Econometrics I
Professor William Greene
Stern School of Business
Department of Economics
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Part 16: Panel Data
Econometrics I
Part 16 – Panel Data
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Part 16: Panel Data
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Part 16: Panel Data
www.oft.gov.uk/shared_oft/reports/Evaluating-OFTs-work/oft1416.pdf
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Part 16: Panel Data
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Part 16: Panel Data
Panel Data Sets

Longitudinal data
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Cross section time series

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Penn world tables
Financial data by firm, by year
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British household panel survey (BHPS)
Panel Study of Income Dynamics (PSID)
… many others
rit – rft = i(rmt - rft) + εit, i = 1,…,many; t=1,…many
Exchange rate data, essentially infinite T, large N
Part 16: Panel Data
Benefits of Panel Data
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Time and individual variation in behavior
unobservable in cross sections or aggregate time
series
Observable and unobservable individual
heterogeneity
Rich hierarchical structures
More complicated models
Features that cannot be modeled with only cross
section or aggregate time series data alone
Dynamics in economic behavior
Part 16: Panel Data
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Part 16: Panel Data
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Part 16: Panel Data
BHPS Has Evolved
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Part 16: Panel Data
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Part 16: Panel Data
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Part 16: Panel Data
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Part 16: Panel Data
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Part 16: Panel Data
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Part 16: Panel Data
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Part 16: Panel Data
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Part 16: Panel Data
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Part 16: Panel Data
Cornwell and Rupert Data
Cornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 Years
(Extracted from NLSY.) Variables in the file are
EXP
WKS
OCC
IND
SOUTH
SMSA
MS
FEM
UNION
ED
BLK
LWAGE
=
=
=
=
=
=
=
=
=
=
=
=
work experience
weeks worked
occupation, 1 if blue collar,
1 if manufacturing industry
1 if resides in south
1 if resides in a city (SMSA)
1 if married
1 if female
1 if wage set by union contract
years of education
1 if individual is black
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.
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Part 16: Panel Data
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Part 16: Panel Data
Balanced and Unbalanced Panels

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
Distinction: Balanced vs. Unbalanced Panels
A notation to help with mechanics
zi,t, i = 1,…,N; t = 1,…,Ti
The role of the assumption
 Mathematical and notational convenience:


Balanced, n=NT
N
Unbalanced: n  i=1 Ti
Is the fixed Ti assumption ever necessary? Almost
never.
Is unbalancedness due to nonrandom attrition from an
otherwise balanced panel? This would require special
considerations.


16-21/135
Part 16: Panel Data
Application: Health Care Usage
German Health Care Usage Data, 7,293 Individuals, Varying Numbers of Periods
This is an unbalanced panel with 7,293 individuals. There are altogether 27,326 observations. The number of
observations ranges from 1 to 7.
(Frequencies are: 1=1525, 2=2158, 3=825, 4=926, 5=1051, 6=1000, 7=987).
(Downloaded from the JAE Archive)
Variables in the file are
DOCTOR = 1(Number of doctor visits > 0)
HOSPITAL = 1(Number of hospital visits > 0)
HSAT
= health satisfaction, coded 0 (low) - 10 (high)
DOCVIS
= number of doctor visits in last three months
HOSPVIS = number of hospital visits in last calendar year
PUBLIC
= insured in public health insurance = 1; otherwise = 0
ADDON
= insured by add-on insurance = 1; otherswise = 0
HHNINC = household nominal monthly net income in German marks / 10000.
(4 observations with income=0 were dropped)
HHKIDS
= children under age 16 in the household = 1; otherwise = 0
EDUC
= years of schooling
AGE
= age in years
MARRIED = marital status
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Part 16: Panel Data
An Unbalanced Panel:
RWM’s GSOEP Data on Health Care
N = 7,293 Households
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Part 16: Panel Data
A Basic Model for Panel Data


Unobserved individual effects in regression: E[yit | xit, ci]
Notation: y it =xit + ci + it
 x i1 
 x 
i2
X i    Ti rows, K columns
 
 
 x iTi 
Linear specification:
Fixed Effects: E[ci | Xi ] = g(Xi). Cov[xit,ci] ≠0
effects are correlated with included variables.
Random Effects: E[ci | Xi ] = 0. Cov[xit,ci] = 0
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Part 16: Panel Data
Convenient Notation

Fixed Effects – the ‘dummy variable model’
yit = i + xit + it
Individual specific constant terms.

Random Effects – the ‘error components model’
yit = xit + it + ui
Compound (“composed”) disturbance
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Part 16: Panel Data
Estimating β

β is the partial effect of interest

Can it be estimated (consistently) in
the presence of (unmeasured) ci?
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Does pooled least squares “work?”
Strategies for “controlling for ci” using the
sample data
Part 16: Panel Data
Assumptions for Asymptotics


Convergence of moments involving cross section Xi.
N increasing, T or Ti assumed fixed.

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


“Fixed T asymptotics” (see text, p. 348)
Time series characteristics are not relevant (may be
nonstationary – relevant in Penn World Tables)
If T is also growing, need to treat as multivariate time series.
Ranks of matrices. X must have full column rank. (Xi
may not, if Ti < K.)
Strict exogeneity and dynamics. If xit contains yi,t-1 then
xit cannot be strictly exogenous. Xit will be correlated with
the unobservables in period t-1. (To be revisited later.)
Empirical characteristics of microeconomic data
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Part 16: Panel Data
The Pooled Regression

Presence of omitted effects
y it =x itβ+c i +εit , observation for person i at time t
y i =X iβ+c ii+ε i , Ti observations in group i
=X iβ+c i +ε i , note c i  (c i , c i ,...,c i )
y =Xβ+c +ε , Ni=1 Ti observations in the sample

Potential bias/inconsistency of OLS – depends
on ‘fixed’ or ‘random’
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Part 16: Panel Data
OLS in the Presence of Individual Effects
b=(X X )-1 X'y
-1
=β + (1/N)ΣNi=1 X i X i  (1/N)ΣNi=1 X ic i  (part due to the omitted c i )
-1
+ (1/N)Σ X i X i  (1/N)ΣNi=1 X iε i  (covariance of X and ε will = 0)
The third term vanishes asymptotically by assumption
N
i=1
-1

1 N
  N Ti
plim b = β + plim  Σ i=1 X i X i  Σ i=1 x ic i  (left out variable formula)
N
N
 

So, what becomes of ΣNi=1 w i x ic i  ?
plim b = β if the covariance of x i and c i converges to zero.
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Part 16: Panel Data
Estimating the Sampling Variance of b

s2(X ́X)-1? Inappropriate because



A ‘robust’ covariance matrix
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Correlation across observations (certainly)
Heteroscedasticity (possibly)
Robust estimation (in general)
The White estimator
A Robust estimator for OLS.
Part 16: Panel Data
Cluster Estimator
Robust variance estimator for Var[b]
Est.Var[b]
Ti
Ti
= ( X'X ) 1 Ni=1 ( t=1
x it ˆ
v it )( t=1
x it ˆ
v it )  ( X'X ) 1


Ti
Ti
ˆ
= ( X'X ) 1 Ni=1  t=1
 s=1
v it ˆ
v is x it x is  ( X'X) 1


ˆ
v it  a least squares residual = it  c i
(If Ti = 1, this is the White estimator.)
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Part 16: Panel Data
Alternative OLS Variance Estimators
Cluster correction increases SEs
+---------+--------------+----------------+--------+---------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
+---------+--------------+----------------+--------+---------+
Constant
5.40159723
.04838934
111.628
.0000
EXP
.04084968
.00218534
18.693
.0000
EXPSQ
-.00068788
.480428D-04
-14.318
.0000
OCC
-.13830480
.01480107
-9.344
.0000
SMSA
.14856267
.01206772
12.311
.0000
MS
.06798358
.02074599
3.277
.0010
FEM
-.40020215
.02526118
-15.843
.0000
UNION
.09409925
.01253203
7.509
.0000
ED
.05812166
.00260039
22.351
.0000
Robust
Constant
5.40159723
.10156038
53.186
.0000
EXP
.04084968
.00432272
9.450
.0000
EXPSQ
-.00068788
.983981D-04
-6.991
.0000
OCC
-.13830480
.02772631
-4.988
.0000
SMSA
.14856267
.02423668
6.130
.0000
MS
.06798358
.04382220
1.551
.1208
FEM
-.40020215
.04961926
-8.065
.0000
UNION
.09409925
.02422669
3.884
.0001
ED
.05812166
.00555697
10.459
.0000
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Part 16: Panel Data
Results of Bootstrap Estimation
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Part 16: Panel Data
Bootstrap variance for a
panel data estimator
 Panel Bootstrap =
Block Bootstrap
 Data set is N groups of
size Ti
 Bootstrap sample is N
groups of size Ti drawn
with replacement.
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Part 16: Panel Data
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Part 16: Panel Data
Using First Differences
yit =xitβ+ci +εit , observation for person i at time t
Eliminating the heterogeneity
y it = y it -y i,t-1 = (x it )β + c i + εit
= (x it )β + w it
Note: Time invariant variables become zero
Time trend becomes the constant term
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Part 16: Panel Data
OLS with First Differences
With strict exogeneity of (Xi,ci), OLS regression of Δyit
on Δxit is unbiased and consistent but inefficient.
 i,2  i,1 


