1. EC220 - Introduction to econometrics (chapter 8)
Christopher Dougherty
EC220 - Introduction to econometrics (chapter 8)
Slideshow: asymptotic and finite-sample distributions of the iv estimator
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Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 8). [Teaching Resource]
© 2012 The Author
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2. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
The asymptotic variance of the IV estimator is given by the expression shown. It is the expression for the variance of the OLS estimator, multiplied by the square of the reciprocal of the correlation between X and Z. 1

3. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
What does this mean? We have seen that the distribution of the IV estimator degenerates to a spike. So how can it have an asymptotic variance? 2

4. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
The contradiction has been caused by compressing several ideas together. We will have to unpick them, taking several small steps. 3

5. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
The application of a central limit theorem (CLT) underlies the assertion. To use a CLT, we must first show that a variable has a nondegenerate limiting distribution. The CLT will then show that, under appropriate conditions, this limiting distribution is normal. 4

6. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
We cannot apply a CLT to b2IV directly, because it does not have a nondegenerate limiting distribution. The expression for the variance may be rewritten as shown. MSD(X) is the mean square deviation of X. 5

7. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
By a law of large numbers, the MSD tends to the population variance of X and so has a well-defined limit. 6

8. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
The variance of b2IV is inversely proportional to n, and so tends to zero. This is the reason that the distribution of b2IV collapses to a spike. 7

9. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
We can deal with the diminishing-variance problem by considering √n b2IV instead of b2IV. This has the variance shown, which is stable. However, √n b2IV still does not have a limiting distribution because its mean increases with n. 8

10. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
So instead, consider √n (b2IV – b2). Since b2IV tends to b2 as the sample size becomes large, this does have a limiting distribution with zero mean and stable variance. 9

11. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
Under conditions that are usually satisfied in regressions using cross-sectional data, it can then be shown that we can apply a central limit theorem and demonstrate that √n (b2IV – b2) has the limiting normal distribution shown. 10

12. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
The arrow with a d over it means ‘has limiting distribution’. 11

13. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
Having established this, we can now start working backwards and say that, for sufficiently large samples, as an approximation, (b2IV – b2) has the distribution shown. (~ means ‘is distributed as’.) 12

14. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
We can then say that, as an approximation, for sufficiently large samples, b2IV is distributed as shown, and use this assertion as justification for performing the usual tests. This is what was intended by equation (8.50). 13

15. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
Of course, we need to be more precise about what we mean by a ‘sufficiently large’ sample, and ‘as an approximation’. 14

16. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
We cannot do this mathematically. This was why we resorted to asymptotic analysis in the first place. 15

17. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
Instead, the usual procedure is to set up a Monte Carlo experiment using a model appropriate to the context. The answers will depend on the nature of the model, the correlation between X and u, and the correlation between X and Z. 16

18. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
Suppose that we have the model shown and the observations on Z, V, and u are drawn independently from a normal distribution with mean zero and unit variance. We will think of Z and V as variables and of u as a disturbance term in the model. l1 and l2 are constants. 17

19. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
By construction, X is not independent of u and so Assumption B.7 is violated when we fit the regression of Y on X. OLS will yield inconsistent estimates and the standard errors and other diagnostics will be invalid. 18

20. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
Z is correlated with X, but independent of u, and so can serve as an instrument. (V is included as a component of X in order to provide some variation in X not connected with either the instrument or the disturbance term.) 19

21. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
We will set b1 = 10, b2 = 5, l1 = 0.5, and l2 = 2.0. 20

22. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
The diagram shows the distributions of the OLS and IV estimators of b2 for n = 25 and n = 100, for 10 million samples in both cases. Given the information above, it is easy to verify that plim b2OLS = Of course, plim b2IV = 5.00 21

23. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
The IV estimator has a greater variance than the OLS estimator and for n = 25 one might prefer the latter. It is biased, but the smaller variance could make it superior, using some criterion such as the mean square error. For n = 100, the IV estimator looks better. 22

24. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
This diagram adds the distribution for n = 3,200. Both estimators are tending to the predicted limits (the IV estimator more slowly than the OLS, because it has a larger variance). Here the IV estimator is definitely superior. 23

25. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
This diagram shows the distribution of √n (b2IV – b2) for n = 25, 100, and 3,200. It also shows, as the dashed red line, the limiting normal distribution predicted by the central limit theorem. 24

26. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
It can be seen that the distribution for n = 3,200 is very close to the limiting normal distribution and so inference would be safe with samples of this magnitude. 25

27. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
However, the distributions for n = 25 and n = 100 are distinctly non-normal. The distribution for n = 25 has fat tails. This means that if you performed a t test, the probability of suffering a Type I error will be much higher than the nominal significance level of the test 26

28. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
The distribution for n = 100 is better, in that the right tail is close to that of the normal distribution, but the left tail is much too fat and, as for n = 25, would give rise to excess instances of Type I error. 27

29. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
The distortion for small sample sizes is partly attributable to the low correlation between X and Z, 0.22. 28

30. ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR
Unfortunately, low correlations (‘weak instruments’) are common in IV estimation. It is difficult to find an instrument that is correlated with X but not the disturbance term. Indeed, it is often difficult to find any credible instrument at all. 29

31. Copyright Christopher Dougherty 2011.
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The content of this slideshow comes from Section 8.5 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre
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