1 MF-852 Financial Econometrics Lecture 9 Dummy Variables, Functional Form, Trends, and Tests for Structural Change Roy J. Epstein Fall 2003

2 Topics 0-1 Dummy Variables Linear Trend Transformations of Variables Tests for Structural Change

3 Dummy Variables H 0 often involves a change in a regression coefficient. Example: Y i is cheese dogs consumed at party by ith person. Use regression to estimate mean number of cheese dogs eaten: Y i =  0 + e i Does the mean differ between men and women?

4 Dummy Variables A dummy variable D has the value 0 or 1. 0 is for a “baseline” group 1 is for a “contrast” group. Suppose women are the baseline. Then D i = 0 if the ith person is female, otherwise D i = 1. What if men were the baseline?

5 Dummy Variables H 0 : men eat same number of cheese dogs on average New regression is Y i =  0 +  1 D i + e i Female mean =  0 ; Male mean =  0 +  1 Test H 0 by testing significance of  1.

6 Dummy Variables Suppose 3 categories: men, women, children. H 0 : same mean for all. Define 2 “dummies”: D 1i = 1 if woman, else D 1i = 0 D 2i = 1 if child, else D 2i = 0 Regression is Y i =  0 +  1 D 1i +  2 D 2i + e i Effects:  0 ;  0 +  1 ;  0 +  2 Test H 0 with F test on  1 and  2.

7 Functional Form We have specified a multiple regression as linear function: Y i =  0 +  1 X 1i +  2 X 2i + … +  k X ki + e i But we have a LOT of flexibility in defining the variables.

8 Transformations of Variables Examples: Z i = ln(X i )Z i = 1/X i Z i = X i 2 Z i = X i – X i –1 (first difference) Z i = (X i – X i –1 )/X i –1 (% change) Z i = ln(X i /X i –1 )(compound g)

9 More Examples of Valid Transformations Suppose Y i = a 0 X i a 1 exp(e i ) where a 0 and a 1 are coefficients. Take logs of both sides: ln(Y i ) =  0 + a 1 ln(X i ) + e i This is a linear regression model!  0 = ln(a 0 )

10 Transformations in General We allow any term with 1 regression coefficient factored out in front. Y i =  0 +  1 [ln(X 1i )*X 2i ] –  2 X 2 3i –1 But not Y i =  0 +  1 ln(X 1i )*X 2i *  2 X 2 3i –1

11 Trend Trend: the average increase (decrease) in Y i each period, after controlling for other factors. Only makes sense for time-series data. Define trend variable T i = i. T 1 = 1, T 2 = 2, etc. Y i =  0 +  1 T i +  2 X i + e i

12 Trend Interpretation: Y changes on average by  1 units each period, after controlling for X. Reflects net effect of omitted variables. Other trend models: Ln(Y i ) =  0 +  1 T i +  2 X i  1 is average percent change in Y each period, after controls.

13 Structural Change We assume that the model describes all of the data but this may not be accurate. The earlier example of a single mean for TV viewing for all populations (men, women, children) is simplest case where assumption might not be valid.

14 Structural Change Testing, Generally H0 defines categories of interest in data, e.g., Genders, age groups, geographic locations (cross-section data) Old vs. recent observations, special time periods (war, different regulatory regime) (time-series data). Define a dummy variable for each category other than the chosen baseline group.

15 Structural Change Testing, Generally Include the dummy variables in the regression. This allows the different categories to have different intercepts. Equivalent to allowing different means. Y i =  0 +  1 D i +  2 X i + e i Test significance of dummies with t or F test, as appropriate.

16 Structural Change Testing, Generally Next level of sophistication is to allow different categories to have different slopes for X i. Create “interaction” term D i X i. Y i =  0 +  1 D i +  2 X i +  3 D i X i + e i Test significance of  1 and  3 with F test. Can do this with categories > 2.

17 Structural Change Examples CAPM (time-series): (A)You estimate model to test if returns were significantly different during a subperiod in the data. This is an “event study.” (B)You estimate model with 20 weekly returns. Beta might have been different for the first 10 weeks.

18 Structural Change Examples Cross-section: Model for prices charged by stores in different locations. Do stores have different prices after controlling for their costs? (from Staples-Office Depot merger) Baseball player salaries depend on years of experience and the square of experience. Does the player’s position also affect salary?

19 Testing for Structural Change CAPM (A). Want to test if returns were higher in weeks Define D i = 0 if i 12. Otherwise D i = 1. Y i =  0 +  1 D i +  2 X i + e i Perform test of significance on  1.

20 Testing for Structural Change CAPM (B). Want to test if beta was different for weeks Define D i = 0 if i > 10. Otherwise D i = 0. Y i =  0 +  1 D i +  2 X i +  3 (D i X i )+ e i Perform F test on  1 and  3.

21 Testing for Structural Change Store model. 50 stores in 3 different cities. Test if average markup is different across cities. Define D 1i =1 if in city 2, else = 0. Define D 2i =1 if in city 3, else = 0. Y i =  0 +  1 D 1i +  2 D 2i +  3 X i + e i Perform F test on  1 and  2.

22 Warning! Amount of data will limit how many structural changes you can test for. Model needs at least 5 data points per estimated coefficient (Epstein’s rule of thumb). So you can’t introduce lots of dummies indiscriminately. Slope changes are harder to measure than intercept changes.