第6讲-内生性问题

1、内生性问题15-118-2假定3不成立：导致OLS不一致的四种情况1.Omitted Variables Bias（遗漏变量）2.Measurement Error（测量误差）3.Simultaneous Causality（内生性）4.Using Lagged Values of the Dependent Variable as Explanators, in the presence of serial correlation（自相关且包含被解释变量滞后项）Any of these conditions will make OLS inconsistent (and biased).遗漏

2、变量 真实模型 估计量b0,b1,b2 估计模型 估计量a0,a115-301122iiiiyXX011iiiyX遗漏变量的后果 1. 假如x1 和x2 是相关的，即r12 0，那么 a1 和a2 都是有偏的和不一致的，即 E (a1) 1,E(a0) 0 ，在大样本的情况下，这种偏差也不消失。15-401122iiiiyXX011iiiyX11221()E abb21为X2对X1的回归系数遗漏变量的后果 2. 即使x1 和x2 是不相关的，a0还是有偏，虽然a1无偏了。 3.误差项方差不正确估计。 按照残差平方和除以n-k-1估计误差项，两个模型不同残差平方和和n-k-1都不同15-5011

3、22iiiiyXX011iiiyX遗漏变量的后果 4. 计算的a1的方差是真实估计量b1方差的有偏估计量15-601122iiiiyXX011iiiyX21211var()()iaXX221222111211var( )() (1)()iibVIFXXrXX11var()var( )ab遗漏变量的后果 5.通常的估计参数的置信区间和假设检验变得不可靠。 6.基于不正确模型的预测和预测区间也不可靠。 注意：假如模型是基于相应的经济理论而构建，不要擅自删除该理论所要求的变量。15-701122iiiiyXX011iiiyX18-8Measurement Error测量误差 解释变量X的测量误差会导

4、致假定三 不成立(also called “errors in variables”).模型设定的X与实际观察的X不同，那么系数估计会错误18-9测量误差 测量错误产生的原因: 数据瑕疵(如：记不清) 使用了不完美的代理变量(如：用应税收入代替总收入) 没有正确理解真正的解释变量（如：年收入变化还是永久收入变化）18-10测量误差 We model Yi01Xiibut instead of observing Xi we observe Mi Xivi where vi is some error E(vi) 0,Cov(Xi,vi) 0We regress Yi01Mii18-11测量误差

5、 We model Yi01XiiWe regress Yi01MiiSubstituting, our regression equation is equivalent to 01(Xivi)i 01Xi1vii18-12测量误差010101111)We model We regress which is, in essence, has to perform two jobs.It estimates the coefficient on ( but it ALSO estimates the coefficient iiiiiiiiiiYXYMXvX.on Since does not

6、 appear in our model, its true coefficient is 0.iivv18-13测量误差01111().In essence, we regress has to perform two jobs.It estimates the coefficient on but it ALSO estimates the coefficient on Since does not appear in our model, iiiiiiiYXvXvv1100. its true coefficient is . will be a weighted average of

7、and 18-14测量误差 Our model is Yi01XiiInstead of Xi, we observe Mi XiiYi01(Mivi)i 01Mi1vii 01Mii where i1vii18-15测量误差010111111(,)(,)( ,)( ,)( )Our model is Instead of , we observe We regress where iiiiiiiiiiiiiiiiiiiiiiiiYXXMXvYMvCov MCov XvvCov vvCov v vVar v 18-16测量误差 Measurement error implies a corre

8、lation between our observed explanator and the error term What bias does this correlation create in OLS?Mi Xivi i -1vii18-17带有测量误差的DGP Yi01XiiE(i) 0Var(i) 2Cov(i,j) 0 for i jE(Xii) 0,1n(xi2) X2Mi XiviE(vi) 0Var(vi) v2Cov(vi,vj) 0 for i jCov(vi,Xi) 018-18测量误差0110111111?)cov(M, )var()cov(M,v)cov(M, )v

9、ar()iiiiiiiYMYMMM是的一致估计量吗（Regress +-v18-19Measurement Error (cont.)12122()()( )iiiXXvVar XVar XVar v18-20Measurement Error (cont.)211122212222211lim()10XXvvXvvXvp 1Notice that The bias will always pull PARTWAY from to .We call this bias ATTENUATION BIAS.18-21Measurement Error (cont.) Mismeasuring X

10、leads to ATTENUATION BIAS（零向偏误）. The estimated coefficient is biased towards 0. The magnitude of the bias depends on the relative variances of X and v. A small amount of random measurement noise will not bias the estimate very much.18-22Measurement Error (cont.) Mismeasuring X leads to ATTENUATION B

11、IAS. The estimated coefficient is biased towards 0. Note: mismeasuring Y does NOT lead to measurement error bias (though it does increase the variance of the error term, thus increasing standard errors).18-23Checking Understanding (cont.) Suppose you are advising policy makers on the effect of a one

