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Econometrics: endogeneit

rollerboller

I've just got the question to solve and unfortunately I'm stuck.

Consider the model

where

for

are independent and identically distributed pairs of random variables (

are scalars here). We don't assume that

and

are independent, but we impose:

The variable

is not observed, we observe:

where

are independent and identically distributed with mean zero and variance

independent of both

and

.

(i) Write down the model that can be estimated, based on observables quantities.
(ii) Show that under the given assumptions the OLS estimator of

is unbiased.
(iii) Show that the OLS estimator of

is consistent (hint: remember the Weak Law of Large Number).
(iv) Discuss the effect of measurement error in the dependent variable compared to the effect of measurement error in the independent variable.

Can anyone possible help?

Reply 1

Classical Liberal

Original post by rollerboller

I've just got the question to solve and unfortunately I'm stuck.

Consider the model

where

for

are independent and identically distributed pairs of random variables (

are scalars here). We don't assume that

and

are independent, but we impose:

The variable

is not observed, we observe:

where

are independent and identically distributed with mean zero and variance

independent of both

and

.

(i) Write down the model that can be estimated, based on observables quantities.
(ii) Show that under the given assumptions the OLS estimator of

is unbiased.
(iii) Show that the OLS estimator of

(i) The regression y = Bx (B should be Beta_hat). Use OLS to find an estimator for B.

(ii) Then take expectations of this estimator to prove it is unbiased.

(iii) This is a bit of a pointless exercise because if an estimator is unbiased then it is consistent. You can use plims to prove consistency or define the variance of the estimator and let n approach infinity.