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chapter 8 [article] - Contents 1 Stochastic regressors 1.1...

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Contents 1 Stochastic regressors 1 1.1 New assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 OLS properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Measurement errors 4 2.1 Sources and consequences . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Friedman and consumption . . . . . . . . . . . . . . . . . . . . . 7 3 Instrumental variables 10 3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1 Stochastic regressors Contents 1.1 New assumptions Contents Everything so far has been in the context of Model A : non-stochastic regressors unrealistic but useful for learning basics Now will move on to Model B : stochastic regressors will assume values of a regressor are random draws from its popula- tion probability distribution Implications for OLS? not necessarily serious, but potential for problems (measurement er- rors. . . ) solution: instrumental variables We will make the following assumptions in Model A : B.1 The model is linear in parameters and correctly specified ( A.1 ) B.2 The values of the regressors are drawn randomly from fixed populations note: we do not assume the regressors are independent of one another two (or more) regressors might have a joint probability distribution, and thus correlated sample values B.3 No exact linear relationship among the regressors ( A.2 )
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B.4 The disturbance term has zero expectation: E ( u i ) = 0 ( A.3 ) B.5 The disturbance term is homoscedastic ( A.4 ) B.6 The values of the disturbance term have independent distributions ( A.5 ) B.7 The disturbance term is distributed independently of the regressors weaker assumptions possible here (p.242), but this one is sufficient very important: if this is violated, problems. . . B.8 The disturbance term has a normal distribution ( A.6 ) 1.2 OLS properties Contents Y i = β 1 + β 2 X i + u i (8.6) We’ve seen that the OLS estimator of β 2 in this simple regression can be written: b OLS 2 = n i =1 ( X i - ¯ X ) ( Y i - ¯ Y ) n i =1 ( X i - ¯ X ) 2 = β 2 + n X i =1 a i u i (8.7) where a i = X i - ¯ X n j =1 ( X j - ¯ X ) 2 When proving unbiasedness of OLS with nonstochastic X i , we used the fact that this implied the a i were also nonstochastic, so that E ( a i u i ) = a i E ( u i ) = a i × 0 Can’t do that now, since the X i and therefore the a i are stochastic this means that we can’t treat the a i as constants and take them out of the expectation E ( a i u i ) Instead, we invoke assumption B.7 of the u i being distributed indepen- dently of the X i Review chapter: if X i and u i are independent, then for any functions f ( . ) and g ( . ), we have: E { f ( X i ) g ( u i ) } = E { f ( X i ) } E { g ( u i ) } (8.10) Applying this result with f ( X i ) = a i and g ( u i ) = u i : 2
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E ( a i u i ) = E ( a i ) E ( u i ) = E ( a i ) × 0 = 0 (8.11) (as long as B.7 holds and the errors u i are distributed independently of the X i ) This result allows us to prove unbiasedness of OLS: E ( b OLS 2 ) = β 2 + n X i =1 E ( a i u i ) = β 2 + n X i =1 E ( a i ) E ( u i ) (8.12) Since E ( u i ) = 0 by B.4 , we then have E ( b OLS 2 ) = β 2 Strictly speaking, need E ( a i ) to exist, so also need some variation in the X i as long as there is a constant term this is guaranteed by assumption B.3 Other finite-sample properties of OLS Conditional on the sample values of the regressors, the expressions for estimator variances are as before, e.g.:
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