1. 1Spring 02 Non-Linear Relationships If a relationship is not linear, how can we deal with it. For example:

2. 2Spring 02 Non-Linear Relationships One possibility is to take the natural logarithm of both sides. The natural logarithm is the inverse of the natural exponential function. Natural exponent:

3. 3Spring 02 Graph of e x

4. 4Spring 02 Graph of ln x

5. 5Spring 02 Non-Linear Relationships Natural logs and exponential functions: e 0 =1 ln 1 = 0 Logarithms have certain properties: ln (XY) = ln X+ ln Y ln(X/Y) = ln X – ln Y ln(1/X) = ln 1 – ln X = - ln X ln a x = x lnA

6. 6Spring 02 Non-Linear Relationships

7. 7Spring 02 Non-Linear Relationships

8. 8Spring 02 Elasticities When you take the natural logs of a non-linear relationship and estimate the equation, the estimated coefficients are also the elasticities.

9. 9Spring 02 Dummy Variables Used to quantify qualitative differences. For example, Consumption as a function of income may be affected by wartime/peacetime Income as a function of education may be affected by gender Quantity demanded of ice cream as a function of price may be affected by season.

10. 10Spring 02 Dummy Variables How does consumption differ in wartime from peacetime? Yd C Cp Cw C Cp Yd Cw C Cp Yd Cw

11. 11Spring 02 Dummy Variables

12. 12Spring 02 Example Salvatore: Below is the quantity supplied of milk by a dairy for 14 months but for 3 of those months there was a strike

13. 13Spring 02 Example If we estimate quantity supplied as a function of price without taking into consideration strike, then we would get these results: If we estimate quantity supplied as a function of price taking into consideration strike (just affecting the constant), then we would get these results: If strike: If no strike: (1.05) (0.76) (-3.26) (13.9) (-21.2)

14. 14Spring 02 Example If we estimate quantity supplied as a function of price taking into consideration strike (just affecting slope), then we would get these results: If strike: If no strike: If we estimate quantity supplied as a function of price taking into consideration strike (affecting both the constant and slope), then we would get these results: If strike: If no strike:

15. 15Spring 02 Wald Test Revisited At least one of the above betas is not zero.

16. 16Spring 02 Example We can test whether the explanatory variables including strike as a set are significant in explaining price by using a Wald Test:

17. 17Spring 02 Test for Structural Change Data 7-19 contains data from 1960-1988 on the demand for cigarettes in Turkey and its determinants. Estimate the whole equation and also estimate whether two anti-smoking campaigns had their desired effect. In late 1981, health warning were issued in Turkey regarding the hazards of cigarette smoking. In 1986, one of the national newspapers launched an antismoking campaign.

18. 18Spring 02 Smoking Problem On levels On natural logs

19. 19Spring 02 Smoking Problem Regression with dummy variables affecting constant for after 1982 and 1986:

20. 20Spring 02 Smoking Problem Regression with dummy variables affecting constant and slope for after 1982 and 1986:

21. 21Spring 02 Smoking Problem Regression with dummy variables affecting just the slope for after 1982 and 1986:

22. 22Spring 02 Coefficients of Different Regressions To test whether the assumptions of two different regressions is correct, we start with the null hypothesis that the regressions are identical and see whether or not we can reject the null hypothesis. Test whether the stock market has changed the relationship between consumption and wealth. Test whether the relationship between years of education and income is different for women and men or for different regions of the country.

23. 23Spring 02 Coefficients of Different Regressions To test the null hypothesis: Run a regression on the whole model, N+M observations. Then run two separate regressions.