第6讲 内生性问题

1、15-1,内生性问题,15-2,假定3不成立：导致OLS不一致的四种情况,Omitted Variables Bias（遗漏变量） Measurement Error（测量误差） Simultaneous Causality（内生性） 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).,1

2、5-3,遗漏变量,真实模型 估计量b0,b1,b2 估计模型 估计量a0,a1,15-4,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。,b21为X2对X1的回归系数,15-5,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,15-6,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,15-7,

3、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。,15-8,Measurement Error测量误差,解释变量X的测量误差会导致假定三 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,15-9,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）,15-10

4、,测量误差,15-11,测量误差,15-12,测量误差,15-13,测量误差,15-14,测量误差,15-15,测量误差,15-16,测量误差,Measurement error implies a correlation between our observed explanator and the error term What bias does this correlation create in OLS?,15-17,带有测量误差的DGP,15-18,测量误差,15-19,Measurement Error (cont.),15-20,Measurement Error (cont.

5、),15-21,Measurement Error (cont.),Mismeasuring X 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.,15-22,Measurement

6、Error (cont.),Mismeasuring X leads to ATTENUATION BIAS. 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).,15-23,Checking Understanding (cont.),Suppose

7、you are advising policy makers on the effect of a one-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 cons

8、umption to justify the policy. (continued on next slide),15-24,Checking 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 “per

9、manent income,” not current income. The resulting measurement error renders your results irrelevant. Assess this argument.,15-25,Checking 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 fou

10、nd. The coefficient with attenuation bias is too small to justify the policy. The larger, true coefficient might or might not be large enough to support the tax rebate.,15-26,Simultaneous Causality（内生性问题）,Another common source of correlation between Xi and ei is SIMULTANEOUS CAUSALITY This complicat

11、ion is also called JOINTLY DETERMINED VARIABLES or ENDOGENEITY,15-27,Simultaneous 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 (e) Thus X wil

12、l be correlated with e,15-28,Simultaneous Causality (cont.),The classic 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.,15-29,Simu

13、ltaneous Causality (cont.),Thus, any attempt to estimate the relationship 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

14、bias is a MAJOR problem.,15-30,Simultaneous Causality (cont.),For example, 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 markets,15-31,Simultaneous Causality (cont.),The quantity of wheat s

15、upplied is a function of the price suppliers receive and the weather (which affects crop yields).,15-32,Simultaneous Causality (cont.),In equilibrium, and Lets focus on the demand equation. Is PiD correlated with eiD ?,15-33,Simultaneous Causality (cont.),Suppose eiD 0 (there is a positive shock to

16、demand). This shock makes QiD greater than 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.,15-34,Simultaneous Causality (cont.),In equilibrium, PiS = PiD. The consumers must pay a highe

17、r price to enjoy the higher quantity of wheat they demand Thus, a positive shock to eiD induces a higher PiD,15-35,Simultaneous 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 correlated.

18、 OLS will be inconsistent.,15-36,Simultaneous 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.,15-37,Simultaneous Causality (cont.),Variables t

19、hat 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.,15-38,Simultaneous Causality (cont.),It is arbitrary which endogenous v

20、ariables 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.,15-39,Simultaneous Causality (cont.),15-40,Simultaneous Caus

21、ality (cont.),15-41,Lagged Dependent Variables （滞后被解释变量）,Using lagged dependent 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.,15-42,Lagged D

22、ependent Variables (cont.),Lagged dependent variables present a problem in the presence of serial correlation. Example: suppose there is first order serial correlation:,15-43,Lagged Dependent Variables (cont.),15-44,Lagged Dependent Variables (cont.),Including lagged dependent variables as an explan

23、ator does NOT lead to inconsistency in the absence of first-order serial correlation.,15-45,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,15-46,Instrumental Variables工具变量,An Instrumental Variable is a variable that

24、 is correlated with X but uncorrelated with e. If Zi is an instrumental variable: E( Zi Xi ) 0 E( Zi ei ) = 0,15-47,Instrumental Variables (cont.),The econometrician can use an instrumental variable Z to estimate the effect on Y of only that part of X that is correlated with Z. Because Z is uncorrel

25、ated with e, any part of X that is correlated with Z must also be uncorrelated with e.,15-48,Instrumental 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）.,15-49,Instrumental Variables (cont.),If our observable control

26、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 e. OLS would be consistent.,15-50,Instrumental Variables (cont.),However, we reasonably believe that a doctors choi

27、ce to perform intensive cardiac procedures are correlated with many other variables.,15-51,Instrumental Variables (cont.),Doctors might select patients to receive treatment based on their underlying health status. If health status is an unobservable determinant of mortality, then it is a component o

28、f e. OLS will give an inconsistent estimate of the benefits of intensive treatment.,15-52,Instrumental Variables (cont.),To eliminate Omitted Variables Bias, we need to find some determinant of a patients receiving intensive cardiac care that is unrelated to mortality. Is there any element of the ca

29、rdiac care process that is reasonably random?,15-53,Instrumental Variables (cont.),Not every hospital is equipped to provide intensive cardiac care. Some patients live near cardiac care centers. When they have a heart attack, they are more likely to be transported to a center, and thus more likely t

30、o receive intensive treatment.,15-54,Instrumental 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 instrumental variable. They argue that geographic location is likely to be uncorrela

31、ted with unobserved traits that influence mortality.,15-55,Instrumental Variables (cont.),When McClellan, McNeil, and Newhouse use their instrumental variable, they find only a small benefit from intensive cardiac care for the marginal patient (some patients may still benefit quite a lot!).,15-56,In

32、strumental 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 are unlikely to be correlated with the mistakes in the first measure, or with

