Econ 399 Chapter3b - 3.2 OLS Fitted Values and Residuals-...

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Unformatted text preview: 3.2 OLS Fitted Values and Residuals- after obtaining OLS estimates, we can then obtain fitted or predicted values for y: (3.20) ˆ ... ˆ ˆ ˆ ˆ 2 2 1 1 ik k i i i x x x y β β β β + + + + =-given our actual and predicted values for y, as in the simple regression case we can obtain residuals: (3.21) ˆ ˆ i i i y y u- =-a positive uhat indicates underprediction (y>yhat) and a negative uhat indicates overprediction (y<yhat) 3.2 OLS Fitted Values and Residuals- We can extend the single variable case to obtain important properties for fitted values and residuals: 1) The sample average of the residuals is zero, therefore: ˆ y y = 2) Sample covariance between each independent variable and the OLS residual is zero…-Therefore the sample covariance between the OLS fitted values and the OLS residual is zero-Since the fitted values come from our independent variables and OLS estimates 3.2 OLS Fitted Values and Residuals 3) The point ) , ,..., , ( 2 1 y x x x n Is always on the OLS regression line: k k x x x y β β β β ˆ ... ˆ ˆ ˆ 2 2 1 1 + + + + = Notes: These properties come from the FOC’s in (3.13):-the first FOC says that the sum of residuals is zero and proves (1)-the rest of the FOC’s imply zero covariance between the independent variables and uhat (2)-(3) follows from (1) 3.2 “Partialling Out”- In multiple regression analysis, we don’t need formulas to obtain OLS’s estimates of B j-However, explicit formulas can give us interesting properties-In the 2 independent variable case: (3.22) ˆ ˆ ˆ 2 1 1 1 ∑ ∑ = i i i r y r β-Where rhat are the residuals from regressing x 1 on x 2-ie: the regression: 2 1 1 x ˆ ˆ ˆ φ φ + = x 3.2 “Partialling Out”- rhat i1 are the part of x i1 that are uncorrelated with x 12-rhat i1 is equivalent to x i1 after x i2 ’s effects have been “partialled out” or “netted out”-thus B 1 hat measures x 1 ’s effect on y after x 2 has been “partialled out”-In a regression with k variables, the residuals come from a regression of x 1 on ALL other x’s-in this case B 1 hat measures x 1 ’s effect on y after all other x’s have been “partialled out” 3.2 Comparing Simple and Multiple Regressions3....
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This note was uploaded on 03/14/2009 for the course ECON ECON 399 taught by Professor Priemaza during the Spring '09 term at University of Alberta.

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Econ 399 Chapter3b - 3.2 OLS Fitted Values and Residuals-...

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