Constrained Estimators and Consistency of a Regression Model on a Lexis Diagram

This article considers a regression model on a Lexis diagram of an a × p table with a single response in each cell following a distribution in the exponential family. A regression model on the fixed effects of a rows, p columns, and a + p − 1 diagonals induces a singular design matrix and yields multiple estimators, leading to parameter identifiability problem in age--period--cohort analysis in social sciences, demography, and epidemiology, where assessment of secular trend in age, period, and birth cohort of social events (e.g., violence) and diseases (e.g., cancer) is of interest. Similar problems also exist in other settings, such as in supersaturated designs. In this article, we study the finite sample properties of the multiple estimators, propose a penalized profile likelihood method to study the consistency and asymptotic bias, and demonstrate the results through simulations and data analysis. As a by-product, the identifiability problem is addressed with consistent estimation for model parameters and secular trend. We conclude that consistent estimation can be identified through estimable function and asymptotics studies in regressions with a singular design. Our method provides a novel approach to studying asymptotics of multiple estimators with a diverging number of nuisance parameters.