 i,3  i,2 
Var 



 i,T  i,T 1 
 i
i

22
 2
 
 0

 0
2
22
2
0
2
2
0 


2


22 
GLS is unpleasantly complicated. Use OLS in first
differences and use Newey-West with one lag.
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Part 16: Panel Data
Application of a Two Period Model
“Hemoglobin and Quality of Life in Cancer
Patients with Anemia,”
 Finkelstein (MIT), Berndt (MIT), Greene (NYU),
Cremieux (Univ. of Quebec)
 1998
 With Ortho Biotech – seeking to change labeling
of already approved drug ‘erythropoetin.’
r-HuEPO

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Part 16: Panel Data
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Part 16: Panel Data
QOL Study

Quality of life study




yit = self administered quality of life survey, scale = 0,…,100
xit = hemoglobin level, other covariates



Treatment effects model (hemoglobin level)
Background – r-HuEPO treatment to affect Hg level
Important statistical issues



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
i = 1,… 1200+ clinically anemic cancer patients undergoing
chemotherapy, treated with transfusions and/or r-HuEPO
t = 0 at baseline, 1 at exit. (interperiod survey by some patients was not
used)
Unobservable individual effects
The placebo effect
Attrition – sample selection
FDA mistrust of “community based” – not clinical trial based statistical
evidence
Objective – when to administer treatment for maximum marginal
benefit
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Part 16: Panel Data
Regression-Treatment Effects Model
QOL it   t + "other covariates"
+ 7Hbit7 + 8Hbit8 + 9Hbit9 + ... 15Hb15
it
+ c i + εit
Hbit  hemoglobin level, grams/deciliter, range 3+ to 15
Hbit7  1(3  Hbit < 7.5) (Base case; 7 = 0)
Hbit8  1(7.5  Hbit < 8.5)
Hb15
it  1(14.5  Hbit  15)
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Part 16: Panel Data
Effects and Covariates


Individual effects that would impact a self reported QOL:
Depression, comorbidity factors (smoking), recent
financial setback, recent loss of spouse, etc.
Covariates





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
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Change in tumor status
Measured progressivity of disease
Change in number of transfusions
Presence of pain and nausea
Change in number of chemotherapy cycles
Change in radiotherapy types
Elapsed days since chemotherapy treatment
Amount of time between baseline and exit
Part 16: Panel Data
First Differences Model
QOL i  QOL i1  QOL i0
j
j
K
= (1  0 )  15

(Hb

Hb
)


j 8 j
i1
i0
k 1k (x ik ,1  x ik ,0 )  i1  i0
Regression to the mean (the "tendency to mediocrity")
i0  i1  ui  (QOL i0  QOL 0 ) Expect 0   < 1
implies
 = 1  0  QOL 0
QOL i  QOL i1  QOL i0
j
j
K
=   15

(Hb

Hb
)


j 8 j
i1
i0
k 1k (x ik ,1  x ik ,0 )  QOL i0 + ui
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Part 16: Panel Data
Dealing with Attrition


The attrition issue: Appearance for the second interview
was low for people with initial low QOL (death or
depression) or with initial high QOL (don’t need the
treatment). Thus, missing data at exit were clearly
related to values of the dependent variable.
Solutions to the attrition problem

Heckman selection model (used in the study)



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Prob[Present at exit|covariates] = Φ(z’θ) (Probit model)
Additional variable added to difference model i = Φ(zi’θ)/Φ(zi’θ)
The FDA solution: fill with zeros. (!)
Part 16: Panel Data
Difference in Differences
With two periods,
y it = y i2 -y i1 = 0 + (x i2 -x i1 )β + ui
Consider a "treatment, Di ," that takes place between
time 1 and time 2 for some of the individuals
y i = 0 + (x i )β + 1Di + ui
Di = the "treatment dummy"
This is a linear regression model. If there are no regressors,
ˆ
1  y | treatment - y | control
= "difference in differences" estimator.
ˆ
0  Average change in y i for the "treated"
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Part 16: Panel Data
Difference-in-Differences Model
With two periods and strict exogeneity of D and T,
y it = 0  1Dit  2 Tt  3 TtDit  it
Dit = dummy variable for a treatment that takes place
between time 1 and time 2 for some of the individuals,
Tt = a time period dummy variable, 0 in period 1,
1 in period 2.
This is a linear regression model. If there are no regressors,
Using least squares,
b3  (y 2  y1 )D1  (y 2  y1 )D0
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Part 16: Panel Data
Difference in Differences
y it = 0  1Dit  2 Tt  3Dit Tt  βx it  it , t  1, 2
y it = 2  3Di 2  (βx it )  it
= 2  3Di 2  β(x it )  ui
 y it | D  1   y it | D  0 
 3  β (x it | D  1)  (x it | D  0) 
If the same individual is observed in both states,
the second term is zero. If the effect is estimated by
averaging individuals with D = 1 and different individuals
with D=0, then part of the 'effect' is explained by change
in the covariates, not the treatment.
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Part 16: Panel Data
A Tale of Two Cities




A sharp change in policy can constitute a natural
experiment
The Mariel boatlift from Cuba to Miami (May-September,
1980) increased the Miami labor force by 7%. Did it
reduce wages or employment of non-immigrants?
Compare Miami to Los Angeles, a comparable
(assumed) city.
Card, David, “The Impact of the Mariel Boatlift on the
Miami Labor Market,” Industrial and Labor Relations
Review, 43, 1990, pp. 245-257.
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Part 16: Panel Data
Difference in Differences
i  individual, T = 0 for no immigration, T=1 for migration
(Yi | T)  Yi,T  1 if unemployed, 0 if employed.
c = city, t = period.
Unemployment rate in city c at time t is E[Yi,0 | c,t] with no migration
Unemploym ent rate in city c at time t is E[Yi,1 | c,t] with migration
Assume E[Yi,0 | c,t]  t   c
E[Yi,1 | c,t]  t   c  
 E[Yi,0 | c,t]  
  the effect of the immigration on the unemployment rate.
16-49/135
Part 16: Panel Data
Applying the Model




c = M for Miami, L for Los Angeles
Immigration occurs in Miami, not Los Angeles
T = 1979, 1981 (pre- and post-)
Sample moment equations: E[Yi|c,t,T]





E[Yi|M,79] = β79 + γM
E[Yi|M,81] = β81 + γM + δ
E[Yi|L,79] = β79 + γL
E[Yi|M,79] = β81 + γL
It is assumed that unemployment growth in the two cities
would be the same if there were no immigration.
16-50/135
Part 16: Panel Data
Implications for Differences

If neither city exposed to migration



If both cities exposed to migration



E[Yi,0|M,81] - E[Yi,0|M,79] = β81 – β79 (Miami)
E[Yi,0|L,81] - E[Yi,0|L,79] = β81 – β79 (LA)
E[Yi,1|M,81] - E[Yi,1|M,79] = β81 – β79 + δ (Miami)
E[Yi,1|L,81] - E[Yi,1|L,79] = β81 – β79 + δ (LA)
One city (Miami) exposed to migration: The difference
in differences is.

