Problem Set 5 ECN 140 Econometrics Professor Oscar Jorda DUE: June 6, 2006 Name 1) Earnings functions, whereby the log of earnings is regressed on years of education, years of on-the-job training, and individual characteristics, have been studied for a variety of reasons. Some studies have focused on the returns to education, others on discrimination, union and non-union differentials, etc. For all these studies, a major concern has been the fact that ability should enter as a determinant of earnings, but that it is close to impossible to measure and therefore represents an omitted variable. Assume that the coefficient on years of education is the parameter of interest. Given that education is positively correlated to ability, since, for example, more able students attract scholarships and hence receive more years of education, the OLS estimator for the returns to education could be upward-biased. To overcome this problem, various authors have used instrumental variables estimation techniques. For each of the instruments potential instruments listed below, briefly discuss instrument validity. (a) The individual's postal zip code. (b) The individual's IQ or test-score on a work-related exam. (c) Years of education for the individual's mother or father. (d) Number of siblings the individual has. 1
2) The figure shows a plot and a fitted linear regression line of the age-earnings profile of 1,744 individuals, taken from the Current Population Survey. (a) Describe the problems in predicting earnings using the fitted line. What would the pattern of the residuals look like for the age category under 40? (b) What alternative functional form might fit the data better? (c) What other variables might you want to consider in specifying the determinants of earnings? 3) Suggest a transformation in the variables that will linearize the deterministic part of the population regression functions below. Write the resulting regression function in a form that can be estimated by using OLS. (a) (b) (c) (d) 2
4) Your textbook gives the following example of simultaneous causality bias of a two equation system: In microeconomics, you studied the demand and supply of goods in a single market. Let the demand ( ) and supply ( ) for the i-th good be determined as follows,,, where P is the price of the good. In addition, you typically assume that the market clears. Explain how the simultaneous causality bias applies in this situation. The textbook explained a positive correlation between and for > 0 through an argument that started from "imagine that is negative." Repeat this exercise here. 3
Answer Key Testname: PS5.TST 1) (a) Instrumental validity has two components, instrument relevance ( ), and instrument exogeneity ( ). The individual's postal zip code will certainly be uncorrelated with the omitted variable, ability, even though some zip codes may attract more able individuals. However, this is an example of a weak instrument, since it is also uncorrelated with years of education. (b) There is instrument relevance in this case, since, on average, individuals who do well in intelligence scores or other work-related test scores will have more years of education. Unfortunately there is bound to be a high correlation with the omitted variable ability, since this is what these tests are supposed to measure. (c) A non-zero correlation between the mother's or father's years of education and the individual's years of education can be expected. Hence this is a relevant instrument. However, it is not clear that the parent's years of education are uncorrelated with parent's ability, which in turn, can be a major determinant of the individual's ability. If this is the case, then years of education of the mother or father is not a valid instrument. (d) There is some evidence that the larger the number of siblings of an individual, the less the number of years of education the individual receives. Hence number of siblings is a relevant instrument. It has been argued that number of siblings is uncorrelated with an individual's ability. In that case it also represents an exogenous instrument. However, there is the possibility that ability depends on the attention an individual receives from parents, and this attention is shared with other siblings. 2) (a) There would be many overpredictions for this age category under 40, and hence more negative residuals. (b) It would be better to fit a quadratic here, i.e., a polynomial regression model, which would produce an inverted U-shape. (c) Answers will vary by students, but education, gender, race, tenure with an employer, professional choice, and ability are typically present in answers. 3) (a) (b) (c) (d) 1
Answer Key Testname: PS5.TST 4) Although quantities appear on the left-hand side of both equations, this is a system of two equations in two unknowns, where quantity and price are determined simultaneously by demand and supply. A negative, call it a "demand shock," decreases the quantity demanded. Since demand equals supply, this results in a lower quantity traded, and hence a lower price. (At the old price level, there would now be excess supply, and hence the price would fall.) The negative has therefore resulted in a lower price, and hence the error term in the demand equation is positively correlated with the price in the same equation. 2