Focussed Model Averaging in Generalised Linear Models

Abstract

Model averaging is often used to allow for uncertainty in the model selection process. In the frequentist setting, a model-averaged point estimate is the weighted mean of the estimates from each of the candidate models. Focussed model averaging is an approach to calculating the model weight which is tailored to the parameter of interest. For the important special case of generalised linear models, we propose a new method for focussed model averaging, in which the weights are chosen to minimise an estimate of the asymptotic mean squared error (MSE) of the model-averaged estimate of the parameter of interest. The procedure we put forward uses standard results for maximum likelihood estimation when the model is misspecified and, unlike existing methods, does not rely on a local-misspecification assumption, which shrinks each model towards the smallest model as the sample size increases. We use the more natural fixed-model framework in which the models do not converge asymptotically.

We first derive a new estimator of the asymptotic MSE for the single-model setting, and compare this with an existing estimator, both analytically and via simulation. We show that the two estimators are identical for the normal linear model, provided that the parameter of interest is a linear function of the model parameters. Our simulation results suggest that our estimator gives estimates that are less biased and has a smaller coefficient of variation.

We then use our estimator of the asymptotic MSE in the context of model selection. The model with the smallest value of the estimated MSE is chosen to be the best from a set of candidate models. Simulation results suggest that for both frameworks, even when we consider the local-misspecification framework, our method performs better than existing methods.

We then propose a new approach to focussed model averaging, using a new estimator of the asymptotic MSE of the model-averaged estimate of the parameter of interest. This approach is compared with existing focussed model averaging procedures. Simulation results suggest that our new procedure generally outperforms the existing methods, with the benefit being greatest when we consider the fixed-model framework and when the sample size is small.