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{E[Yi,1|M,81] - E[Yi,1|M,79]} – {E[Yi,0|L,81] - E[Yi,0|L,79]}
= δ (Miami)
Part 16: Panel Data
UK Office of Fair Trading, May 2012; Stephen Davies
http://dera.ioe.ac.uk/14610/1/oft1416.pdf
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Part 16: Panel Data
Outcome is the fees charged.
Activity is collusion on fees.
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Part 16: Panel Data
Treatment Schools:
Treatment is an
intervention by the
Office of Fair Trading
Control Schools were
not involved in the
conspiracy
Treatment is not
voluntary
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Part 16: Panel Data
Apparent Impact of the Intervention
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Part 16: Panel Data
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Part 16: Panel Data
Treatment (Intervention)
Effect = 1 +
2 if SS school
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Part 16: Panel Data
In order to test robustness two versions of the fixed effects model were run. The first is
Ordinary Least Squares, and the second is heteroscedasticity and auto-correlation robust
(HAC) standard errors in order to check for heteroscedasticity and autocorrelation.
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Part 16: Panel Data
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Part 16: Panel Data
The cumulative impact of the intervention is the
area between the two paths from intervention to
time T.
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Part 16: Panel Data
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Part 16: Panel Data
The Fixed Effects Model
yi = Xi + diαi + εi, for each individual
 y1 
 
 y2  
 
 
 yN 
 X1
X
 2


 X N
d1
0
0
d2
0
0
0
0
0
0
0   β 
ε
  α 

dN 
β
= [X, D]    ε
 α
= Zδ  ε
E[ci | Xi ] = g(Xi); Effects are correlated with included variables.
Cov[xit,ci] ≠0
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Part 16: Panel Data
The Within Groups Transformation
Removes the Effects
y it  xitβ  ci +εit
y i  x iβ  ci +εi
y it  y i  ( x it - x i )β  (εit  εi )
Use least squares to estimate β.
16-63/135
Part 16: Panel Data
Useful Analysis of Variance Notation
Decomposition of Total variation:
N
i=1
Σ Σ
Ti
t=1
2
(zit  z)  Σ
N
i=1
Σ

Ti
t=1
  Σ Ti  z.i  z 
(zit  z.)
i



2
N
i=1
2
Total variation = Within groups variation
+ Between groups variation
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Part 16: Panel Data
WHO Data
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Part 16: Panel Data
Baltagi and Griffin’s Gasoline Data
World Gasoline Demand Data, 18 OECD Countries, 19 years
Variables in the file are
COUNTRY = name of country
YEAR = year, 1960-1978
LGASPCAR = log of consumption per car
LINCOMEP = log of per capita income
LRPMG = log of real price of gasoline
LCARPCAP = log of per capita number of cars
See Baltagi (2001, p. 24) for analysis of these data. The article on which the
analysis is based is Baltagi, B. and Griffin, J., "Gasolne Demand in the OECD: An
Application of Pooling and Testing Procedures," European Economic Review, 22,
1983, pp. 117-137. The data were downloaded from the website for Baltagi's
text.
16-66/135
Part 16: Panel Data
Analysis of Variance
16-67/135
Part 16: Panel Data
Analysis of Variance
+--------------------------------------------------------------------------+
| Analysis of Variance for
LGASPCAR
|
| Stratification Variable
_STRATUM
|
| Observations weighted by
ONE
|
| Total Sample Size
342
|
| Number of Groups
18
|
| Number of groups with no data
0
|
| Overall Sample Mean
4.2962420
|
| Sample Standard Deviation
.5489071
|
| Total Sample Variance
.3012990
|
|
|
| Source of Variation
Variation
Deg.Fr.
Mean Square |
| Between Groups
85.68228007
17
5.04013 |
| Within Groups
17.06068428
324
.05266 |
| Total
102.74296435
341
.30130 |
| Residual S.D.
.22946990
|
| R-squared
.83394791
MSB/MSW
21.96425 |
| F ratio
95.71734806
P value
.00000 |
+--------------------------------------------------------------------------+
16-68/135
Part 16: Panel Data
Estimating the Fixed Effects Model
The FEM is a plain vanilla regression model but
with many independent variables
 Least squares is unbiased, consistent, efficient,
but inconvenient if N is large.

1
 b   X X X D   X y 
  
 Dy 

a
D
X
D
D
  
 

Using the Frisch-Waugh theorem
b
16-69/135
=[X MD X ]1 X MD y 
Part 16: Panel Data
Fixed Effects Estimator (cont.)
M1D 0
0 


2
0
M
0
D
 (The dummy variables are orthogonal)
MD  



N
0
MD 
 0
MDi  I Ti  di (didi ) 1 d = I Ti  (1/Ti )did
X MD X = Ni=1 X iMDi X i ,
X MD y = Ni=1 X iMDi y i ,
16-70/135


XM y 
X iMDi X i
i
i
D
i k
k,l
T
i
  t=1
(x it,k -x i.,k )(x it,l -x i.,l )
T
i
  t=1
(x it,k -x i.,k )(y it -y i. )
Part 16: Panel Data
Least Squares Dummy Variable Estimator

b is obtained by ‘within’ groups least squares
(group mean deviations)

a is estimated using the normal equations:
D’Xb+D’Da=D’y
a = (D’D)-1D’(y – Xb)
Ti
ai=(1/Ti )Σ t=1
(yit -xitb)=ei
16-71/135
Part 16: Panel Data
Inference About OLS



Assume strict exogeneity: Cov[εit,(xjs,cj)]=0. Every
disturbance in every period for each person is
uncorrelated with variables and effects for every person
and across periods.
Now, it’s just least squares in a classical linear
regression model.
2
N
N
N
i
1

(

/

T
)plim[(1
/

T
)

X
M
X
]

i=1 i
i=1 i
i=1 i D i
Asy.Var[b] =
which is the usual estimator for OLS
2


ˆ 
Ti
Ni=1 t=1
(y it -ai -x it b)2

N
i=1
Ti - N - K

(Note the degrees of freedom correction)
16-72/135
Part 16: Panel Data
Application Cornwell and Rupert
16-73/135
Part 16: Panel Data
LSDV Results
Note huge changes in
the coefficients. SMSA
and MS change signs.
Significance changes
completely!
Pooled OLS
16-74/135
Part 16: Panel Data
The Effect of the Effects
16-75/135
Part 16: Panel Data
The Within (LSDV) Estimator is an IV Estimator
y = Xβ+(Dα+ε)
= Xβ+ w
Regression of y on X is inconsistent because X is
correlated with w. The data in group mean deviations is
Z = MD X = X - D(DD)-1 DX
The inconsistent OLS estimator is b = (X X)-1 X y (omits D)
The IV estimator bLSDV = (ZX)-1 Zy = (X MD X)-1 X MD y.
=[(X MD )(MD X)]-1 (X MD )(MD y)
This is OLS using data in mean deviations, i.e., LSDV.
16-76/135
Part 16: Panel Data
LSDV – As Usual
16-77/135
Part 16: Panel Data
2SLS Using Z=MDX as Instruments
16-78/135
Part 16: Panel Data
A Caution About Stata and R2
Residual Sum of Squares
Total Sum of Squares
Or is it? What is the total sum of squares?
R squared = 1 -
For the FE model above,
Conventional: Total Sum of Squares =
  y
"Within Sum of Squares"
  y
=
N
Ti
i 1
t 1
N
Ti
i 1
t 1
it
it
 y
2
 yi 
2
R2 = 0.90542
R2 = 0.65142
Which should appear in the denominator of R 2
The coefficient estimates and standard errors are the same. The calculation of the R2 is
different. In the areg procedure, you are estimating coefficients for each of your covariates
plus each dummy variable for your groups. In the xtreg, fe procedure the R2 reported is
obtained by only fitting a mean deviated model where the effects of the groups (all of the
dummy variables) are assumed to be fixed quantities. So, all of the effects for the groups are
simply subtracted out of the model and no attempt is made to quantify their overall effect on
the fit of the model.
Since the SSE is the same, the R2=1−SSE/SST is very different. The difference is real in that
we are making different assumptions with the two approaches. In the xtreg, fe approach, the
effects of the groups are fixed and unestimated quantities are subtracted out of the model
before the fit is performed. In the areg approach, the group effects are estimated and affect the
total sum of squares of the model under consideration.
16-79/135
Part 16: Panel Data
Examining the Effects with a KDE
Fixed Effects from Cornwell and Rupert Wage Model
.3 4 5
.2 7 6
De ns ity
.2 0 7
.1 3 8
.0 6 9
Fixed E ffects fr om C or nw ell and R uper t W age Model
.0 0 0
0
1
2
3
4
5
6
7
AI
AI
Frequency
Ke rn e l d e n s i ty e s ti m a te fo r
Mean = 4.819,
Standard deviation = 1.054.
.8 5 6
1 .6 8 8
2 .5 2 0
3 .3 5 1
4 .1 8 3
5 .0 1 5
5 .8 4 7
6 .6 7 8
AI
16-80/135
Part 16: Panel Data
Robust Covariance Matrix for LSDV
Cluster Estimator for Within Estimator
+--------+--------------+----------------+--------+--------+----------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+--------+--------------+----------------+--------+--------+----------+
|OCC
|
-.02021
.01374007
-1.471
.1412
.5111645|
|SMSA
|
-.04251**
.01950085
-2.180
.0293
.6537815|
|MS
|
-.02946
.01913652
-1.540
.1236
.8144058|
|EXP
|
.09666***
.00119162
81.114
.0000
19.853782|
+--------+------------------------------------------------------------+
+---------------------------------------------------------------------+
| Covariance matrix for the model is adjusted for data clustering.
|
| Sample of
4165 observations contained
595 clusters defined by |
|
7 observations (fixed number) in each cluster.
|
+---------------------------------------------------------------------+
+--------+--------------+----------------+--------+--------+----------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+--------+--------------+----------------+--------+--------+----------+
|DOCC
|
-.02021
.01982162
-1.020
.3078
.00000|
|DSMSA
|
-.04251
.03091685
-1.375
.1692
.00000|
|DMS
|
-.02946
.02635035
-1.118
.2635
.00000|
|DEXP
|
.09666***
.00176599
54.732
.0000
.00000|
+--------+------------------------------------------------------------+
16-81/135
Part 16: Panel Data
Time Invariant Regressors
16-82/135

Time invariant xit is defined as
invariant for all i. E.g., sex dummy
variable, FEM and ED (education in
the Cornwell/Rupert data).

