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Unformatted text preview: hat is the mean grade in the population of college students? Melissa Tartari (Yale) Econometrics 60 / 93 Failure of SLR.2: An Example (Questions) We will try to answer two questions with our possibly nonrandom sample:
Q.1: what is the mean grade in the population of college students?
Q.2: what is the e¤ect of attending a private high school on college
grades, other things (namely, ability and drive) equal? Melissa Tartari (Yale) Econometrics 60 / 93 Failure of SLR.2: An Example (Q.1) The answer to Q.1 is E [Y ]. The problem is that E [Y ] is a function
of unknown parameters hence it is itself an unknown parameter. Melissa Tartari (Yale) Econometrics 61 / 93 Failure of SLR.2: An Example (Q.1) The answer to Q.1 is E [Y ]. The problem is that E [Y ] is a function
of unknown parameters hence it is itself an unknown parameter.
An idea: Can we use the sample average y to estimate E [Y ]? Is the
sample average an unbiased estimator of E [Y ]? Melissa Tartari (Yale) Econometrics 61 / 93 Failure of SLR.2: An Example (Q.1) The answer to Q.1 is E [Y ]. The problem is that E [Y ] is a function
of unknown parameters hence it is itself an unknown parameter.
An idea: Can we use the sample average y to estimate E [Y ]? Is the
sample average an unbiased estimator of E [Y ]?
Look at the object of interest E [Y ] and write it out using the LIE as:
E [Y ] = Pr (A) E [Y jA] + Pr (B ) E [Y jB ] = fA E [ Y j A ] + ( 1 Melissa Tartari (Yale) Econometrics fA ) E [ Y j B ] 61 / 93 Failure of SLR.2: An Example (Q.1)
Look at the sample average  the candidate estimator of E [Y ]  and
write it out:
y =
=
= 1
N
1
N N ∑ Yi i =1 1 ∑ Yi + N ∑ Yi i 2A NA
N i 2B 1
NA = sA y A + ( 1 Melissa Tartari (Yale) ∑ Yi i 2A ! + N sA ) y B Econometrics NA
N 1
N NA ∑ Yi i 2B ! 62 / 93 Failure of SLR.2: An Example (Q.1)
The candidate estimator y is unbiased for E [Y ] i¤ E [y ]
that is, i¤
0 = E [ sA y A + ( 1 E [Y ] = 0, (fA E [Y jA] + (1 fA ) E [Y jB ])
= (sA E [y A ] + (1 sA ) E [y B ]) (fA E [Y jA] + (1 fA ) E [Y jB ])
= (sA E [Y jA] + (1 sA ) E [Y jB ]) (fA E [Y jA] + (1 fA ) E [...
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This note was uploaded on 02/13/2014 for the course ECON 350 taught by Professor Donaldbrown during the Fall '10 term at Yale.
 Fall '10
 DonaldBrown
 Econometrics

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