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Unformatted text preview: e true yi is the same as the sum of the fitted yi : n n i =1 i =1 ˆ
∑ yi = ∑ yi . ˆ
o Thus, the mean of the true yi is the same as the mean of the fitted yi : ˆ
yi = yi . (On average, you do not make any mistakes.) • The sample covariance between the regressor and the OLS residual is zero 1n
ˆ ( n ∑ i =1 xi μi =0) This is the first order condition (2) • The OLS regression line always goes through the mean of the sample ( y , x ) , ˆˆ y = α + β x . Some comments: 9 ˆ
1. β is an estimate of how much Y (e.g., wage) is expected to change in response to one unit increase in X (e.g., year of schooling, what if X is ˆ
measured in quarters, or months?). β is positive means that wage and education are positively correlated; the expected wage will increase given one more year of schooling. ˆ
2. β = 2.75 means that an additional year of schooling is predicted to increase hourly wage by 2.75 dollars (what if Y is logarithm of wage?). ˆ
3. α = an estimate of yi when xi = 0. But regression models give statistically meaningful estimates only near th...
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