Nothing Special   »   [go: up one dir, main page]
IDEAS home Printed from https://ideas.repec.org/a/sae/jedbes/v39y2014i6p524-549.html
   My bibliography  Save this article

Handling Correlations Between Covariates and Random Slopes in Multilevel Models

Author

Listed:
  • Michael David Bates

    (Michigan State University)

  • Katherine E. Castellano

    (Educational Testing Service)

  • Sophia Rabe-Hesketh

    (University of California, Berkeley)

  • Anders Skrondal

    (Norwegian Institute of Public Health)

Abstract
This article discusses estimation of multilevel/hierarchical linear models that include cluster-level random intercepts and random slopes. Viewing the models as structural, the random intercepts and slopes represent the effects of omitted cluster-level covariates that may be correlated with included covariates. The resulting correlations between random effects (intercepts and slopes) and included covariates, which we refer to as “cluster-level endogeneity,†lead to bias when using standard random effects (RE) estimators such as (restricted) maximum likelihood. While the problem of correlations between unit-level covariates and random intercepts is well known and can be handled by fixed-effects (FE) estimators, the problem of correlations between unit-level covariates and random slopes is rarely considered. When applied to models with random slopes, the standard FE estimator does not rely on standard cluster-level exogeneity assumptions, but requires an “uncorrelated variance assumption†that the variances of unit-level covariates are uncorrelated with their random slopes. We propose a “per-cluster regression†(PC) estimator that is straightforward to implement in standard software, and we show analytically that it is unbiased for all regression coefficients under cluster-level endogeneity and violation of the uncorrelated variance assumption. The PC, RE, and an augmented FE estimator are applied to a real data set and evaluated in a simulation study that demonstrates that our PC estimator performs well in practice.

Suggested Citation

  • Michael David Bates & Katherine E. Castellano & Sophia Rabe-Hesketh & Anders Skrondal, 2014. "Handling Correlations Between Covariates and Random Slopes in Multilevel Models," Journal of Educational and Behavioral Statistics, , vol. 39(6), pages 524-549, December.
    • Handle: RePEc:sae:jedbes:v:39:y:2014:i:6:p:524-549
    DOI: 10.3102/1076998614559420
    as

    Download full text from publisher

    File URL: https://journals.sagepub.com/doi/10.3102/1076998614559420
    Download Restriction: no
    File URL: https://libkey.io/10.3102/1076998614559420?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
    ---><---

    References listed on IDEAS

    as
    1. Hausman, Jerry A & Taylor, William E, 1981. "Panel Data and Unobservable Individual Effects," Econometrica, Econometric Society, vol. 49(6), pages 1377-1398, November.
    2. Frees,Edward W., 2004. "Longitudinal and Panel Data," Cambridge Books, Cambridge University Press, number 9780521828284, October.
    3. Borjas, George J. & Sueyoshi, Glenn T., 1994. "A two-stage estimator for probit models with structural group effects," Journal of Econometrics, Elsevier, vol. 64(1-2), pages 165-182.
    4. Jeffrey M. Wooldridge, 2005. "Fixed-Effects and Related Estimators for Correlated Random-Coefficient and Treatment-Effect Panel Data Models," The Review of Economics and Statistics, MIT Press, vol. 87(2), pages 385-390, May.
    5. Verbeke G. & Spiessens B. & Lesaffre E., 2001. "Conditional Linear Mixed Models," The American Statistician, American Statistical Association, vol. 55, pages 25-34, February.
    6. Jeffrey M Wooldridge, 2010. "Econometric Analysis of Cross Section and Panel Data," MIT Press Books, The MIT Press, edition 2, volume 1, number 0262232588, April.
    7. Jay Teachman & Greg J. Duncan & W. Jean Yeung & Dan Levy, 2001. "Covariance Structure Models for Fixed and Random Effects," Sociological Methods & Research, , vol. 30(2), pages 271-288, November.
    8. Jee-Seon Kim & Edward Frees, 2007. "Multilevel Modeling with Correlated Effects," Psychometrika, Springer;The Psychometric Society, vol. 72(4), pages 505-533, December.
    9. Frees,Edward W., 2004. "Longitudinal and Panel Data," Cambridge Books, Cambridge University Press, number 9780521535380, October.
    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. Michael Bates & Seolah Kim, 2019. "Per-Cluster Instrumental Variables Estimation: Uncovering the Price Elasticity of the Demand for Gasoline," Working Papers 202003, University of California at Riverside, Department of Economics.
    2. Milla, J. & San Martin , E. & Van Bellegem, S., 2015. "Higher education value added using multiple outcomes," LIDAM Discussion Papers CORE 2015045, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Michael Bates & Seolah Kim, 2024. "Estimating the price elasticity of gasoline demand in correlated random coefficient models with endogeneity," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 39(4), pages 679-696, June.
    4. Garritt L. Page & Ernesto San Martín & Javiera Orellana & Jorge González, 2017. "Exploring complete school effectiveness via quantile value added," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 180(1), pages 315-340, January.

