Using follow-up data to avoid omitted variable bias: an application to cardiovascular epidemiology

Omitted variable bias is discussed in the context of linear models. It is shown that the effect of omitted variables can be controlled in linear models for metric dependent variables by using data from follow-up studies. Two different models for analysing such data are proposed. In the first model the omitted variables are assumed to be uncorrelated with the explanatory variables in the model and to be constant over time. These assumptions lead to a special structure of the covariance matrix of the errors over time. Efficient estimation of the parameters in the linear model has to take this special covariance matrix of the errors into account by using appropriate generalized least squares or maximum likelihood methods. In the second model the omitted variables are assumed to be time constant. Additionally, they are allowed to be correlated with the explanatory variables, that is these variables are omitted confounders in the usual epidemiological sense. It is shown that even in this case the parameters of the linear model can be estimated consistently with ordinary least squares if a follow-up study is available. The differences between the parameter estimates under the first and the second model may be used to construct a Hausman test for misspecification. The models, the estimation methods and the Hausman test are illustrated by the example that explores the determinants of serum cholesterol in German adoloscents of both sexes.