1 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). structural equations reduced form equation

Here the reduced form equation for w reveals that u p 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 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS structural equations reduced form equation

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 u p. Further, it is not an explanatory variable in its own right. 3 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS structural equations reduced form equation

structural equations Thus it satisfies the three requirements for acting as an instrument for w and we will obtain a consistent estimate of  2 if we use the IV estimator shown. 4 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS

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 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS structural equations reduced form equation `

structural equations However, we obtain the result more quickly if we use its structural equation. 6 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS reduced form equation `

The  1 terms cancel. We rearrange the remaining terms in the second factor. 7 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS structural equations

The expression decomposes into the true value  2 plus an error term. 8 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS structural equations

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 u p. 9 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS structural equations

Instead we will demonstrate that the IV estimator is consistent. We will then use a Monte Carlo experiment to demonstrate that it may be free from serious bias in finite samples. 10 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS structural equations

11 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. INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS structural equations IV estimator if A and B have probability limits and plim B is not 0.

structural equations IV estimator if A and B have probability limits and plim B is not However, as the expression stands, the numerator and the denominator of the error term do not have limits. INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS

13 To deal with this problem, we divide both the numerator and the denominator by n. INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS structural equations IV estimator if A and B have probability limits and plim B is not 0.

14 It can be shown that the limit of the numerator is the covariance of U and u p and the limit of the denominator is the covariance of U and w. INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS structural equations IV estimator

Hence the numerator and the denominator of the error term have limits and we can implement the plim quotient rule. 15 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS structural equations IV estimator

reduced form equation structural equations Of course, we have to check that cov(U,w) is not zero. This follows from the reduced form, which shows that w is a function of U. 16 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS IV estimator

cov(U,u p ) is zero under the assumption that U is exogenous. 17 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS IV estimator structural equations

18 Thus we have shown that the IV estimator is consistent. INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS IV estimator structural equations

structural equations The table shows the results of 10 replications of the Monte Carlo experiment described in the previous sequence. 19 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS OLS IV b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) –

OLS IV b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) – When we used OLS to fit the equation for p, the slope coefficient was overestimated, as predicted. 20 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS structural equations

OLS IV b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) – 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. 21 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS structural equations

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. 22 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS OLS IV b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) – structural equations

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. 23 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS OLS IV b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) – structural equations

Returning to the estimates of the slope coefficients, note that the dispersion of the OLS estimates is smaller than that of the IV estimates. 24 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS OLS IV b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) – structural equations

The chart shows the distributions of the slope coefficient estimated with OLS and IV in a Monte Carlo experiment with 1 million samples million samples n = 20 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS OLS mean = 0.95 IV mean = 0.46 plim OLS = 0.91 plim IV = 0.50 b2b2

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. 26 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS 1 million samples n = 20 OLS mean = 0.95 IV mean = 0.46 plim OLS = 0.91 plim IV = 0.50 b2b2

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. 27 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS 1 million samples n = 20 OLS mean = 0.95 IV mean = 0.46 plim OLS = 0.91 plim IV = 0.50 b2b2

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. 28 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS 1 million samples n = 20 OLS mean = 0.95 IV mean = 0.46 plim OLS = 0.91 plim IV = 0.50 b2b2

Certainly in this case it was an improvement on the OLS estimator, which was subject to a much larger positive bias. 29 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS 1 million samples n = 20 OLS mean = 0.95 IV mean = 0.46 plim OLS = 0.91 plim IV = 0.50 b2b2

However an IV estimator is not always preferable to an OLS estimator, even though it is consistent and the OLS estimator is biased. 0.5 OLS IV 30 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS

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. 31 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS 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. 32 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS 0.5 OLS IV

The equation for p is described as identified because we can use IV to obtain consistent estimates of its parameters, U being the instrument. 33 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS structural equations reduced form equation

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. 34 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS structural equations reduced form equation

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. 35 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS structural equations reduced form equation

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 u w. However, it is already in the equation in its own right and this prevents it from being used. 36 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS structural equations reduced form equation

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). 37 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS structural equations reduced form equation

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 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 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 EC2020 Elements of Econometrics