计量经济学工具变量IVSLS

1、计量经济学（研究生,Week 14,Instrument Variable,Regression Models,Simultaneous Equation Using 2SLS (Chapter 16,IV Estimation in Multiple Regression models (15.1-3,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 lea

2、d 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 does ch

3、ange with time we need,a new approach,Instrumental Variables Regression,Three important threats to internal validity are,1,2,3,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,Y,c

4、auses,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,uncorrelated with,

5、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,errors-in-variabl

6、e bias, not just to simultaneous,causality bias,What is Simultaneous Causality,Suppose we have two endogenous variables,Y,1,Y,2,and,two exogenous variables,X,1,X,2,such that,Y,1,i,0,1,X,1,i,2,Y,2,i,u,1,i,1,Y,2,i,0,1,Y,1,i,2,X,2,i,u,2,i,2,Lets see why,Y,2,or,Y,1,is endogenous,Suppose,u,1,i,0 and,u,2,

7、i,0,then we have,Y,1,i,E,Y,1,i,from,1,But in (2), if,2,0, this will cause a change in,Y,2,i,so,Y,2,i,is correlated with,u,1,i,through,2,The same is true for,Y,1,i,and,u,2,i,in (2,through,1,Simultaneous Bias,Can we estimate these two equations,consistently,y,1,a,1,y,2,1,z,1,u,1,y,2,a,2,y,1,2,z,2,u,2,

8、For consistency, we need cov(y,2,u,1,0, and,cov(y,1,u,2,0,However, a large,u,2,means a larger,y,2,which,implies,a,larger,y,1,if,a,1,0), so cov(y,1,u,2,0,The same is true for cov(y,2,u,1,due to the,circular effect of,u,1,The IV Estimator with a Single,Regressor and a Single Instrument,y,i,0,1,x,i,u,i

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,z,i,which is,uncorrelated with,u,i,The instrumental variable det

10、ects movements in,x,i,that,are uncorrelated with,u,i,and use these to estimate,1,Two conditions for a valid instrument,y,i,0,1,x,i,u,i,For an instrumental variable (an,instrument,z,to be valid, it must satisfy two conditions,1,Instrument relevance,cov,z,i,x,i,0,2,Instrument exogeneity,cov,z,i,u,i,=

11、0,In other words, IV variable,z,i,must be an,exogenous variable that is correlated with,x,Or,z,i,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 1,st,but have to,assume,the 2,nd,Example: Labor Economics,Suppose log,wage,0,1,educ,u,

12、 u,2,abil,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 siblings mi

13、ght,be an instrument for,educ,Instrument Variable Regression,Suppose we have,y,i,0,1,x,i,u,i,cov,x,u,i,0,Our estimate of,1,will be inconsistent,Either we find the omitted variable in,u,i,and add it,into our model to overcome the inconsistency,Or we find an instrument,z,i,for the included variable,Su

14、ppose for now that you have such a,z,i,we,ll,discuss how to find instrumental variables later,How can you use,z,i,to estimate,1,We will explain this in two ways,The IV Estimator, one,x,and one,z,Explanation #1:Two Stage Least Squares (TSLS,As it sounds, TSLS has two stages,two regressions,1) First i

15、solates the part of,x,that is uncorrelated with,u,regress,x,on,z,using OLS,x,i,0,1,z,i,v,i,1,Because,z,i,is uncorrelated with,u,i,0,1,z,i,is uncorrelated,with,u,i,We dont know,0,or,1,but we have estimated them, so,Compute the predicted values of,x,i,x,i,where,x,i,0,1,z,i,i,1,n,2) Replace,x,i,by,x,i,

16、in the regression of interest,regress,y,on,x,i,using OLS,y,i,0,1,x,i,u,i,Because,x,i,is uncorrelated with,u,i,2,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

17、regression (1,This the resulting estimator is called the,Two,T,S,L,S,Stage Least Squares,TSLS) estimator,1,The IV Estimator, one,x,and one,z,ctd,Explanation #2: (only) a little algebra,y,i,0,1,x,i,u,i,But,x,i,0,1,z,i,v,i,Thus,cov,y,i,z,i,= cov,0,1,x,i,u,i,z,i,cov,0,z,i,+ cov,1,x,i,z,i,+ cov,u,i,z,i,

18、0 + cov,1,x,i,z,i,+ 0,1,cov,x,i,z,i,where cov,u,i,z,i,= 0 (instrument exogeneity); thus,cov,Y,i,Z,i,1,cov,X,i,Z,i,s,Y,Z,s,X,Z,in large samples,The instrument relevance condition, cov,x,z,0,ensures that you don,t divide by zero,Supply and Demand Example,Start with an equation you,d like to,estimate,

