Alternatively if cant measure all sources of bias say

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Alternatively, if can’t measure all sources of bias  (say, unmeasurable “ability”) can at least discuss the  likely sign of the bias: If correlation between (X 2  , X 1 ) and (X , Y) is the same  direction, bias will be positive If correlation between (X 2  , X 1 ) and (X , Y) is the opposite  direction, bias will be negative
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18 Aside: More general cases of short and long regression What is the effect on our estimate of  β 1  of failing to  include the “K+1 th ” X,  X k+1 ? Technically, should examine the relationships controlling for  the other x’s (using Frisch-Waugh theorem) Typically, we just pretend we are in the one X versus two Xs  case to sign the bias ... 1 1 2 2 1 1 i ik k ik k i i i u x x x x y + + + + + = + + β β β β ~ ... ~ ~ 2 2 1 1 i ik k i i i u x x x y + + + + = β β β
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19 Sample Variation in estimates What about sample variation? Even if on average correct (=unbiased), estimates  will still vary from sample to sample Asking one randomly chosen American provides an  unbiased estimate of average income in the U.S. We want to know how reliable  our estimates  are:  Roughly speaking, what is the chance that our  estimate is close to the true population number?
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20 Sample Variation in Estimates How to we measure the sample variation in our  estimates? With the standard error  = estimated standard  deviation of the estimate: How much     is expected to vary from sample to sample  (given sample size / other features in our sample) Can only be computed under assumptions about  distribution of the errors, which I will turn to next… ( 29 ( 29 1 1 ˆ var ˆ β β = se 1 ˆ β
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21 Additional Assumptions about Error  Variation MLR.5 (Homoskedasticity). Var( u | x ) =  σ 2 This says at all different values of all of the Xs  (shorthand,  x ), the variance of the errors is the  same It is an empirically verifiable  assumption. If it does not hold, we may able to detect if from  observing patterns in our residuals (future lecture) Opposite of homoskedasticity: Heteroskedasticity 
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22 Example of Heteroskedasticity 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Error variance is bigger down here... … than up here.
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23 Additional Assumptions about  Error Variation Another assumption not written out explicitly  in the textbook – no autocorrelation Errors are not related across observations Plausibly holds in cross-sectional random sample Cases where it does not hold Time-series data Samples with “clusters” – survey several members of the  same family, for example We will discuss cases like these when we get to panel  data.
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24 Variance of OLS (cont) Under MLR.1-MLR.5 (and no  autocorrelation of errors): ( 29 ( 29 ( 29 ( 29 s. ' other
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