If xit,k is invariant for all t, then the
group mean deviations are all 0.
Part 16: Panel Data
FE With Time Invariant Variables
+----------------------------------------------------+
| There are 2 vars. with no within group variation. |
| FEM
ED
|
+----------------------------------------------------+
+--------+--------------+----------------+--------+--------+----------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+--------+--------------+----------------+--------+--------+----------+
EXP
|
.09671227
.00119137
81.177
.0000
19.8537815
WKS
|
.00118483
.00060357
1.963
.0496
46.8115246
OCC
|
-.02145609
.01375327
-1.560
.1187
.51116447
SMSA
|
-.04454343
.01946544
-2.288
.0221
.65378151
FEM
|
.000000
......(Fixed Parameter).......
ED
|
.000000
......(Fixed Parameter).......
+--------------------------------------------------------------------+
|
Test Statistics for the Classical Model
|
+--------------------------------------------------------------------+
|
Model
Log-Likelihood
Sum of Squares R-squared |
|(1) Constant term only
-2688.80597
886.90494
.00000 |
|(2) Group effects only
27.58464
240.65119
.72866 |
|(3) X - variables only
-1688.12010
548.51596
.38154 |
|(4) X and group effects
2223.20087
83.85013
.90546 |
+--------------------------------------------------------------------+
16-83/135
Part 16: Panel Data
Drop The Time Invariant Variables
Same Results
+--------+--------------+----------------+--------+--------+----------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+--------+--------------+----------------+--------+--------+----------+
EXP
|
.09671227
.00119087
81.211
.0000
19.8537815
WKS
|
.00118483
.00060332
1.964
.0495
46.8115246
OCC
|
-.02145609
.01374749
-1.561
.1186
.51116447
SMSA
|
-.04454343
.01945725
-2.289
.0221
.65378151
+--------------------------------------------------------------------+
|
Test Statistics for the Classical Model
|
+--------------------------------------------------------------------+
|
Model
Log-Likelihood
Sum of Squares R-squared |
|(1) Constant term only
-2688.80597
886.90494
.00000 |
|(2) Group effects only
27.58464
240.65119
.72866 |
|(3) X - variables only
-1688.12010
548.51596
.38154 |
|(4) X and group effects
2223.20087
83.85013
.90546 |
+--------------------------------------------------------------------+
No change in the sum of squared residuals
16-84/135
Part 16: Panel Data
Fixed Effects Vector Decomposition
Efficient Estimation of Time Invariant and
Rarely Changing Variables in Finite Sample
Panel Analyses with Unit Fixed Effects
Thomas Plümper and Vera Troeger
Political Analysis, 2007
16-85/135
Part 16: Panel Data
Introduction
[T]he FE model … does not allow the estimation of
time invariant variables. A second drawback of
the FE model … results from its inefficiency in
estimating the effect of variables that have very
little within variance.
This article discusses a remedy to the related
problems of estimating time invariant and rarely
changing variables in FE models with unit effects
16-86/135
Part 16: Panel Data
The Model
yit = αi +  k=1βk xkit +  m=1 γm zmi + ε it
K
M
where αi denote the N unit effects.
16-87/135
Part 16: Panel Data
Fixed Effects Vector Decomposition
Step 1: Compute the fixed effects regression to
get the “estimated unit effects.” “We run this FE
model with the sole intention to obtain estimates
of the unit effects, αi.”
ˆαi = yi - K bFE
xki
k=1 k
16-88/135
Part 16: Panel Data
Step 2
Regress ai on zi and compute residuals
ai = m=1 γm zim +hi
M
hi is orthogonal to zi (since it is a residual)
Vector hi is expanded so each element
hi is replicated Ti times - h is the length of
the full sample.
16-89/135
Part 16: Panel Data
Step 3
Regress yit on a constant, X, Z and h using
ordinary least squares to estimate α, β, γ, δ.
yit = α +  k=1βk xkit +  m=1 γm zmi + δhi + ε it
K
M
Notice that i in the original model has
become  +h i in the revised model.
16-90/135
Part 16: Panel Data
Step 1 (Based on full sample)
These 2 variables have no within group variation.
FEM
ED
F.E. estimates are based on a generalized inverse.
--------+--------------------------------------------------------|
Standard
Prob.
Mean
LWAGE| Coefficient
Error
z
z>|Z|
of X
--------+--------------------------------------------------------EXP|
.09663***
.00119
81.13 .0000
19.8538
WKS|
.00114*
.00060
1.88 .0600
46.8115
OCC|
-.02496*
.01390
-1.80 .0724
.51116
IND|
.02042
.01558
1.31 .1899
.39544
SOUTH|
-.00091
.03457
-.03 .9791
.29028
SMSA|
-.04581**
.01955
-2.34 .0191
.65378
UNION|
.03411**
.01505
2.27 .0234
.36399
FEM|
.000
.....(Fixed Parameter).....
.11261
ED|
.000
.....(Fixed Parameter).....
12.8454
--------+---------------------------------------------------------
16-91/135
Part 16: Panel Data
Step 2 (Based on 595 observations)
--------+--------------------------------------------------------|
Standard
Prob.
Mean
UHI| Coefficient
Error
z
z>|Z|
of X
--------+--------------------------------------------------------Constant|
2.88090***
.07172
40.17 .0000
FEM|
-.09963**
.04842
-2.06 .0396
.11261
ED|
.14616***
.00541
27.02 .0000
12.8454
--------+---------------------------------------------------------
16-92/135
Part 16: Panel Data
Step 3!
--------+--------------------------------------------------------|
Standard
Prob.
Mean
LWAGE| Coefficient
Error
z
z>|Z|
of X
--------+--------------------------------------------------------Constant|
2.88090***
.03282
87.78 .0000
EXP|
.09663***
.00061
157.53 .0000
19.8538
WKS|
.00114***
.00044
2.58 .0098
46.8115
OCC|
-.02496***
.00601
-4.16 .0000
.51116
IND|
.02042***
.00479
4.26 .0000
.39544
SOUTH|
-.00091
.00510
-.18 .8590
.29028
SMSA|
-.04581***
.00506
-9.06 .0000
.65378
UNION|
.03411***
.00521
6.55 .0000
.36399
FEM|
-.09963***
.00767
-13.00 .0000
.11261
ED|
.14616***
.00122
120.19 .0000
12.8454
HI|
1.00000***
.00670
149.26 .0000 -.103D-13
--------+---------------------------------------------------------
16-93/135
Part 16: Panel Data
16-94/135
Part 16: Panel Data
What happened here?
yit = αi +  k=1βk xkit +  m=1 γm zmi + ε it
K
M
where αi denote the N unit effects.
An assumption is added along the way
Cov(αi ,Zi ) = 0. This is exactly the number of
orthogonality assumptions needed to
identify γ. It is not part of the original model.
16-95/135
Part 16: Panel Data
http://davegiles.blogspot.com/2012/06/fixed-effects-vector-decomposition.html
16-96/135
Part 16: Panel Data
16-97/135
Part 16: Panel Data
The Random Effects Model

The random effects model
y it =x itβ+c i +εit , observation for person i at time t
y i =X iβ+c ii+ε i , Ti observations in group i
=X iβ+c i +ε i , note c i  (c i , c i ,...,c i )
y =Xβ+c +ε , Ni=1 Ti observations in the sample
c=(c1 , c2 ,...cN ), Ni=1 Ti by 1 vector

ci is uncorrelated with xit for all t;
E[ci |Xi] = 0
E[εit|Xi,ci]=0
16-98/135
Part 16: Panel Data
Notation
 y1   X1 
 ε1   u1i1  T1 observations
y  X 
 ε   u i  T observations
2
2
2
    β   2   2 2
   
  

   
  

y
X
ε
u
i
 N  N
 N   N N  TN observations
= Xβ+ε+u Ni=1 Ti observations
= Xβ+w
In all that follows, except where explicitly noted, X, X i
and x it contain a constant term as the first element.
To avoid notational clutter, in those cases, x it etc. will
simply denote the counterpart without the constant term.
Use of the symbol K for the number of variables will thus
be context specific but will usually include the constant term.
16-99/135
Part 16: Panel Data
Error Components Model
A Generalized Regression Model
y it  x it b+εit +ui
E[εit | X i ]  0
E[εit2 | X i ]  σ 2
E[ui | X i ]  0
σ 2ε + σ u2

2
σ
u
Var[ε i +uii ]  
 ...

