ECON Interactions and Dummies

Transcription

1 ECON Interactions and Dummies Maggie Jones 1 / 25

2 Readings Chapter 6: Section on Models with Interaction Terms Chapter 7: Full Chapter 2 / 25

3 Interaction Terms with Continuous Variables In some regressions we might expect the partial effect of one variable to depend on the magnitude of another variable An example of this is the effect of adding another bedroom to a house on the price of a house might vary based on the original size of the house: large houses should have a higher increase in house price when adding a new bedroom 3 / 25

4 Interaction Terms with Continuous Variables We can formalize this as (for example) y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 1 x 2 + u Where now, the partial effect of x 2 on y is: dy dx 2 = β 2 + β 3 x 1 So that the effect of a one unit increase in x 2 on y depends on what the value of x 1 is 4 / 25

5 Dummy Variables Sometimes we wish to incorporate qualitative information into our regressions Examples of qualitative information could be the colour of a car, city in which someone lives, whether a person identifies as male or female, etc. A dummy variable (also called a binary variable) is one that contains information on two outcomes (male or female, owns a computer or does not, is white or non-white, etc) These variables are represented in regression analysis as a 0 or 1 5 / 25

6 Why do we use the values zero and one to describe qualitative information? In a sense, these values are arbitrary: any two different values would do. The real benefit of capturing qualitative information using zero-one variables is that it leads to regression Example: models where Female the parameters have very and natural interpretations, Married as we will see now. are Dummy Variables Table 7.1 A Partial Listing of the Data in WAGE1.RAW person wage educ exper female married / 25

7 Dummy Variables When we include dummy variables in the regression model, we interpret the effect of the dummy variables as a change in intercept. E.g., y = β 0 + δ 0 D + β 1 x + u where now D = 1 if the individual belongs to a specific group and 0 otherwise, x is a continuous variable as before We can evaluate the change in intercept by taking the expected value of y conditional on x and D: E[y D = 0, x] = β 0 E[y D = 1, x] = β 0 + δ 0 The difference: E[y D = 0, x] E[y D = 1, x] is the change in intercept from belonging to the group identified by D: δ 0 7 / 25

8 Dummy Variables Example: effect on wages from being female and controlling for education: wage = β 0 + δ 0 female + β 1 education + u where female = 1 if the individual identifies as female and 0 otherwise, education is a continuous variable as before We can show the change in intercept graphically for a hypothetical wage regression: 8 / 25

9 Figure 7.1 Graph of wage = 0 0 female 1 educ for 0 0. wage men: wage = 0 1 educ women: wage = ( 0 0 ) + 1 educ slope = educ 9 / 25

10 Dummy Variables The graph on the previous slide shows an example of the wage regressions when there is a positive intercept term, β 0 and when women have a lower average wage compared to men, conditional on their level of education (δ 0 < 0) Note that by including a dummy variable for female in the regression model, we have indirectly selected a base group i.e. a group whose average wage (given that education is 0) is equal to the intercept term All other dummy variables in the regression are measured with respect to the base group e.g., δ0 measures the change in intercept given that you are female, relative to male (or non-female) 10 / 25

11 Dummy Variables for Multiple Categories It is often the case that we have a qualitative variable that has more than two groups e.g., race: black, white, hispanic, indigenous, asian, etc e.g., highest level of schooling: no school, college, bachelor s degree, master s, etc e.g., geographic location: Western Canada, Eastern Canada, Northern Canada 11 / 25

12 Dummy Variables for Multiple Categories We can use dummy variables for multiple categories in the same way that we use dummy variables for two categories Note that if we have g groups, we can only include g 1 dummy variables in the regression model If we include g dummy variables then the constant term will be an exact linear combination of the dummy variables and thus we will have perfect collinearity - this is called the dummy variable trap 12 / 25

13 Dummy Variables for Multiple Categories For example, suppose we are interested in whether or not different races have different wages, conditional on education We can split race into: black, white, hispanic, indigenous, asian, other Then our regression model may look like the following: wage = α 0 + δ 1 black + δ 2 hisp + δ 3 indig + δ 4 asian + δ 5 other + α 1 educ + u where we have omitted white so that all other dummy variables are measured with respect to white α 0 is thus the intercept term when all other dummies are equal to 0 - i.e., the intercept for white 13 / 25

14 Dummy Variables for Multiple Categories We can work this out for each category with expectations as follows: E(wage white, educ = 0) = α0 E(wage black, educ = 0) = α0 + δ 1 E(wage hisp, educ = 0) = α0 + δ 2 E(wage indig, educ = 0) = α0 + δ 3 E(wage asian, educ = 0) = α0 + δ 4 E(wage other, educ = 0) = α0 + δ 5 14 / 25

