chapter 3 [article]

chapter 3 [article] - Contents 1 OLS and multiple...

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Unformatted text preview: Contents 1 OLS and multiple regression 1 1.1 OLS estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Multicollinearity 4 2.1 Estimator properties . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Multicollinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Testing 9 3.1 t tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 F tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1 OLS and multiple regression Contents 1.1 OLS estimators Contents • So far, simple regression: one regressor • It will usually be the case that we think multiple variables will help explain the dependent variable: Y i = β 1 + β 2 X 2 i + β 3 X 3 i + ··· + β k X ki + u i ∼ true model • k parameters to estimate Model specification • With many potential explanatory variables, we must decide which to in- clude on the ‘right-hand-side’ of our regression model • Will deal with this later • For this chapter, we will assume that our assumption regarding the form of the true model is correct – i.e.we know which regressors should be included Example: explaining individual earnings • We might think that work experience as well as education might help determine earnings: EARNINGS i = β 1 + β 2 S i + β 3 EXP i + u i (3.1) • EARNINGS is hourly earnings and S is highest grade completed as before; EXP is years spent working after education • Just as with simple regression, we have OLS estimators of our β i param- eters • Conceptually identical to before: the OLS estimators b i of the (assumed) true model parameters β i are those that minimise the residual sum of squares RSS : RSS = n X i =1 e 2 i = n X i =1 ˆ Y i- Y i 2 = n X i =1 ([ b 1 + b 2 X 2 i + ··· + b k X ki ]- Y i ) 2 • Minimising the RSS with respect to the k (unknown) b i coefficients gives us k first-order conditions: δRSS δb 1 = 0 , δRSS δb 2 = 0 , ..., δRSS δb k = 0 • We have k equations in k unknowns (the b i ) ⇒ can solve for the b i to get our OLS estimators in terms of the X and Y sample data IMPORTANT While the principle underlying the derivation of OLS is unchanged when we move from simple to multiple regression, the expressions for the estimators are NOT the same • For example, with two regressors X 2 and X 3 , it is not the case that b 2 is the same as before: b 2 6 = ∑ n i =1 ( X 2 i- ¯ X 2 )( Y i- ¯ Y ) ∑ n i =1 ( X 2 i- ¯ X 2 ) 2 2 • For actual b 2 expression with two regressors, see equation (3 . 11) in book • Nasty, and you do not need to know it • Better done using matrix algebra, which you don’t need to know either... • Why do our OLS estimator expressions differ?...
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chapter 3 [article] - Contents 1 OLS and multiple...

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