CORNELL ECON 3120 - Omitted Variable Bias with Many Regressors (2 pages)

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Omitted Variable Bias with Many Regressors



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Econ 3120 1st Edition Lecture 15 Outline of Current Lecture I Omitted Variable Bias with Many Regressors Current Lecture II Dummy Variables Dummy Variables Dummy variables aka binary variables indicator variables or dichotomous variables are simply variables that take on a value of 0 or 1 They indicate a single status of the observation Some examples female 1 for female 0 for male non white 1 if race is non white 0 if white urban 1 if the person lives in an urban area 0 if lives in a rural area Note that we could also define our dummy variables to indicate male white or rural but it turns out not to matter more on this below Dummy variables change the intercept of the regression equation For example suppose we want to examine the relationship between test scores and class sizes in primary schools We think that the gender of the child also has an effect on test scores so we include it in the model We therefore model the relationship as score 0 1 f emale 2clsize u 1 How do we interpret 1 1 actually represents a shift in the intercept associated with the gender of the child To see this take the conditional expectation for females and for males E score f emale 0 clsize 0 2clsize E score f emale 1 clsize 0 1 2clsize The difference between these two equations is simply a shift in the intercept from 0 to 0 1 1 score Slope 2 1 0 female male class size This interpretation easily generalizes to situations with more independent variables The coeffi cients on the continuous variables i e slope coefficients remain the same for different values of the dummy variable but the dummy variable shifts the intercept What would happen if you included the dummy variable male in the equation where male 1 if the child is a male and 0 if she is female You would therefore be running the regression score 0 1 f emale 2clsize 3male u It is not possible to run this regression because male is simply a linear combination of f emale male 1 f emale This violates Assumption MLR 3 If you tried to do this in Stata the program would drop one of these dummy variables for you Thus you could include either male or f emale but not both It turns out not to matter which one you include If you ran the regression score 0 1male 2clsize u 2 Then using male 1 f emale you can show that 2 becomes 1 when you set 0 0 1 and 1 1 Note that we can use dummy variables if we have more than two categories Suppose that we have 3 categories for race white black and other We run the regression including two of these variables score 0 2size 3white 4black u Where again we have to exclude other since other 1 white black Interactions between dummy variables 2 We can interact dummy variables to create individual intercepts for each sub category within the two dummy variables Suppose we interact f emale with a variable public indicating whether the student is in a public school The new var ables are constructed as f emale public 1 if f emale 1 public 1 0 These notes represent a detailed interpretation of the professor s lecture GradeBuddy is best used as a supplement to your own notes not as a substitute otherwise We can therefore run the regression score 0 1 f emale 2clsize 3 public 4 f emale public u 3 This regression implies a separate intercept for each gender x school type category To see this take conditional expectations to yield the following intercepts private public male 0 0 3 f emale 0 1 0 1 3 4 Suppose we hadn t included the interaction and instead ran the regression score 0 1 f emale 2clsize 3 public u 4 In equation 4 we are not letting the intercept vary by each individual gender x school type category We are assuming that the difference in mean test scores for females is the same in both public and private school In equation 3 on the other hand the effect of being female in a private school is 1 while the effect of being female in a public school is 1 4 Thus 4 allows the effect of gender to vary by school type The parameter represents the difference in test scores between females and males in public schools relative to that difference in private schools holding class size constant 1 Interactions between dummy variables and continuous variables We can interact dummy variables and other variables to change a slope coefficient in our regression Suppose we would like to test whether the effect of class size on test scores differs by 1This is sometimes called a difference in difference estimator because it can be written as follows 4 E score f emale 1 public 1 clsize E score f emale 0 public 1 clsize E score f emale 1 public 0 clsize E score f emale 0 public 0 clsize 3 gender We generate a new variable that equals f emale clsize and regress score 0 1 f emale 2clsize 3 f emale clsize u Differentiating with respect to class size for each value of f emale and holding u constant yields the following slope coefficients score clsize f emale 0 2 score clsize f emale 1 2 3 Thus 3 represents the difference in the effect of class size on test scores for females relative to males score 1 female slope 2 3 0 male l i slope 2 class size


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