ec20-newlec6-v1 - Economics20 Lecture#6:TowardsMultivariate...

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1 Economics 20 Lecture #6: Towards Multivariate  Regression Statistical Inference
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2 From Last Time  MLR Assumptions MLR.1: Population Relationship… MLR.2: Random sample of data MLR.3: No perfect linear relationship between  the X’s If this fails to hold, can’t estimate (more below) MLR.4: Conditonial mean zero error, i.e. Critical for interpreting OLS (least squares)  estimates as causal. ( 29 0 , , | 1 = Ki i i X X u E i Ki K i i i i u X X X X Y + + + + + + = β β β β β 3 3 2 2 1 1 0
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3 Today “Solution” to multivariate least squares which  supports the “holding constant” interpretation  of the slopes “Partialling Out” – Frisch-Waugh Theorem “Bivariate” vs multivariate” or “short vs long”  regression Too few X’s   “omitted variables” bias Towards Statistical inference: Standard Errors
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4 Multivariate OLS estimates Estimated OLS relationship:                                                     i.e., a partial derivative: treats all the other  variables as constants A one unit increase in x 1  is associated with  a     unit increase in y, holding constant x 2 x 3 …x K Ki K i i i x x x y β β β β ˆ ˆ ˆ ˆ ˆ 2 2 1 1 0 + + + + = i i x y 1 1 ˆ ˆ = β 1 ˆ β
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5 A “Partialling Out” Interpretation Frisch-Waugh Theorem .  The OLS  estimator for     can be written as where     is the residuals from a regression  of x 1  on all of the other x’s In other words, the following two steps  deliver the same slope as multiple  regression: 1. Regress x 1  on x 2  … x k , get residuals 2. Regress y on residuals     (bivariate) 1 ˆ β 1 ˆ i r ( 29 ( 29 1 1 1 ˆ var , ˆ cov ˆ i i i r y r = β 1 ˆ i r 1 ˆ i r
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STATA Example… Also good for creating partial scatter plots 6
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7 A “Partialling Out” Interpretation In other words, the slope on x 1  from a  multiple regression literally “takes out” the  influence of (a linear function of) x 2 …x k Only the part of x 1 ’s variation unrelated to x 2 x k  is being related to y Why we need assumption MLR.3: if there were a  perfect linear relationship between x 1  and rest of  the x’s, then there would be no variation left over  to relate to y!   ( 29 0 ˆ 1 = i r Var
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8 Too Many or Too Few Variables What happens if we include variables in our specification that do not belong? There is no effect on our parameter estimate, and OLS remains unbiased However later we will learn this is inefficient (makes standard errors bigger) What if we exclude a variable from our specification that does belong?
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