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Christopher Dougherty EC220 - Introduction to econometrics (chapter 8) Slideshow: asymptotic and finite-sample distributions of the iv estimator Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 8). [Teaching Resource] © 2012 The Author This version available at: Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms.

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 1 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.

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 2 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?

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

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 4 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.

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 5 We cannot apply a CLT to b 2 IV 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.

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

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 7 The variance of b 2 IV is inversely proportional to n, and so tends to zero. This is the reason that the distribution of b 2 IV collapses to a spike.

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 8 We can deal with the diminishing-variance problem by considering √n b 2 IV instead of b 2 IV. This has the variance shown, which is stable. However, √n b 2 IV still does not have a limiting distribution because its mean increases with n.

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 9 So instead, consider √n (b 2 IV – 2 ). Since b 2 IV tends to 2 as the sample size becomes large, this does have a limiting distribution with zero mean and stable variance.

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 10 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 (b 2 IV – 2 ) has the limiting normal distribution shown.

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

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 12 Having established this, we can now start working backwards and say that, for sufficiently large samples, as an approximation, (b 2 IV – 2 ) has the distribution shown. (~ means ‘is distributed as’.)

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 13 We can then say that, as an approximation, for sufficiently large samples, b 2 IV is distributed as shown, and use this assertion as justification for performing the usual tests. This is what was intended by equation (8.50).

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

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

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 16 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.

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 17 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. 1 and 2 are constants.

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 18 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.

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 19 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.)

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 20 We will set 1 = 10, 2 = 5, 1 = 0.5, and 2 = 2.0.

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 21 The diagram shows the distributions of the OLS and IV estimators of 2 for n = 25 and n = 100, for 10 million samples in both cases. Given the information above, it is easy to verify that plim b 2 OLS = Of course, plim b 2 IV = 5.00

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 22 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.

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 23 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.

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 24 This diagram shows the distribution of √n (b 2 IV – 2 ) 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.

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 25 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.

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 26 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

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 27 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.

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

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 29 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.

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Copyright Christopher Dougherty These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. 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 Individuals studying econometrics on their own and who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics or the University of London International Programmes distance learning course 20 Elements of Econometrics

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