1. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. Lecture 6: Supplemental A Slightly More Technical Discussion of Logarithms (Including the Chain Rule) (Chapter 4.5)

2. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-2 Example: The Phillips Curve The US data from 1958–1969 suggest a trade-off between inflation and unemployment.

3. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-3 Figure 4.2 U.S. Unemployment and Inflation, 1958–1969

4. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-4 TABLE 4.1 The Phillips Curve

5. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-5 Example: The Phillips Curve We no longer need to assume our regression line goes through the origin. We have learned how to estimate an intercept. A straight line doesn’t seem to do a great job here. Can we use a linear estimator to fit a nonlinear relationship?

6. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-6 Logarithms in Econometrics (Chapter 4.5) How can we use a linear estimator to fit a non-linear relationship? A trick: do the “nonlinear” part BEFORE plugging into OLS.

7. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-7 Logarithms in Econometrics (cont.) A trick: we can do something “nonlinear” BEFORE plugging into OLS. OLS doesn’t know (or care) where the Y ’s and X ’s came from. Instead of plugging X in directly, we can plug in f ( X )

8. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-8 Logarithms in Econometrics (cont.) We can apply functions to X and Y before we plug them into our OLS formula. If we do something nonlinear to Y and/or X, then our linear regression of f ( Y ) on f ( X ) will plot a nonlinear relationship between Y and X. What f (·) can we apply to X and Y to fit a curve?

9. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-9 Logarithms in Econometrics (cont.) Regress How do we interpret our estimated coefficients? Our fitted value for  1 no longer tells us the effect of a 1-unit change in X on Y. It tells us the effect of a 1-unit change in ln(X) on ln(Y). Huh?

10. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-10 Logarithms in Econometrics (cont.) Regress Our fitted value for  1 no longer tells us the effect of a 1-unit change in X on Y. It tells us the effect of a 1-unit change in ln(X) on ln(Y). Unit changes in log- X translate into PERCENTAGE changes in X.

11. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-11 Logarithms in Econometrics (cont.) Regress Apply the chain rule of calculus.

12. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-12 Regress Unit changes in ln(X) translate into PERCENTAGE changes in X. Our estimate tells us the percentage change in Y we predict from a 1-percent change in X. doesn’t have a natural interpretation in a logarithm framework. Logarithms in Econometrics (cont.)

13. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-13 Regress Unit changes in log- X translate into PERCENTAGE changes in X Our estimate tells us the percentage change in Y we predict from a 1-percent change in X The percentage change in Y from a 1-percent change in X has a special name in economics—elasticity! Taking logs of both Y and X lets us estimate elasticities! Logarithms in Econometrics (cont.)

14. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-14 Example: The Phillips Curve If we log both Unemployment and Inflation, then we can predict the percentage change in unemployment resulting from a one percent change in inflation. Percentage changes are nonlinear. Changing inflation from 0.01 to 0.02 is a 100% increase. Changing inflation from 0.02 to 0.03 is only a 50% increase. If we plot our regression line estimated on lnY and lnX on a graph of Y and X, our line will look nonlinear.

15. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-15 Figure 4.6 A Logarithmic Phillips Curve

16. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-16 TABLE 4.2 A Logarithmic Phillips Curve

17. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-17 Example: The Phillips Curve A 1-percent increase in Inflation leads to a -0.30% change in Unemployment The inflation elasticity of unemployment is -0.30

18. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-18 Checking Understanding What if we log only one variable? Regress

19. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-19 Checking Understanding (cont.) What if we log only one variable? Regress  1 is the percentage change in Y for a 1-unit change in X. We call this specification “semi-log.”

20. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-20 Checking Understanding (cont.) What if we log only one variable?  1 is the percentage change in Unemployment from a 1-unit change in Inflation. We call this specification “semi-log.” For example, a 1 point increase in inflation causes a 100·  1 % change in unemployment.

21. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-21 Figure 4.7 A Semilog Phillips Curve

22. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-22 TABLE 4.3 A Semilog Phillips Curve

23. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6S-23 Semi-Log Specifications Regress – A 1-unit increase in X leads to a 100·  1 % change in Y Regress – A 1% increase in X leads to a 0.01·  1 unit change in Y