Bài 12: Nội sinh và biến công cụ

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ENDOGENEITY AND

INSTRUMENTAL VARIABLE

REGRESSION

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Endogeneity



OLS assumption



When

:

Endogeneity problem





|



0 







2 V











|



0 or

 



0

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Reasons for Endogeneity



errors in variables

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Consequences of endogeneity



If we use OLS in a regression with endogeneity:

BIASED AND INCONSISTENT ESTIMATES

x

y

ε







 



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Endogeneity: errors in variables



Consider a regression



We can’t observe , but



Then the regression becomes









 

,



*

,

*



*





*









*

ˆ

*

ˆ



















 



*

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Endogeneity: Endogenous variables



Consider a (market) demand equation

is not exogenous by theory



Instead, it should be the supply-demand system

1

2 <i>d</i>











1

2 <i>d</i>











1 <i>s</i>









2

1 <i>d</i>



 













2

1

0



 







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Endogeneity: Omitted variables



Suppose the true model is



If we regress

0 1 1

2 <i>y</i>

















0 1 1

omitted variable:

2 <i>y</i>













2 2

then

 







 

1 

1

2 

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Solution to Endogeneity:

Instruments



Instrumental variables (instruments) Z must satisfy



exogeneity (uncorrelated with or )



relevance (correlated with )

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Identification problem



If is the number of endogenous variables, and

is the number of instruments, then



If

the model is unidentified



If

the model is just-identified



If

the model is over-identified

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IV Estimation



If is the number of endogenous variables, and

is the number of instruments, then



If

find the instrument!!!



If

use IV estimator



If

use

2SLS

or GMM

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Two-Stage Least Square (2SLS)



Consider a regression

where is endogenous



if is used as instruments. Then the procedure is



Step 1: Regress each endogenous variable on

and



Step 2: Compute the fitted values



Step 3: Regress

1 1 2 2

 

 









2 0 1 1 2















ˆ

1 1

ˆ













1 1 2

ˆ

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The wage equation



<i>ed: education</i>



<i>X: other control variables</i>



Endogeneity: missing important variable of

<i>ability</i>



<i>ability is believed to be correlated with ed.</i>





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Summary statistics

year 20306 2001.088 1.61576 1999 2003

h 20306 2022.203 706.4409 1 5508

married 20306 .660002 .4737198 0 1

nch 20306 .9591746 1.137898 0 8

race 20306 1.410618 .6499018 1 3

mo_ed 20306 1.844726 .6290755 1 3

fa_ed 20306 1.83857 .6961686 1 3

ed 20306 13.4512 2.488962 0 17

union 20306 .1518763 .3589098 0 1

tenure 20306 6.359746 7.725706 0 42

wage 20306 20.08589 19.17634 5 491

age 20306 39.01532 9.901983 21 59

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OLS Regression

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Testing for endogeneity



<i>regress ed on X and IV variables</i>



<i>predict error terms e</i>



regress

<i>with e included</i>



<i>endogeneity if e is statistically significant</i>





ln



,







var,











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Testing for endogeneity



. quietly regress ed age age2 tenure union nch married

white black

fa_ed1 fa_ed2 mo_ed1 mo_ed2

year2001

year2003



. predict ed_hat, xb /* find the fitted value of ed*/



. predict r, resid /* find the error variance of the

model*/



. regress lnwage ed age age2 tenure union nch married

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Testing for endogeneity

_cons -.2893795 .0781816 -3.70 0.000 -.4426218 -.1361372
r -.0745455 .0048828 -15.27 0.000 -.0841163 -.0649748
year2003 -.0092245 .0092467 -1.00 0.318 -.0273487 .0088997
year2001 -.000035 .0092487 -0.00 0.997 -.0181632 .0180932
black -.1986132 .0159947 -12.42 0.000 -.229964 -.1672624
white -.0707782 .0163623 -4.33 0.000 -.1028496 -.0387068
married .0142878 .008775 1.63 0.103 -.002912 .0314876
nch .0253419 .0038428 6.59 0.000 .0178097 .0328742
union .1061971 .0107717 9.86 0.000 .0850837 .1273104
tenure .011755 .0005423 21.68 0.000 .0106921 .0128179
age2 -.0004601 .0000409 -11.26 0.000 -.0005401 -.00038
age .0444652 .0032132 13.84 0.000 .038167 .0507634
ed .1527935 .0045929 33.27 0.000 .143791 .161796

lnwage Coef. Std. Err. t P>|t| [95% Conf. Interval]

Prob > F = 0.0000

F( 1, 20293) = 233.08

( 1) r = 0

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2SLS IV Regression [Manually]

