Econometric Analysis of Panel Data

1
Econometric Analysis of Panel Data

• William Greene
• Department of Economics
• Stern School of Business

2
Econometric Analysis of Panel Data

• 8. Instrumental Variables Estimation

3
Structure and Regression
4
Agenda

• Single equation instrumental variable estimation
• Exogeneity
• Instrumental Variable (IV) Estimation
• Two Stage Least Squares (2SLS)
• Generalized Method of Moments (GMM)
• Panel data
• Fixed effects
• Hausman and Taylors formulation
• Application
• Arellano/Bond/Bover framework

5
Exogeneity
6
The Effect of Education on LWAGE
7
What Influences LWAGE?
8
An Exogenous Influence
9
An Experimental Treatment Effect
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The Measurement Error Problem
How general is this result?
11
The Endogeneity Problem

• Regression y ßx e Changes in x are
  associated with changes in y, but not e
• dy/dx is measured by Cov(x,y), dx/dx is
  measured by Var(x)dy/dx ß dx/dx de/dx ß
  ß Cov(x,y)/Var(x)
• If x is correlated with e, then changes in x are
  associated with changes in e.There are now two
  sources of change in y, direct change in x,
  change in e associated with change in x
• dy/dx measured in the data is not equal to ß
  dx/dx ß.dy/dx ß dx/dx de/dx ß de/dx

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Instrumental Variables

• Instrumental variable associated with changes in
  x, not with e
• dy/dz ß dx/dz de /dz. The second term is 0.
• ß cov(y,z)/cov(x,z)This is the IV estimator
• Example Corporate earnings in year t
  Earnings(t) ß RD(t) e(t)
  RD(t) responds directly to Earnings(t) thus
  e(t) A likely valid instrumental
  variable would be RD(t-1)
  which probably does not respond to current year
  shocks to earnings.

13
The First IV Study(Snow, J., On the Mode of
Communication of Cholera, 1855)

• London Cholera epidemic, ca 1853-4
• Cholera f(Water Purity,u)e.
• Effect of water purity on cholera?
• Purityf(cholera prone environment (poor, garbage
  in streets, rodents, etc.). Regression does not
  work.
• Two London water companies
• Lambeth Southwark
• Main sewage discharge

Paul Grootendorst A Review of Instrumental
Variables Estimation of Treatment
Effectshttp//individual.utoronto.ca/grootendors
t/pdf/IV_Paper_Sept6_2007.pdf
14
IV Estimation

• Choleraf(Purity,u)e
• Z water company
• Cov(Cholera,Z)dCov(Purity,Z)
• Z is randomly mixed in the population (two full
  sets of pipes) and uncorrelated with behavioral
  unobservables, u)
• CholeraadPurityue
• Purity Meanrandom variation?u
• Cov(Cholera,Z) dCov(Purity,Z)

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Instrumental Variable Estimation

• One problem variable the last one
• yit ?1x1it ?2x2it ?KxKit eit
• EeitxKit ? 0. (0 for all others)
• There exists a variable zit such that
• ExKit x1it, x2it,, xK-1,it,zit g(x1it,
  x2it,, xK-1,it,zit)
• In the presence of the other variables, zit
  explains xit
• Eeit x1it, x2it,, xK-1,it,zit 0
• In the presence of the other variables, zit
  and eit are uncorrelated.
• A projection interpretation In the projection
• Xkt ?1x1it, ?2x2it ?k-1xK-1,it ?K
  zit,
• ?K ? 0.

