Two Wave Missing Data with Refreshment Sample and Free Knot Spline in Semiparametric Regression

Spline techniques are applied to construct confidence bands in generalized regression model with flexible knot location and to estimate the component function specified in the missingness probability in a 2-wave attrition model. In spline approximation, knot location is critical to capture turning points and changing trends in the data. A free knot spline with flexible location parameter is proposed to optimize in generalized regression. Free knots may lead to the so called lethargy property, the slow convergence to replicate knots. A Jupp transformation is applied as well as a multiplicative penalty term, penalizing overlapping knots. Simultaneous confidence bands of the free knot spline are used as a diagnostic tool and better understanding of the data. With asymptotic distribution being unknown, we rely on the wild bootstrap resampling method to deal with heteroskedasticity. Freeing up knot location and number of knots adds additional computational cost. Automatic differentiation is applied in the C++/R framework Automatic Differentiation Model Builder. With participants dropping out of survey panels, missing data is a common problem in longitudinal studies. The misspecified underlying missingness mechanism can lead to biased inference. Correctly understanding the attrition process can help picking the appropriate statistical tool. Based on Hirano's identification equations, we propose an additive attrition model with refreshment sample in the second wave, with components in missingness estimated by B-spline. We show that the objective function converges to its probability limit and show consistency of the spline estimator minimizing the objective function.