Econometric analysis Instrumental Variables.ppt

1、Applied Econometrics,William Greene Department of Economics Stern School of Business,Applied Econometrics,13. Instrumental Variables,Instrumental Variables,Framework: y = X + , K variables in X. There exists a set of K variables, Z such that plim(ZX/n) 0 but plim(Z/n) = 0 The variables in Z are call

2、ed instrumental variables. An alternative (to least squares) estimator of is bIV = (ZX)-1Zy We consider the following: Why use this estimator? What are its properties compared to least squares? We will also examine an important application,IV Estimators,Consistent bIV = (ZX)-1Zy = (ZX/n)-1 (ZX/n)+ (

3、ZX/n)-1Z/n = + (ZX/n)-1Z/n Asymptotically normal (same approach to proof as for OLS) Inefficient to be shown.,LS as an IV Estimator,The least squares estimator is (X X)-1Xy = (X X)-1ixiyi = + (X X)-1ixii If plim(XX/n) = Q nonzero plim(X/n) = 0 Under the usual assumptions LS is an IV estimator X is i

4、ts own instrument.,IV Estimation,Why use an IV estimator? Suppose that X and are not uncorrelated. Then least squares is neither unbiased nor consistent. Recall the proof of consistency of least squares: b = + (XX/n)-1(X/n). Plim b = requires plim(X/n) = 0. If this does not hold, the estimator is in

5、consistent.,A Popular Misconception,A popular misconception. If only one variable in X is correlated with , the other coefficients are consistently estimated. False. The problem is “smeared” over the other coefficients.,The General Result,By construction, the IV estimator is consistent. So, we have

6、an estimator that is consistent when least squares is not.,Asymptotic Covariance Matrix of bIV,Asymptotic Efficiency,Asymptotic efficiency of the IV estimator. The variance is larger than that of LS. (A large sample type of Gauss-Markov result is at work.) (1) Its a moot point. LS is inconsistent. (

7、2) Mean squared error is uncertain: MSEestimator|=Variance + square of bias. IV may be better or worse. Depends on the data,Two Stage Least Squares,How to use an “excess” of instrumental variables (1) X is K variables. Some (at least one) of the K variables in X are correlated with . (2) Z is M K va

8、riables. Some of the variables in Z are also in X, some are not. None of the variables in Z are correlated with . (3) Which K variables to use to compute ZX and Zy?,Choosing the Instruments,Choose K randomly? Choose the included Xs and the remainder randomly? Use all of them? How? A theorem: (Brundy

9、 and Jorgenson, ca. 1972) There is a most efficient way to construct the IV estimator from this subset: (1) For each column (variable) in X, compute the predictions of that variable using all the columns of Z. (2) Linearly regress y on these K predictions. This is two stage least squares,Algebraic E

10、quivalence,Two stage least squares is equivalent to (1) each variable in X that is also in Z is replaced by itself. (2) Variables in X that are not in Z are replaced by predictions of that X with all the variables in Z that are not in X.,2SLS Algebra,Asymptotic Covariance Matrix for 2SLS,2SLS Has La

11、rger Variance than LS,Estimating 2,Measurement Error,y = x* + all of the usual assumptions x = x* + uthe true x* is not observed (education vs. years of school) What happens when y is regressed on x? Least squares attenutation:,Why Is Least Squares Attenuated?,y = x* + x = x* + u y = x + ( - u) y =

12、x + v, cov(x,v) = - var(u) Some of the variation in x is not associated with variation in y. The effect of variation in x on y is dampened by the measurement error.,Measurement Error in Multiple Regression,Twins,Application from the literature: Ashenfelter/Kreuger: A wage equation that includes “sch

13、ooling.”,Orthodoxy,A proxy is not an instrumental variable Instrument is a noun, not a verb,Applied Econometrics,William Greene Department of Economics Stern School of Business,Applied Econometrics,14. Nonlinear Regression and Nonlinear Least Squares,Nonlinear Regression,What makes a regression mode

14、l “nonlinear?” Nonlinear functional form? Regression model: yi = f( xi , ) + i Not necessarily: yi = exp() + 2*xi + i 1 = exp() yi = exp()xiexp(i) is “loglinear” Models can be nonlinear in the functional form of the relationship between y and x, and not be nonlinear for purposes here. We will redefi

