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A Parameter Dimension-Split Based Asymptotic Regression Estimation Theory for a Multinomial Panel Data Model

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  • Brajendra C Sutradhar

    (Carleton University
    Memorial University)

Abstract

In this paper we revisit the so-called non-stationary regression models for repeated categorical/multinomial data collected from a large number of independent individuals. The main objective of the study is to obtain consistent and efficient regression estimates after taking the correlations of the repeated multinomial data into account. The existing (1) ‘working’ odds ratios based GEE (generalized estimating equations) approach has both consistency and efficiency drawbacks. Specifically, the GEE-based regression estimates can be inconsistent which is a serious limitation. Some other existing studies use a MDL (multinomial dynamic logits) model among the repeated responses. As far as the estimation of the regression effects and dynamic dependence (i.e., correlation) parameters is concerned, they use either (2) a marginal or (3) a joint likelihood approach. In the marginal approach, the regression parameters are estimated for known correlation parameters by solving their respective marginal likelihood estimating equations, and similarly the correlation parameters are estimated by solving their likelihood equations for known regression estimates. Thus, this marginal approach is an iterative approach which may not provide quick convergence. In the joint likelihood approach, the regression and correlation parameters are estimated simultaneously by searching the maximum value of the likelihood function with regard to these parameters together. This approach may encounter computational drawback, specially when the number of correlation parameters gets large. In this paper, we propose a new estimation approach where the likelihood function for the regression parameters is developed from the joint likelihood function by replacing the correlation parameter with a consistent estimator involving unknown regression parameters. Thus the new approach relaxes the dimension issue, that is, the dimension of the correlation parameters does not affect the estimation of the main regression parameters. The asymptotic properties of the estimates of the main regression parameters (obtained based on consistent estimating functions for correlation parameters) are studied in detail.

Suggested Citation

  • Brajendra C Sutradhar, 2018. "A Parameter Dimension-Split Based Asymptotic Regression Estimation Theory for a Multinomial Panel Data Model," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(2), pages 301-329, August.
    • Handle: RePEc:spr:sankha:v:80:y:2018:i:2:d:10.1007_s13171-017-0120-8
    DOI: 10.1007/s13171-017-0120-8
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    References listed on IDEAS

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    1. Chen, Baojiang & Yi, Grace Y. & Cook, Richard J., 2010. "Weighted Generalized Estimating Functions for Longitudinal Response and Covariate Data That Are Missing at Random," Journal of the American Statistical Association, American Statistical Association, vol. 105(489), pages 336-353.
    2. Alan Agresti & Ranjini Natarajan, 2001. "Modeling Clustered Ordered Categorical Data: A Survey," International Statistical Review, International Statistical Institute, vol. 69(3), pages 345-371, December.
    3. Ludwig Fahrmeir & Heinz Kaufmann, 1987. "Regression Models For Non‐Stationary Categorical Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 8(2), pages 147-160, March.
    4. J. C. Loredo‐Osti & Brajendra C. Sutradhar, 2012. "Estimation of regression and dynamic dependence paremeters for non‐stationary multinomial time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 33(3), pages 458-467, May.
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    Cited by:

    1. Brajendra C. Sutradhar, 2022. "Multinomial Logistic Mixed Models for Clustered Categorical Data in a Complex Survey Sampling Setup," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(2), pages 743-789, August.
    2. R. Prabhakar Rao & Brajendra C. Sutradhar, 2020. "Multiple Categorical Covariates-Based Multinomial Dynamic Response Model," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 82(1), pages 186-219, February.

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