15 Interactions Involving Dummy Variables We can interact two dummy variables to see if the effect of one category depends on belonging to the other category e.g. does the effect of race on wage vary by gender? e.g. does the effect of gender on wage vary by marital status? e.g. does the effect of parental education on educational achievement vary by single parent households? 15 / 25

16 Interactions Involving Dummy Variables For instance, the question does the effect of gender on wage vary by marital status can be addressed with the following regression: wage = β 0 + β 1 female + β 2 married + β 3 female married + u The interpretation of each of the dummy variables is as an intercept shift relative to the omitted category Here, the omitted category is the value of the intercept when all dummies are equal to 0: f = 0, m = 0, f m = 0, i.e. the intercept for non-female, non-married people 16 / 25

17 Interactions Involving Dummy Variables and Continuous Variables We have established that dummy variables shift the intercept up or down depending on group membership Interacting two dummy variables translates to another shift in intercept Interacting a dummy variable with a continuos variable is interpreted as a change in slope There are many relationships that might vary based on group membership e.g. does the number of weeks worked increase wages more for people with higher levels of schooling? e.g. is the return to education greater for men or women? is it greater for minorities or non-minorities? 17 / 25

18 Interactions Involving Dummy Variables and Continuous Variables We ll consider as an example whether the effect of education on wages varies by gender: wage = β 0 + δ 0 female + β 1 education + δ 1 female education + u E(wage fem = 0, edu = 0) = β 0 is the intercept term for non-female E(wage fem = 1, edu = 0) = β 0 + δ 0 is the itnercept for female dwage dedu = β 1 is the change in wage associated with a one unit change in education for non-female dwage dedu = β 1 + δ 1 is the change in wage associated with a one unit change in education for female 18 / 25

19 Figure 7.2 Graphs of equation (7.16). (a) 0 0, 1 0; (b) 0 0, 1 0. wage wage men women men women (a) educ (b) educ / 25

20 Note that all the standard t statistics and F statistics are computed as before. Nothing changes with statistical testing for interactions or dummy variables. The only difference between interactions and dummy variables from what we saw in the previous chapters is our interpretation of the regression coefficients. 20 / 25

21 Testing for Differences in Regression Functions Across Groups We may be faced with situations in which we hypothesize that the entire regression function is different for two groups Start with the standard regression function y = β 0 + β 1 x β k x k + u If the entire regression function were different based on some group membership, e.g. D = 1 if belong to group 1, D = 0 otherwise, then we can rewrite as: y = β 0 + δ 0 D + β 1 x β k x k + δ 1 x 1 D + + δ k x k D + e Here we have allowed the intercept to vary based on group membership and all of the slope parameters to vary based on group membership 21 / 25

22 Testing for Differences in Regression Functions Across Groups We may be faced with situations in which we hypothesize that the entire regression function is different for two groups Start with the standard regression function y = β 0 + β 1 x β k x k + u (1) If the entire regression function were different based on some group membership, e.g. D = 1 if belong to group 1, D = 0 otherwise, then we can rewrite as: y = β 0 +δ 0 D+β 1 x 1 + +β k x k +δ 1 x 1 D+ +δ k x k D+e (2) Here we have allowed the intercept to vary based on group membership and all of the slope parameters to vary based on group membership 22 / 25

23 Testing for Differences in Regression Functions Across Groups If the regression function were exactly the same for group 1 and group 2, then δ 0 = δ 1 = = δ k = 0 This suggests that a natural way to test this hypothesis would be to use an F statistic H 0 : δ 0 = δ 1 = = δ k = 0 H a : at least one δ k 0 (2) is the unrestricted model (1) is the restricted model 23 / 25

24 Testing for Differences in Regression Functions Across Groups Then the F statistic is: F = (SSR R SSR UR )/(k + 1) SSR UR /(n 2k 2) Which follows a F k+1,n 2k 2 distribution This statistic is called the Chow statistic 24 / 25

25 Testing for Differences in Regression Functions Across Groups The Chow statistic can also be computed as: F = (SSR R SSR 1 SSR 2 )/(k + 1) (SSR 1 + SSR 2 )/(n 2k 2) Where SSR 1 is the SSR obtained from running the restricted model using only the sample of data from group 1 and SSR 2 is the SSR obtained from running the restricted model using only the sample of data from group 2 This suggests that there are two ways to compute the F -statistic: one using the restricted and unrestricted models, and one using the restricted model on the full sample, group 1 sample, and group 2 sample 25 / 25