Estimating multi-way error components models with unbalanced data structures

I develop simple matrix algebra techniques that simplify and unify much of the previous literature on estimating error components models (ECMs). In fact, the simple analytic results provided here are useful for analyzing a very broad set of models with complex error structures. To illustrate the techniques, I develop the algebra for three- and four-way ECMs explicitly. In addition, I provide Monte Carlo simulation evidence on the performance of several estimators for the three-way ECM and estimate the model using data from a retail market where the three dimensions of data variation are products selling in many locations over time.