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Unformatted text preview: 93 The Variance of the OLS Estimator  Remarks f y x The Simple LRM under Homoschedasticity y x1 x2 E [y  x ]= β o + β1 x x3
x Melissa Tartari (Yale) Econometrics 81 / 93 The Variance of the OLS Estimator  Remarks
ˆ
How does Var β1 jx depend on σ2 , on the total variation in
2
fx1 , ..., xn g (denoted sx ), and on sample size n? Melissa Tartari (Yale) Econometrics 82 / 93 The Variance of the OLS Estimator  Remarks
ˆ
How does Var β1 jx depend on σ2 , on the total variation in
2
fx1 , ..., xn g (denoted sx ), and on sample size n? ˆ
the larger σ2 the larger is Var β1 jx
INTUITION: more variation in the unobservables a¤ecting y makes it
more di¢ cult to precisely estimate β1 Melissa Tartari (Yale) Econometrics 82 / 93 The Variance of the OLS Estimator  Remarks
ˆ
How does Var β1 jx depend on σ2 , on the total variation in
2
fx1 , ..., xn g (denoted sx ), and on sample size n? ˆ
the larger σ2 the larger is Var β1 jx
INTUITION: more variation in the unobservables a¤ecting y makes it
more di¢ cult to precisely estimate β1
2
ˆ
the larger sx the smaller is Var β1 jx
INTUITION: the more spread out is the sample of eVars the easier it is
to trace out the relationship between E [y jx ] and x hence to estimate
β1 ; if there is little variation in the xi then it can be hard to pin point
how E [y jx ] varies with x Melissa Tartari (Yale) Econometrics 82 / 93 The Variance of the OLS Estimator  Remarks
ˆ
How does Var β1 jx depend on σ2 , on the total variation in
2
fx1 , ..., xn g (denoted sx ), and on sample size n? ˆ
the larger σ2 the larger is Var β1 jx
INTUITION: more variation in the unobservables a¤ecting y makes it
more di¢ cult to precisely estimate β1
2
ˆ
the larger sx the smaller is Var β1 jx
INTUITION: the more spread out is the sample of eVars the easier it is
to trace out the relationship between E [y jx ] and x hence to estimate
β1 ; if there is little variation in the xi then it can be hard to pin point
how E [y jx ] varies with x
2
ˆ
the larger n the larger is sx and (as in b.) the smaller is Var β1 jx
INTUITION: as n " so does total variation in xi hence by b. it is
easier to e...
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 Fall '10
 DonaldBrown
 Econometrics

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