Topic 2: Endogeneity and IV TSLS

 Endogeneity refers to the fact that an independent variable IV included in the model is a choice variable not exogenous Structure E[ ] =...  An independent variable IV included in the

Dr Pham Thi Bich Ngoc

Hoa Sen University

ngoc.phamthibich@hoasen.edu.vn

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 Endogeneity refers to the fact that an independent variable (IV) included in the model is a choice

variable (not exogenous)

Structure E[ ] =

 Omitted variable bias

 Measurement error

 Simultaneity

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Omitting a variable (X2) creates a bias only if:

1 X2 is an explanator of Y (so, when omitted, it

becomes a component of the error term)

2 X2 is correlated with X1 (so that X2 creates a

correlation between X1 and the error term).

be an increase in The estimate of picks up

the effect of and the hidden effect of

 Measurement error also induces a

correlation between our included

explanator and the error term.

 Instead of observing Xi , we observe X*

 An independent variable (IV) included in the model

is a choice variable, potentially affected by the

dependent variable (DV)

 Examples:

◦ IV = Exports; DV = GDP

◦ IV = education; DV = income

 Given: both X and Y are jointly determined

Because X and Y are determined simultaneously, X

can adjust in response to shocks to Y ()

Thus X will be correlated with 

 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.

 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 bias is a MAJOR problem.

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 Suppose that Y2it is an endogenous explanatory

variable:

◦ Y1it = a0 + a1 Y2it + a2 Xit + uit (1)

◦ Y2it = b0 + b1 Xit + b2 Zit + vit (2)

 Equations (1) and (2) have a “triangular” structure

 Given this triangular structure, the OLS estimate of a1

in equation (1) is unbiased only if vit is uncorrelated with uit

 If vit is correlated with uit, then Y2it is correlated

with uit which means that the OLS estimate of a1

would be biased

 To avoid this bias, we must estimate equation (1)

“instrumental variables” (IV) regression rather than OLS

Endogeneity bias

 Instrumental Variables (IV) estimation is used when your model has endogenous x’s

 That is, whenever Cov(x,u) ≠ 0

 Thus, IV can be used to address the problem

of omitted variable bias

 Additionally, IV can be used to solve the

classic errors-in-variables problem

 Suppose that Y2it is an endogenous

explanatory variable:

◦ Y1it = a0 + a1 Y2it + a2 Xit + uit (1)

◦ Y2it = b0 + b1 Xit + b2 Zit + vit (2)

A Triangle Relationship:

 Substituting eq (2) into eq (1):

◦ Y1it = a0 + a1 (b0 + b1 Xit + b2 Zit + vit) + a2 Xit + uit (3)

◦ All the explanatory variables (Xit and Zit) are

exogenous

 The basic idea underlying IV regression is to

remove vit from the Y1it model so that our

estimate of a1 is unbiased.

Instrumental Variable:

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 Note that vit is removed from the Y1it model

if we use the predicted rather than the

actual values of Y2it on the right hand side

◦ Y1it = a0 + a1 (b0^ + b1^ Xit + b2^ Zit ) + a2 Xit + uit (4)

 The a1 estimate is biased in eq (3) but it is

unbiased in eq (4) because the vit term has been removed.

 Z : instrumental variable

Instrumental Variable:

 Instrument Relevance: The instrument must

be correlated with the endogenous variable

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 Two stage least square?

- Stage 1: Regress eq (2)

 Using the ivregress command

◦ The models can be estimated using two-stage least squares (2SLS), limited-information maximum

likelihood (LIML) or generalized method of

ivreg2 depvar [varlist1] (varlist2=varlist_iv)

[weight] [if exp] [in range] [, options]

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 The most-up-to-date implementation of

ivreg2 requires Stata version 11 or later.

data

varlist1 are the exogenous regressors or "included

instruments"

varlist_iv are the exogenous variables excluded from

the regression or "excluded instruments"

varlist2 the endogenous regressors that are being

"instrumented"

Used for panel data:

 ssc install xtivreg2

xtivreg2 depvar [varlist1] (varlist2=varlist_iv)

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 IV estimation can be extended to the

multiple regression case

 Call the model we are interested in

estimating the structural model

 Our problem is that one or more of the

variables are endogenous

 We need an instrument for each endogenous variable

 If there is just one instrument for our

endogenous variable, we can’t test whether

the instrument is uncorrelated with the error

 We say the model is just identified

 If we have multiple instruments, it is possible

to test the overidentifying restrictions – to

see if some of the instruments are correlated with the error

 In the instrumental variable regression, if we have multiple endogenous regressors x1, …,

xk and multiple instruments z1, …, zm, the

coefficients on the endogenous

regressors are said to be:

 Exactly identified if m = k

 Overidentified if m > k

 Underidentified if m < k  can not identify the coefficients.

 The Sargan-Hansen test is a test of overidentifyingrestrictions.

 The joint null hypothesis (H0) is that the instruments arevalid instruments, i.e., uncorrelated with the error term,and that the excluded instruments are correctly excludedfrom the estimated equation

 Under the null, the test statistic is distributed as squared in the number of (L-K) overidentifyingrestrictions

chi- A rejection casts doubt on the validity of the instruments

If p-value >5%  instruments are valid

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 For the 2SLS estimator, the test statistic is Sargan's

statistic, typically calculated as N*R-squared from a

regression of the IV residuals on the full set of instruments

 Under the assumption of conditional homoskedasticity,

Hansen's J statistic becomes Sargan's statistic The J

statistic is consistent in the presence of heteroskedasticityand (for HAC-consistent estimation) autocorrelation;

 Sargan's statistic is consistent if the disturbance is

homoskedastic and (for AC-consistent estimation) if it is

also autocorrelated With robust, bw and/or cluster,

Hansen's J statistic is reported

 Endogeneity tests of one or more endogenous regressors

can implemented using the endog option

 Under the null hypothesis (H0) that the specified

endogenous regressors can actually be treated as

exogenous, the test statistic is distributed as chi-squared with degrees of freedom equal to the number of regressorstested

 Unlike the Durbin-Wu-Hausman tests reported by ivendog, the endog option of ivreg2 can report test statistics that are robust to various violations of conditional homoskedasticity

 If p-value <5%  endogenous regressors

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 The underidentification test is an LM test of whether the equation is identified, i.e., that the excluded instruments are "relevant", meaning correlated with the endogenous regressors

 Under the null hypothesis (H0) that the equation is

underidentified A rejection of the null indicates that the

matrix is full column rank, i.e., the model is identified

 If p-value <5%  instruments are relevant

Dependent Variable: LWAGE=Log of wage

 EXP =Work experience,

 WKS =Weeks worked,

 OCC =Occupation, 1 if blue collar,

 IND =1 if manufacturing industry,

 SOUTH =1 if resides in south,

 SMSA =1 if resides in a city (SMSA),

LWAGE=β1+ β2 EXP + β3 EXPsq + β4OCC + β5 SOUTH + β6 SMSA + β7 WKS + ε

Weeks worked (WKS) is believed to be endogenous

(1) use 1 instrumental variable: the Marital Status dummy

variable (MS)

(2) use 2 instrumental variables: the Marital Status dummy

variable (MS) and dummy variable (BLK)

Please replicate for the second case!