Otherwise model 2 contains one quantitative variable

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otherwise Model (2) contains one quantitative variable (years of teaching experience) and one qualitative variable (sex) that has two classes, namely, male and female. Assuming, as usual, that E( u t ) = 0, we see that Mean salary of a female college professor: E (Y t \ X t , D t = 0 ) = 1 + X t Mean salary of a male college professor: E(Y t \X t , D t =1) = ( 1 + 2 )+ X t
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ECON 301 - Introduction to Econometrics I May 2013 METU - Department of Economics Instructor: H. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 4 So, model (2) postulates that the male and female college professors’ salary functions in relation to the years of teaching experience have the same slope ( ) but different intercepts. In other words, it is assumed that the level of the male professor’s mean salary is different from that of the female professors’ mean salary (by 2 ) but the rate of change in the mean annual salary by years of experience is the same for both sexes. II. Dummy Variable Trap To distinguish the two categories, male and female, we have introduced only one dummy variable D t . For if D t = 1 always denotes a male, when D t =0 we know that it is a female since there are only two possible outcomes. Hence, one dummy variable suffices to distinguish two categories (this is the second parameterization). Let us assume that the regression model contains an intercept term; if we were to write model (2) as: Y t = 1 + 2 D t1 + 3 D t2 + X t + u t (3) where Y and X are as defined before D t1 = 1 if male professor 0 otherwise
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ECON 301 - Introduction to Econometrics I May 2013 METU - Department of Economics Instructor: H. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 5 D t2 = 1 if female professor 0 otherwise then model (3), as it stands, can not be estimated because of perfect collinearity between D t1 and D t2 . To see this, suppose we have a sample of three male professors and two female professors. The data matrix will look something like that following: Y t X t0 D 1 D 2 X t1 Male Y 1 1 1 0 X 11 Male Y 2 1 1 0 X 21 Female Y 3 1 0 1 X 31 Male Y 4 1 1 0 X 41 Female Y 5 1 0 1 X 51 The first column on the right hand side of the preceding data matrix represents the common intercept term 1 . Now it can be seen readily that D 2 = 1 - D 1 or D 1 = 1 - D 2 , that is, they are perfectly collinear. As known, in the case of perfect multicollinearity the usual OLS estimation is not possible. Then simplest way of solution is to obtain second parametrization by substituting D 2 = 1 - D 1 or D 1 = 1 - D 2 . In this case, the data matrix will not have the column labeled D 2 , thus avoiding the perfect multicollinearity problem. The general rule is if a qualitative variable has m categories, introduce only m-1 dummy variables (second parameterization). So if we have m categories and an intercept term in the model, in the case
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ECON 301 - Introduction to Econometrics I May 2013 METU - Department of Economics Instructor: H. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 6 of using m dummy variables we fall into what might be called the dummy variable trap , that is, the situation of perfect multicollinearity.
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