In Sports Economics, Production Functions Are Often Estimated by Relating $\hat { \beta }$

Essay

In Sports Economics, production functions are often estimated by relating the winning percentage of teams (Y)to inputs indicating performance in certain aspects of the game. However, this omits the quality of management. Assume that you could measure the quality of pitching and hitting by a single index L, and that managerial ability is represented by M, which is assumed to be constant over time. The production function would then be specified as follows:
Y_it = β₀ +β₁ L_it + β₂M_i + u_it
where i is an index for the baseball team, and t indexes time and all variables are in logs.
(a)Assume that managerial ability is unobservable but is positively related, in a linear way, to L. Explain why the OLS estimator $\hat { \beta }$ ₁ is inconsistent in the case of a single cross-section, i.e., if you attempt to estimate the above regression for a single year. Do you expect this coefficient to over- or under-estimate β₁?
(b)If you had data for two years, indicate the transformation, which allows you to obtain a consistent estimator for β₁.