2
 σ u
σ u2
σ 2ε + σ u2
...
σ u2


...
σ u2 
 Ωi

...

… σ 2ε + σ u2 
...
σ u2
E[ui2 | X i ]  σ u2
y i =X iβ+ε i +uii for Ti observations
16-100/135
Part 16: Panel Data
Notation
 2   u2
 u2

 u2
 2   u2

Var[ε i +uii ] 


2
 u2
  u





 2   u2 
 u2
 u2
=  2I Ti   u2ii Ti  Ti
=  2I Ti   u2ii
= Ωi
Ω1
0
Var[w | X ]  


 0
16-101/135
0
Ω2
0
0
0  (Note these differ only
 in the dimension Ti )

ΩN 
Part 16: Panel Data
Convergence of Moments
X i X i
X X
N


f
 a weighted sum of individual moment matrices
i1 i
N
i1 T
Ti
X iΩi X i
X ΩX
N


f
 a weighted sum of individual moment matrices
i1 i
N
i1 T
Ti
=  2 Ni1fi
X i X i
  u2 Ni1fi x i x i
Ti
Note asymptotics are with respect to N. Each matrix
X i X i
is the
Ti
moments for the Ti observations. Should be 'well behaved' in micro
level data. The average of N such matrices should be likewise.
T or Ti is assumed to be fixed (and small).
16-102/135
Part 16: Panel Data
Random vs. Fixed Effects

Random Effects




Small number of parameters
Efficient estimation
Objectionable orthogonality assumption (ci  Xi)
Fixed Effects


Robust – generally consistent
Large number of parameters
16-103/135
Part 16: Panel Data
Ordinary Least Squares

Standard results for OLS in a GR model




Consistent
Unbiased
Inefficient
True variance of the least squares estimator
1
1
 XX  XΩX  XX 
Var[b | X]  N  N 


i1 Ti  i1 Ti  Ni1 Ti  Ni1 Ti 
 0   Q-1   Q *   Q-1
 0 as N  
1
16-104/135
Part 16: Panel Data
Estimating the Variance for OLS
1
 X X   X ΩX   X X 
Var[b | X ]  N  N   N
 N 
i1 Ti  i1 Ti   i1 Ti   i1 Ti 
In the spirit of the White estimator, use
1
1
ˆ
ˆ iw
ˆ i X i
X i w
Ti
X ΩX
N
ˆ i = y i - X ib, fi  N
 i1 fi
, w
N
Ti
i1 Ti
i1 Ti
Hypothesis tests are then based on Wald statistics.
THIS IS THE 'CLUSTER' ESTIMATOR
16-105/135
Part 16: Panel Data
OLS Results for Cornwell and Rupert
+----------------------------------------------------+
| Residuals
Sum of squares
=
522.2008
|
|
Standard error of e =
.3544712
|
| Fit
R-squared
=
.4112099
|
|
Adjusted R-squared
=
.4100766
|
+----------------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Constant
5.40159723
.04838934
111.628
.0000
EXP
.04084968
.00218534
18.693
.0000
19.8537815
EXPSQ
-.00068788
.480428D-04
-14.318
.0000
514.405042
OCC
-.13830480
.01480107
-9.344
.0000
.51116447
SMSA
.14856267
.01206772
12.311
.0000
.65378151
MS
.06798358
.02074599
3.277
.0010
.81440576
FEM
-.40020215
.02526118
-15.843
.0000
.11260504
UNION
.09409925
.01253203
7.509
.0000
.36398559
ED
.05812166
.00260039
22.351
.0000
12.8453782
16-106/135
Part 16: Panel Data
Alternative Variance Estimators
+---------+--------------+----------------+--------+---------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
+---------+--------------+----------------+--------+---------+
Constant
5.40159723
.04838934
111.628
.0000
EXP
.04084968
.00218534
18.693
.0000
EXPSQ
-.00068788
.480428D-04
-14.318
.0000
OCC
-.13830480
.01480107
-9.344
.0000
SMSA
.14856267
.01206772
12.311
.0000
MS
.06798358
.02074599
3.277
.0010
FEM
-.40020215
.02526118
-15.843
.0000
UNION
.09409925
.01253203
7.509
.0000
ED
.05812166
.00260039
22.351
.0000
Robust – Cluster___________________________________________
Constant
5.40159723
.10156038
53.186
.0000
EXP
.04084968
.00432272
9.450
.0000
EXPSQ
-.00068788
.983981D-04
-6.991
.0000
OCC
-.13830480
.02772631
-4.988
.0000
SMSA
.14856267
.02423668
6.130
.0000
MS
.06798358
.04382220
1.551
.1208
FEM
-.40020215
.04961926
-8.065
.0000
UNION
.09409925
.02422669
3.884
.0001
ED
.05812166
.00555697
10.459
.0000
16-107/135
Part 16: Panel Data
Generalized Least Squares
ˆ=[X Ω-1 X ]1 [X Ω-1 y ]
β
=[Ni1 XiΩi-1 X i ]1 [Ni1 X iΩi-1 y i ]
2



1
-1
Ωi  2 I Ti  2
ii
2
 
  Tiu 
(note, depends on i only through Ti )
16-108/135
Part 16: Panel Data
Generalized Least Squares
GLS is equivalent to OLS regression of
y it *  y it  i y i. on x it *  x it  i x i .,
where i  1 

2  Tiu2
ˆ]  [XΩ-1 X]-1  2 [X * X*]-1
Asy.Var[β

16-109/135
Part 16: Panel Data
Estimators for the Variances
y it  x it β  it  ui
Using the OLS estimator of β, bOLS ,
Ni1 tTi 1 (y it - a - x it b)2
  T  -1-K
N
i1
estimates 2  U2
i
With the LSDV estimates, ai and bLSDV ,
Ni1 tTi 1 (y it - ai - x it b)2
  T  -N-K
N
i1
estimates 2
i
Using the difference of the two,
 N  Ti (y - a - x  b)2   N  Ti (y - a - x  b)2 
it
it
 i1 t 1 it
   i1 t 1 it i
 estimates U2

 

Ni1 Ti -1-K
Ni1 Ti -N-K

 


16-110/135



Part 16: Panel Data
Practical Problems with FGLS
 The preceding regularly produce negative estimates of u2 .
 Estimation is made very complicated in unbalanced panels.
A bulletproof solution (originally used in TSP, now NLOGIT and others).
Ti
N
2



2
i1 t 1 (y it  ai  x it bLSDV )
From the robust LSDV estimator: 
ˆ 
Ni1 Ti
Ni1 tTi 1 (y it  aOLS  x itbOLS )2
2
From the pooled OLS estimator: Est(   ) 


ˆ

Ni1 Ti
2

2
u
Ni1 tTi 1 (y it  aOLS  x itbOLS )2  Ni1 tTi 1 (y it  ai  x itbLSDV )2

0
ˆ 
Ni1 Ti
2
u
16-111/135
Part 16: Panel Data
Stata Variance Estimators
Ni1 tTi 1 (y it  ai  x it bLSDV )2

> 0 based on FE estimates
ˆ 
Ni1 Ti  K  N
2

2


(N

K)

SSE(group
means)
ˆ
2



ˆ u  Max 0,
  0
NA
(N  A)T 

2
where A = K or if 
ˆ u is negative,
A=trace of a matrix that somewhat resembles IK .
Many other adjustments exist. None guaranteed to be
positive. No optimality properties or even guaranteed consistency.
16-112/135
Part 16: Panel Data
Other Variance Estimators
Ni1 (y it  a  x ibMEANS )2
From the group means regression:  / T   
N  K 1
ˆ it w
ˆ is
Ni1 tTi 11 sTi t 1 w
2
2
(Wooldridge) Based on E[w it w is | X i ]  u if t  s, 

