EC220 - Introduction to econometrics (chapter 9)

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Description: EC220 - Introduction to econometrics (chapter 9)

EC220 - Introduction to econometrics (chapter 9)
Christopher Dougherty EC220 - Introduction to econometrics (chapter 9) Slideshow: instrumental variable estimation of simultaneous equations Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 9). [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.

INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS
In the previous sequence it was asserted that the reduced form equations have two important roles. One is that they reveal violations of Assumption B.7, that the disturbance term be distributed independently of the explanatory variable(s). 1

Here the reduced form equation for w reveals that up is a determinant of it, so we would obtain inconsistent estimates if we used OLS to fit the structural equation for p. We would have a parallel problem if we used OLS to fit the structural equation for w. 2

However the reduced form equation for w also provides a solution to the problem. U is a determinant of w, and by virtue of being exogenous, it is distributed independently of up. Further, it is not an explanatory variable in its own right. 3

Thus it satisfies the three requirements for acting as an instrument for w and we will obtain a consistent estimate of b2 if we use the IV estimator shown. 4

We will demonstrate that it is consistent. The first step is to substitute from the true model for p. We now have two equations for p, the structural equation and the reduced form equation, and in principle we could use either. 5

However, we obtain the result more quickly if we use its structural equation. 6

The b1 terms cancel. We rearrange the remaining terms in the second factor. 7

The expression decomposes into the true value b2 plus an error term. 8

We would like to demonstrate that the expected value of the error term is 0 and hence that the estimator is unbiased. However it is impossible to obtain a closed-form analytical expression for the error term because it is a complex nonlinear function of up. 9

Instead we will demonstrate that the IV estimator is consistent. We will then use a Monte Carlo experiment to demonstrate that it appears to be free from serious bias in finite samples. 10

We focus on the error term. We would like to use the plim quotient rule. The plim of a quotient is the plim of the numerator divided by the plim of the denominator, provided that both of these limits exist. 11

However, as the expression stands, the numerator and the denominator do not have limits. Both increase indefinitely as the sample size increases. 12

To deal with this problem, we divide both the numerator and the denominator by n. 13

Now they do have limits and we can apply the plim quotient rule. 14

It can be shown that the limit of the numerator is the covariance of U with up and the limit of the denominator is the covariance of U with w. 15

Cov(U, up) = 0 if U is exogenous, as we are assuming. 16

We should check that Cov(U, w) is not equal to 0. Otherwise the limiting value of the error term would not be defined. 17

The covariance is not 0 because U is a determinant of w, as shown by the reduced form equation. 18

Thus we have shown that the IV estimator is consistent. 19

OLS IV b1 s.e.(b1) b2 s.e.(b2) b1 s.e.(b1) b2 s.e.(b2) 9 – The table shows the results of 10 replications of the Monte Carlo experiment described in the previous sequence. 20

OLS IV b1 s.e.(b1) b2 s.e.(b2) b1 s.e.(b1) b2 s.e.(b2) 9 – When we used OLS to fit the equation for p, the slope coefficient was overestimated, as predicted. 21

OLS IV b1 s.e.(b1) b2 s.e.(b2) b1 s.e.(b1) b2 s.e.(b2) 9 – In the case of the IV estimates, there is no obvious sign of bias. However, we should repeat the experiment many more times to be sure on this point. 22

OLS IV b1 s.e.(b1) b2 s.e.(b2) b1 s.e.(b1) b2 s.e.(b2) 9 – The OLS estimator of the intercept was biased downwards because the estimator of the slope coefficient was biased upwards. However, as far as we can tell, the IV estimates are not subject to obvious bias. 23

OLS IV b1 s.e.(b1) b2 s.e.(b2) b1 s.e.(b1) b2 s.e.(b2) 9 – The IV standard errors are greater than the OLS counterparts, but the OLS estimates are invalid and so there is no basis for a comparison. 24

OLS IV b1 s.e.(b1) b2 s.e.(b2) b1 s.e.(b1) b2 s.e.(b2) 9 – Returning to the estimates of the slope coefficients, note that the dispersion of the OLS estimates is smaller than that of the IV estimates. 25

The chart shows the distributions of the slope coefficient estimated with OLS and IV in a Monte Carlo experiment with 1 million samples. 26

The mean value of the IV estimates is It should be remembered that the IV estimator is consistent, meaning that it will tend to the true value in large samples, but there is no claim that it is unbiased in finite samples. 27

If the sample size had been very large in each of the 1 million samples, the IV distribution would have collapsed to a spike at 0.5, the true value. 28

However the sample size was only 20. The Monte Carlo experiment reveals that the IV estimator was biased downwards, but the bias was quite small. 29

Certainly in this case it was an improvement on the OLS estimator, which was subject to a much larger positive bias. 30

0.5 OLS IV However an IV estimator is not always preferable to an OLS estimator, even though it is consistent and the OLS estimator is biased. 31

0.5 OLS IV The variance of the IV estimator will be greater than that of the OLS estimator. If the instrument is weak, the IV variance may be much greater, as in the diagram above. 32

0.5 OLS IV In this case, if the OLS bias is small, the OLS estimator could be superior, according to some criterion such as the mean square error. 33

The equation for p is described as identified because we can use IV to obtain consistent estimates of its parameters, U being the instrument. 34

We have focused on the slope coefficient, but we can also derive a consistent estimator of the intercept. If you can obtain a consistent estimator of one parameter in an equation, you can obtain consistent estimators for all of them. 35

How about the equation for w? It includes the endogenous variable p as an explanatory variable. We therefore need to find a variable to act as an instrument for it. 36

Again, U is a candidate. It is correlated with p as one can see from the reduced form equation for p and, because it is exogenous, it is distributed independently of uw. However, it is already in the equation in its own right and this prevents it from being used. 37

If you did try to use it as an instrument for p, you would encounter a form of exact multicollinearity. There is no solution to this problem and the wage inflation equation is said to be underidentified (or not identified). 38

Copyright Christopher Dougherty 2011.
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 9.3 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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