19、say a supply function in a,market,q,s,a,1,p + b,1,z + u,1,where,p,is the price and,z,is a supply shifter,Call this a structural equation,it,s,derived from economic theory and has a,causal interpretation where,p,directly,affects,q,s,Example (cont,Problem that can,t just regress,observed quantity on p

20、rice, since,observed quantity are determined by the,equilibrium of supply and demand,Consider a second structural equation,in this case the demand function,q,d,a,2,p + u,2,So quantity are determined by a SEM,Example (cont,Both,q,and,p,are endogenous because,they are both determined by the,equilibriu

21、m of supply and demand,z,is exogenous, and it,s the availability,of this exogenous supply shifter that,allows us to identify the structural,demand equation,With no observed demand shifters,supply is not identified and cannot be,estimated,Identification of Demand,Equation,p,D,S (z=z,1,S (z=z,2,S (z=z

22、,3,q,Using IV to Estimate Demand,Given,q,s,a,1,p + b,1,z + u,1,q,d,a,2,p + u,2,So, we can estimate the structural demand,equation, using,z,as an instrument for,p,First stage equation is,p,0,1,z,v,2,u,Second stage equation is,q,a,2,p,2,Thus, 2SLS provides a consistent estimator,of,a,2,the slope of th

23、e demand curve,We cannot estimate,a,1,the slope of the,supply curve,The General SEM,Suppose our structural equations are,y,1,a,1,y,2,1,z,1,u,1,y,2,a,2,y,1,2,z,2,u,2,Thus,y,2,a,2,a,1,y,2,1,z,1,u,1,2,z,2,u,2,So, (1,a,2,a,1,y,2,a,2,1,z,1,2,z,2,a,2,u,1,u,2,which can,be rewritten,if,a,2,a,1,1,as,y,2,1,z,

24、1,2,z,2,v,2,v,2,a,2,u,1,u,2,(1,a,2,a,1,This is the so called,reduced,form,However, in the reduced form, we don,t know what is the,value of,a,1,or,a,2,Example #1: Supply and demand for butter,IV regression was originally developed to estimate,demand elasticities for agricultural goods, for,example bu

25、tter,butter,butter,log,Q,0,1,log,P,u,i,1,price elasticity of butter = percent change in,quantity for a 1% change in price (recall log-log,specification discussion,Data: observations on price and quantity of butter,for different years,The OLS regression of log,Q,butter,on log,P,butter,suffers from si

26、multaneous causality bias,why,Simultaneous causality bias in the OLS regression of,log,Q,butter,on log,P,butter,arises because price and quantity are,determined by the interaction of demand,and,supply,A side note,What is the relationship between, say Marxian,concept of,labor theory of value,and the

27、Microeconomics theory,of,price formation,What is the long-term supply curve and its determination,A Quick Note on Marxian Economics,At Q,1,the production is less then,socially necessary, and is,causing a shortage,The competition will drive the,price above it value, until more,producers enters the ma

28、rket or,more product is being produced,This leads to an increase in the,level of output, all the way to Q,At Q,then socially necessary, and is,2,the production is more,causing a surplus,The competition will drive the,price below it value, until some,producers leaves the market or,less product is bei

29、ng produced,This leads to a drop in the level,of output, all the way to Q,S,L,is the long-term supply curve that is,consistent with the Marxian concept of,socially necessary labor time,Is it true that main stream economic has,no theory to explain why it is at S,L,rather,then some other level,Back to

30、 our supply and demand for butter,This interaction of demand and supply produces,Would a regression using these data produce the,demand curve,A) Demand,B) Supply,C) Neither,What would you get if only supply shifted,TSLS estimates the demand curve by isolating shifts in,price and quantity that arise

31、from shifts in supply,Z,is a variable that shifts supply but not demand,TSLS in the supply-demand example,log,Q,butter,0,1,log,P,butter,u,i,Let,Z,rainfall in dairy-producing regions,Is,Z,a valid instrument,Lets check 2 conditions,1) Exogenous? corr,rain,i,u,i,= 0,A) Yes,B) No,C) In sufficient inform

32、ation,Plausibly,whether it rains in dairy-producing,regions shouldn,t affect demand,butter,2) Relevant? corr,rain,i,log,P,0,A) Yes,B) No,C) In sufficient information,Plausibly,insufficient rainfall means less grazing,means less butter,TSLS in the supply-demand example, ctd,butter,log,Q,1,i,Z,rain,i,

33、rainfall in dairy-producing regions,Stage 1: regress log,P,butter,on,rain,get log,P,butter,0,butter,log,P,u,log,P,butter,isolates changes in log price that arise from,supply (part of supply, at least,butter,on log,P,butter,Stage 2: regress log,Q,The regression counterpart of using shifts in the supp