ˆu
Ni1 Ti  K  N
2

2
u
There are many others. Generally if the original, standard choices fail,
these will also.
x´ does not contain a constant term in the preceding.
16-113/135
Part 16: Panel Data
Fixed Effects Estimates
---------------------------------------------------------------------Least Squares with Group Dummy Variables..........
LHS=LWAGE
Mean
=
6.67635
Residuals
Sum of squares
=
82.34912
Standard error of e =
.15205
These 2 variables have no within group variation.
FEM
ED
F.E. estimates are based on a generalized inverse.
--------+------------------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
Mean of X
--------+------------------------------------------------------------EXP|
.11346***
.00247
45.982
.0000
19.8538
EXPSQ|
-.00042***
.544864D-04
-7.789
.0000
514.405
OCC|
-.02106
.01373
-1.534
.1251
.51116
SMSA|
-.04209**
.01934
-2.177
.0295
.65378
MS|
-.02915
.01897
-1.536
.1245
.81441
FEM|
.000
......(Fixed Parameter).......
UNION|
.03413**
.01491
2.290
.0220
.36399
ED|
.000
......(Fixed Parameter).......
--------+-------------------------------------------------------------
16-114/135
Part 16: Panel Data
Computing Variance Estimators
Using the full list of variables (FEM and ED are time invariant)
OLS sum of squares = 522.2008.
2 +u2 = 522.2008 / (4165 - 9) = 0.12565.
Using full list of variables and a generalized inverse (same
as dropping FEM and ED), LSDV sum of squares = 82.34912.
2 = 82.34912 / (4165 - 8-595) = 0.023119.
u2  0.12565 - 0.023119 = 0.10253
Both estimators are positive. We stop here. If u2 were
negative, we would use estimators without DF corrections.
16-115/135
Part 16: Panel Data
Application
---------------------------------------------------------------------Random Effects Model: v(i,t)
= e(i,t) + u(i)
Estimates: Var[e]
=
.023119
Var[u]
=
.102531
Corr[v(i,t),v(i,s)] =
.816006
Lagrange Multiplier Test vs. Model (3) =3713.07
( 1 degrees of freedom, prob. value = .000000)
(High values of LM favor FEM/REM over CR model)
Fixed vs. Random Effects (Hausman)
=
.00 (Cannot be computed)
( 8 degrees of freedom, prob. value = 1.000000)
(High (low) values of H favor F.E.(R.E.) model)
Sum of Squares
1411.241136
R-squared
-.591198
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
EXP
.08819204
.00224823
39.227
.0000
19.8537815
EXPSQ
-.00076604
.496074D-04
-15.442
.0000
514.405042
OCC
-.04243576
.01298466
-3.268
.0011
.51116447
SMSA
-.03404260
.01620508
-2.101
.0357
.65378151
MS
-.06708159
.01794516
-3.738
.0002
.81440576
FEM
-.34346104
.04536453
-7.571
.0000
.11260504
UNION
.05752770
.01350031
4.261
.0000
.36398559
ED
.11028379
.00510008
21.624
.0000
12.8453782
Constant
4.01913257
.07724830
52.029
.0000
16-116/135
Part 16: Panel Data
Testing for Effects: An LM Test
Breusch and Pagan Lagrange Multiplier statistic
 0   u2
y it  x it  ui  it , ui and it ~ Normal   , 
 0   0
0 
2 
  
H0 : u2  0
General
2
 Ni1 (Ti ei )2

( Ti )
2
LM =

1



[1]
 N T 2

N
2i1 Ti (Ti  1)  i1 t 1eit

Balanced Panel
N
i1
2
NT   [(Te )  eiei ] 
LM 


2(T-1) 
Ni1eiei

N
i1
16-117/135
2
i
2
Part 16: Panel Data
Application: Cornwell-Rupert
16-118/135
Part 16: Panel Data
Testing for Effects
Regress; lhs=lwage;rhs=fixedx,varyingx;res=e$
Matrix ; tebar=7*gxbr(e,person)$
Calc
; list;lm=595*7/(2*(7-1))*
(tebar'tebar/sumsqdev - 1)^2$
LM
16-119/135
= 3797.06757
Part 16: Panel Data
A Hausman Test for FE vs. RE
Estimator
Random Effects
E[ci|Xi] = 0
Fixed Effects
E[ci|Xi] ≠ 0
FGLS
(Random Effects)
Consistent and
Efficient
Inconsistent
LSDV
(Fixed Effects)
Consistent
Inefficient
Consistent
Possibly Efficient
16-120/135
Part 16: Panel Data
Computing the Hausman Statistic
 N

1  
2
ˆ

Est.Var[βFE ]  
ˆ  i1 X i  I  ii  X i 
Ti  


1
-1
2
 N

Ti
ˆ i  
ˆu
2
ˆ
Est.Var[βRE ]  
 1
ˆ  i1 Xi  I  ii  X i  , 0  ˆ i = 2
2
Ti  

ˆ   Ti
ˆu


2
2
ˆ ]  Est.Var[β
ˆ ]
As long as 
ˆ  and 
ˆ u are consistent, as N  , Est.Var[β
FE
RE
will be nonnegative definite. In a finite sample, to ensure this, both must
2
be computed using the same estimate of 
ˆ  . The one based on LSDV will
generally be the better choice.
ˆ ] if there are time
Note that columns of zeros will appear in Est.Var[β
FE
invariant variables in X.
β does not contain the constant term in the preceding.
16-121/135
Part 16: Panel Data
Hausman Test
+--------------------------------------------------+
| Random Effects Model: v(i,t) = e(i,t) + u(i)
|
| Estimates: Var[e]
=
.235368D-01 |
|
Var[u]
=
.110254D+00 |
|
Corr[v(i,t),v(i,s)] =
.824078
|
| Lagrange Multiplier Test vs. Model (3) = 3797.07 |
| ( 1 df, prob value = .000000)
|
| (High values of LM favor FEM/REM over CR model.) |
| Fixed vs. Random Effects (Hausman)
= 2632.34 |
| ( 4 df, prob value = .000000)
|
| (High (low) values of H favor FEM (REM).)
|
+--------------------------------------------------+
16-122/135
Part 16: Panel Data
Fixed Effects
+----------------------------------------------------+
| Panel:Groups
Empty
0,
Valid data
595 |
|
Smallest
7,
Largest
7 |
|
Average group size
7.00 |
| There are 2 vars. with no within group variation. |
| ED
FEM
|
| Look for huge standard errors and fixed parameters.|
| F.E. results are based on a generalized inverse.
|
| They will be highly erratic. (Problematic model.) |
| Unable to compute std.errors for dummy var. coeffs.|
+----------------------------------------------------+
+--------+--------------+----------------+--------+--------+----------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+--------+--------------+----------------+--------+--------+----------+
|WKS
|
.00083
.00060003
1.381
.1672
46.811525|
|OCC
|
-.02157
.01379216
-1.564
.1178
.5111645|
|IND
|
.01888
.01545450
1.221
.2219
.3954382|
|SOUTH
|
.00039
.03429053
.011
.9909
.2902761|
|SMSA
|
-.04451**
.01939659
-2.295
.0217
.6537815|
|UNION
|
.03274**
.01493217
2.192
.0283
.3639856|
|EXP
|
.11327***
.00247221
45.819
.0000
19.853782|
|EXPSQ
|
-.00042***
.546283D-04
-7.664
.0000
514.40504|
|ED
|
.000
......(Fixed Parameter).......
|
|FEM
|
.000
......(Fixed Parameter).......
|
+--------+------------------------------------------------------------+
16-123/135
Part 16: Panel Data
Random Effects
+--------------------------------------------------+
| Random Effects Model: v(i,t) = e(i,t) + u(i)
|
| Estimates: Var[e]
=
.235368D-01 |
|
Var[u]
=
.110254D+00 |
|
Corr[v(i,t),v(i,s)] =
.824078
|
| Lagrange Multiplier Test vs. Model (3) = 3797.07 |
| ( 1 df, prob value = .000000)
|
| (High values of LM favor FEM/REM over CR model.) |
+--------------------------------------------------+
+--------+--------------+----------------+--------+--------+----------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+--------+--------------+----------------+--------+--------+----------+
|WKS
|
.00094
.00059308
1.586
.1128
46.811525|
|OCC
|
-.04367***
.01299206
-3.361
.0008
.5111645|
|IND
|
.00271
.01373256
.197
.8434
.3954382|
|SOUTH
|
-.00664
.02246416
-.295
.7677
.2902761|
|SMSA
|
-.03117*
.01615455
-1.930
.0536
.6537815|
|UNION
|
.05802***
.01349982
4.298
.0000
.3639856|
|EXP
|
.08744***
.00224705
38.913
.0000
19.853782|
|EXPSQ
|
-.00076***
.495876D-04
-15.411
.0000
514.40504|
|ED
|
.10724***
.00511463
20.967
.0000
12.845378|
|FEM
|
-.24786***
.04283536
-5.786
.0000
.1126050|
|Constant|
3.97756***
.08178139
48.637
.0000
|
+--------+------------------------------------------------------------+
16-124/135
Part 16: Panel Data
The Hausman Test, by Hand
--> matrix; br=b(1:8) ; vr=varb(1:8,1:8)$
--> matrix ; db = bf - br ; dv = vf - vr $
--> matrix ; list ; h =db'<dv>db$
Matrix H
has 1 rows and
1
+-------------1| 2523.64910
1 columns.
--> calc;list;ctb(.95,8)$
+------------------------------------+
| Listed Calculator Results
|
+------------------------------------+
Result =
15.507313
16-125/135
Part 16: Panel Data
Hello, professor greene.
I’ve taken the liberty of attaching some LIMDEP output in order to ask your
view on whether my Hausman test stat is “large,” requiring the FEM, or not,
allowing me to use the (much better for my research) REM.
Specifically, my test statistic, corrected for heteroscedasticity, is about 34
and significant with 6 df.
I considered this a large value until I found your “assignment 2” on the
internet which shows a value of 2554 with 4 df. Now, I’d like to assert that
34/6 is a small value.
16-126/135
Part 16: Panel Data
Variable Addition
A Fixed Effects Model
yit  i  xit  it
LSDV estimator - Deviations from group means:
To estimate , regress (yit  yi ) on (xit  xi )
Algebraic equivalent: OLS regress yit on (xit , xi )
Mundlak interpretation: i    xi  u i
Model becomes yit    xi  u i  xit  it
=   xi  xit  it  u i
 a random effects model with the group means.
Estimate by FGLS.
16-127/135
Part 16: Panel Data
A Variable Addition Test
Asymptotic equivalent to Hausman
 Also equivalent to Mundlak formulation
 In the random effects model, using FGLS