    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. Dalina-Maria Andrei, 2021. "Determinants of New Companies’ Formation in Romania at Regional Level. A Fixed Effects Model (FEM) Approach," Ovidius University Annals, Economic Sciences Series, Ovidius University of Constantza, Faculty of Economic Sciences, vol. 0(1), pages 18-27, August.
    2. Anders Skrondal & Sophia Rabe-Hesketh, 2022. "The Role of Conditional Likelihoods in Latent Variable Modeling," Psychometrika, Springer;The Psychometric Society, vol. 87(3), pages 799-834, September.
    3. Antonio Ruiz Porras, 2016. "La investigación econométrica mediante paneles de datos:historia, modelos y usos en México," Archivos Revista Economía y Política., Facultad de Ciencias Económicas y Administrativas, Universidad de Cuenca., vol. 24, pages 11-32, Julio.
    4. Pillai N., Vijayamohanan, 2016. "Panel Data Analysis with Stata Part 1 Fixed Effects and Random Effects Models," MPRA Paper 76869, University Library of Munich, Germany.
    5. Aleksey Oshchepkov & Anna Shirokanova, 2020. "Multilevel Modeling For Economists: Why, When And How," HSE Working papers WP BRP 233/EC/2020, National Research University Higher School of Economics.
    6. Fukui, Hideki & Nagata, Koki, 2014. "Flight cancellation as a reaction to the tarmac delay rule: An unintended consequence of enhanced passenger protection," Economics of Transportation, Elsevier, vol. 3(1), pages 29-44.
    7. Marta Spreafico, 2013. "Institutions, the resource curse and the transition economies: further evidence," DISCE - Quaderni del Dipartimento di Politica Economica ispe0064, Università Cattolica del Sacro Cuore, Dipartimenti e Istituti di Scienze Economiche (DISCE).
    8. Yang, Yimin & Schmidt, Peter, 2021. "An econometric approach to the estimation of multi-level models," Journal of Econometrics, Elsevier, vol. 220(2), pages 532-543.
    9. Lars Jensen & Teit Lüthje, 2009. "Driving forces of vertical intra-industry trade in Europe 1996–2005," Review of World Economics (Weltwirtschaftliches Archiv), Springer;Institut für Weltwirtschaft (Kiel Institute for the World Economy), vol. 145(3), pages 469-488, October.
    10. Cotte Poveda Alexander, 2011. "Socio-Economic Development and Violence: An Empirical Application for Seven Metropolitan Areas in Colombia," Peace Economics, Peace Science, and Public Policy, De Gruyter, vol. 17(1), pages 1-23, September.
    11. Jee-Seon Kim & Edward Frees, 2007. "Multilevel Modeling with Correlated Effects," Psychometrika, Springer;The Psychometric Society, vol. 72(4), pages 505-533, December.
    12. Purnima PUROHIT & Katsushi S. Imai & Kunal Sen, 2017. "Do Agricultural Marketing Laws Matter for Rural Growth? Evidence from the Indian States," Discussion Paper Series DP2017-17, Research Institute for Economics & Business Administration, Kobe University.
    13. Katherine E. Castellano & Sophia Rabe-Hesketh & Anders Skrondal, 2014. "Composition, Context, and Endogeneity in School and Teacher Comparisons," Journal of Educational and Behavioral Statistics, , vol. 39(5), pages 333-367, October.
    14. Judith Offerhaus, 2013. "The Type to Train?: Impacts of Personality Characteristics on Further Training Participation," SOEPpapers on Multidisciplinary Panel Data Research 531, DIW Berlin, The German Socio-Economic Panel (SOEP).
    15. Kerstens, Kristiaan & Van de Woestyne, Ignace, 2014. "Comparing Malmquist and Hicks–Moorsteen productivity indices: Exploring the impact of unbalanced vs. balanced panel data," European Journal of Operational Research, Elsevier, vol. 233(3), pages 749-758.
    16. Valentina Alvarez-Saavedra & Pierre Levasseur & Suneha Seetahul, 2023. "The Role of Gender Inequality in the Obesity Epidemic: A Case Study from India," Journal of Development Studies, Taylor & Francis Journals, vol. 59(7), pages 980-996, July.
    17. Fulvia Pennoni & Beata Bal-Domańska, 2022. "NEETs and Youth Unemployment: A Longitudinal Comparison Across European Countries," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 162(2), pages 739-761, July.
    18. Rodrigo V. Ventura & Manoela Cabo & Rafael Caixeta & Elton Fernandes & Vicente Aprigliano Fernandes, 2020. "Air Transportation Income and Price Elasticities in Remote Areas: The Case of the Brazilian Amazon Region," Sustainability, MDPI, vol. 12(15), pages 1-18, July.
    19. Lynn, Peter & Bosch, Oriol, 2021. "Methodological lessons from the pilot longitudinal survey on debt advice," ISER Working Paper Series 2021-03, Institute for Social and Economic Research.
    20. Woutersen, Tiemen & Hausman, Jerry A., 2019. "Increasing the power of specification tests," Journal of Econometrics, Elsevier, vol. 211(1), pages 166-175.

    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:sae:jedbes:v:39:y:2014:i:6:p:524-549. 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: SAGE Publications (email available below). General contact details of provider: .

    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.