15、ne “nonlinear” shortly, as we proceed.,Nonlinear Least Squares,Least squares: Minimize wrt i yi - f(xi,)2 = i ei2 First order conditions: iyi- f(xi,)2 / = i(-2)yi- f(xi,) f(xi,)/ = -i ei xi0 = 0(familiar?) There is no explicit solution, b = f(data) like LS. (Nonlinearity of the FOC defines nonlinear

16、 model),Example: NIST,How to solve this kind of set of equations: Example, yi = 0 + 1xi2 + i. i ei2/0 = i (-1) (yi - 0 - 1xi2) 1 = 0 i ei2/1 = i (-1) (yi - 0 - 1 xi2) xi2 = 0 i ei2/2 = i (-1) (yi - 0 - 1 xi2) 1 xi2lnxi = 0 Nonlinear equations require a nonlinear solution. Well return to that problem

17、 shortly. This defines a nonlinear regression model. I.e., when the first order conditions are not linear in . (!) Check your understanding. What does this produce if f( xi , ) = xi? (I.e., a linear model),The Linearized Regression Model,Linear Taylor series: y = f(xi,) + . Expand the regression aro

18、und some point, 0. f(xi,) f(xi,0) + kf(xi,0)/k0( k - k0) = f(xi,0) + k xi0 ( k - k0) = f(xi,0) - k xi0k0 + k xi0k = f0 + k xi0k which looks linear. The pseudo-regressors are the derivative functions in the linearized model.,Estimating Asy.Varb,Computing the asymptotic covariance matrix for the nonli

19、near least squares estimator using the pseudo regressors and the sum of squares.,Gauss-Marquardt Algorithm,Given a coefficient vector at step m, find the vector for step m+1 by b(m), b(m+1) = b(m) + X0(m)X0(m)-1X0(m)e0(m) Columns of X0(m) are the derivatives, f(xi,b(m)/b(m) e0 = vector of residuals,

20、 y - fx,b(m) “Update” vector is the slopes in the regression of the residuals on the pseudo-regressors. Update is zero when they are orthogonal. (Just like LS),A NIST Application,Y X 2.138 1.309 3.421 1.471 3.597 1.490 y = 0 + 1x2 + . 4.340 1.565 4.882 1.611 xi0 = 1, x2, 1x2logx 5.660 1.680,Iteratio

21、ns,NLSQ;LHS=Y ;FCN=b0+B1*XB2 ;LABELS=b0,B1,B2 ;MAXIT=500;TLF;TLB;OUTPUT=1;DFC ;START=0,1,5 $ Begin NLSQ iterations. Linearized regression. Iteration= 1; Sum of squares= 149.719219 ; Gradient= 149.718223 Iteration= 2; Sum of squares= 5.04072877 ; Gradient= 5.03960538 Iteration= 3; Sum of squares= .13

22、7768222E-01; Gradient= .125711747E-01 Iteration= 4; Sum of squares= .186786786E-01; Gradient= .174668584E-01 Iteration= 5; Sum of squares= .121182327E-02; Gradient= .301702148E-08 Iteration= 6; Sum of squares= .121182025E-02; Gradient= .134513256E-15 Iteration= 7; Sum of squares= .121182025E-02; Gra

23、dient= .644990175E-20 Convergence achieved,Stata Version 9,Most maximum likelihood estimators now test for convergence using the Hessian-scaled gradient, g*inv(H)*g. This criterion ensures that the gradient is close to zero when scaled by the Hessian (the curvature of the likelihood or pseudolikelih

24、ood surface at the optimum) and provides greater assurance of convergence for models whose likelihoods tend to be difficult to optimize, such as those for arch, asmprobit, and scobit. See R maximize.,Results,+-+ | User Defined Optimization | | Nonlinear least squares regression Weighting variable =

25、none | | Number of iterations completed = 30 | | Dep. var. = Y Mean= 4.006333333 , S.D.= 1.233983576 | | Model size: Observations = 6, Parameters = 3, Deg.Fr.= 3 | | Residuals: Sum of squares= .1211820252D-02, Std.Dev.= .02010 | | Fit: R-squared= .999841, Adjusted R-squared = .99973 | | (Note: Not u

