Influence diagnostics for simultaneous equations and measurement error models

In recent years, the study of regression diagnostics has advanced considerably due to their important role in statistical modeling. Influence assessment is the diagnostic method to monitor the changes in the outcome of a statistical analysis when perturbations are introduced in the statistical model being used. This thesis develops influence diagnostic methods for simultaneous equations models and measurement error models. In econometrics and other relevant areas, linear simultaneous equations models have been widely used. However, little work has been done specifically on influence diagnostics for model assessment. In this thesis, the influence methodology is extended to linear simultaneous equations models. The residuals, leverage and case-deletion measures are proposed. A missing data method is adopted to minimize the masking effect due to case deletion when lagged variable exists. We discuss and compare four different measures of influence. The assessment of local influence is also considered. We show how to evaluate the effect that perturbations to the endogenous variables, predetermined variables, case weights and model structure may have on the parameter estimates. The direction cosines are found to be informative in assessing parameter sensitivity. The diagnostics are applied to a macroeconomic model. Measurement error problems often occur in biological and medical studies. For instance, urinary sodium chloride varies from day to day is subject to measurement error (Liu and Liang, 1992). This thesis concentrates on linear, nonlinear and generalized linear models when covariates are observed with random measurement errors. A known measurement error covariance structure and no error in the equation are typically assumed, except where specified otherwise. For linear measurement error models, we discuss residuals, leverage and the influence function, and examine various measures of influence. A diagnostic display with simulated envelope, similar to the technique proposed by Atkinson (1981), is suggested to assess the influence measures and other diagnostics. The simulation procedure correspondmg to structural models differs from that of functional models. We also consider the assessment of local influence for these models. The schemes of perturbations to the response, covariates, case weights and measurement error covariance matrix are studied in detail. Based on the likelihood displacement functions, direction cosines and other local influence diagnostics can be derived. Perturbation of the measurement error covariance matrix has been found especially useful to assess the underlying model assumptions. A data set is used to illustrate the techniques. For nonlinear measurement error models, the nonlinear least squares method (Fuller. 1987) is used to obtain the parameter estimates when both coefficients and covariates are assumed to he unknown parameters. Residuals, tangent plane leverage, Jacobian leverage and case-deletion diagnostics are examined. Influence measures for a subset of parameters are considered when the regression coefficients are of direct interest. We then extend the local influence method to study the sensitivity of a fit-ted model to minor modifications including the perturbation of measurement error variances. The diagnostics are demonstrated in an example from nonlinear regression. Finally, the influence diagnostics developed for linear measurement error models are extended to the class of generalized linear measurement error models. We first review various parameter estimation methods proposed in the literature. Using a simulation study on logistic regression, five different estimators are compared for their relative performances. Based on our limited simulation results, none of the estimators involved always performs better than the others. Given its asymptotic results are readily available, the bias-corrected estimator of Stefanski (1985, 1989) is subsequently adopted in the influence analysis. Studentized residual and case leverage are defined. Influence functions and measures of case-deletion influence are derived. To assess these influence diagnostics, the simulated envelope approach is again recommended. Since the true likelihood function does not exist, two strategies to extend the local influence methodology are outlined, resulting in separate perturbation diagnostics.