Only applies to time varying variables
Add expanded group means to the regression (i.e.,
observation i,t gets same group means for all t.
Use Wald test to test for coefficients on means
equal to 0. Large chi-squared weighs against
random effects specification.
16-128/135
Part 16: Panel Data
Means Added to REM - Mundlak
+--------+--------------+----------------+--------+--------+----------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+--------+--------------+----------------+--------+--------+----------+
|WKS
|
.00083
.00060070
1.380
.1677
46.811525|
|OCC
|
-.02157
.01380769
-1.562
.1182
.5111645|
|IND
|
.01888
.01547189
1.220
.2224
.3954382|
|SOUTH
|
.00039
.03432914
.011
.9909
.2902761|
|SMSA
|
-.04451**
.01941842
-2.292
.0219
.6537815|
|UNION
|
.03274**
.01494898
2.190
.0285
.3639856|
|EXP
|
.11327***
.00247500
45.768
.0000
19.853782|
|EXPSQ
|
-.00042***
.546898D-04
-7.655
.0000
514.40504|
|ED
|
.05199***
.00552893
9.404
.0000
12.845378|
|FEM
|
-.41306***
.03732204
-11.067
.0000
.1126050|
|WKSB
|
.00863**
.00363907
2.371
.0177
46.811525|
|OCCB
|
-.14656***
.03640885
-4.025
.0001
.5111645|
|INDB
|
.04142
.02976363
1.392
.1640
.3954382|
|SOUTHB |
-.05551
.04297816
-1.292
.1965
.2902761|
|SMSAB
|
.21607***
.03213205
6.724
.0000
.6537815|
|UNIONB |
.08152**
.03266438
2.496
.0126
.3639856|
|EXPB
|
-.08005***
.00533603
-15.002
.0000
19.853782|
|EXPSQB |
-.00017
.00011763
-1.416
.1567
514.40504|
|Constant|
5.19036***
.20147201
25.762
.0000
|
+--------+------------------------------------------------------------+
16-129/135
Part 16: Panel Data
Wu (Variable Addition) Test
--> matrix ; bm=b(12:19);vm=varb(12:19,12:19)$
--> matrix ; list ; wu = bm'<vm>bm $
Matrix WU
has 1 rows and
1
+-------------1| 3004.38076
16-130/135
1 columns.
Part 16: Panel Data
A Hierarchical Linear Model
Interpretation of the FE Model
y it  x it β  c i +εit , (x does not contain a constant)
E[εit|X i , c i ]  0, Var[ε it|X i , c i ]=2
c i  +ziδ + ui ,
E[u|i zi ]  0, Var[u|i zi ]  u2
y it  x it β  [  ziδ  ui ]  εit
16-131/135
Part 16: Panel Data
Hierarchical Linear Model as REM
+--------------------------------------------------+
| Random Effects Model: v(i,t) = e(i,t) + u(i)
|
| Estimates: Var[e]
=
.235368D-01 |
|
Var[u]
=
.110254D+00 |
|
Corr[v(i,t),v(i,s)] =
.824078
|
|
Sigma(u)
= 0.3303
|
+--------------------------------------------------+
+--------+--------------+----------------+--------+--------+----------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+--------+--------------+----------------+--------+--------+----------+
OCC
|
-.03908144
.01298962
-3.009
.0026
.51116447
SMSA
|
-.03881553
.01645862
-2.358
.0184
.65378151
MS
|
-.06557030
.01815465
-3.612
.0003
.81440576
EXP
|
.05737298
.00088467
64.852
.0000
19.8537815
FEM
|
-.34715010
.04681514
-7.415
.0000
.11260504
ED
|
.11120152
.00525209
21.173
.0000
12.8453782
Constant|
4.24669585
.07763394
54.702
.0000
16-132/135
Part 16: Panel Data
Evolution: Correlated Random Effects
Unknown parameters
yit   i  xit  it ,   [1 ,  2 ,...,  N , ,  2 ]
Standard estimation based on LS (dummy variables)
Ambiguous definition of the distribution of yit
Effects model, nonorthogonality, heterogeneity
yit   i  xit  it , E[ i | Xi ]  g( Xi )  0
Contrast to random effects E[i | X i ]  
Standard estimation (still) based on LS (dummy variables)
Correlated random effects, more detailed model
yit   i  xit  it , P[ i | Xi ]  g( Xi )  0
Linear projection?  i  xi  u i Cor(u i , xi )  0
16-133/135
Part 16: Panel Data
Mundlak’s Estimator
Mundlak, Y., “On the Pooling of Time Series and Cross Section
Data, Econometrica, 46, 1978, pp. 69-85.
Write c i = x iδ  ui , E[c i | x i1 , x i1 ,...x iTi ] = x iδ
Assume c i contains all time invariant information
y i =X iβ+c ii+ε i , Ti observations in group i
=X iβ+ix iδ+ε i + uii
Looks like random effects.
Var[ε i + uii]=Ωi +σ 2uii
This is the model we used for the Wu test.
16-134/135
Part 16: Panel Data
Correlated Random Effects
Mundlak
c i = x iδ  ui , E[c i | x i1 , x i1 ,...x iTi ] = x iδ
Assume c i contains all time invariant information
y i =X iβ+c ii+ε i , Ti observations in group i
=X iβ+ix iδ+ε i + uii
Chamberlain / Wooldridge
c i = x i1δ1  x i2 δ2  ...  x iT δ T  ui
y i =X iβ  ix i1δ1  ix i1δ2  ...  ix iT δ T  iui +ε i
TxK  TxK  TxK 
TxK etc.
Problems: Requires balanced panels
Modern panels have large T; models have large K
16-135/135
Part 16: Panel Data
Mundlak’s Approach for an FE Model with
Time Invariant Variables
y it  x itβ+ziδ  c i +εit , (x does not contain a constant)
E[εit|X i , c i ]  0, Var[ε it|X i , c i ]=2
c i  + x iθ + w i ,
E[w i|X i , zi ]  0, Var[w i|X i , zi ]  2w
y it  x itβ  ziδ    x iθ  w i  εit
 random effects model including group means of
time varying variables.
16-136/135
Part 16: Panel Data
Mundlak Form of FE Model
+--------+--------------+----------------+--------+--------+----------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+--------+--------------+----------------+--------+--------+----------+
x(i,t)=================================================================
OCC
|
-.02021384
.01375165
-1.470
.1416
.51116447
SMSA
|
-.04250645
.01951727
-2.178
.0294
.65378151
MS
|
-.02946444
.01915264
-1.538
.1240
.81440576
EXP
|
.09665711
.00119262
81.046
.0000
19.8537815
z(i)===================================================================
FEM
|
-.34322129
.05725632
-5.994
.0000
.11260504
ED
|
.05099781
.00575551
8.861
.0000
12.8453782
Means of x(i,t) and constant===========================================
Constant|
5.72655261
.10300460
55.595
.0000
OCCB
|
-.10850252
.03635921
-2.984
.0028
.51116447
SMSAB
|
.22934020
.03282197
6.987
.0000
.65378151
MSB
|
.20453332
.05329948
3.837
.0001
.81440576
EXPB
|
-.08988632
.00165025
-54.468
.0000
19.8537815
Variance Estimates=====================================================
Var[e]|
.0235632
Var[u]|
.0773825
16-137/135
Part 16: Panel Data
Panel Data Extensions
Dynamic models: lagged effects of the
dependent variable
 Endogenous RHS variables
 Cross country comparisons– large T
 More general parameter heterogeneity – not
only the constant term
 Nonlinear models such as binary choice

