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IV_B_Binary_Variables_2011

IV_B_Binary_Variables_2011 - IV B 1 James B McDonald...

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IV B 1 James B. McDonald Brigham Young University 10/11/2011 IV. Miscellaneous Topics B. Binary Variables (Dummy Variables) Many variables, which we may want to include in an econometric model, may not be quantitative (measurable), but rather are qualitative in nature. For example, an individual will be a homeowner, or will not; will be married or not. Such characteristics may have a bearing on an individual's behavior, but are not quantifiable. One way to include the effect of such characteristics is to introduce binary or dummy variables. For example, let the binary variable D t indicate whether a given individual is married or not by defining D t = 0 if the t th individual is single and D t = 1 if the t th individual is married. We now consider several models which make use of dummy variables, discuss the dummy variable trap, indicate some interesting generalizations, and investigate applications of these techniques to several problems in economics. 1. Models with binary explanatory variables a. An example: the relationship between salary and a college degree Let Y t = Annual salary of the t th person in the sample, D 1t = 1 if the t th person is a college graduate = 0 otherwise, D 2t = 1 if the t th person isn't a college graduate = 0 otherwise. Note that D 2t = 1 - D 1t Consider the following two models which can be used to study the impact of a college degree on annual salary. Model 1: Y t = α 1 + α 2 D 1t + ε t Model 2: Y t = β 1 D 1t + β 2 D 2t + ε t .
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IV B 2 The coefficients in the two representations have different interpretations as summarized in the following table. E(Y t ) E(Y t D1=1) Model 1 α 1 + α 2 Model 2 β 1 E(Y t D2=1) α 1 β 2 In the model with one fewer dummy variables than categories (model 1; categories = college graduate, not a college graduate) the coefficient of the binary variable represents the expected difference or differential between the income levels associated with state of the included dummy variable and the state (bench mark) associated with the deleted dummy variable, i.e., α 2 = E(Y t graduate) - E(Y t not a college graduate) The coefficients in the representation which includes the same number of binary variables as categories (model 2) represent the expected income level associated with each category. b. Estimation : Assume that we have a total of n observations with the first n 1 (n 1 + n 2 = n) having college degrees. The two different models can be written in matrix notation as Model 1: n 2 1 2 1 n 2 1 + 0 1 0 1 1 1 1 1 = Y Y Y or Y = X α + ε
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IV B 3 Model 2: n 2 1 2 1 n 2 1 + 1 0 1 0 0 1 0 1 = Y Y Y or Y = X*β + ε . The least squares estimators of the vectors α and β are given by ˆ = (X'X) -1 X'Y ˆ ˆ = Y - Y Y = 2 1 2 1 2 and ˆ = (X*'X*) -1 X*'Y ˆ ˆ = Y Y = 2 1 2 1 where Y 1 and Y 2 respectively, denote the sample mean income for those having college degrees and those without a degree. Note that these are sample estimates (sample means) of the population means.
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