12、-time tax rebate on consumption. Using a cross section of 50,000 households, you regress reported current consumption against reported current income. Your results suggest that the tax rebate in question will NOT have a large enough impact on consumption to justify the policy. (continued on next sli

13、de)18-24Checking Understanding (cont.) Suppose your results found that the marginal propensity to consume was too small to justify the tax rebate. A proponent of the measure argues that you should be regressing consumption against Friedmans “permanent income,” not current income. The resulting measu

14、rement error renders your results irrelevant. Assess this argument.18-25Checking Understanding (cont.) Answer: If your regression suffers from measurement error, then the true effect of the tax rebate on consumption is larger than what you found. The coefficient with attenuation bias is too small to

15、 justify the policy. The larger, true coefficient might or might not be large enough to support the tax rebate.18-26Simultaneous Causality（内生性问题） Another common source of correlation between Xi and i is SIMULTANEOUS CAUSALITY This complication is also called JOINTLY DETERMINED VARIABLES or ENDOGENEI

16、TY18-27Simultaneous Causality (cont.) Both X and Y are jointly determined The process that generates Y also generates X at the same time Because X and Y are determined simultaneously, X can adjust in response to shocks to Y () Thus X will be correlated with 18-28Simultaneous Causality (cont.) The cl

17、assic example of simultaneous causality in economics is supply and demand. Both prices and quantities adjust until supply and demand are in equilibrium. A shock to demand or supply causes BOTH prices and quantities to move.18-29Simultaneous Causality (cont.) Thus, any attempt to estimate the relatio

18、nship between prices and quantities (say, to estimate a demand elasticity) suffers from SIMULTANEITY BIAS. Econometricians have a frequent interest in estimating elasticities resulting from such an equilibrium process. Simultaneity bias is a MAJOR problem.18-30Simultaneous Causality (cont.) For exam

19、ple, consider the market for wheat. The quantity demanded for wheat is a function of the price consumers pay and the income of the population: i indexes separate marketsQiD01PiD2IiiD18-31Simultaneous Causality (cont.) The quantity of wheat supplied is a function of the price suppliers receive and th

20、e weather (which affects crop yields).QiS01PiS2WiiS18-32Simultaneous Causality (cont.) In equilibrium, and Lets focus on the demand equation. Is PiD correlated with iD ?QiSQiDPiS PiD18-33Simultaneous Causality (cont.) Suppose iD 0 (there is a positive shock to demand). This shock makes QiD greater t

21、han usual. In equilibrium, QiD = QiS To balance the supply equation, PiS must increase. Suppliers must be paid a higher price to supply the greater demanded quantity.QiS01PiS2WiiSQiD01PiD2IiiD18-34Simultaneous Causality (cont.) In equilibrium, PiS = PiD. The consumers must pay a higher price to enjo

22、y the higher quantity of wheat they demand Thus, a positive shock to iD induces a higher PiDQiS01PiS2WiiSQiD01PiD2IiiD18-35Simultaneous Causality (cont.) A positive demand shock increases the quantity demanded. In order to increase supply, the price must go up. The demand shock and the price are cor

23、related. OLS will be inconsistent.E(PiDiD) 018-36Simultaneous Causality (cont.) When we have a system of equations (as with supply and demand), all the variables that are jointly determined are called endogenous variables. Price and quantity are endogenous variables.18-37Simultaneous Causality (cont

24、.) Variables that are determined outside the system of equations are called exogenous variables. The weather is an exogenous variable. In partial equilibrium (such as the supply and demand for wheat), the populations income is also exogenous.18-38Simultaneous Causality (cont.) It is arbitrary which

25、endogenous variables we write on the left-hand side. We could write both equations with either Price or Quantity on the left-hand side. For convenience, let us use Price on the LHS of the Supply equation and Quantity on the LHS of the Demand equation.18-39Simultaneous Causality (cont.) (1) QiD01PiD2

26、IiiD(2) PiS01QiS2WiiS iD determines QiD. QiDQiS, so Cov(iD,QiS) 0 QiS determines PiS PiS PiD, so Cov(PiD,QiS) 0 Therefore, Cov(PiD,iD) 018-40Simultaneous Causality (cont.) (1) QiD01PiD2IiiD(2) PiS01QiS2WiiSiD QiD QiS PiS PiD18-41 InftInft1tLagged Dependent Variables （滞后被解释变量） Using lagged dependent

27、variables as explanators is another potential source of correlation between an explanator and the error term. For example, you try to predict next periods inflation as a function of this periods inflation.18-42Lagged Dependent Variables (cont.) InftInft1ttt1vt0 |1 Lagged dependent variables present

