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intro-ects-handout-4

# intro-ects-handout-4 - Introductory Econometrics...

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Introductory Econometrics ECON0701 (2009) 47 4. Multiple regression analysis: Estimation The simple linear regression model, with one explanatory vari- able, is useful in a range of situations however, most problems involve 2 or more explanatory variables that in fl uence the dependent variable e.g., the simple regression model: wage = α + β × education + error does not capture the fact that most people have higher wage income when they have more relevant experience (in current or related jobs) than when they have less experience, other things (such as education level) being equal a better model may be: wage = α + β × educ + γ × experience + ε compared with the simple regression model above, this equation e ff ectively takes experience out of the error term and puts it explicitly in the equation because this equation contains experience explicitly, we will be able to measure the e ff ect of education on wage, holding experience constant (on the other hand, the simple regression model above puts experience in the error term; thus, the e ff ect of education on wage is measured by assuming that experience is uncorrelated with education, a tenuous assumption) moreover, since experience appears in the equation, its coe - cient, γ , measures the ceteris paribus e ff ect of experience on wage , which is also of some interest

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Introductory Econometrics ECON0701 (2009) 48 most of the results for the simple regression model can be ex- tended naturally to this general case, known as the multiple regression model Multiple regression model is also useful for generalizing func- tional relationships between variables for example, suppose wage income is modelled as a quadratic function of education (see Figure 3.15 for an example of nonlin- ear relationship between wage and educ ): wage = α + β × educ + γ × ( educ ) 2 + ε this fi ts the multiple linear regression model with educ and ( educ ) 2 being the 2 explanatory variables 4.1. The multiple regression model a multiple regression model: y = β 0 + β 1 x 1 + β 2 x 2 + ... + β k x k + ε (4.1) interpretation: parameter β j ( j = 1 , 2 , ..., k ) measures the e ff ect of a change in the variable x j upon E ( y ) , all other variables held constant; parameter β 0 is the intercept term (the ‘variable’ to which β 0 is attached to is x 0 = 1 ) e.g. a model with 2 explanatory variables: for i = 1 , ..., n y = β 0 + β 1 x 1 + β 2 x 2 + ε
Introductory Econometrics ECON0701 (2009) 49 interpretation: x ’s a ff ect y separately β 1 = E ( y ) x 1 β 2 = E ( y ) x 2 graphically, (4.1) describes a k + 1 -dimensional “plane”, not a line; for example, (4.1) with k = 2 gives a 3-dimensional plane (see Figure 4.1) note that β 0 = β 0 × 1 = β 0 × x 0 for all observations; therefore, the following 2 regressions are equivalent: (a) regress y on a constant, x 1 , ..., x k ; & (b) generate x 0 = 1 ; regress y on x 0 , x 1 , ..., x k .

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intro-ects-handout-4 - Introductory Econometrics...

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