Practice Q1 (i) There are omitted variba;es that are correlated with education, for example, ability. Thus, the ZCM assumption does not hold; (|) ≠ 0. (ii) Cov(,) = 0 Validity (cannot be tested) Cov(, ) is ‘large’ Relevance (can be tested) (iii) First stage: Regress the endogenous variable on all the exogenous variables including our IV:
= 0 + 1 + 22 + 3ℎ + 4ℎ + 5 + 6 + (*) Form the prediction of :
� = �0 + �1 + �22 + ⋯+ �6 Substitute � for in the structural model (8.1): log() = 0 + 1IV� + ⋯+ Our 1IV is our IV estimate. The limitation of the procedure is that the standard errors from doing 2SLS in two steps are incorrect. (iv)
. . = ̂1IV ± × �̂1IV� = 0.2791 ± 1.96 × (0.1255) Where we use = 1.96 as = 947, so ≈ ∞. Hence . . = (0.0331, 0.5251) and the OLS estimate is within this . . (v) Run the reduced form regression (*) in (iii) above. The exogenous variables, call them 1, … , are all uncorrelated with in (8.1). Thus, any part of that is correlated with in (8.1) will be left in �. Then the idea of the test for endogeneity is to take � and use it to check for endogeneity by adding it to the structural model. We are looking to see if is correlated with
. We run the regression: log() = 0 + 1 + 2 + ⋯+ 6 + 0� +
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by OLS and test if 0 is statistically different from zero or not. If it is, we have evidence to suggest is endogenous. (vi) Your choice should hinge on whether the IV is valid and relevant. Given vary in the Australian case by degree, maybe Cov(,) = 0 is not necessarily going to hold if degree choice is a function of ability. Then, will be related to ability. If IV is invalid then OLS would be a better choice. Practice Q2
Practice Q3 (i)
0:1 = 2 = 3 = 4 = 5 = 0
1:0 is false
-stat = (0.056/5) / ((1-0.056)/(20050-5-1)) = 237.81.
5,20044 = 3.02 with 1% significance level. Since -stat > critical value, we reject 0. Conclude that the model for birth weight in (4.1) has overall significance.
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(ii) Var(|) = 2 If we have heteroskedasticity, our estimates are unaffected. But our inference is invalid because our standard errors are invalid. (iii) Either the BP, White, or modified White tests would be appropriate. Refer to Lecture 8 for the steps that must be followed for each test. You would need to describe / list the steps to get full marks.