Multiple Regression Analysisy = b0 + b1x1 + b2x2 + . . . bkxk + u5. Dummy Variables1Dummy Variables A dummy variable is a variable that takes on the value 1 or 0 Examples: male (= 1 if are male, 0 otherwise), south (= 1 if in the south, 0 otherwise), etc. Dummy variables are also called binary variables, for obvious reasons2A Dummy Independent Variable Consider a simple model with one continuous variable (x) and one dummy (d) y = b0 + d0d + b1x + u This can be interpreted as an intercept shift If d = 0, then y = b0 + b1x + u If d = 1, then y = (b0 + d0) + b1x + u The case of d = 0 is the base group3Example of d0 0xyd0 b0y = (b0 + d0) + b1xy = b0 + b1xslope = b1d = 0d = 14Dummies for Multiple Categories We can use dummy variables to control for something with multiple categories Suppose everyone in your data is either a HS dropout, HS grad only, or college grad To compare HS and college grads to HS dropouts, include 2 dummy variables hsgrad = 1 if HS grad only, 0 otherwise; and colgrad = 1 if college grad, 0 otherwise5Multiple Categories (cont) Any categorical variable can be turned into a set of dummy variables Because the base group is represented by the intercept, if there are n categories there should be n 1 dummy variables If there are a lot of categories, it may make sense to group some together Example: top 10 ranking, 11 25, etc.6Interactions Among Dummies Interacting dummy variables is like subdividing the group Example: have dummies for male, as well as hsgrad and colgrad Add male*hsgrad and male*colgrad, for a total of 5 dummy variables 6 categories Base group is female HS dropouts hsgrad is for female HS grads, colgrad is for female college grads The interactions reflect male HS grads and male college grads7More on Dummy Interactions Formally, the model is y = b0 + d1male + d2hsgrad + d3colgrad + d4male*hsgrad + d5male*colgrad + b1x + u, then, for example: If male = 0 and hsgrad = 0 and colgrad = 0 y = b0 + b1x + u If male = 0 and hsgrad = 1 and colgrad = 0 y = b0 + d2hsgrad + b1x + u If male = 1 and hsgrad = 0 and colgrad = 1 y = b0 + d1male + d3colgrad + d5male*colgrad + b1x + u8Other Interactions with Dummies Can also consider interacting a dummy variable, d, with a continuous variable, x y = b0 + d1d + b1x + d2d*x + u If d = 0, then y = b0 + b1x + u If d = 1, then y = (b0 + d1) + (b1+ d2) x + u This is interpreted as a change in the slope9yxy = b0 + b1xy = (b0 + d0) + (b1 + d1) xExample of d0 0 and d1 0d = 1d = 010Testing for Differences Across Groups Testing whether a regression function is different for one group versus another can be thought of as simply testing for the joint significance of the dummy and its interactions with all other x variables So, you can estimate the model with all the interactions and without and form an F statistic, but this could be unwieldy11The Chow Test Turns out you can compute the proper F statistic without running the unrestricted model with interactions with all k continuous variables If run the restricted model for group one and get SSR1, then for group two and get SSR2 Run the restricted model for all to get SSR, then12The Chow Test (continued) The Chow test is really just a simple F test for exclusion restrictions, but weve realized that SSRur = SSR1 + SSR2 Note, we have k + 1 restrictions (each of the slope coefficients and the intercept) Note the unrestricted model would estimate 2 different intercepts and 2 different slope coefficients, so the df is n 2k 2 13Linear Probability Model P(y = 1|x) = E(y|x), when y is a binary variable, so we can write our model as P(y = 1|x) = b0 + b1x1 + + bkxk So, the interpretation of bj is the change in the probability of success when xj changes The predicted y is the predicted probability of success Potential problem that can be outside 0,114Linear Probability Model (cont) Even without predictions outside of 0,1, we may estimate effects that imply a change in x changes the probability by more than +1 or 1, so best to use changes near mean This model will violate assumption of homoskedasticity, so will affect inference Despite drawbacks, its usually a good place to start when y is binary15Caveats on Program Evaluation A typical use of a dummy variable is when we are looking for a program effect For example, we may have individuals that received job training, or welfare, etc We need to remember that usually individuals choose whether to participate in a program, which may lead to a self- selection problem16Self-selection Problems If we can control for everything that is correlated with both participation and the outcome of interest then its not a problem Often, though, there are unobservables that are correlated with participation In this case, the estimate of the program effect is biased, and we dont want to set policy based on it!17