1. 1 In a second variation, we shall consider the model shown above. x is the rate of growth of productivity, assumed to be exogenous. w is now hypothesized to be a function of p, U, and x. For simplicity, p has reverted to being a simple function of w. TWO-STAGE LEAST SQUARES structural equations

2. reduced form equation structural equations 2 The reduced form equations are now as shown. The equation for w is underidentified because there is no exogenous variable to act as an instrument for p. Both U and x are already in the equation in their own right and are therefore unavailable. TWO-STAGE LEAST SQUARES reduced form equation

3. 3 However, the equation for p is now overidentified. An equation is said to be overidentified if the number of available instruments exceeds the number of endogenous explanatory variables used as explanatory variables. TWO-STAGE LEAST SQUARES structural equations reduced form equation

4. 4 We could use U as an instrument. TWO-STAGE LEAST SQUARES structural equations reduced form equation

5. 5 Equally, we could use x as an instrument. Both estimators are consistent. Hence, in large samples they will yield the same estimate — the true value. But in finite samples they will be different because they have different error components. TWO-STAGE LEAST SQUARES structural equations reduced form equation

6. 6 If we had to choose between them, it would be rational to choose the more efficient, that is, that with the smaller population variance. We would therefore choose the instrument more highly correlated with w. TWO-STAGE LEAST SQUARES structural equations reduced form equation

7. structural equations 7 But we do not need to choose between them. Instead, we will combine them into a super- instrument more highly correlated with w than either individually. TWO-STAGE LEAST SQUARES reduced form equation

8. 8 Let us denote this super-instrument Z. It will be a linear function of U and x. How do we choose h 1, h 2, and h 3 so as to maximize the correlation between Z and w? TWO-STAGE LEAST SQUARES structural equations reduced form equation

9. 9 Very simple. Regress the reduced form equation for w using OLS. When you do this, the coefficients are chosen so as to minimize the sum of the squares of the residuals. TWO-STAGE LEAST SQUARES structural equations reduced form equation

10. 10 But, as we saw in Chapter 1, the OLS coefficients are optimal with respect to two other criteria of goodness of fit. They maximize R 2, and they maximize the correlation between the actual and fitted values of the dependent variable. TWO-STAGE LEAST SQUARES structural equations reduced form equation

11. 11 Since the fitted values are calculated as the linear function of U and x most highly correlated with w, they are exactly what we need. TWO-STAGE LEAST SQUARES structural equations reduced form equation

12. 12 This is the Two-Stage Least Squares estimator. The first stage is to regress the reduced form equation for the endogenous variable being used as an explanatory variable. The second is to save the fitted values from this regression and use them as the instrument. TWO-STAGE LEAST SQUARES structural equations reduced form equation

13. 13 The TSLS estimator satisfies the usual requirements. It is correlated with w — indeed it is the function of U and x most highly correlated with w. Since U and x are both exogenous, it is independent of u p. And since U and x are not regressors, it is available as an instrument. TWO-STAGE LEAST SQUARES structural equations reduced form equation

14. 14 Finally, note that the TSLS estimator, and indeed any IV estimator, will yield consistent estimates only if the instruments are truly exogenous. If they turn out to be endogenous, they are unlikely to be independent of the disturbance terms in the model. TWO-STAGE LEAST SQUARES structural equations reduced form equation

15. 15 In the previous example, is m likely to be exogenous? Probably not. In the long run, at least, an increase in prices is likely to be one of the determinants of an increase in the money supply. TWO-STAGE LEAST SQUARES structural equations reduced form equation

16. 16 Finding truly exogenous variables to act as instruments is often a serious problem when fitting simultaneous equations models. TWO-STAGE LEAST SQUARES structural equations reduced form equation

17. Copyright Christopher Dougherty 2012. 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 http://www.oup.com/uk/orc/bin/9780199567089/http://www.oup.com/uk/orc/bin/9780199567089/. 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 http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx or the University of London International Programmes distance learning course EC2020 Elements of Econometrics www.londoninternational.ac.uk/lsewww.londoninternational.ac.uk/lse. 2012.11.24