28、a problem in the presence of serial correlation. Example: suppose there is first order serial correlation:18-43Lagged Dependent Variables (cont.) InftInft1tSubstituting in tt1vt: InftInft1t1vtHowever, Inft1Inft2t1t1 is a determinant of BOTH t and Yt1Cov(Inft1,t) 018-44Lagged Dependent Variables (con

29、t.) Including lagged dependent variables as an explanator does NOT lead to inconsistency in the absence of first-order serial correlation.假定3不成立 When an explanator is correlated with the error term, we call the explanator a “troublesome variable.” 处理方法： 工具变量法 联立方程 面板数据15-4519-46Instrumental Variable

30、s工具变量 An Instrumental Variable is a variable that is correlated with X but uncorrelated with . If Zi is an instrumental variable:1.E( Zi Xi ) 02.E( Zi i ) = 019-47Instrumental Variables (cont.) The econometrician can use an instrumental variable Z to estimate the effect on Y of only that part of X t

31、hat is correlated with Z. Because Z is uncorrelated with , any part of X that is correlated with Z must also be uncorrelated with .19-48Instrumental Variables (例1) For example, lets revisit the question of how much mortality can be reduced by intensive cardiac treatment（重症心脏监护）.19-49 Mortalityi01DiI

32、ntensivelyTreated2DiFemale .kUrbaniiInstrumental Variables (cont.) If our observable control variables, such as DFemale, were the only differences between patients who received intensive treatment and those who did not, then DIntensivelyTreated would not tell us anything about . OLS would be consist

33、ent.19-50Instrumental Variables (cont.) However, we reasonably believe that a doctors choice to perform intensive cardiac procedures are correlated with many other variables.19-51Instrumental Variables (cont.) Doctors might select patients to receive treatment based on their underlying health status

34、. If health status is an unobservable determinant of mortality, then it is a component of . OLS will give an inconsistent estimate of the benefits of intensive treatment.19-52Instrumental Variables (cont.) To eliminate Omitted Variables Bias, we need to find some determinant of a patients receiving

35、intensive cardiac care that is unrelated to mortality. Is there any element of the cardiac care process that is reasonably random?19-53Instrumental Variables (cont.) Not every hospital is equipped to provide intensive cardiac care. Some patients live near cardiac care centers. When they have a heart

36、 attack, they are more likely to be transported to a center, and thus more likely to receive intensive treatment.19-54Instrumental Variables (cont.) McClellan, McNeil, and Newhouse argue that the distance from a patients home to the nearest hospital equipped for intensive cardiac care is a valid ins

37、trumental variable. They argue that geographic location is likely to be uncorrelated with unobserved traits that influence mortality.19-55Instrumental Variables (cont.) When McClellan, McNeil, and Newhouse use their instrumental variable, they find only a small benefit from intensive cardiac care fo

38、r the marginal patient (some patients may still benefit quite a lot!).19-56Instrumental Variables (测量误差时) When the economist is worried about measurement error, a good choice of instrument is simply a different measure of the same variable. The new measure may have its own errors, but these errors a

39、re unlikely to be correlated with the mistakes in the first measure, or with any other component of .19-57Instrumental Variables (cont.) For example, Ashenfelter and Rouse were studying the effect of education on earnings. Their data came from a survey of twins. They were concerned that individuals

40、might mis-report their own years of schooling, leading to measurement error biases.19-58Instrumental Variables (cont.) However, Ashenfelter and Rouse had two separate measures for each individuals years of schooling. The survey asked each individual to list both his/her own years of schooling, and a

41、lso the years of schooling for his/her twin. The twins report of an individuals schooling served as an instrumental variable for the individuals self-report.19-59Instrumental Variables (例3) Another example: policy makers are greatly interested in the effects of tax rates on labor force participation

42、 (and other taxpayer behaviors). They would like to run regressions with an individuals tax rate as an explanator. However, an individual has some choice over his/her tax rate.19-60Instrumental Variables (cont.) Taxpayers who are close to the income threshold for a new tax bracket can choose to limi

43、t their taxable income. For example, they might take more of their pay in the form of untaxed benefits or deferred 401(k) compensation rather than pay higher taxes on the extra compensation. The ability and desire to adjust taxable income may well be correlated with .19-61Instrumental Variables (con

44、t.) When the government changes the tax rates, the individuals new tax rate is determined by two elements:1. The change in tax rates (which is uncorrelated with anything else about the individual), and2. The individuals decisions about how to respond to the tax change (which could well be correlated

45、 with ).19-62Instrumental Variables (cont.) Public finance economists construct an instrumental variable that captures only the change in tax rates, not the change in behavior. They use the new tax tables to look up the tax rate individuals would face IF they did NOT change their behavior from befor