16
Least Squares
17
The IV Estimator
18
A Moment Based Estimator
19
Consistency and Asymptotic Normality of the IV
Estimator
20
Least Squares Revisited
21
Comparing OLS and IV
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Cornwell and Rupert Data
Cornwell and Rupert Returns to Schooling Data,
595 Individuals, 7 YearsVariables in the file
are EXP work experience, EXPSQ EXP2WKS
weeks workedOCC occupation, 1 if blue collar,
IND 1 if manufacturing industrySOUTH 1 if
resides in southSMSA 1 if resides in a city
(SMSA)MS 1 if marriedFEM 1 if
femaleUNION 1 if wage set by unioin
contractED years of educationLWAGE log of
wage dependent variable in regressions These
data were analyzed in Cornwell, C. and Rupert,
P., "Efficient Estimation with Panel Data An
Empirical Comparison of Instrumental Variable
Estimators," Journal of Applied Econometrics, 3,
1988, pp. 149-155.  See Baltagi, page 122 for
further analysis.  The data were downloaded from
the website for Baltagi's text.
23
Wage Equation with Endogenous Weeks
logWageß1 ß2 Exp ß3 ExpSq ß4OCC ß5 South
ß6 SMSA ß7 WKS e Weeks worked is believed
to be endogenous in this equation. We use the
Marital Status dummy variable MS as an exogenous
variable. Wooldridge Condition (5.3) CovMS, e
0 is assumed. Auxiliary regression For MS to
be a valid instrumental variable, In the
regression of WKS on 1,EXP,EXPSQ,OCC,South,SM
SA,MS, MS significantly explains WKS. A
projection interpretation In the
projection XitK ?1 x1it ?2 x2it ?K-1
xK-1,it ?K zit , ?K ? 0. (One normally
doesnt check the variables in this fashion.
24
Auxiliary Projection
-------------------------------------------------
--- Ordinary least squares regression
LHSWKS Mean
46.81152 ----------------------------------
------------------ ---------------------------
--------------------------------------- Varia
ble Coefficient Standard Error
b/St.Er.PZgtz Mean of X ----------------
----------------------------------------------
---- Constant 45.4842872 .36908158
123.236 .0000 EXP .05354484
.03139904 1.705 .0881 19.8537815 EXPSQ
-.00169664 .00069138 -2.454
.0141 514.405042 OCC .01294854
.16266435 .080 .9366 .51116447 SOUTH
.38537223 .17645815 2.184
.0290 .29027611 SMSA .36777247
.17284574 2.128 .0334 .65378151 MS
.95530115 .20846241 4.583
.0000 .81440576
25
Application IV for WKS in Rupert
-------------------------------------------------
--- Ordinary least squares regression
Residuals Sum of squares
678.5643 Fit R-squared
.2349075 Adjusted
R-squared .2338035 ------------------
---------------------------------- ------------
--------------------------------------------
Variable Coefficient Standard Error
b/St.Er.PZgtz --------------------------
------------------------------ Constant
6.07199231 .06252087 97.119 .0000 EXP
.04177020 .00247262 16.893
.0000 EXPSQ -.00073626 .546183D-04
-13.480 .0000 OCC -.27443035
.01285266 -21.352 .0000 SOUTH
-.14260124 .01394215 -10.228 .0000 SMSA
.13383636 .01358872 9.849
.0000 WKS .00529710 .00122315
4.331 .0000
26
Application IV for wks in Rupert
-------------------------------------------------
--- LHSLWAGE Mean
6.676346 Standard deviation
.4615122 Residuals Sum of squares
13853.55 Standard
error of e 1.825317 Fit
R-squared -14.64641
Adjusted R-squared -14.66899
Not using OLS or no constant. Rsqd F may be lt
0. --------------------------------------------
-------- -------------------------------------
------------------- Variable Coefficient
Standard Error b/St.Er.PZgtz
--------------------------------------------
------------ Constant -9.97734299
3.59921463 -2.772 .0056 EXP
.01833440 .01233989 1.486 .1373 EXPSQ
-.799491D-04 .00028711 -.278
.7807 OCC -.28885529 .05816301
-4.966 .0000 SOUTH -.26279891
.06848831 -3.837 .0001 SMSA
.03616514 .06516665 .555 .5789 WKS
.35314170 .07796292 4.530
.0000 OLS----------------------------------------
-------------- WKS .00529710
.00122315 4.331 .0000
27
Generalizing the IV Estimator-1
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Generalizing the IV Estimator - 2
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Generalizing the IV Estimator
30
The Best Set of Instruments
31
Two Stage Least Squares
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2SLS Estimator
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2SLS Algebra
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A General Result for IV

• We defined a class of IV estimators by the set of
  variables
• The minimum variance (most efficient) member in
  this class is 2SLS (Brundy and Jorgenson(1971))
  (rediscovered JW, 2000, p. 96-97)

