finalquestÆo1

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Question 1
Suppose that the relation between yi and xi is of the form:
yNon-standardi = α˜+ β˜xi + γ
2x2i + θi non-standard model
We suppose in all questions that xi are non-stochastic.
We also admit that θi˜N(0, σ
2) and that θi is such that E[θi] = 0.
However an econometrician uses the regression of yi on xi with an intercept.
He thinks that the true model is the standard-linear model.
Which means that he thinks that yi = y
Standard
i = α + βxi + εi , with
E[εi] = 0. He computes the OLS estimators αˆ, βˆ.
a) Give the formulas for αˆ, βˆ the OLS estimators of α and β in the standard-
linear model.
βˆ =
S
(yi−y¯)(xi−x¯)S
(xi−x¯)2 and αˆ = y¯− βˆ1x¯
If the standard-linear model is the true one, are the OLS estimators αˆ, βˆ
unbiased estimators of α and β resp.? No proof is needed, just answer yes or no
and state what it means for αˆ, βˆ .
Yes they are unbiased and E[βˆ] = α and E[αˆ] = β
b) What is the estimated (forecasted) value yˆhi this econometrician would
give using his OLS estimators if given only xi for an individual i. Just use αˆ
and βˆ you don’t need to plug the formulas you found in a).
yˆhi = αˆ+ βˆxi
c) If the econometrician’s model (the standard-linear model) was true (yi =
yStandardi ) what would be the error yi − yˆhi in terms of xi,α,β,εi,αˆ and βˆ. What
would be its expectation.
yStandardi − yˆhi = α+ βxi + εi − αˆ− βˆxi
E[yStandardi − yˆhi ] = α+ βxi + 0−E[αˆ]−E[βˆ]xi = 0
d) Find the true systematic error yi− yˆhi , which is the difference between the
true model of yi (using the non-standard model yi = y
Non-standard
i ) and what
the econometrician did forecast given the xi (your result in question b) yˆ
h
i ) in
terms of xi,α˜,β˜,γ, εi,αˆ and βˆ.
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yNon-standardi − yˆhi = α˜+ β˜xi + γ2x2i + θi − αˆ− βˆxi
e) What is the expectation of this error in terms of xi,α˜,β˜,γ, εi,E[αˆ] and
E[βˆ]? (You don’t need to compute E[αˆ] and E[βˆ]).
E[yNon-standardi − yˆhi ] = α˜+ β˜xi + γ2x2i + 0−E[αˆ]−E[βˆ]xi
f) Suppose α˜ = E[αˆ], β˜ = E[βˆ], 0 < γ and 0 < xi. What is the sign of
E[yi − yˆhi ]? Compare your result to c).
E[yNon-standardi − yˆhi ] = α˜+ β˜xi + γ2x2i + 0−E[αˆ]− E[βˆ]xi = γ2x2i >
E[yStandardi − yˆhi ] = 0
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