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Testing for good instruments



quietly regress ed age age2 tenure union nch

married white black

fa_ed1 fa_ed2 mo_ed1

mo_ed2

year2001 year2003

Prob > F = 0.0000

F( 4, 20291) = 660.56

( 4) mo_ed2 = 0

( 3) mo_ed1 = 0

( 2) fa_ed2 = 0

( 1) fa_ed1 = 0

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Implement IV reg in Stata



. ivreg lnwage age age2 tenure union nch married

white black year2001 year2003 (ed = fa_ed1

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Implement IV reg in Stata

_cons 10.03607 .2575517 38.97 0.000 9.531245 10.54089
mo_ed2 1.221048 .0654893 18.65 0.000 1.092684 1.349412
mo_ed1 .5029502 .0450191 11.17 0.000 .4147092 .5911912
fa_ed2 1.833566 .0582525 31.48 0.000 1.719386 1.947746

fa_ed1 .6310663 .0429161 14.70 0.000 .5469473 .7151854
year2003 -.0024599 .0391737 -0.06 0.950 -.0792435 .0743237
year2001 -.0107218 .0391791 -0.27 0.784 -.0875159 .0660724
black .8421189 .0659016 12.78 0.000 .7129464 .9712914
white 1.072611 .0633436 16.93 0.000 .9484524 1.19677
married .3649581 .0366104 9.97 0.000 .2931988 .4367174
nch -.2159402 .0156491 -13.80 0.000 -.2466137 -.1852667
union .074779 .0456544 1.64 0.101 -.0147073 .1642652
tenure .0054745 .0022971 2.38 0.017 .0009721 .009977
age2 -.0004542 .0001726 -2.63 0.009 -.0007926 -.0001158
age .053069 .0135771 3.91 0.000 .0264568 .0796812

ed Coef. Std. Err. t P>|t| [95% Conf. Interval]

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Implement IV reg in Stata

SECOND STAGE

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Hausman test OLS agaisnt IV



regress lnwage ed age age2 tenure union nch

married white black year2001 year2003



est store OLS



ivreg lnwage age age2 tenure union nch married

white black year2001 year2003 (ed = fa_ed1

fa_ed2 mo_ed1 mo_ed2), first



est store IV

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Hausman test OLS agaisnt IV

year2003 -.0092245 -.006866 -.0023585 .0027505

year2001 -.000035 .0009202 -.0009552 .0027474
black -.1986132 -.1106727 -.0879405 .0074915
white -.0707782 .0655964 -.1363746 .0102137
married .0142878 .0362749 -.0219871 .0029815
nch .0253419 .010029 .0153129 .0015232
union .1061971 .1102531 -.004056 .0032101
tenure .011755 .0120222 -.0002671 .000162
age2 -.0004601 -.0005048 .0000447 .0000125
age .0444652 .047922 -.0034568 .0009811
ed .1527935 .0868367 .0659568 .004554

IV OLS Difference S.E.

(b) (B) (b-B) sqrt(diag(V_b-V_B))
Coefficients

. hausman IV OLS /*note the order of IV and OLS*/

Prob>chi2 = 0.0000
= 209.77

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OLS vs. IV – the contribution of ed

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Relationship and credit limit

Chakraborty et al. (2010) The Importance of Being

Known: Relationship Banking and Credit Limits.

<i>Quarterly J of Finance and Accounting 49(2) 27-48.</i>



Objective: investigate the effect of relationship on

credit limits given to firms

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Relationship and credit limit

Chakraborty et al. (2010)



Indep var:



contract’s characteristics (prices, collateral, loan terms)



relationship (bank-firm years of relationship)



bank’s characteristics



Endogeneity: credit limit (dep var) and contract’s

characteristics are determined simultaneously



Istrumented vars: contract’s characteristics (interest rate

and collateral)

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Bank loan and trade credit

Du et al. (2012) Bank Loan vs. Trade Credit –

<i>Evidence from China. Economics of Transition </i>

20(3): 457-80



Objective: effects of bank loan and trade credit on

firm performance and growth

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Bank loan and trade credit



Dep var:



labor productivity: output per worker [in log]



ROA



change in employment [in log]



reinvestment rate [share of profit reinvested]



Indep var



bank loan [ratio of bank loan to total asset]



trade credit [% purchased with credit of two main

inputs]

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Bank loan and trade credit



Instrumented variables: bank loan and trade credit



Endogeneity:



reverse causality



spurious correlation



Instrumental variables:



for trade credit: relationship [dummy, 1 if the two main inputs

are supplied by relatives or friends]



previous studies showed that suppliers are more likely to offer trade

credit when customers are in the same network



for bank loan: British administration [dummy, 1 if the located

city is administered by GB in the Qing dynasty]



<sub>reason: GB during their administration develop their own bank </sub>

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Incentive Contracts and Bank

Performance

Li et al. (2007) Incentive Contracts and Bank

Performance – Evidence from Rural China.

<i>Economics of Transition 15(1): 109-24.</i>



Objective: the effect of incentive to bank’s

manager to bank performance.



Data: bank branches in rural China



Dep var:



deposit growth

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Incentive Contracts and Bank

Performance



Indep var:



the amount of money given to manager per

performance point



branch size [asset value]



town’s industrial development [per capita industrial

output]

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Bài 12: Nội sinh và biến công cụ

ENDOGENEITY AND

INSTRUMENTAL VARIABLE

REGRESSION

Endogeneity



OLS assumption



When

:

Endogeneity problem





|

<i>X</i>



0









2

V



<i>| X</i>









|

<i>X</i>



0 or

 



<i>X</i>

0

Reasons for Endogeneity



errors in variables

Consequences of endogeneity



If we use OLS in a regression with endogeneity:

BIASED AND INCONSISTENT ESTIMATES

x

y

ε

<i>y</i>

<i>x</i>

<i>x</i>









 





Endogeneity: errors in variables



Consider a regression



We can’t observe , but



Then the regression becomes

<i>y</i>





<i>x</i>





 

<i>y x</i>

,



<i>y x</i>

*

,

*



*