slides_Ch2_W[1]

2 an example q1a to see that the unbiased estimator y

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Unformatted text preview: le) fA ) ( E [ Y j A ] E [Y jB ]) fA ) is known. Econometrics 66 / 93 Failure of SLR.2: An Example (Q.1) Use what we showed to “guess” an unbiased estimator: E [ y ] = E [ Y ] + ( sA fA ) ( E [ Y j A ] E [Y jB ]) Observe that by C.1 (sA fA ) is known. Observe that by C.2 we have unbiased estimators of E [Y jA] and E [Y jB ]: E [y A ] = E [Y jA] E [y B ] = E [Y jB ] Melissa Tartari (Yale) Econometrics 66 / 93 Failure of SLR.2: An Example (Q.1) Use what we showed to “guess” an unbiased estimator: E [ y ] = E [ Y ] + ( sA fA ) ( E [ Y j A ] E [Y jB ]) Observe that by C.1 (sA fA ) is known. Observe that by C.2 we have unbiased estimators of E [Y jA] and E [Y jB ]: E [y A ] = E [Y jA] E [y B ] = E [Y jB ] Thus, C.1 and C.2 imply that the following estimator, called y W , is unbiased for E [Y ]: yW Melissa Tartari (Yale) y ( sA fA ) ( y A Econometrics yB ) . 66 / 93 Failure of SLR.2: An Example (Q.1) We will next answer two questions: Melissa Tartari (Yale) Econometrics 67 / 93 Failure of SLR.2: An Example (Q.1) We will next answer two questions: Q.1.a Does the unbiased estimator y W make intuitive sense? Melissa Tartari (Yale) Econometrics 67 / 93 Failure of SLR.2: An Example (Q.1) We will next answer two questions: Q.1.a Does the unbiased estimator y W make intuitive sense? Q.1.b What does this have to do with regression analysis in the SLRM? Melissa Tartari (Yale) Econometrics 67 / 93 Failure of SLR.2: An Example (Q.1.a) To see that the unbiased estimator y W is intuitively sensible it is useful to do some algebra ... yW y = ... 1 = N = Melissa Tartari (Yale) 1 N ( sA N fA ) ( y A fA N ∑ sA yi + N ∑ 1 fA yi sA ∑ ωi yi fj ( i ) sj ( i ) i =1 N i =1 Econometrics 1 yB ) 1 i =1 with ω i = 68 / 93 Failure of SLR.2: An Example (Q.1.a) To see that the unbiased estimator y W is intuitively sensible it is useful to do some algebra ... yW y = ... 1 = N = 1 N ( sA N fA ) ( y A fA N ∑ sA yi + N ∑ 1 fA yi sA ∑ ωi yi fj ( i ) sj ( i ) i =1 N i =1 1 yB ) 1 i =1 with ω i = We see that y W is a weighted sample aver...
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