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Econometrics Lecture 8

Lecture 8: Simultaneous Equations Instrumedntal variables estimation. 2. Instrumental Variable Estimation

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Econometrics Lecture 8

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    1. Lecture 8: Simultaneous Equations Instrumedntal variables estimation 1 Econometrics Lecture 8 Instrumental Variable Estimation & 2SLS & Simultaneous Equation Models

    2. Lecture 8: Simultaneous Equations Instrumedntal variables estimation 2 Instrumental Variable Estimation & Two-Stage Lest squares Overview of the lecture Stat commands Motivation IV-Estimation Applying 2SLS Simultaneous Equation Models Identifying and Estimating Structural Equations Systems with 2 or more equations.

    3. Lecture 8: Simultaneous Equations Instrumedntal variables estimation 3 Stata-commands (1) ivreg (cross-sectional) reg 2 (cross-sectional) xtivreg (panel) xthtylor (panel)

    4. Lecture 8: Simultaneous Equations Instrumedntal variables estimation 4 Note: As the following lecture (and the related exercise) will show: Instrumental variable is much more difficult to describe than to perform! - But, without some theorertical understanding, you can not properly apply instrumental variable estimation!

    5. Lecture 8: Simultaneous Equations Instrumedntal variables estimation 5 Gauss-Markov (1) 1. We have the correct model; that is the true data generating process (the real world is of the linear form yi=b0+b1Xi+e 2. X does not take the same value for all observations 3. Given the values of Xi, the variance of e is s2 for all observations: that is, the variance of the error term for each observation is the same. 4. The error terms are not correlated with each other. 5. The error term are not correlated with the value of Xi; the mean value of e is 0 no matter that the value of X is.

    6. Lecture 8: Simultaneous Equations Instrumedntal variables estimation 6 Gauss-Markov (2) If all of the assumptions of the Gauss-Markov theorem are true except (5), and one of the rigth-hand-side variables of the regression equation is endogenous, then (1) The OLS estimates of the b^S become biased; (2) The OLS estimates also become inconsistent; (3) The OLS estimates also become inefficient; (4) The usual standard error estimates also become biased and inconsistent.

    7. Lecture 8: Simultaneous Equations Instrumedntal variables estimation 7 (Remember) A variable is endogenous to an economic model ih its value is determined within the model. A model is exogenous to the model if its value is determined outside the model

    8. Lecture 8: Simultaneous Equations Instrumedntal variables estimation 8 Gauss-Markov (3) If a rigth-hand-side variable is endogenous, then it is correlated with the error term. In that case, the error term do not have an average value of 0; the average will be higher or lower depending on the value of the endogenous variable.

    9. Lecture 8: Simultaneous Equations Instrumedntal variables estimation 9 Gauss-Markov (4) Solving the problem: One solution to the problem of an endogenous rigth-hand side variable is to use the fact that, if a right-hand variable is endogenous, the there must be some second relationship between it and the dependent variable. This implies that hhere must be a second equation relating the endogenous variables that can be used to eliminate on the rigth side from the model

    10. Lecture 8: Simultaneous Equations Instrumedntal variables estimation 10 The idea behind Instrumental variable estimation y = b0 + b1x1 + b2x2 + . . . bkxk + u x1 = p0 + p1z + p2x2 + . . . pkxk + v

    11. Lecture 8: Simultaneous Equations Instrumedntal variables estimation 11 Gauss-Markov (5) Reduced form: The system of equations that is produced by solving the model for endogenous variables is called for the reduced form of the model. The original equations are called structural strauctural equations, and the solved equations, which only have exogenous variables on the rigth hand side, are called reduced-form equations.

    12. Lecture 8: Simultaneous Equations Instrumedntal variables estimation 12 Definition and Motivation Instrumental variable (IV): In an equation with an endogenous explonatory variable, an IV is a variable that does not appear in the equation, is uncorrelated with the error in the equation, and is (partially) correlated with the endogenous explonatory variable.

    13. Lecture 8: Simultaneous Equations Instrumedntal variables estimation 13 IV estimator Instrumental Variables (IV) Estimator: An estimator in a linear model used when instrumental variables are available for one or more endogenous explanator variable

    14. Lecture 8: Simultaneous Equations Instrumedntal variables estimation 14 Hence: Why Use Instrumental Variables? Instrumental Variables (IV) estimation is used when your model has endogenous xs That is, whenever Cov(x,u) ? 0 Thus, IV can be used to address the problem of omitted variable bias (Additionally, IV can be used to solve the classic errors-in-variables problem)

    15. Lecture 8: Simultaneous Equations Instrumedntal variables estimation 15 What Is an Instrumental Variable?; In compact form In order for a variable, z, to serve as a valid instrument for x, the following must be true The instrument must be exogenous That is, Cov(z,u) = 0 The instrument must be correlated with the endogenous variable x That is, Cov(z,x) ? 0

    16. Lecture 8: Simultaneous Equations Instrumedntal variables estimation 16 More on Valid Instruments We have to use common sense and economic theory to decide if it makes sense to assume Cov(z,u) = 0 We can test if Cov(z,x) ? 0 Just testing H0: p1 = 0 in x = p0 + p1z + v Sometimes refer to this regression as the first-stage regression

    17. Lecture 8: Simultaneous Equations Instrumedntal variables estimation 17 IV Estimation in the Simple Regression Case For y = b0 + b1x + u, and given our assumptions Cov(z,y) = b1Cov(z,x) + Cov(z,u), so b1 = Cov(z,y) / Cov(z,x) Then the IV estimator for b1 is

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