34、ly,curve to trace out the demand curve,TSLS (2 stage lest squares) in EViews,Everything the same as in OLS except,In,Estimation Methods,select,TSLS,Two,stage lest squares (TSNLS and ARMA,Provide a list of instrument variables, be sure to,include all exogenous variables as well,Only the variables on

35、the right hand side not in,the list of instruments are considered,endogenous,In Options, select,Heteroskedasiticity consistent,coefficient covariance,Example 15.5 using 2SLS,Dependent Variable: LOG(WAGE,Note,Red are instruments,Method: Two-Stage Least Squares,Blue are exogenous,Sample: 1 753 IF INLF

36、,Green is endogenous,Included observations: 428,Instrument list,EXPER,EXPERSQ,FATHEDUC MOTHEDUC,Variable,Coefficient,Std. Error,t-Statistic Prob,EDUC,0.061397,0.031437,1.953024,0.0515,EXPER,0.044170,0.013432,3.288329,0.0011,EXPERSQ,0.000899,0.000402,2.237993 0.0257,C,0.048100,0.400328,0.120152,0.904

37、4,R-squared,0.135708,Mean dependent var,1.190173,Adjusted R-squared,0.129593,S.D. dependent var,0.723198,S.E. of regression,0.674712,Sum squared resid,193.0200,F-statistic,8.140709,Durbin-Watson stat,1.945659,Prob(F-statistic,0.000028,Example,Demand for Cigarettes,How much will a hypothetical cigare

38、tte tax,reduce cigarette consumption,To answer this, we need the elasticity of,demand for cigarettes, that is,1,in the,regression,cigarettes,cigarettes,log,Q,0,1,log,P,u,i,Will the OLS estimator plausibly be,unbiased,Why or why not,Example,Cigarette demand, ctd,log,Q,cigarettes,0,1,log,P,cigarettes,

39、u,i,Panel data,Annual cigarette consumption and average prices,paid (including tax,48 continental US states, 1985-1995,Proposed instrumental variable,Z,i,general sales tax per pack in the state,GSTax,i,Is this a valid instrument,1) Relevant? corr,GSTax,i,log,P,cigarettes,0,2) Exogenous? corr,GSTax,i

40、,u,i,= 0,Example,Cigarette demand, two instruments,Dependent Variable: LOG(PACKPC,Method: Two-Stage Least Squares,Sample: 1 528 IF YEAR=1995,Included observations: 48,White Heteroskedasticity-Consistent Standard Errors & Covariance,Instrument list,LOG(INCOME/POP,TAX-TAXS)/CPI TAXS/CPI,Variable,Coeff

41、icient,Std. Error,t-Statistic Prob,LOG(INCOME/POP,0.280405,0.253890,1.104436,0.2753,LOG(AVGPRS/CPI,1.277424,0.249610,5.117680 0.0000,C,9.776810,0.961763,10.16551,0.0000,R-squared,0.429422,Mean dependent var,4.538837,Adjusted R-squared,0.404063,S.D. dependent var,0.243346,S.E. of regression,0.187856,

42、Sum squared resid,1.588044,F-statistic,13.28079,Durbin-Watson stat,1.946351,Prob(F-statistic,0.000029,Identification,The general IV regression model, ctd,Y,1,0,1,Y,2,k,Y,k,1,k,1,Z,1,k+r,Z,r,u,We need to introduce some new concepts and to extend some,old concepts to the general IV regression model,Te

43、rminology,identification,and,overidentification,TSLS with included exogenous variables,one endogenous regressor,multiple endogenous regressors,Assumptions that underlie the normal sampling distribution of,TSLS,Instrument validity (relevance and exogeneity,General IV regression assumptions,Identifica

44、tion, ctd,The coefficients,1,k,are said to be,exactly,identified,if,m,k,There are just enough instruments to estimate,1,k,overidentified,if,m,k,There are more than enough instruments to estimate,1,k,If so, you can test whether the instruments are,valid,a test of the,overidentifying,restrictions,we,l

45、l,return to this later,underidentified,if,m,k,There are too few enough instruments to estimate,1,k,If so, you need to get more instruments,Identification,In general, a parameter is said to be,identified,if,different values of the parameter would produce,different distributions of the data,In IV regr

46、ession, whether the coefficients are,identified depends on the relation between the,number of instruments,m,and the number of,endogenous regressors,k,Intuitively, if there are fewer instruments than,endogenous regressors, we can,t estimate,1,k,For example, suppose,k,1 but,m,0 (no,instruments),Identi

47、fication of General SEM,Once again, our structural equations are,y,1,a,1,y,2,1,z,1,u,1,y,2,a,2,y,1,2,z,2,u,2,Let,z,1,be all the exogenous variables in the,first equation, and,z,2,be all the exogenous,variables in the second equation,It,s okay for there to be overlap in,z,1,and,z,2,How are we able to

