Problem Set 5-Solution.pdf - Introductory Econometrics...

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Introductory Econometrics - Problem Set V (Suggested Solutions) 1 . Stock and Watson, Exercise 9 . 2 (a) When Y i is measured with error, we have ˜ Y i = Y i + w i , or Y i = ˜ Y i - w i . Substituting the 2 nd equation into the regression model Y i = β 0 + β 1 X i + u i gives ˜ Y i - w i = β 0 + β 1 X i + u i , or ˜ Y i = β 0 + β 1 X i + u i + w i . Thus v i = u i + w i . (b) i. The error term vi has conditional mean zero given X i : E ( v i | X ) = E ( u i + w i | X i ) = E ( u i | X i ) + E ( w i | X i ) = 0 + 0 = 0. ii. ˜ Y i = Y i + w i is i.i.d since both Y i and w i are i.i.d. and mutually inde- pendent; X i and ˜ Y j ( i 6 = j ) are independent since X i is independent of both Y j and w j . Thus, ( X i , ˜ Y i ) , i = 1, · · · , n are i.i.d. draws from their joint distribution. iii. v i = u i + w i has a finite fourth moment because both u i and w i have finite fourth moments and are mutually independent. So ( X i , v i ) have nonzero finite fourth moments. (c) The OLS estimators are consistent because the least squares assumptions hold. (d) Because of the validity of the least squares assumptions, we can construct the confidence intervals in the usual way. (e) The answer here is the economists’ “On the one hand, and on the other hand.” On the one hand, the statement is true: i.i.d. measurement error in X means that the OLS estimators are inconsistent and inferences based on OLS are invalid.
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