1 We will continue with a variation on the basic model. We will now hypothesize that p is a function of m, the rate of growth of the money supply, as well.

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Introduction to Econometrics, 5th edition

Introduction to Econometrics, 5th edition

Introduction to Econometrics, 5th edition

Introduction to Econometrics, 5th edition

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1 We will continue with a variation on the basic model. We will now hypothesize that p is a function of m, the rate of growth of the money supply, as well as w. INSTRUMENTAL VARIABLES ESTIMATION: VARIATION structural equations

2 An increase in the money supply is likely to increase price inflation and hence we would anticipate  3  > 0. INSTRUMENTAL VARIABLES ESTIMATION: VARIATION structural equations

3 The reduced form equations are now as shown. Again, we see that we would obtain inconsistent estimates if we used OLS to fit the structural equations. INSTRUMENTAL VARIABLES ESTIMATION: VARIATION reduced form equation structural equations reduced form equation

4 The equation for p remains identified. It contains the endogenous variable w as an explanatory variable, but we can use U as an instrument for it. U satisfies the three conditions of being correlated with w, independent of u p, and not already in the equation. INSTRUMENTAL VARIABLES ESTIMATION: VARIATION structural equations reduced form equation

5 As an exogenous variable, m was also a potential instrument but, since it appeared in the equation in its own right, it could not be used. INSTRUMENTAL VARIABLES ESTIMATION: VARIATION structural equations reduced form equation

6 Note that, since this is now a multiple regression, the mathematical expression for the IV estimator of the slope coefficient will now be different from that in the simple regression model. INSTRUMENTAL VARIABLES ESTIMATION: VARIATION structural equations reduced form equation

7 The equation for w is now identified. p is an endogenous variable correlated with u w, so we need to find an instrument to act for it. U is ruled out, as before, because it appears in the equation in its own right. INSTRUMENTAL VARIABLES ESTIMATION: VARIATION structural equations reduced form equation

8 However, m is now available as an instrument. INSTRUMENTAL VARIABLES ESTIMATION: VARIATION structural equations reduced form equation

9 m is correlated with p, by virtue of the reduced form equation for p, it is exogenous and therefore distributed independently of u w, and it is not already in the equation in its own right. INSTRUMENTAL VARIABLES ESTIMATION: VARIATION structural equations reduced form equation

10 Both equations are now exactly identified. An equation is said to be exactly identified if the number of exogenous variables available as instruments is equal to the number of endogenous variables that require instruments. INSTRUMENTAL VARIABLES ESTIMATION: VARIATION structural equations reduced form equation

In each case there was one endogenous variable on the right side and one exogenous variable was available to act as an instrument. 11 INSTRUMENTAL VARIABLES ESTIMATION: VARIATION 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