1. 1 Research Method Lecture 5 (Ch6) Multiple regression: Further issues ©

2. Effects of data scaling on OLS statistics. 4Consider the following labor supply equation for married women. Hours = β 0 + β 1 (non wife income)+ β 2 (experience)+ β 3 (education)+ β 4 (# kids aged less than 6)+ u Hours : the woman’s annual hours worked Non wife income : wife’s non labor income (such as husband’ s income) in $1000. Estimate the model using MROZ.dta 2

3. 4Question: What is the effect of measuring non-wife-income in $1 instead of in $1000? 4What is the effect of measuring wife’s labor supply in days-equivalent (i.e., days=hours/8)? 3

4. Standardizing a variable 4The effects of some variables, such as test scores, are often difficult to interpret: For example, what does it mean by the effect of increasing the test score by 1 point on wage? 4In such case, it is often better to standardize the variable first, then include it in the model. That is; include the following variable instead. 4

5. 4Then coefficient on the variable will tell you the effect when test score increases by one standard deviation from the mean. 5

6. 4Example Using HPRICE2.dta, estimate the effect of environment (measured by nox) on housing price. Standardize nox before including in the model. In the model, include (crime), (dist), (rooms) and (stratio). 6

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8. Including quadratic terms 4Often you would like to capture ‘diminishing marginal returns’. For example, the effect of experience on wage may face diminishing marginal returns. 4To capture such effects, include quadratic term. Log(wage) = β 0 +β 1 (experience)+β 2 (experience) 2 +u If β 2 is negative, experience faces diminishing marginal returns. 8

9. 4Note that the effect of education is given by ∂log(wage)/∂(experience) =β 1 + 2β 2 (experience) So, the effect changes with experience. Exercise 1: Use MROZ.dta, estimate the above equation. What is the effect of increasing experience by 1 year on wage evaluated at the average experience? Exercise 2: What is the effect of increasing experience by 1 year on wage for those with experience equal to 1? 9

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11. Interaction terms 4Consider you estimate the following production function Log(Q)= β 0 +β 1 log(labor)+β 2 log(capital)+u This model assumes that the effect of labor on output is independent of the effect of capital. But, workers may be more productive if they have more capital. So in reality, there is an interaction effect between labor and capital. 11

12. 4To capture such interaction effects, you can include an interaction term, like: Log(Q)= β 0 +β 1 log(labor)+β 2 log(capital) +β 3 log(labor)log(capital)+ u The effect of labor on output is now give as ∂log(Q)/∂log(labor) =β 1 +β 3 log(capital) So the effect of labor depends on the amount of capital. 12

13. 4Exercise: using HPRICE1.dta, estimate the following model Price= β 0 +β 1 sqrft+β 2 bdrms+β 3 (sqrft)(bdrms)+ u Question 1: Is there positive or negative interaction effect between the size of the house and the # of bedrooms? Question 2: What is the effect (sqrft) on price of house evaluated at the average # of bedrooms? Question 3: What is the effect of (bdrms) on price of house evaluated at the average size of house? 13

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15. Adjusted R-squared 4The usual R-squared is given by 15 4The adjusted R-squared is given by Almost but different It is equal to

16. 4The usual R squared always increases if you add additional variables (even if it does not make sense to add some variables). 4The adjusted R squared imposes a penalty to adding additional variables because it divides SSR by (n-k-1). 4The adjust R squared can also be written as 16

17. Controlling for too many factors in regression 4After learning the omitted variable bias, one may be tempted to control for as many factors as possible. 4 But this often lead students to control for factors that shouldn’t be controlled for. 4Next slide shows the example. 17

18. 4Suppose you would like to see the effect of beer tax on traffic fatalities. The idea is that beer tax will reduce beer consumption, which would lead to fewer fatalities. So you may estimate (Fatalities)= β0+β1(beer tax) +β2(Percentage of male in town) +β3(percentage of young drivers) +β(other variables)+u 18

19. 4Question is whether you should also control beer consumption. Like: (Fatalities)= β0+β1(beer tax) +β2(beer consumption) +β3(Percentage of male in town) +β4(percentage of young drivers) +β(other variables)+u 4The answer to this question is NO. Beer tax would affect fatalities mainly by reducing beer consumption. So, it does not make sense to hold beer consumption constant when you examine the effect of beer tax on fatalities. 19

20. Different models serve different purposes 4Suppose that you would like to estimate the gender salary gap among academic economists. One possible model is to include female dummy together with rank dummies. Log(salary)= β0+β1(Female) +β2(FullProf) +β3(AssocProf) +(other variables)+u 20

21. 4The above model estimates the gender salary gap while holding rank constant. 4Thus, the female coefficient captures the gender salary gap within each rank. 4However, females may be discriminated in terms of promotions as well, and this would indirectly cause gender salary gap. 21

22. 4If you would like to evaluate the gender salary gap that is caused by (i) discrimination in terms of salary and (ii) the discrimination in terms of promotion combined, it makes sense to drop rank variables from the model. 4When you drop rank variables (i.e., FullProf and AssocProf), then female coefficient will show the salary gap that is caused by salary discrimination and promotion discrimination combined. 22