35
Inference with IV Estimators
36
Robust estimation of VC
Predicted X
Actual X
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2SLS vs. Robust Standard Errors
-------------------------------------------------
- Robust Standard Errors
----------------------------------------
------- Variable Coefficient Standard
Error b/St.Er. ------------------------------
----------------- B_1 45.4842872
4.02597121 11.298 B_2 .05354484
.01264923 4.233 B_3
-.00169664 .00029006 -5.849 B_4
.01294854 .05757179 .225 B_5
.38537223 .07065602 5.454 B_6
.36777247 .06472185 5.682
B_7 .95530115 .08681261 11.000
-----------------------------------------------
--- 2SLS Standard Errors
---------------------------------------
-------- Variable Coefficient Standard
Error b/St.Er. ------------------------------
----------------- B_1 45.4842872
.36908158 123.236 B_2 .05354484
.03139904 1.705 B_3
-.00169664 .00069138 -2.454 B_4
.01294854 .16266435 .080 B_5
.38537223 .17645815 2.184 B_6
.36777247 .17284574 2.128
B_7 .95530115 .20846241 4.583

38
Weak Instruments
39
Weak Instruments (cont.)
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Testing for Endogeneity(?)
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Regression Based Endogeneity Test
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Testing Endogeneity of WKS
(1) Regress WKS on 1,EXP,EXPSQ,OCC,SOUTH,SMSA,MS.
Uresidual, WKSHATprediction (2) Regress
LWAGE on 1,EXP,EXPSQ,OCC,SOUTH,SMSA,WKS, U or
WKSHAT ---------------------------------------
--------------------------- Variable
Coefficient Standard Error b/St.Er.PZgtz
Mean of X ----------------------------------
-------------------------------- Constant
-9.97734299 .75652186 -13.188 .0000
EXP .01833440 .00259373 7.069
.0000 19.8537815 EXPSQ -.799491D-04
.603484D-04 -1.325 .1852 514.405042 OCC
-.28885529 .01222533 -23.628
.0000 .51116447 SOUTH -.26279891
.01439561 -18.255 .0000 .29027611 SMSA
.03616514 .01369743 2.640
.0083 .65378151 WKS .35314170
.01638709 21.550 .0000 46.8115246 U
-.34960141 .01642842 -21.280
.0000 -.341879D-14 ---------------------------
--------------------------------------- Varia
ble Coefficient Standard Error
b/St.Er.PZgtz Mean of X ----------------
----------------------------------------------
---- Constant -9.97734299 .75652186
-13.188 .0000 EXP .01833440
.00259373 7.069 .0000 19.8537815 EXPSQ
-.799491D-04 .603484D-04 -1.325
.1852 514.405042 OCC -.28885529
.01222533 -23.628 .0000 .51116447 SOUTH
-.26279891 .01439561 -18.255
.0000 .29027611 SMSA .03616514
.01369743 2.640 .0083 .65378151 WKS
.00354028 .00116459 3.040
.0024 46.8115246 WKSHAT .34960141
.01642842 21.280 .0000 46.8115246
43
General Test for Endogeneity
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GMM Estimation Orthogonality Conditions
45
GMM Estimation - 1
46
GMM Estimation - 2
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IV Estimation
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An Optimal Weighting Matrix
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The GMM Estimator
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GMM Estimation
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Application - 2SLS
-------------------------------------------------
--- Two stage least squares regression
LHSLWAGE Mean
6.676346 Model size Parameters
7 Degrees of
freedom 4158 Instruments for
WKS Residuals Sum of squares
638.5818 are MS,UNION,ED
Standard error of e .4054646 Fit
R-squared .2279527
Not using OLS or no constant. Rsqd F may be lt
0. --------------------------------------------
-------- -------------------------------------
----------------------------- Variable
Coefficient Standard Error b/St.Er.PZgtz
Mean of X ----------------------------------
-------------------------------- Constant
6.41895193 .29411045 21.825 .0000
EXP .04227684 .00251697 16.797
.0000 19.8537815 EXPSQ -.00075045
.560650D-04 -13.385 .0000 514.405042 OCC
-.27411851 .01290268 -21.245
.0000 .51116447 SOUTH -.14000277
.01415810 -9.889 .0000 .29027611 SMSA
.13594785 .01375050 9.887
.0000 .65378151 WKS -.00222272
.00634746 -.350 .7262 46.8115246