16-138/135
Part 16: Panel Data
The Hausman and Taylor Model
y it  x1it β1  x2it β2  z1i α1  z2i α2  it  ui
Model: x2 and z2 are correlated with u.
Deviations from group means removes all time invariant variables
y it  y i  ( x1it - x1i )'β1  ( x2it - x2i )'β2  it
Implication: β1 , β2 are consistently estimated by LSDV.
( x1it - x1i ) = K 1 instrumental variables
( x2it - x2i ) = K 2 instrumental variables
z1i
?
= L1 instrumental variables (uncorrelated with u)
= L 2 instrumental variables (where do we get them?)
H&T: x1i = K 1 additional instrumental variables. Needs K 1  L 2 .
16-139/135
Part 16: Panel Data
H&T’s 4 Step FGLS Estimator
(1) LSDV estimates of β1 , β2 , 2
(2) (e*)' = (e1 , e1 ,..., e1 ), (e2 , e2 ,..., e 2 ),..., (eN , eN ,..., eN )
IV regression of e * on Z * with instruments
Wi consistently estimates α1 and α2 .
(3) With fixed T, residual variance in (2) estimates u2  2 / T
With unbalanced panel, it estimates u2  2 (1/T) or something
resembling this. (1) provided an estimate of 2 so use the two
to obtain estimates of u2 and 2 . For each group, compute
2
2
2
ˆ
i  1  
ˆ  / (
ˆ   Ti
ˆu )
(4) Transform
[x it1 , x it2 , zi1 , zi2 ] to
Wi * = [x it1 , x it2 , zi1 , zi2 ] - ˆ
i [x i1 , x i2 , zi1 , zi2 ]
and
16-140/135
y it to y it * = y it - ˆ
i y i.
Part 16: Panel Data
H&T’s 4 STEP IV Estimator
Instrumental Variables Vi 
( x1it - x1i ) = K1 instrumental variables
( x2it - x2i ) = K 2 instrumental variables
z1i
= L1 instrumental variables (uncorrelated with u)
x1i
= K1 additional instrumental variables.
Now do 2SLS of y * on W * with instruments V to estimate
all parameters. I.e.,
ˆ * W*
ˆ )-1W
ˆ * y * .
[β1 , β2 , α1 , α2 ]=(W
16-141/135
Part 16: Panel Data
16-142/135
Part 16: Panel Data
Arellano/Bond/Bover’s Formulation Builds
on Hausman and Taylor
y it  x1it β1  x2it β2  z1i α1  z2i α2  it  ui
Instrumental variables for period t
( x1it - x1i ) = K 1 instrumental variables
( x2it - x2i ) = K 2 instrumental variables
z1i
= L1 instrumental variables (uncorrelated with u)
x1i
= K 1 additional instrumental variables. K 1  L 2 .
Let v it  it  ui
Let zit  [( x1it - x1i )',( x2it - x2i )',z1i , x1']
Then E[zit v it ]  0
We formulate this for the Ti observations in group i.
16-143/135
Part 16: Panel Data
Arellano/Bond/Bover’s Formulation Adds a
Lagged DV to H&T
y it  y i,t 1 +x1it β1  x2it β2  z1i α1  z2i α2  it  ui
Parameters : θ = [,β1 , β2 , α1 , α2 ]'
The data
 y i,2 
 y i,1 x1i2 x2i2 z1i z2i 








y
y
x1
x2
z1
z2
i3
i3
i
i
 i,3 
 i,2
yi  
,
X

i


 , Ti -1 rows




 y i,Ti 
 y i,T-1 x1iTi x2iTi z1i z2i 
1 K1
K2
L1 L2
columns
This formulation is the same as H&T with yi,t-1 contained in x2it .
16-144/135
Part 16: Panel Data
Dynamic (Linear) Panel
Data (DPD) Models
Application
 Bias in Conventional Estimation
 Development of Consistent Estimators
 Efficient GMM Estimators

16-145/135
Part 16: Panel Data
Dynamic Linear Model
Balestra-Nerlove (1966), 36 States, 11 Years
Demand for Natural Gas
Structure
New Demand: G*i,t  Gi,t  (1  )Gi,t 1
Demand Function G*i,t  1  2Pi,t  3 Ni,t  4Ni,t  5 Yi,t  6 Yi,t  i,t
G=gas demand
N = population
P = price
Y = per capita income
Reduced Form
Gi,t  1  2Pi,t  3 Ni,t  4Ni,t  5 Yi,t  6 Yi,t  7 Gi,t 1  i  i,t
16-146/135
Part 16: Panel Data
A General DPD model
y i,t  x i,t β  y i,t 1  c i  i,t
E[i,t | X i ,c i ]  0
2
E[i,t
| X i , c i ]  2 , E[i,t i,s | X i , c i ]  0 if t  s.
E[c i | X i ]  g( X i )
No correlation across individuals
OLS and GLS are both inconsistent.
16-147/135
Part 16: Panel Data
Arellano and Bond Estimator
Base on first differences
y i,t  y i,t 1  ( x i,t  x i,t 1 )'β+(y i,t 1  y i,t 2 )  (i,t  i,t 1 )
Instrumental variables
y i,3  y i,2  ( x i,3  x i,2 )'β+(y i,2  y i,1 )  (i,3  i,2 )
Can use y i1
y i,4  y i,3  ( x i,4  x i,3 )'β+(y i,3  y i,2 )  (i,4  i,3 )
Can use y i,1 and y i2
y i,5  y i,4  ( x i,5  x i,4 )'β+(y i,4  y i,3 )  (i,5  i,4 )
Can use y i,1 and y i2 and y i,3
16-148/135
Part 16: Panel Data
Arellano and Bond Estimator
More instrumental variables - Predetermined X
y i,3  y i,2  ( x i,3  x i,2 )'β+(y i,2  y i,1 )  (i,3  i,2 )
Can use y i1 and x i,1 , x i,2
y i,4  y i,3  ( x i,4  x i,3 )'β+(y i,3  y i,2 )  (i,4  i,3 )
Can use y i,1 , y i2 , x i,1 , x i,2 , x i,3
y i,5  y i,4  ( x i,5  x i,4 )'β+(y i,4  y i,3 )  (i,5  i,4 )
Can use y i,1 , y i2 , y i,3 , x i,1 , x i,2 , x i,3 , x i,4
16-149/135
Part 16: Panel Data
Arellano and Bond Estimator
Even more instrumental variables - Strictly exogenous X
y i,3  y i,2  ( x i,3  x i,2 )'β+(y i,2  y i,1 )  (i,3  i,2 )
Can use y i1 and x i,1 , x i,2 ,..., x i,T (all periods)
y i,4  y i,3  ( x i,4  x i,3 )'β+(y i,3  y i,2 )  (i,4  i,3 )
Can use y i,1 , y i2 , x i,1 , x i,2 ,..., x i,T
y i,5  y i,4  ( x i,5  x i,4 )'β+(y i,4  y i,3 )  (i,5  i,4 )
Can use y i,1 , y i2 , y i,3 , x i,1 , x i,2 ,..., x i,T
The number of potential instruments is huge.
These define the rows of Zi . These can be used for
simple instrumental variable estimation.
16-150/135
Part 16: Panel Data
Application: Maquiladora
http://www.dallasfed.org/news/research/2005/05us-mexico_felix.pdf
16-151/135
Part 16: Panel Data
Maquiladora
16-152/135
Part 16: Panel Data
Estimates
16-153/135
Part 16: Panel Data