In a certain industry, firrms relate their stocks of finished goods, Y , to their expected annual sales, X^e, according to a linear relationship
Yi =β1 + β2Xi^e Actual sales, X , differ from expected sales by random quantity u:

Xi = Xi^e + ui

where u is distributed independently of X^e with mean zero and constant variance. An investigator has data on Y and X (but not on X^e ) for a cross section of firms in the industry.

a) Describe analytically the problems that would be encountered if we estimate β2 regressing Y on X using OLS. (i.e. Derive the expression for measurement error bias mathematically.)

b) Suppose the amount of labor, L , employed by the firms is also a linear function of expected sales: Li = α1 + α2Xi^e

Explain how the relationship between L and X^e can be used to overcome the measurement error bias, and justify your suggestion.

BONUS: Show analytically that doing what you suggest results in a consistent estimator for β2 .