with linked data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(1), pages 78-107, February.
    37. Annamaria Bianchi & Nicola Salvati, 2015. "Asymptotic Properties and Variance Estimators of the M-quantile Regression Coefficients Estimators," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 44(11), pages 2416-2429, June.
    38. Merlo, Luca & Petrella, Lea & Raponi, Valentina, 2021. "Forecasting VaR and ES using a joint quantile regression and its implications in portfolio allocation," Journal of Banking & Finance, Elsevier, vol. 133(C).
    39. Raúl Torres & Carlo De Michele & Henry Laniado & Rosa E. Lillo, 2017. "Directional multivariate extremes in environmental phenomena," Environmetrics, John Wiley & Sons, Ltd., vol. 28(2), March.
    40. Fraiman, Ricardo & Pateiro-López, Beatriz, 2012. "Quantiles for finite and infinite dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 108(C), pages 1-14.
    41. Liya Fu & Yangyang Hao & You-Gan Wang, 2018. "Working correlation structure selection in generalized estimating equations," Computational Statistics, Springer, vol. 33(2), pages 983-996, June.
    42. Kokic, Philip, et al, 1997. "A Measure of Production Performance," Journal of Business & Economic Statistics, American Statistical Association, vol. 15(4), pages 445-451, October.
    43. Newey, Whitney K & Powell, James L, 1987. "Asymmetric Least Squares Estimation and Testing," Econometrica, Econometric Society, vol. 55(4), pages 819-847, July.
    44. Luke B. Smith & Brian J. Reich & Amy H. Herring & Peter H. Langlois & Montserrat Fuentes, 2015. "Multilevel quantile function modeling with application to birth outcomes," Biometrics, The International Biometric Society, vol. 71(2), pages 508-519, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Ochoa Arellano, Maicol Jesús & Cascos Fernández, Ignacio, 2022. "Data depth and multiple output regression, the distorted M-quantiles approach," DES - Working Papers. Statistics and Econometrics. WS 35465, Universidad Carlos III de Madrid. Departamento de Estadística.
    2. Valeria Bignozzi & Luca Merlo & Lea Petrella, 2022. "Inter-order relations between moments of a Student $t$ distribution, with an application to $L_p$-quantiles," Papers 2209.12855, arXiv.org.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Otto-Sobotka, Fabian & Salvati, Nicola & Ranalli, Maria Giovanna & Kneib, Thomas, 2019. "Adaptive semiparametric M-quantile regression," Econometrics and Statistics, Elsevier, vol. 11(C), pages 116-129.
    2. Luca Merlo & Lea Petrella & Nikos Tzavidis, 2022. "Quantile mixed hidden Markov models for multivariate longitudinal data: An application to children's Strengths and Difficulties Questionnaire scores," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 71(2), pages 417-448, March.
    3. Francesco Schirripa Spagnolo & Nicola Salvati & Antonella D’Agostino & Ides Nicaise, 2020. "The use of sampling weights in M‐quantile random‐effects regression: an application to Programme for International Student Assessment mathematics scores," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 69(4), pages 991-1012, August.
    4. Maria Marino & Alessio Farcomeni, 2015. "Linear quantile regression models for longitudinal experiments: an overview," METRON, Springer;Sapienza Università di Roma, vol. 73(2), pages 229-247, August.
    5. Marco Alfò & Maria Francesca Marino & Maria Giovanna Ranalli & Nicola Salvati & Nikos Tzavidis, 2021. "M‐quantile regression for multivariate longitudinal data with an application to the Millennium Cohort Study," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 70(1), pages 122-146, January.
    6. Klaus Herrmann & Marius Hofert & Melina Mailhot, 2017. "Multivariate Geometric Expectiles," Papers 1704.01503, arXiv.org, revised Jan 2018.
    7. Torres Díaz, Raúl Andrés & Michele, Carlo de & Lillo Rodríguez, Rosa Elvira & Laniado Rodas, Henry, 2016. "Directional multivariate extremes in environmental phenomena," DES - Working Papers. Statistics and Econometrics. WS 23419, Universidad Carlos III de Madrid. Departamento de Estadística.
    8. Fu, Liya & Wang, You-Gan, 2016. "Efficient parameter estimation via Gaussian copulas for quantile regression with longitudinal data," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 492-502.
    9. Bernardi, Mauro & Bottone, Marco & Petrella, Lea, 2018. "Bayesian quantile regression using the skew exponential power distribution," Computational Statistics & Data Analysis, Elsevier, vol. 126(C), pages 92-111.
    10. Paolo Frumento & Nicola Salvati, 2020. "Parametric modelling of M‐quantile regression coefficient functions with application to small area estimation," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 183(1), pages 229-250, January.
    11. G. Bertarelli & R. Chambers & N. Salvati, 2021. "Outlier robust small domain estimation via bias correction and robust bootstrapping," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(1), pages 331-357, March.
    12. N. Salvati & N. Tzavidis & M. Pratesi & R. Chambers, 2012. "Small area estimation via M-quantile geographically weighted regression," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(1), pages 1-28, March.
    13. Brenda López Cabrera & Franziska Schulz, 2017. "Forecasting Generalized Quantiles of Electricity Demand: A Functional Data Approach," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(517), pages 127-136, January.
    14. Fabrizi, Enrico & Salvati, Nicola & Trivisano, Carlo, 2020. "Robust Bayesian small area estimation based on quantile regression," Computational Statistics & Data Analysis, Elsevier, vol. 145(C).
    15. Daouia, Abdelaati & Paindaveine, Davy, 2019. "Multivariate Expectiles, Expectile Depth and Multiple-Output Expectile Regression," TSE Working Papers 19-1022, Toulouse School of Economics (TSE), revised Feb 2023.
    16. Hamidi, Benjamin & Maillet, Bertrand & Prigent, Jean-Luc, 2014. "A dynamic autoregressive expectile for time-invariant portfolio protection strategies," Journal of Economic Dynamics and Control, Elsevier, vol. 46(C), pages 1-29.
    17. Giulia Romano & Nicola Salvati & Andrea Guerrini, 2014. "Factors Affecting Water Utility Companies’ Decision to Promote the Reduction of Household Water Consumption," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 28(15), pages 5491-5505, December.
    18. Paindaveine, Davy & Šiman, Miroslav, 2012. "Computing multiple-output regression quantile regions," Computational Statistics & Data Analysis, Elsevier, vol. 56(4), pages 840-853.
    19. Kneib, Thomas & Silbersdorff, Alexander & Säfken, Benjamin, 2023. "Rage Against the Mean – A Review of Distributional Regression Approaches," Econometrics and Statistics, Elsevier, vol. 26(C), pages 99-123.
    20. Daouia, Abdelaati & Padoan, Simone A. & Stupfler, Gilles, 2023. "Extreme expectile estimation for short-tailed data, with an application to market risk assessment," TSE Working Papers 23-1414, Toulouse School of Economics (TSE).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:csdana:v:173:y:2022:i:c:s0167947322000809. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/csda .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.