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ECON301_Handout_11_1213_02

# To see this suppose we have a sample of three male

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. 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: 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. IV. Some Illustrations
ECON 301 - Introduction to Econometrics I May 2013 METU - Department of Economics Instructor: H. Ozan ERUYGUR e-mail: Lecture Notes 7 V. Model Parametrizations Consider the following equations; Y t = B 01 + B 11 X t1 + B 21 X t2 + u t t T 1 (1 st Subsample) Y t = B 02 + B 12 X t1 + B 22 X t2 + u t t T 2 (2 nd Subsample) Let us now define dummy variables for these two sub periods; 1 if t T 1 D t1 = 0 otherwise while

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ECON 301 - Introduction to Econometrics I May 2013 METU - Department of Economics Instructor: H. Ozan ERUYGUR e-mail: Lecture Notes 8 1 if t T 2 D t2 = 0 otherwise 1 st Parametrization : We are combining two models by using dummy variables; Y t = B 01 D t1 +B 11 D t1 X t1 +B 21 D t1 X t2 +B 02 D t2 +B 12 D t2 X t1 +B 22 D t2 X t2 2 nd Parametrization 1 : D t1 + D t2 = 1 D t1 = 1 - D t2 , putting this in the model above we get; Y t = B 01 (1-D t2 )+ B 11 (1-D t2 )X t1 + B 21 (1-D t2 )X t2 + B 02 D t2 + B 12 D t2 X t1 + B 22 D t2 X t2 and we obtain Y t = B 01 +B 11 X t1 + B 21 X t2 + (B 02 - B 01 ) D t2 + (B 12 - B 11 )D t2 X t1 + (B 22 - B 21 )D t2 X t2 + u t 1.subsample is base 02 12 22 ( D t1 = 1 - D t2 ) Intercept Slope Slope dummy dummy dummy 1 We are using (p -1) Dummy variables (p number of subsamples, or categories ) in the 2 nd parametrization. .
ECON 301 - Introduction to Econometrics I May 2013 METU - Department of Economics Instructor: H. Ozan ERUYGUR e-mail: Lecture Notes 9 A. An Example Model Frequently researchers need to include qualitative variables in their regressions to explain variations in the dependent variable. For example household consumption (the dependent variable) cannot be

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