Linear Models of Different Scales

Many statistical models for environmental studies can be run at different scales, e.g., for daily, weekly or monthly data. It is important to know how these models differ in terms of prediction. We provide some theoretical results to compare these models. We show that when there is no high order terms of explanatory variables, the small scale model yields more efficient estimators for model parameters and also produces better prediction. However, when there are high order terms of explanatory variables, a larger scale model can be run in two different ways: productof-sum or sum-of-product. Current practice made it hard to compare directly the larger scale model with the small scale model and no explicit conclusions are drawn in general. We provide a case study on gross primary production of terrestrial ecosystems in the conterminous United States to demonstrate our results. Key-words: Gauss-Markov theorem, gross primary production, multiple linear regression, prediction, best linear unbiased estimator.