Endogeneity. Fall Environmental Econometrics () Endogeneity Fall / 27

Transcription

1 Endogeneity Fall 2008 Environmental Econometrics () Endogeneity Fall / 27

2 De nition Endogeneity is said to occur in a multiple regression model if E (X j u) 6= 0, for some j = 1,..., k Environmental Econometrics (GR03) Endogeneity Fall / 27

3 De nition Endogeneity is said to occur in a multiple regression model if Examples: E (X j u) 6= 0, for some j = 1,..., k Environmental Econometrics (GR03) Endogeneity Fall / 27

4 De nition Endogeneity is said to occur in a multiple regression model if E (X j u) 6= 0, for some j = 1,..., k Examples: Omitted variables Environmental Econometrics (GR03) Endogeneity Fall / 27

5 De nition Endogeneity is said to occur in a multiple regression model if Examples: Omitted variables Measurement error E (X j u) 6= 0, for some j = 1,..., k Environmental Econometrics (GR03) Endogeneity Fall / 27

6 De nition Endogeneity is said to occur in a multiple regression model if Examples: E (X j u) 6= 0, for some j = 1,..., k Omitted variables Measurement error Simultaneity in simultaneous equations models Environmental Econometrics (GR03) Endogeneity Fall / 27

7 Omitted Variable and Proxy Variable Suppose that a regression model excludes a key variable, due to data unavailability. For example, consider a wage equation explicitly recognizing that ability a ects wage: log(wage i ) = β 0 + β 1 Educ i + β 2 Exper i + β 3 Abil i + u i. Our primary interest is to measure the e ects of education and job experience on wage, holding the e ect of ability constant. But ability is usually not available in the data. One remedy is to obtain a proxy variable that is correlated to the omitted variable. In the wage equation, we may want to use the intelligence quotient (IQ) as a proxy for ability: log(wage i ) = β 0 + β 1 Educ i + β 2 Exper i + β 3 IQ i + v i. Environmental Econometrics (GR03) Endogeneity Fall / 27

8 Example: Wage Equation The data contains 935 men in 1980 from the Young Men s Cohort of the National Longitudinal Survey (NLSY), USA. Environmental Econometrics (GR03) Endogeneity Fall / 27

9 Example: Wage Equation The data contains 935 men in 1980 from the Young Men s Cohort of the National Longitudinal Survey (NLSY), USA. The results from the regression with omitting ability variable are Log(wage) Coe. Std. Err. Education Experience Constant The estimated return to education is 7.8%. Environmental Econometrics (GR03) Endogeneity Fall / 27

10 Example: Wage Equation The data contains 935 men in 1980 from the Young Men s Cohort of the National Longitudinal Survey (NLSY), USA. The results from the regression with the proxy variable (IQ) for ability are Log(wage) Coe. Std. Err. Coe. Std. Err. Education Experience IQ Constant The estimated return to education changes from 7.8% to 5.7%. Environmental Econometrics (GR03) Endogeneity Fall / 27

11 Measurement Error Data is often measured with error: reporting errors. coding errors. When the measurement error is in the dependent variable, the zero conditional mean assumption is not violated and thus no endogeneity. In contrast, when the measure error is in the independent variable, the problem of endogeneity arises. Environmental Econometrics (GR03) Endogeneity Fall / 27

12 Measurement Error in an Independent Variable Consider a simple regression model: Y i = β 0 + β 1 X i + u i. X i is measured with errors. That is, we observe ex i = X i + e i instead of X i. We assume that e i is uncorrelated with X i, E (X i e i ) = 0. Then, the regression equation we use is Y i = β 0 + β 1 ex i + u i β 1 e i = β 0 + β 1 ex i + v i. It can be seen that the problem of endogeneity occurs: E X ei v i = E ((X i + e i ) (u i β 1 e i )) = β 1 Var (e i ) 6= 0. Environmental Econometrics (GR03) Endogeneity Fall / 27

