~
x x x x x
i1 1 i2 2 i1 1
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Omitted Variable Bias (cont)
Consider t he regressionof x2 on x1 ~ x2 0 1 x1 then 1 so E b1 b1 b 2 1
2
ˆ y ˆ y y y y y yˆ yˆ
2 i i 2 2 i i
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More about R-squared
R2 can never decrease when another independent variable is added to a regression, and usually will increase
2 2
x1 and x2 are uncorrelat ed in thesample
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Goodness-of-Fit
We can thinkof each observatio n as being made up of an explainedpart,and an unexplaine d part, ˆi u ˆi Wethendefine thefollowing: yi y
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Goodness-of-Fit (continued)
How do we think about how well our sample regression line fits our sample data?
Can compute the fraction of the total sum of squares (SST) that is explained by the model, call this the R-squared of regression R2 = SSE/SST = 1 – SSR/SST
1 1 2 2 k k
so holding x2 ,..., xk fixed implies that ˆ x , thatis each b has ˆ b y
1 1
a ceteris paribus interpreta tion
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Interpreting Multiple Regression
ˆ b ˆ x b ˆ x ... b ˆ x , so ˆb y 0 1 1 2 2 k k ˆ x b ˆ x ... b ˆ x , ˆ b y
y y is the totalsum of squares (SST ) ˆ y is theexplainedsum of squares (SSE) y ˆ is theresidual sum of squares (SSR) u
2 2 i i 2 i
T henSST SSE SSR
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Omitted Variable Bias Summary
Two cases where bias is equal to zero

b2 = 0, that is x2 doesn’t really belong in model
x x b b x x
i1 1 1 i1 1
0 2
b1 xi1 b 2 xi 2 ui b 2 xi1 x1 xi 2 xi1 x1 ui
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Omitted Variable Bias (cont)
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Simple vs Multiple Reg Estimate
~ ~ ~ Comparethesimple regression y b 0 b1 x1 ˆ b ˆ x b ˆ x ˆb wit h themultipleregression y 0 1 1 2 2 ~ ˆ unless : Generally,b1 b 1 ˆ 0 (i.e.no partialeffectof x ) OR b
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x1 and x2 are uncorrelated in the sample
If correlation between x2 , x1 and x2 , y is the same direction, bias will be positive If correlation between x2 , x1 and x2 , y is the opposite direction, bias will be negative
b1
~
x x y x x
i1 1 2 i1 1
0
1 1
i
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Omitted Variable Bias (cont)
Recall the true model,so that yi b 0 b1 xi1 b 2 xi 2 ui , so the numeratorbecomes
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Omitted Variable Bias
Suppose the true modelis given as y b 0 b1 x1 b 2 x2 u, but we ~ ~ ~ estimatey b b x u, then
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~
~
~

~
~
x x x x x
i1 1 i2 2 i1 1
Summary of Direction of Bias
Corr(x1, x2) > 0 Corr(x1, x2) < 0
b2 > 0 b2 < 0 Positive bias Negative bias Negative bias Positive bias
u is still the error term (or disturbance) Still need to make a zero conditional mean assumption, so now assume that E(u|x1,x2, …,xk) = 0 Still minimizing the sum of squared residuals, so have k+1 first order conditions
Population model is linear in parameters: y = b0 + b1x1 + b2x2 +…+ bkxk + u We can use a random sample of size n, {(xi1, xi2,…, xik, yi): i=1, 2, …, n}, from the population model, so that the sample model is yi = b0 + b1xi1 + b2xi2 +…+ bkxik + ui E(u|x1, x2,… xk) = 0, implying that all of the explanatory variables are exogenous None of the x’s is constant, and there are no exact linear relationships among them
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Goodness-of-Fit (continued)
We can also thinkof R 2 as being equal to thesquared correlatio n coefficien t between ˆi theactual yi and the values y R
Because R2 will usually increase with the number of independent variables, it is not a good way to compare models
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Assumptions for Unbiasedness
Multiple Regression Analysis
y = b0 + b1x1 + b2x2 + . . . bkxk + u 1. Estimation
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Parallels with Simple Regression
b0 is still the intercept b1 to bk all called slope parameters
b b1 b 2
~
x x x x x x x
i1 1 i2 2 i1 1
i1 i1 Βιβλιοθήκη 1 ui x1 2