Question

(d) Suppose that the true population model, which satisfies the Gauss-Markov assumptions, is given as [ Y=eta_{0}+eta_{1} X_{1}+eta_{2} X_{2}+U ] that is, ( eta_{2} eq 0 ). However, ( X_{2} ) is omitted as there is no proper variable to measure ( X_{2} ). As a result, we end up estimating the following model with ( X_{2} ) omitted: [ Y=eta_{0}+eta_{1} X_{1}+U ] and obtained OLS estimator ( widetilde{eta}_{1} ). i. Let ( widehat{eta}_{1} ) refer to the OLS estimator of ( eta_{1} ) from the true population model (1), that is, the model with ( X_{2} ) included. Then there is a tradeoff between bias and variance, in the sense that (i) ( widetilde{eta}_{1} ) is biased, but (ii) ( widetilde{eta}_{1} ) is more efficient than ( widehat{eta}_{1} ). (Hint: Consider the condition(s) for the tradeoff. Under what conditions do we have a tradeoff?) ii. If the sample size is very large, then the potential multicolinearity problem can disappear. iii. If the sample size is large, the potential bias can disappear.

          (d) Suppose that the true population model, which satisfies the Gauss-Markov assumptions, is given as
[
Y=eta_{0}+eta_{1} X_{1}+eta_{2} X_{2}+U
]
that is, ( eta_{2} 
eq 0 ). However, ( X_{2} ) is omitted as there is no proper variable to measure ( X_{2} ). As a result, we end up estimating the following model with ( X_{2} ) omitted:
[
Y=eta_{0}+eta_{1} X_{1}+U
]
and obtained OLS estimator ( widetilde{eta}_{1} ).
i. Let ( widehat{eta}_{1} ) refer to the OLS estimator of ( eta_{1} ) from the true population model (1), that is, the model with ( X_{2} ) included. Then there is a tradeoff between bias and variance, in the sense that (i) ( widetilde{eta}_{1} ) is biased, but (ii) ( widetilde{eta}_{1} ) is more efficient than ( widehat{eta}_{1} ). (Hint: Consider the condition(s) for the tradeoff. Under what conditions do we have a tradeoff?)
ii. If the sample size is very large, then the potential multicolinearity problem can disappear.
iii. If the sample size is large, the potential bias can disappear.

(d) Suppose that the true population model, which satisfies the Gauss-Markov assumptions, is given as
[
Y=eta0+eta1 X1+eta2 X2+U
]
that is, ( eta2
eq 0 ). However, ( X2 ) is omitted as there is no proper variable to measure ( X2 ). As a result, we end up estimating the following model with ( X2 ) omitted:
[
Y=eta0+eta1 X1+U
]
and obtained OLS estimator ( widetildeeta1 ).
i. Let ( widehateta1 ) refer to the OLS estimator of ( eta1 ) from the true population model (1), that is, the model with ( X2 ) included. Then there is a tradeoff between bias and variance, in the sense that (i) ( widetildeeta1 ) is biased, but (ii) ( widetildeeta1 ) is more efficient than ( widehateta1 ). (Hint: Consider the condition(s) for the tradeoff. Under what conditions do we have a tradeoff?)
ii. If the sample size is very large, then the potential multicolinearity problem can disappear.
iii. If the sample size is large, the potential bias can disappear.

Added by Niterso P.

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