13 Attenuation Bias If we perform the regression of Y i on ex i, then the measurement error leads to a biased OLS estimate towards zero. This is called attenuation bias. Environmental Econometrics (GR03) Endogeneity Fall / 27

14 Attenuation Bias If we perform the regression of Y i on ex i, then the measurement error leads to a biased OLS estimate towards zero. This is called attenuation bias. The OLS estimator of β 1 is bβ 1 = β 1 + N i=1 X ei ex (u i β 1 e i ) 2 N i=1 X ei ex! p β 1 β 1 Var (e) Var (X ) + Var (e). Environmental Econometrics (GR03) Endogeneity Fall / 27

15 Attenuation Bias If we perform the regression of Y i on ex i, then the measurement error leads to a biased OLS estimate towards zero. This is called attenuation bias. The OLS estimator of β 1 is bβ 1 = β 1 + N i=1 X ei ex (u i β 1 e i ) 2 N i=1 X ei ex! p β 1 β 1 Var (e) Var (X ) + Var (e). Thus, the OLS estimator is inconsistent p lim b Var (X ) β 1 = β 1 Var (X ) + Var (e) β 1. Environmental Econometrics (GR03) Endogeneity Fall / 27

16 Simultaneity Simultaneity arises when one or more of the independent variables, X j s, is jointly determined with the dependent variable, Y, typically through an equilibrium mechanism. This arises in many economic contexts: quantity and price by demand and supply investment and productivity sales and advertizement Environmental Econometrics (GR03) Endogeneity Fall / 27

17 Simultaneous Equations Model Suppose that the equilibrium relation between X and Y is expressed by the following simultaneous equations: Y i = β 0 + β 1 X i + u i, X i = α 0 + α 1 Y i + v i. Each one is called a structural equation since it has a ceteris paribus, causal interpretation. By solving two equations, we have Y i = β 0 + β 1 α 0 1 α 1 β 1 + β 1 v i + u i 1 α 1 β 1, X i = α 0 + α 1 β 0 1 α 1 β 1 + v i + α 1 u i 1 α 1 β 1. Environmental Econometrics (GR03) Endogeneity Fall / 27

18 These are called the reduced form of the model. It is easy to see that, if one perform the regression with just one equation, it will lead to a biased OLS estimator, called simultaneity bias. Cov (X i, u i ) = vi + α 1 u i Cov, u i 1 α 1 β 1 = α 1 Var (u i ). 1 α 1 β 1 Environmental Econometrics (GR03) Endogeneity Fall / 27

19 Endogenous and Exogenous Variables Suppose a more general model: Yi = β 0 + β 1 X i + β 2 T i + u i We have two kinds of variables: X i = α 0 + α 1 Y i + α 2 Z i + v i Endogenous variables (X i and Y i ) are determined within the system. Exogenous variables (T i and Z i ) are exogenously given outside of the model. Example: wage and labor supply for married women 8 log(hours i ) = β >< 0 + β 1 log(wage i ) + β 2 Educ i +β 3 Age i + β 4 Kidslt6 i + β 5 Nwinc i + u i log(wage >: i ) = α 0 + α 1 log(hours i ) + α 2 Educ i +α 3 Exper i + α 4 Experi 2 + v i Environmental Econometrics (GR03) Endogeneity Fall / 27

20 Identi cation I The reduced form of the model is 8 Y i = β 0 +β 1 α 0 1 α >< 1 β + β 1 α α 1 β Z i + β α 1 β T i + eu i 1 = B 0 + B 1 Z i + B 2 T i + eu i X i = >: α 0+α 1 β 0 1 α 1 β + α α 1 β Z i + α 1β α 1 β T i + ev i 1 = A 0 + A 1 Z i + A 2 T i + ev i We can OLS estimate both equations of the reduced form to get consistent estimates of the recuded form parameters: B 0, B 1, B 2, A 0, A 1, and A 2. Note that B 1 = β A 1, B A 2 = α 1, A 1 1 B 2 B 1 A 2 = β A 1 B 2 2 B 1 A 2 = α 2 A 1 B 2 Environmental Econometrics (GR03) Endogeneity Fall / 27

