Aggregation of Variables in Least-Squares Regression

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Abstract
I consider the properties of the estimator when the true model is y = β1 x 1 + β2 x 2 + u, but the restriction β1 = β2 = β is incorrectly imposed. I show that the probability limit of is a weighted sum of β1 and β2; the weights sum to 1 but do not necessarily lie in the unit interval, so plim need not be bounded by β1 and β2. Sufficient conditions for such bounding are derived. Certain changes in the moments of x 1 and x 2 have “perverse” effects on the weights. I illustrate the consequences of inappropriate aggregation of variables with an empirical example of the effect of research and development investment on productivity.

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