26、sing OLS. R-squared is not bounded in 0,1 | | Model test: F 2, 3 = 9422.64, Prob value = .00000 | | Diagnostic: Log-L = 17.0085, Restricted(b=0) Log-L = -9.2282 | | LogAmemiyaPrCrt.= -7.409, Akaike Info. Crt.= -4.670 | +-+ +-+-+-+-+-+-+ |Variable | Coefficient | Standard Error |b/St.Er.|P|Z|z | Mean

27、 of X| +-+-+-+-+-+-+ B0 -.5455928058 .22460069 -2.429 .0151 B1 1.080717551 .13697694 7.890 .0000 B2 3.372865575 .17846759 18.899 .0000,NLS Solution,The pseudo regressors and residuals at the solution are: 1 x2 1x2 lnx e0 12.479830.721624 .0036 13.675661.5331 -.0058 13.838261.65415-.0055 14.529722.19

28、255-.0097 14.994662.57397 .0298 15.753583.22585-.0124 X0e0 = .3375078D-13 .3167466D-12 .1283528D-10,Application: Doctor Visits,German Individual Health Care data: N=27,236 Model for number of visits to the doctor,Conditional Mean and Projection,Notice the problem with the linear approach. Negative p

29、redictions.,Most of the data are in here,This area is outside the range of the data,Nonlinear Model Specification,Nonlinear Regression Model y=exp(x) + X =one,age,health_status, married, education, household_income, nkids,NLS Iterations,- nlsq;lhs=docvis;start=0,0,0,0,0,0,0;labels=k_b;fcn=exp(b1x);m

30、axit=25;out. Begin NLSQ iterations. Linearized regression. Iteration= 1; Sum of squares= 1014865.00 ; Gradient= 257025.070 Iteration= 2; Sum of squares= .130154610E+11; Gradient= .130145942E+11 Iteration= 3; Sum of squares= .175441482E+10; Gradient= .175354986E+10 Iteration= 4; Sum of squares= 23536

31、9144. ; Gradient= 234509185. Iteration= 5; Sum of squares= 31610466.6 ; Gradient= 30763872.3 Iteration= 6; Sum of squares= 4684627.59 ; Gradient= 3871393.70 Iteration= 7; Sum of squares= 1224759.31 ; Gradient= 467169.410 Iteration= 8; Sum of squares= 778596.192 ; Gradient= 33500.2809 Iteration= 9; S

32、um of squares= 746343.830 ; Gradient= 450.321350 Iteration= 10; Sum of squares= 745898.272 ; Gradient= .287180441 Iteration= 11; Sum of squares= 745897.985 ; Gradient= .929823308E-03 Iteration= 12; Sum of squares= 745897.984 ; Gradient= .839914514E-05 Iteration= 13; Sum of squares= 745897.984 ; Grad

33、ient= .991471058E-07 Iteration= 14; Sum of squares= 745897.984 ; Gradient= .132954206E-08 Iteration= 15; Sum of squares= 745897.984 ; Gradient= .188041512E-10,Nonlinear Regression Results,+-+ | Nonlinear least squares regression | | LHS=DOCVIS Mean = 3.183525 | | Standard deviation = 5.689690 | | WT

34、S=none Number of observs. = 27326 | | Model size Parameters = 7 | | Degrees of freedom = 27319 | | Residuals Sum of squares = 745898.0 | | Standard error of e = 5.224584 | | Fit R-squared = .1567778 | | Adjusted R-squared = .1568087 | | Info criter. LogAmemiya Prd. Crt. = 3.307006 | | Akaike Info. C

35、riter. = 3.307263 | | Not using OLS or no constant. Rsqd k=col(x)$ nlsq;lhs=docvis;start=0,0,0,0,0,0,0 ;labels=k_b;fcn=exp(b1x); matr;xbar=mean(x)$ calc;mean=exp(xbarb)$ matr;me=b*mean$ matr;g=mean*iden(k)+mean*b*xbar$ matr;vme=g*varb*g$ matr;stat(me,vme)$,Partial Effects at the Means of X,+-+ |Number of observations in current sample = 27326 | |Number of parameters computed here = 7 | |Number of degrees of freedom