21 Identi cation II Thus, we can back out the estimates of structural parameters from the reduced form coe cients. In this case it is said that the model is identi ed. Environmental Econometrics (GR03) Endogeneity Fall / 27

22 Rules for Identi cation I M (K ) is the number of endogenous (exogenous) variables in the model. m (k) is the number of endogenous (exogenous) variables in a given equation. Order Condition (necessary but not su cient): In order to have identi cation in a given model, we must have K k m 1 Example1: M = 2, K = 0 Yi = β 0 + β 1 X i + u i m = 2, k = 0 not identi ed X i = α 0 + α 1 Y i + v i m = 2, k = 0 not identi ed Example2: M = 2, K = 1 Yi = β 0 + β 1 X i + β 2 T i + u i m = 2, k = 1 not identi ed X i = α 0 + α 1 Y i + v i m = 2, k = 0 identi ed Environmental Econometrics (GR03) Endogeneity Fall / 27

23 Estimation of an Identi ed Equation I Consider the following system of equations. Yi = β 0 + β 1 X i + u i, X i = α 0 + α 1 Y i + α 2 Z i + v i. Note that the rst equation is identi ed. Thus, we are interested in estimating β 1. The reduced form is Yi = B 0 + B 1 Z i + eu i, X i = A 0 + A 1 Z i + ev i, where B 1 /A 1 = β 1. Environmental Econometrics (GR03) Endogeneity Fall / 27

24 Estimation of an Identi ed Equation II The OLS from the reduced form model gives us bb 1 = N i=1 Z i Z Y i Y N i=1 Z i Z, ba 2 1 = N i=1 Z i Z X i X N i=1 Z i Z 2 Hence, the estimator of β 1 is bβ 1,IV = N i=1 Z i Z Y i Y N i=1 Z i Z X i X = Cov \ (Z, Y ). Cov \(Z, X ) In fact, this is the instrumental variable (IV) estimator, which can be obtained in just one step. Environmental Econometrics (GR03) Endogeneity Fall / 27

25 Instrumental Variables (IVs) De nition: An instrument for the model, Y i = β 0 + β 1 X i + u i, is a variable Z i such that Cov (Z, X ) 6= 0 and Cov (Z, u) = 0. The IV estimation can be seen as a two step estimator within a simultaneous equations model as seen just before. Another way of deriving an IV estimator is from its de nition: 0 = Cov (Z, u) = Cov (Z, Y β 0 β 1 X i ) = Cov (Z, Y ) β 1 Cov (Z, X ) And so Cov bβ 1,IV = \ (Z, Y ). Cov \(Z, X ) Environmental Econometrics (GR03) Endogeneity Fall / 27

26 Properties of IV Estimator Under the maintained assumptions, the IV estimator is consistent: bβ 1,IV = β 1 + N i=1 Z i Z u i N i=1 Z i Z X i X Since N i=1 Z i Z u i /N! p 0 as N!, p lim b β 1,IV = β 1 The IV estimator can have a substantial bias in small samples and thus large samples are preferred. The asymptotic variance of the IV estimator is Var b β 1,IV p σ 2 Var (Z ) u N Cov (Z, X ) 2 Environmental Econometrics (GR03) Endogeneity Fall / 27

27 Another Example: Lagged dependent variable Consider the following time-series model: Y t = β 0 + β 1 Y t 1 + β 2 X t + u t, where u t = v t + λv t 1 and v t is a iid noise and E (u t jx ) = 0. It can be easily seen that Cov (Y t 1, u t ) = λvar (v t 1 ) 6= 0. A valid instrument is X t 1 since it is correlated with Y t 1 but not with u t. Therefore, the IV estimator is bβ 1,IV = T t=2 X t 1 X Y t Y T t=2 X t 1 X X t X. Environmental Econometrics (GR03) Endogeneity Fall / 27