46、e the tax change.19-63Instrumental Variables (cont.) The constructed tax rate is correlated with the tax rate the individuals face after the tax change. The constructed tax rate is uncorrelated with the behavioral adjustments individuals make in response to the tax rate. Such a constructed instrumen

47、t is called a simulated instrumental variable.19-64Checking Understanding（例4） Suppose you are studying the effect of price on the demand for cigarettes, using a cross-section of different states cigarette consumption and average price. You would like to regress CigarettesSoldi01Priceiiwhere i indexe

48、s each state19-65Checking Understanding (cont.) Because Pricei is endogenous, you need to instrument. Which of these variables would be suitable?1.Each states cigarette excise tax2.A measure of each states anti-smoking laws3.Each states sales tax CigarettesSoldi01Priceiiwhere i indexes each state19-

49、66Checking Understanding (cont.)1. Each states cigarette excise tax Cigarette excise taxes are surely correlated with cigarette prices. However, they also reflect the level of anti-smoking sentiment in the state (MA has a tax of $1.51 per pack, NC has a tax of $0.05 per pack). Anti-smoking sentiment

50、 is an omitted determinant of consumption, so excise taxes are correlated with . Excise taxes are not a valid instrument.19-67Checking Understanding (cont.)2. A measure of state anti-smoking laws State anti-smoking laws might be correlated with price, but only through their effect on cigarette deman

51、d in the state. Such measures are an explanator of cigarette consumption; moreover, they are also a proxy for state anti-smoking sentiment. Anti-smoking laws are a component of , and would make a terrible instrument.19-68Checking Understanding (cont.)3. Each states sales tax State sales taxes are co

52、rrelated with cigarette prices. Higher sales taxes raise the prices of all goods. There is no reason to expect sales taxes to have any other effect on cigarette consumption, or to be correlated with any other determinant of consumption. State sales taxes are a reasonable instrument.19-69Using Instru

53、mental Variables（IV的作用） Instrumental variables are NOT the explanator of interest. We want to know the effect of intensive cardiac care on mortality, not just the effect of living near a cardiac care center.19-70Using Instrumental Variables (cont.) We do NOT simply use instrumental variables as prox

54、ies for the explanator of interest. Instead, we use IVs as a tool to tease out the “random” (or at least uncorrelated) component of X.19-71DGP with E(Xii ) 0 Yi01XiiE(i) 0Var(i) 2Cov(i,j) 0 for i jE(Xii) 0,1n(xi2) X2Mi XiviE(vi) 0Var(vi) v2Cov(vi,vj) 0 for i jCov(vi,Xi) 0Cov(Zi,Xi) 0Cov(Zi,i) 019-72

55、Using Instrumental Variables（IV估计量原理） If Xi were uncorrelated with i , we would want to weight more heavily observations with a high xi value. We know that Zi is correlated with the “clean” part of Xi , so now we want to weight more heavily observations with a high zi value.19-73Using Instrumental V

56、ariables (cont.)IViiiizYz m19-74Using Instrumental Variables (cont.) Generalizing away from measurement error: Yi01XiiE(Xii) 0E(Zii) 0E(zixi) 019-75Using Instrumental Variables (cont.)IViiiizYz x19-76Checking Understanding01How does differ from the that would result from regressingIViiiiIVOLSiiizYz

57、xYZ19-77Checking Understanding (cont.)22The estimators differ in the denominators.Only appears in the denominator of , in the form of .Both and appear in the denominator of , in the IVOLSiiiiiiiOLSiiiiIVzYzYz xzzzzxform of . iiz x19-7801111()()IViiiiiiiiiiiiiiiiiiiiizYzXEEEz xz xz XzEEz xz xzEz xUsi

58、ng Instrumental Variables What is the expectation of IV?19-791()(,)0Because , thebias term cannot be eliminated.IV is biased in the same directionas the bias in OLS.IViiiiiizEEz xCov XUsing Instrumental Variables (cont.) What is the expectation of IV?19-80011limlim()lim1lim1lim()1lim iiIViiiiiiiiiii

59、pzYzYnppz xpz xnpzXnpz xnUsing Instrumental Variables (cont.) What is the probability limit of IV?19-81111111limlimlim()1lim(,)(,)(,)By the Law of Large Numbers.iiiiIViiiiiiiipz xpznnppz xnCov Z XCov ZCov Z XUsing Instrumental Variables (cont.) What is the probability limit of IV?19-82111(,)(,)lim()

60、(,)(,)(,)00lim(,)0If , the denominator equals , and the does not exist.If , then is inconsistent.IViiiiiiiiiiIViiCov Z XCov ZpCov Z XCov Z XCov Z XpCov ZUsing Instrumental Variables (cont.) What is the probability limit of IV?19-83Using Instrumental Variables (cont.) The asymptotic variance of IV is