Model-Checking Techniques for Linear Models With Parametric Variance Functions

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We present graphical and numerical methods for assessing the adequacy of the log-linear model for variances. The proposed methods are derived from the cumulative sum of residuals over the covariate or linear predictor. Under the assumed model, the cumulative residual process converges weakly to a 0-mean Gaussian process whose distribution can be approximated via Monte Carlo simulation. The observed cumulative residual pattern can then be compared both visually and analytically to a number of simulated realizations from the approximate null distribution. These comparisons enable one to examine the functional form of each covariate, the link function as well as the overall model adequacy of the variance model. Simulation studies demonstrate that the proposed methods perform well in practical settings. Illustrations with three datasets are provided.