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1、Week 14 Instrument Variable Regression Models,Simultaneous Equation Using 2SLS (Chapter 16), IV Estimation in Multiple Regression models (15.1-3),计量经济学（研究生）,A New Approach to the Omitted Variable Problem,We have talked about the problem of omitted variable bias (in Ch.3), and have shown that it will
2、 lead to inconsistency, for If we have a suitable proxy, we can minimize the bias, to some degree. (see Chapter 9) Furthermore, if the omitted variable is time invariant, then we can use a panel data model without much hesitation. Without a suitable proxy, no panel data, or if the omitted variable d
3、oes change with time we need a new approach,Instrumental Variables Regression,Three important threats to internal validity are: omitted variable bias from a variable that is correlated with X but is unobserved, so cannot be included in the regression;(遗留变量偏差) simultaneous causality bias (X causes Y,
4、 Y causes X);(联立因果) errors-in-variables bias (X is measured with error)(变量误差) Instrumental variables regression can eliminate bias from these three sources.,Terminology: endogeneity and exogeneity,An endogenous variable is one that is correlated with u. An exogenous variable is one that is uncorrela
5、ted with u. Historical note: “Endogenous” literally means “determined within the system,” that is, a variable that is jointly determined with y. In other words, it is a variable subject to simultaneous causality. However, this definition is narrow and IV regression can be used to address OV bias and
6、 errors-in-variable bias, not just to simultaneous causality bias.,What is Simultaneous Causality,Suppose we have two endogenous variables Y1, Y2 and two exogenous variables X1, X2 such that Y1i = 0 + 1X1i+2Y2i + u1i (1) Y2i = 0 + 1Y1i+2X2i + u2i (2) Lets see why Y2 (or Y1) is endogenous Suppose u1i
7、 0 and u2i =0, then we have Y1i E(Y1i) from (1) But in (2), if 20, this will cause a change in Y2i , so Y2i is correlated with u1i through (2) The same is true for Y1i and u2i in (2) through (1),Simultaneous Bias,Can we estimate these two equations consistently? y1 = a1y2 + b1z1 + u1 y2 = a2y1 + b2z
8、2 + u2 For consistency, we need cov(y2,u1)=0, and cov(y1,u2)=0 However, a large u2 means a larger y2, which implies a larger y1 (if a10), so cov(y1,u2)0 The same is true for cov(y2,u1) due to the circular effect of u1,The IV Estimator with a Single Regressor and a Single Instrument,yi = 0 + 1xi + ui
9、 Loosely, IV regression breaks x into two parts: a part that might be correlated with u, and a part that is not. By isolating the part that is not correlated with u, it is possible to estimate 1. This is done using an instrumental variable, zi, which is uncorrelated with ui. The instrumental variabl
10、e detects movements in xi that are uncorrelated with ui, and use these to estimate 1.,Two conditions for a valid instrument,yi = 0 + 1xi + ui For an instrumental variable (an “instrument”) z to be valid, it must satisfy two conditions: Instrument relevance: cov(zi,xi) 0 Instrument exogeneity: cov(zi
11、,ui) = 0 In other words, IV variable zi must be an exogenous variable that is correlated with x Or, zi s effect on y is only through x Which condition can we test? A)1 B) 2 C) Both D) Neither E) Dont know We can test the 1st but have to assume the 2nd,Example: Labor Economics,Suppose log(wage) = 0+1
12、educ+u, u=2abil+v When abil is unobserved, how can we estimate 1 consistently if cov(educ, abil) 0? If we have a proxy for abil, such as IQ and substitute it into our model, then we are fine Otherwise, we need something that is correlated with educ but not with abil Parents education, or number of s
13、iblings might be an instrument for educ,Suppose we have: yi = 0 + 1xi + ui cov(x,ui)0 Our estimate of 1 will be inconsistent Either we find the omitted variable in ui and add it into our model to overcome the inconsistency Or we find an instrument zi for the included variable Suppose for now that yo
14、u have such a zi (well discuss how to find instrumental variables later) How can you use zi to estimate 1? We will explain this in two ways,Instrument Variable Regression,The IV Estimator, one x and one z,Explanation #1:Two Stage Least Squares (TSLS) As it sounds, TSLS has two stages two regressions
15、: (1) First isolates the part of x that is uncorrelated with u: regress x on z using OLS xi = 0 + 1zi + vi (1) Because zi is uncorrelated with ui, 0 + 1zi is uncorrelated with ui. We dont know 0 or 1 but we have estimated them, so Compute the predicted values of xi, xi, where xi = 0 + 1 zi, i = 1,n.
16、, ,(2) Replace xi by xi in the regression of interest:,regress y on xi using OLS: yi = 0 + 1 xi + ui (2) Because xi is uncorrelated with ui in large samples, so the first least squares assumption holds Thus 1 can be estimated by OLS using regression (2) This argument relies on large samples (so 0 and 1 are well estimated using regression (1) This the resulting estimator is called the “Two Stage Least Squares” (TSLS) e
