Econometrics Multiple Regression Analysis with Qualitative Information: Binary (or Dummy) Variables

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1 Econometrics Multiple Regression Analysis with Qualitative Information: Binary (or Dummy) Variables João Valle e Azevedo Faculdade de Economia Universidade Nova de Lisboa Spring Semester João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 11

2 A dummy (or binary) variable takes on the value 0 or 1 and are often used as regressors Dummy variables are also called binary variables, for obvious reasons Examples: wage = β 0 + β 1 Education + β 2 Experience + β 3 Male + u Male i (= 1 if individual i is male, 0 if female) If β 3 > 0, there is evidence of discrimination against women wage = β 0 + β 1 Education + β 2 Experience + β 3 Sporting + u Sporting i (= 1 if individual i supports Sporting) Can use all the inference tools learned so far, as long as the necessary assumptions hold! João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 11

3 Consider a simple model with one continuous variable (x) and one dummy (d) y = β 0 + δ 0 d + β 1 x + u This can be interpreted as an intercept shift If d=0, then y = β0 + β 1 x + u If d=1, then y = (β0 + δ 0 ) + β 1 x + u We call the group of individuals with d=0 is the base group João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 11

4 Example with δ 0 > 0 where Gauss-Markov assumptions hold E[y x,d] E[y x,d=1] = (b 0 + d 0 ) + b 1 x d = 1 d 0 slope = b 1 d = 0 b 0 E[y x,d=0] = b 0 + b 1 x x João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 11

5 : Alternative Formulations Consider a model with one dummy variable (d) y = β 0 + δ 0 Male + β 1 x + u, where Male is either 0 or 1 The intercept for females is β0 The intercept for males is β0 + δ 0 Could write instead: y = β 0 Female + δ 0 Male + β 1 x + u The intercept for females is β 0 The intercept for males is δ 0 But never ever write a model like: y = β 0 + δ 1 Female + δ 0 Male + β 1 x + u, Why? Violates Assumption MLR.3 (No perfect Collinearity) João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 11

6 Dummies for Multiple Categories Can use dummy variables to control for multiple categories wage = β 0 +β 1 Education+β 2 Experience +β 3 Sporting +β 4 Benfica+u Sporting i (=1 if individual i supports Sporting, 0 otherwise) Benficai (=1 if individual i supports Benfica, 0 otherwise) As long as our sample contains supporters of both teams and supporters of other teams (or not supporting any team) Could include additionally a dummy Porto, as long as our sample contains supporters of the three teams and supporters of other teams (or not supporting any team). Why? Otherwise we would violate Assumption MLR.3 (No perfect Collinearity) João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 11

7 Multiple Categories If we have n categories we should have n-1 dummy variables since the base group is represented by the intercept β 0 Example: In our wage regression, we want to include dummies for physical attractiveness. Categories are: Below average, Average and Above average. We have 3 categories, so use 2 dummies wage = β 0 + β 1 BelowAverage + β 2 AboveAverage + otherfactors + u So, we chose Average individuals as the base group Can transform continuous variable into categories (e.g., transform years of schooling into dummies Elementary School, Middle School, High-School, College, Post-Graduate Studies etc.). Be sure to leave one group as reference group (e.g., No Schooling) João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 11

8 Interactions among Dummies Suppose we want to interact the dummies for physical attractiveness with the dummy for gender: have a total of 6 categories (must have 5 parameters besides the intercept) wage = β 0 + β 1BelowAverage + β 2AboveAverage + β 3BelowAverage Male + +β 4AboveAverage Male + β 5Male + otherfactors + u Interacting dummy variables is like subdividing the group Here the base group is Female with Average looks (measured by β 0 ) João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 11

9 More on Interactions among Dummies wage = β 0 + β 1BelowAverage + β 2AboveAverage + β 3BelowAverage Male + β 4AboveAverage Male + β 5Male + otherfactors + u If Male=0 and BelowAverage=0 and AboveAverage=1 we are talking about a Female in the category AboveAverage. The measured effect on this group is β 0 + β 2 If Male=1 and BelowAverage=0 and AboveAverage=0 we are talking about a Male in the category Average. The measured effect on this group is β 0 + β 5 (...) João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 11

10 Other Interactions among Dummies Can also consider interacting a dummy variable, Male, with a continuous variable, Education measured in years y = β 0 + δ 1 Male + β 1 Education + δ 2 Male Education + u If Male=0, then y = β 0 + β 1 Education + u If Male=1, then y = (β0 + δ 1 ) + (β 1 + δ 2 )Education + u Allows us to investigate whether the effect of Education is different across the two groups João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 11

11 Example of δ 0 > 0 and δ 1 < 0 y y = b 0 + b 1 Education Male = 0 Male = 1 y = (b 0 + d 0 ) + (b 1 + d 1 ) Education Education João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 11

Econometrics Multiple Regression Analysis: Heteroskedasticity

Econometrics Multiple Regression Analysis: João Valle e Azevedo Faculdade de Economia Universidade Nova de Lisboa Spring Semester João Valle e Azevedo (FEUNL) Econometrics Lisbon, April 2011 1 / 19 Properties