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Application - GMM
NAMELIST Xone,exp,expsq,occ,south,smsa,wks NAM
ELIST Zone,exp,expsq,occ,south,smsa,ms,union,e
d 2SLS LHS lwage RHS X INST
Z NLSQ FCN lwage-b1'x
LABELS b1,b2,b3,b4,b5,b6,b7
START b INST
Z PDS 0
53
GMM Estimates
---------------------------------------------
Instrumental Variables (NL2SLS)
GMM Estimator - Lags 0 Periods
Value of the GMM criterion
e(b)tZ inv(ZtWZ) Zte(b) 537.3916
Sum of functions f(x,b) -221.9274
Estimation problem for 7 parameters.
Sample size is 4165 observations.
---------------------------------------------
----------------------------------------------
---------- Variable Coefficient Standard
Error b/St.Er.PZgtz ---------------------
----------------------------------- B1
6.98356848 .27787583 25.132 .0000
B2 .04080996 .00259196 15.745
.0000 B3 -.00075277 .588278D-04
-12.796 .0000 B4 -.24671927
.01276555 -19.327 .0000 B5
-.14393303 .01461203 -9.850 .0000 B6
.14449428 .01358328 10.638
.0000 B7 -.01346160 .00601608
-2.238 .0252
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GMM vs. 2SLS
----------------------------------------------
-------------------- Variable Coefficient
Standard Error b/St.Er.PZgtz Mean of
X -------------------------------------------
----------------------- TWO STAGE LEAST
SQUARES Constant 6.41895193 .29411045
21.825 .0000 EXP .04227684
.00251697 16.797 .0000 19.8537815 EXPSQ
-.00075045 .560650D-04 -13.385
.0000 514.405042 OCC -.27411851
.01290268 -21.245 .0000 .51116447 SOUTH
-.14000277 .01415810 -9.889
.0000 .29027611 SMSA .13594785
.01375050 9.887 .0000 .65378151 WKS
-.00222272 .00634746 -.350
.7262 46.8115246 GENERALIZED METHOD OF
MOMENTS WITH HETEROSCEDASTICITY B1
6.98356848 .27787583 25.132 .0000 B2
.04080996 .00259196 15.745
.0000 B3 -.00075277 .588278D-04
-12.796 .0000 B4 -.24671927
.01276555 -19.327 .0000 B5
-.14393303 .01461203 -9.850 .0000 B6
.14449428 .01358328 10.638
.0000 B7 -.01346160 .00601608
-2.238 .0252
55
Testing the Overidentifying Restrictions
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GMM Estimation
57
Inference About the Parameters
58
Specification Test Based on the Criterion
59
Extending the Form of the GMM Estimator to
Nonlinear Models
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A Nonlinear Conditional Mean
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Nonlinear Regression/GMM
NAMELIST Xone,exp,expsq,occ,south,smsa,wks NAM
ELIST Zone,exp,expsq,occ,south,smsa,ms,union,e
d ? Get initial values to use for optimal
weighting matrixNLSQ LHS lwage
fcnexp(b1'x)labelsb
1,b2,b3,b4,b5,b6,b7
INST Z start7_0 ? GMM using previous
estimates to compute weighting matrix NLSQ (GMM)
FCN lwage-exp(b1'x)
LABELS b1,b2,b3,b4,b5,b6,b7
START b INST Z
PDS 0
62
Nonlinear Wage Equation Estimates
-------------------------------------------------
--- Instrumental Variables (NL2SLS)
Residuals Sum of squares
690.1236 ----------------------------------
------------------ ---------------------------
----------------------------- Variable
Coefficient Standard Error b/St.Er.PZgtz
--------------------------------------------
------------ B1 1.87846422
.04366585 43.019 .0000 B2
.00640621 .00038344 16.707 .0000 B3
-.00011391 .851508D-05 -13.378
.0000 B4 -.04106490 .00193784
-21.191 .0000 B5 -.02095476
.00215243 -9.735 .0000 B6
.02065632 .00208495 9.907 .0000 B7
-.00075796 .00094230 -.804
.4212 -------------------------------------------
-- Instrumental Variables (NL2SLS)
Value of the GMM criterion
e(b)tZ inv(ZtWZ) Zte(b) 530.2036
---------------------------------------------
----------------------------------------------
---------- Variable Coefficient Standard
Error b/St.Er.PZgtz ---------------------
----------------------------------- B1
1.95181383 .04066956 47.992 .0000
B2 .00612089 .00039476 15.506
.0000 B3 -.00011276 .894520D-05
-12.606 .0000 B4 -.03671530
.00190644 -19.259 .0000 B5
-.02133895 .00221339 -9.641 .0000 B6
.02166118 .00205502 10.541
.0000 B7 -.00218988 .00088200
-2.483 .0130 Value of the GMM criterion for
the linear model (Basis for a test?) e(b)tZ
inv(ZtWZ) Zte(b) 537.3916
63
IV for Panel Data