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1、Chapter 12Instrumental Variables Regression12.1 The IV estimator with a single regressor and a single instrumentVariables correlated with the error term are called endogenous variables, cov(X,u)0Variables uncorrelated with the error term are called exogenous variables, cov(W,u)=0Thinking the variati

2、on in X has two parts: one part is correlated with error term, which causes the problem, a second part is uncorrelated with error term12.1 The IV estimator with a single regressor and a single instrumentTwo conditions for a valid instrumentInstrument relevance: corr(Z,X)0, variation in the instrumen

3、t Z is related to variation in XInstrument exogeneity: corr(Z,u)=0, part of the variation of X captured by the instrumental variable is exogenous12.1 The IV estimator with a single regressor and a single instrumentThe two stage least squares estimatorFirst stage, posing X into two components, the pr

4、oblematic component that may be correlated with the regression error (v) and another problem-free component that is uncorrelated with the error term (Z)X = 0+ 1Z + vSecond stage, using the problem-free component to estimate coefficient12.1 The IV estimator with a single regressor and a single instru

5、mentWhy does IV regression work?Philip Wrights problemZ = weather, X= price, Y = quantityEstimating the effect on test scores of class sizeZ = distance from the epicenter, X = class size, Y = test score12.1 The IV estimator with a single regressor and a single instrumentFormula for the TSLS estimato

6、r (12.4), showApplication to the Demand for CigZ = SaleTax, X = Price, Y = Quantity of demand12.2 the General IV regression Model12.13 & 12.14 formulax1,x2,xk are k endogenous regressorsw1,w2,wr are r included exogenous regressorsz1,z2,zm are m instrumental variablesIf mk, overidentification; if m=k

7、, exact identification; if m1)12.2 the General IV regression ModelThe IV regression AssumptionE(u|W)=0i.i.dLarge outliers are unlikelyConditions for a valid instrument12.2 the General IV regression ModelApplicationZ1 = SaleTax, X = Price, W = e, Y = Quantity of demand intuition explanation: if the s

8、tate sales tax is related to state e, then it is correlated with a variable in the error term of the cig demand equation, Z2 = states levy special taxes to cigAre all previous estimates credible?12.3 Checking Instrument ValidityAssumption #1, instrument relevance the more the variation in X is expla

9、ined by the instruments, the more information is available for use in IV regression. A more relevant instrument produces a more accurate estimator.Instruments that explain little of the variation in X are called weak instruments12.3 Checking Instrument ValidityWhy weak instruments are a problem?cov(

10、Z,X)=0Checking for weak instruments when there is a single endogenous regressorcomputing the F-statistic testing the hypothesis that the coefficients on the coefficients on the instruments are all zero in the first-stage regression of TSLSThe more information content, the larger is the expected valu

11、e of the F statistics.Simple rule of thumb is that you do not need to worry about weak IV if the first stage F-statistic exceeds 10.12.3 Checking Instrument ValidityWhat do I do if I have weak IV?mk, using the most relevant subset for your TSLS analysis. But your TSLS standard errors might increase

12、when you drop your weak IVm=k, (1) finding additional stronger IV; (2)employing methods other than TSLS with using the weak IV12.3 Checking Instrument ValidityAssumption #2, Instrument ExogeneityCan you test statistically the assumption that the IV are exogenous?m=k, nomk, yes.12.3 Checking Instrume

13、nt ValidityThe overidentification restrictions testexogeneity of the instruments means that they are uncorrelated with the error term. This suggests that the instruments should be approximatesly uncorrelated with u_hat_TSLS. (key concept 12.6;12.17)u_hat_TSLS= y-X-W12.4 ApplicationAre the instruments relevant? The first stage F-statisticsAre the instruments exogenous? The J-statistics (this is a chi-squared distribution)12.5 where do valid instruments come from?Using econ