Semi-parametric Dynamic Models for Longitudinal Ordinal Categorical Data

The over all regression function in a semi-parametric model involves a partly specified regression function in some primary covariates and a non-parametric function in some other secondary covariates. This type of semi-parametric models in a longitudinal setup has recently been discussed extensively both for repeated Poisson and negative binomial count data. However, when it is appropriate to interpret the longitudinal binary responses through a binary dynamic logits model, the inferences for semi-parametric Poisson and negative binomial models cannot be applied to such binary models as these models unlike the count data models produce recursive means and variances containing the dynamic dependence or correlation parameters. In this paper, we consider a general multinomial dynamic logits model in a semi-parametric setup first to analyze nominal categorical data in a semi-parametric longitudinal setup, and then modify this model to analyze ordinal categorical data. The ordinal responses are fitted by using a cumulative semi-parametric multinomial dynamic logits model. For the benefits of practitioners, a step by step estimation approach is developed for the non-parametric function, and for both regression and dynamic dependence parameters. In summary, a kernel-based semi-parametric weighted likelihood approach is used for the estimation of the non-parametric function. This weighted likelihood estimate for the non-parametric function is shown to be consistent. The regression and dynamic dependence parameters of the model are estimated by maximizing an approximate semi-parametric likelihood function for the parameters, which is constructed by replacing the non-parametric function with its consistent estimate. Asymptotic properties including the proofs for the consistency of the likelihood estimators of the regression and dynamic dependence parameters are discussed

A parameter dimension-split based asymptotic regression estimation theory for a multinomial panel data model

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

An Overview on Econometric Models for Linear Spatial Panel Data

When spatial data are repeatedly collected from the same spatial locations over a short period of time, a spatial panel/longitudinal data set is generated. Thus, this type of spatial longitudinal data must exhibit both spatial and longitudinal correlations, which are not easy to model. This work is motivated by existing studies in statistics and econometrics literature but the proposed model and inference procedures should be applicable to the spatial panel data encountered in other fields as well such as environmental and/or ecological setups. Specifically, unlike the existing studies, we propose a new dynamic mixed model to accommodate both spatial and panel correlations. A complete theoretical analysis is given for the estimation of regression effects, and spatial and panel correlations by exploiting second and higher order moments based quasi-likelihood methods. Asymptotic properties are also studied in details. The step by step estimation results developed in the paper should be useful to the practitioners dealing with spatial panel data

Inferences in binary dynamic fixed models in a semi-parametric setup

In a longitudinal setup, the so-called generalized estimating equations approach was a popular inference technique to obtain efficient regression estimates until it was discovered that this approach may in fact yield less efficient estimates than an independence assumption-based estimating equation approach. In this paper, we revisit this inference issue in a semi-parametric longitudinal setup for binary data and find that the semi-parametric generalized estimating equations also encounter similar efficiency drawbacks when compared with independence assumption-based approach. This makes the generalized estimating equations approach unacceptable for correlated data analysis. We analyze the repeated binary data by fitting a semi-parametric binary dynamic model. The non-parametric function and the regression parameters involved in the semi-parametric regression function are estimated by using a semi-parametric generalized quasi-likelihood and a semi-parametric quasi-likelihood approach, respectively, whereas the dynamic dependence, that is, the correlation index parameter of the model is estimated by a semi-parametric method of moments. Asymptotic and finite sample properties of the estimators are discussed. The proposed model and the estimation methodology are also illustrated by reanalyzing the well-known respiratory disease data