33、any other component of e.,15-57,Instrumental 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 might mis-report their own years of schooling, leading to measurement erro

34、r biases.,15-58,Instrumental 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 also the years of schooling for his/her twin. The twins report of an indi

35、viduals schooling served as an instrumental variable for the individuals self-report.,15-59,Instrumental Variables (例3),Another example: policy makers are greatly interested in the effects of tax rates on labor force participation (and other taxpayer behaviors). They would like to run regressions wi

36、th an individuals tax rate as an explanator. However, an individual has some choice over his/her tax rate.,15-60,Instrumental Variables (cont.),Taxpayers who are close to the income threshold for a new tax bracket can choose to limit their taxable income. For example, they might take more of their p

37、ay 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 e.,15-61,Instrumental Variables (cont.),When the government changes the tax rates, the individuals ne

38、w tax rate is determined by two elements: The change in tax rates (which is uncorrelated with anything else about the individual), and The individuals decisions about how to respond to the tax change (which could well be correlated with e).,15-62,Instrumental Variables (cont.),Public finance economi

39、sts 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 before the tax change.,15-63,Instrumental Variables (cont.),The constru

40、cted 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 instrument is called a simulated instrumental variable.,15-64,Checking Un

41、derstanding（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,15-65,Checking Understanding (cont.),Because Pricei is endogenous, you need to instrument. Which of

42、these variables would be suitable? Each states cigarette excise tax A measure of each states anti-smoking laws Each states sales tax,15-66,Checking Understanding (cont.),Each states cigarette excise tax Cigarette excise taxes are surely correlated with cigarette prices. However, they also reflect th

43、e 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 is an omitted determinant of consumption, so excise taxes are correlated with e. Excise taxes are not a valid instrument.,15-67,Checking Understanding (cont.),A mea

44、sure of state anti-smoking laws State anti-smoking laws might be correlated with price, but only through their effect on cigarette demand 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

45、component of e, and would make a terrible instrument.,15-68,Checking Understanding (cont.),Each states sales tax State sales taxes are correlated 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 co

46、nsumption, or to be correlated with any other determinant of consumption. State sales taxes are a reasonable instrument.,15-69,Using Instrumental 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 th

47、e effect of living near a cardiac care center.,15-70,Using Instrumental Variables (cont.),We do NOT simply use instrumental variables as proxies for the explanator of interest. Instead, we use IVs as a tool to tease out the “random” (or at least uncorrelated) component of X.,15-71,DGP with E(Xiei )

48、0,15-72,Using Instrumental Variables（IV估计量原理）,If Xi were uncorrelated with ei , 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.,15-73,Using Ins

49、trumental Variables (cont.),15-74,Using Instrumental Variables (cont.),15-75,Using Instrumental Variables (cont.),15-76,Checking Understanding,15-77,Checking Understanding (cont.),15-78,Using Instrumental Variables,What is the expectation of IV?,15-79,Using Instrumental Variables (cont.),What is the

50、 expectation of IV?,15-80,Using Instrumental Variables (cont.),What is the probability limit of IV?,15-81,Using Instrumental Variables (cont.),What is the probability limit of IV?,15-82,Using Instrumental Variables (cont.),What is the probability limit of IV?,15-83,Using Instrumental Variables (cont

51、.),The asymptotic variance of b IV is The greater the covariance between X and Z, the lower the asymptotic variance.,15-84,Using Instrumental Variables (多元回归时),In the multiple regression case, we may have more than one explanator that is correlated with the error term (i.e. more than one troublesome

52、 variable). However, an instrumental variable only needs to instrument for one of the troublesome variables.,15-85,Using Instrumental Variables,A variable Zi can instrument for a particular troublesome explanator, XRi, if: Cov( Zi,XRi ) 0 Cov( Zi,ei ) = 0 Zi must be correlated with the troublesome v

53、ariable for which it instruments, but need not be correlated with all of the troublesome variables.,15-86,Using Instrumental Variables (cont.),To estimate a multiple regression consistently, we need at least one instrumental variable for each troublesome explanator.,15-87,Using Instrumental Variable

54、s (cont.),When we have just enough instruments for consistent estimation, we say the regression equation is exactly identified. When we have more than enough instruments, the regression equation is over identified. When we do not have enough instruments, the equation is under identified (and inconsi

55、stent).,15-88,Example: Public Housing（例5）,Does living in a housing project increase childrens chances of being held back in school（留级）? Currie and Yelowitz estimated a childs chance of being held back in school as a function of living in a housing project and a variety of control variables (househol

56、d heads age, gender, race, education, and marital status).,15-89,Example: Public Housing (cont.),The coefficient on DiProject was positive and statistically significant, suggesting that children who live in housing projects are more likely to be held back in school than other children from similar h

57、ouseholds. But is our OLS regression misleading?,15-90,Example: Public Housing (cont.),Curry and Yelowitz argued that families choosing to move into public housing are likely to differ in unobserved ways from other families. Such families are likely to have poorer alternative housing options than fa

58、milies that choose not to enter a housing project.,15-91,Example: Public Housing (cont.),We would expect a family with worse outside housing options to have fewer resources in general. Such families would be less equipped to support their childrens academic efforts. As such, we should expect a bias

59、towards finding that children in housing projects do worse in school.,15-92,Example: Public Housing (cont.),Currie and Yelowitz used public housing rules to construct an instrumental variable. First, they restricted their attention to families with two children. According to the public housing rules, boys and girls cannot share a room. Two-child families with one boy and one girl are assigned a three-bedroom apartment; otherwise, they receive a two-bedroom apartment.,15-93,Example: Public Housing (cont.),The gender composition of a two-child family is essentially random and is unlikely