48、 identify which equation is,which,We need to state the,rank,condition,Identification of General SEM,Given our two equations,y,1,a,1,y,2,1,z,1,u,1,y,2,a,2,y,1,2,z,2,u,2,To identify equation 1, there must be some,variables (at least 1) in,z,2,that are not in,z,1,To identify equation 2, there must be s

49、ome,variables (at least 1) in,z,1,that are not in,z,2,We refer to this as the,rank,condition,We are able to identify the two equations if,the rank condition is satisfied,Example: Labor Market,Suppose the structural equations for the labor market are,hours,a,1,log,wage,nwifeinc,10,11,educ,12,age,13,k

50、idslt6,14,15,exper,16,exper,2,u,1,log,wage,a,2,hours,20,21,educ,22,age,23,kidslt6,24,nwifeinc,25,exper,26,exper,2,u,2,Can we identify which is the supply / demand equation,for labor,No,That is the reason for the rank condition,1,2,Example: Labor Market,Suppose the structural equations for the labor

51、market,instead are as follows,1,hours,a,1,log,wage,10,11,educ,12,age,13,kidslt6,14,nwifeinc,u,1,2,log,wage,a,2,hours,20,21,educ,22,exper,2,23,exper,u,2,Which is the supply / demand equation for labor,1. is supply and 2. is demand equations for labor, for,age,kidslt6,and,nwifeinc,affects supply but n

52、ot demand,for labor, while experience affects demand but not,supply of labor,Order Condition,Note that the exogenous variable excluded from,the first equation must have a,non-zero,coefficient,in the second equation for the,rank,condition to,hold,Order,condition states that there must be at least,as

53、many exogenous variables excluded in the first,equation as there are endogenous variables in the,first equation (see page 529,Note that the order condition clearly holds if the,rank condition does,there will be an exogenous,variable for the endogenous one,Rank condition,Order condition,Order vs. Ran

54、k Conditions,Order condition only counts the number of,variables, while the,Rank condition is concerned with the,significance of the coefficients of the,excluded variable from the first equation on,the second equation,We can check the order condition by,counting the number of exogenous,variables exc

55、luded in each equation, while,the rank condition requires,t,or,F,test,2,Note,R,from IV can be misleading,Given our linear model,y,0,1,x,u,i,When,x,and,u,are correlated, then,v,y,1,2,v,x,v,u,So we can,t break the,SST,into,SSE,and,SSR,For reasons that I don,t fully understand yet,EView uses,SSR,from I

56、V that might be larger then,SST,of,y,to compute the,R,2,Thus,R,2,might be less than 0, and do not have,the same meaning as in the case for OLS,See example 15.3 using bwght data,Multicolinearity with 2SLS,Multicolinearity with 2SLS,For our original,structural,model,reduced,form equation,y,1,0,1,y,2,2

57、,z,1,u,1,y,2,0,1,z,1,2,z,2,3,z,3,v,2,In 2SLS, we have,Var,2,1,2,SST,2,1,R,2,where,SST,2,is the sum squared total of,2,and,R,2,2,is from regression of,2,on,z,1,Multicolinearity with 2SLS,For our original,structural,model,reduced,form equation,y,1,0,1,y,2,2,z,1,u,1,y,2,0,1,z,1,2,z,2,3,z,3,v,2,2,2,Sinc

58、e,Var,1,SST,2,1,R,2,Is,SST,2,SST,2,Is Corr,2,z,1,Corr,y,2,z,1,2,2,R,2,R,2,A) Yes,B) No,C) In sufficient information,Multicolinearity problem has to do with 2 things,1,SST,2,SST,2,2,2,2. Since Corr,2,z,1,Corr,y,2,z,1,R,2,R,2,Multicolinearity with 2SLS,Again, our original,structural,reduced,form equat

59、ions,y,1,0,1,y,2,2,z,1,u,1,y,2,0,1,z,1,2,z,2,3,z,3,v,2,We have,Var,1,2,SST,2,2,1,R,2,Lets see why,1,SST,2,SST,2,2. Since Corr,Corr,y,2,2,z,1,2,z,1,R,2,R,2,2,Testing for,Endogeneity,Testing for Endogeneity,Since OLS is preferred to IV if we do not,have an endogeneity problem, then we,d,like to be abl

60、e to test for endogeneity,If we do not have endogeneity, both OLS,and IV are consistent,Idea of,Hausman,test is to see if the,estimates from OLS and IV are different,Testing for Endogeneity (cont,While it,s a good idea to see if IV and OLS have,different implications, it,s easier to use a,regression