Question

Consider an earnings function with the dependent variable y monthly usual earnings and as independent variables years of education x1, gender x2 coded as 1 if female and 0 if male, and work experience in years x3. We are interested in the partial effect of years of education on earnings. We consider the following possible relations (that are assumed to be exact) y =β0 + β1x1 + β2x2 + β3x3 (1) y =β0 + β1x1 + β2x2 + β3x3 + β4x 2 1 (2) y =β0 + β1x1 + β2x2 + β3x3 + β4x1x2 (3) We are interested in the partial, i.e. ceteris paribus, effect of x1 on earnings y. (i) Use partial differentiation to find the partial effect in the three specifications above. (ii) For which specifications are the partial effects constant, i.e. independent of the level of x1, x2, x3? If not constant how does the partial effect change with x1, x2, x3? (iii) If we have data that allow us to estimate the regression coefficients, how would you report the partial effects if they are not constant and you still want to report a single number? (iv) Can you use partial differentiation to find the partial effect of x2? Why (not)? (v) Often work experience is not directly observed, but measured as AGE YEARS OF EDUCATION - 6. Does this change your answers to (i) and (ii)?

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Answer 1