28 More Than One Instrument So far we showed how to use one variable as an instrument. Sometimes, we can think more than one variable as an instrument. Suppose that Z 1 and Z 2 are two possible instruments for a variable X. Cov (Z 1, u) = 0 = Cov (Z 2, u) Cov (Z 1, X ) 6= 0 and Cov (Z 2, X ) 6= 0. Rather than using just one instrument, it will be more e cient to use two instruments at the same time. How? Environmental Econometrics (GR03) Endogeneity Fall / 27

29 Two-stage Least Squares (2SLS) We can use a linear combination of both instruments: Z i = α 1 Z 1i + α 2 Z 2i, which is still a valid instrument since Cov (Z, u) = 0. In order to choose α 1 and α 2 so that the correlation between Z i and X i is maximal, we perform the OLS from the regression equation: Z i = α 0 + α 1 Z 1i + α 2 Z 2i + w i. Once we have obtained the tted value, bz i = bα 0 + bα 1 Z 1i + bα 2 Z 2i, we are back to the case with a single IV: bβ 1,2SLS = N i=1 Z bi bz Y i Y N i=1 Z bi bz X i X. This entire procedure is called two-stage least squares (2SLS) estimation. Environmental Econometrics (GR03) Endogeneity Fall / 27

30 Example: Wage and Labor Supply of Married Woman Suppose that the wage and labor supply are determined by 8 log(hours i ) = β >< 0 + β 1 log(wage i ) + β 2 Educ i +β 3 Age i + β 4 Kidslt6 i + β 5 Nwinc i + u i log(wage >: i ) = α 0 + α 1 log(hours i ) + α 2 Educ i +α 3 Exper i + α 4 Experi 2 + v i Is each equation in the model identi ed? Environmental Econometrics (GR03) Endogeneity Fall / 27

31 Example: Wage and Labor Supply of Married Woman Using the 2SLS estimation, we have the following results: Log(hours) Coe. Std. Err. Log(wage) Coe. Std. Err. Log(wage) Log(hours) Educ Educ Age Exper Kidslt (Exper)^ Nwinc Constant Constant Environmental Econometrics (GR03) Endogeneity Fall / 27

32 Example: Wage and Labor Supply of Married Woman I For comparison, we perform the OLS regression for the model: Log(hours) Coe. Std. Err. Log(wage) Coe. Std. Err. Log(wage) Log(hours) Educ Educ Age Exper Kidslt (Exper)^ Nwinc Constant Constant The coe cient on log(wage) is statistically insigni cant in OLS, while signi cant in 2SLS. Environmental Econometrics (GR03) Endogeneity Fall / 27

33 Exogeneity Test When the independent variables are exogenous, the 2SLS is less e cient than OLS since the 2SLS estimates can have very large standard errors. Hauseman s exogeneity test is as follows H 0 : Cov (X, u) = 0, H 1 : Cov (X, u) 6= 0 An idea is to compare both the OLS estimator, bβ 1,OLS, and the 2SLS estimator, bβ 1,2SLS. To test whether the di erences are statistically signi cant, it is easier to use the following regression test: First, regress X i on Z i and get the residual bv i : Regress X i = α 0 + α 1 Z i + v i. Y i = β 0 + β 1 X i + γbv i + u i. Test for γ = 0. If γ is statistically di erent from zero, then we conclude that X i is endogeneous. Environmental Econometrics (GR03) Endogeneity Fall / 27

34 Example: Wage and Labor Supply of Married Women In the rst equation (labor supply), we want to test whether log(wage) is endogenous. First, we regress the following equation to get the residual bv i : log(wage i ) = α 0 + α 1 Exper i + α 1 Exper 2 i Then add bv i in the rst equation and do OLS: + v i log(hours i ) = β 0 + β 1 log(wage i ) + β 2 Educ i + β 3 Age i +β 4 Kidslt6 i + β 5 Nwinc i + γbv i + u i The estimation and t-statistic on bv i are as follows: Coe. Std. Err. t-statistic bv i Environmental Econometrics (GR03) Endogeneity Fall / 27