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the smaller is var 1 jx intuition as n so does total

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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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