Suppose you want to estimate the labor supply of the following form HOURS = alph

Question

Suppose you want to estimate the labor supply of the following form HOURS = alpha_1 + alpha_2 In(WAGE) + u, where HOURS is the number of hours worked per week, WAGE is hourly wage and u is an error term. Your classmate however points out that the wage is likely to be affected by other workers' characteristics such as education and experience. Therefore, there may exist an auxiliary relationship such as ln( WAGE) = beta_1 + beta_2 HOURS + beta_3 EDUC + beta_4 EXPER + e, where EDUC is years of education, EXPER is years of experience and e is the error term. Recognize that (2) is nothing but the labor demand equation in which EDUC and EXPER are exogenous. Can equation (1) be estimated satisfactorily using the ordinary least squares estimator? If not, why? Make sure to explicitly show the source of the problem (if any at all). Are the labor supply equation (1) and the demand equation in (2) "identified"? Answer using the order condition. Write down the reduced-form expressions of the HOURS and ln(WAGE) in (1) and (2). Sketch the derivation of the IV estimator for equation (1) where both exogenous variables from (2) are used as instruments for In(WAGE). What complication arises as a result of the over-identification of the labor supply? Suggest an alternative estimation procedure to estimate the labor supply equation (1) that will take care of endogeneity of In(WAGE) as w ell as will overcome the complication in the IV estimator from part (d). Show how you will carry it out (step by step and NOT a computer command). Suppose that the two instruments [namely, EDUC and EXPER] which you have planned on using for In(WAGE) are "weak". First, explain in what sense can instruments be "weak"? Then, elaborate on the problem associated with such "weak instruments"?

Explanation / Answer

(a) No,Ordinary least squares is a statistical technique that uses sample data to estimate the true population relationship between two variables.

The Method of least squares: (OLS) produces a line that minimizes the sum of the squared vertical distances from the line to the observed data points.

i.e. it minimizes S ei2 = e12 + e22 + e32 +.........+ en2 , where n is the sample size

                                    (hats over all of the e's)

The sum of the residuals (unsquared) is exactly zero.  

(b)

The order condition is the state of a set of simultaneous equations in an econometric system such that all its parameters may be identified.

For an equation in a system of equations to be identified, the number of excluded exogenous variables in that equation must be at least as great as the number of includedendogenous variables, less one. However, a stronger argument is the rank condition which is both necessary and sufficient for identification.

(c) Wi = p10 + p11Xi + p12NKi + e1i

  

a1 + b1Wi + ui = a2 + b2Wi + c2Xi + d2NKi + vi

•Working with the equation, we have:

•

•

–This gives us an expression for wages in terms of the exogenous variables

–Here, our exogenous variables are: Xi, age, and NKi no. of kids

(d) IV estimator Keep considering the model yi = x 0 i + ui Y = X + u Now, assume that we have a K 1 instrumental vector zi (that contains instrumental variable zi1;:::;ziK). that has following properties (1) zi is uncorrelated with ui : (2) zi is correlated with xi Mathematically, we need the condition 1 N P N i=1 ziui p! 0K1: Then, the instrumental variable (IV) estimator is deÖned as ^ IV = (Z 0X) 1 Z 0Y = 1 N P N i=1 zix 0 i 1 1 N P N i=1 ziyi Consider the consistency of ^ IV : We can rewrite the IV estimator and checking asymptotic behavior ^ IV = 1 N P N i=1 zix 0 i 1 1 N P N i=1 ziyi = 1 N P N i=1 zix 0 i 1 1 N P N i=1 zi (xi + ui) = + 1 N P N i=1 zix 0 i 1 1 N P N i=1 ziui : p! + E [zixi ] 1 | {z } KK E [ziui ] | {z } =OK1 = Thus, the IV estimator is consistent (large sample property).

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