Transcript Class 3. Models with Individual Effects

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Econometric Analysis of Panel Data
William Greene
Department of Economics
Stern School of Business
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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
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Short Term Agenda for Simple Effects Models
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Models with individual effects
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Extensions
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Interpretation of models
Computation (practice) and estimation (theory)
Nonstandard panels: Rotating, Pseudo-, Nested
Generalizing the regression model
Alternative estimators
Methods
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Least squares: OLS, GLS, FGLS
MLE and Maximum Simulated Likelihood
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Fixed and Random Effects

Unobserved individual effects in regression: E[yit | xit, ci]
Notation:
yit =xit + ci + it

 xi1 
 x 
i2
X i    Ti rows, K columns
 
 

Linear specification: x iTi 
Fixed Effects: E[ci | Xi ] = g(Xi). Cov[xit,ci] ≠0
effects are correlated with included variables.

Random Effects: E[ci | Xi ] = μ; effects are uncorrelated with
included variables. If Xi contains a constant term, μ=0 WLOG.
Common: Cov[xit,ci] =0, but E[ci | Xi ] = μ is needed for the
full model
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Convenient Notation

Fixed Effects – the ‘dummy variable model’
yit = i + xit + it
Individual specific constant terms.
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Random Effects – the ‘error components model’
yit = xit + it + ui
Compound (“composed”) disturbance
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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
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The role of the assumption
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Mathematical and notational convenience:
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Balanced, n=NT
N
Unbalanced: n   i=1 Ti
Is the fixed Ti assumption ever necessary? Almost
never. (Baltagi chapter 9 is about algebra, not
different models!)
Is unbalancedness due to nonrandom attrition
from an otherwise balanced panel? This will
require special considerations.
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An Unbalanced Panel: RWM’s
GSOEP Data on Health Care
N = 7,293 Households
Some households exited then returned
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Exogeneity
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Contemporaneous exogeneity
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Strict exogeneity – the most common assumption
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E[εit|xi1, xi2,…,xiT,ci]=0
Can use first difference or fixed effects
Cannot hold if xit contains lagged values of yit
Sequential exogeneity?
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E[εit|xit,ci]=0  Not sufficient for regression
Doesn’t imply how to estimate β
E[εit|xi1, xi2,…,xit,ci] = 0
These assumptions are not testable. They are part of the
model.
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Assumptions for Asymptotics
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Convergence of moments involving cross section Xi.
N increasing, T or Ti assumed fixed.
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“Fixed T asymptotics” (see text, p. 175)
Time series characteristics are not relevant (may be
nonstationary)
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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Estimating β
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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
Using a proxy variable.
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The Pooled Regression
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Presence of omitted effects
y it =x itβ+c i +εit , observation for person i at time t
y i =X iβ+cii+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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Most Helpful Customer Reviews
31 of 39 people found the following review helpful Too theoretical and poorly written
By Doktor Faustus on May 7, 2013
Format: Hardcover Econometric Analysis" by William Greene is one of the more widely use graduate-level
textbooks in econometrics. I used it in my first year PhD econometrics course. This is unfortunate for several
reasons. The book states that its first objective is to introduce students to applied econometrics, especially the basic
techniques of linear regression. When reading the book, however, what the reader notices first is that the applications
are essentially just footnotes; the meat of each chapter is dense econometric theory. An applied textbook would focus
on working with data, but Greene's book has exercises that focus on proving obscure statistical properties (i.e. prove
that the asymptotic variance of various estimators goes to zero). Useful for theorists, but not for applied work, which
is what the book advertises itself as.
Another problem with the book is its impenetrable text. Reading this book is drudgery even when not trying to make
sense of the absurdly huge matrix equations. Greene uses academic, elevated language that does not belong in a
technical textbook. Where the student needs clear explanation, he instead reads sentences like the following found in
a chapter introduction: "We first consider the consequences for the least squares estimator of the more general form
of the regression model. This will include assessing the effect of ignoring the complication of the generalized model
and of devising an appropriate estimation strategy, still based on least squares". After reading that second sentence
several times I still don't understand what Greene is trying to convey.
Finally the book is much too large and expensive for a class textbook. The book is 1200 pages long and includes
numerous asides in every chapter. If the objective of the book is to teach econometrics to graduate students (as it
says in the book), then it would be better off focusing on important topics and applications, not on topics that are
never used by the vast majority of economists. I do not recommend this book for anyone; there are better
econometrics textbooks available for undergraduates, graduate students, and professionals.
[Part 3: Common Effects ] 14/57
October 13, 2014
By Daniel Pulido
This review is from: Econometric Analysis (7th Edition) (Hardcover)
The delivery was fine. But the book itself is the worst Econometric Analysis book
I have ever come across. No examples. Only a continuous list of theorems.
I would not recommend anyone this book.
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A Popular Misconception
If only one variable in X is correlated with , the other coefficients are
consistently estimated. False.
Suppose only the first variable is correlated with ε
 1 
 
0
Under the assumptions, plim( X'ε /n) =   . Then
 ... 
 
 . 
 q11 
 1 
 21 
 
0
q 
plim b - β = plim(X'X /n)-1    1 
 ... 
 ... 
 K 1 
 
.
 
q 
 1 times the first column of Q-1
The problem is “smeared” over the other coefficients.
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OLS with Individual Effects
b=(X X )-1 X'y = (X X )-1 X'(Xβ+c+ε)
-1
=β + (1/N)Σ X iX i  (1/N)Σ Ni=1 X ic i  (part due to the omitted c i )
N
i=1
-1
+ (1/N)Σ X iX i  (1/N)Σ Ni=1 X iε i  (covariance of X and ε will = 0)
The third term vanishes asymptotically by assumption
N
i=1
-1
T

1
 
plim b = β + plim  ΣNi=1 X iX i  ΣNi=1 i x ic i  (left out variable formula)
N
N
 

So, what becomes of ΣNi=1 wi x i c i  ?
plim b = β if the covariance of x i and ci converges to zero.
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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
May be estimable by 2 step FGLS.
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Chamberlain’s (1982) Approach
Use a linear projection, not necessarily the conditional
mean.
P[ci | xi1 , xi1 ,...xiTi ] = xi11 + xi22  ...  xiT  T
ci  P[ci | xi1 , xi1 ,...xiTi ]  ui , cov[ui ,xit ]=0
y it =xitβ+xi11 + xi22  ...  xiT  T + εit  ui
This “regression” can be computed T times, using one year at a time.
How would we reconcile the multiple estimators of each parameter?.
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Chamberlain’s (1982) Approach
P[ci | xi1 , xi1 ,...xiTi ] = xi11 + xi22  ...  xiT  T
ci  P[ci | x i1 , xi1 ,...xiTi ]  ui , cov[ui ,xit ]=0
y it =xitβ+xi11 + xi22  ...  xiT  T + εit  ui
Period 1
y i1=xi1 (β+1 ) + xi22  ...  xiT  T + εi1  ui
Period 2
y i2=xi11 + xi2 (β+2 )  ...  xiT  T + εi2  ui
and so on...
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Proxy Variables
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Proxies for unobserved effects: e.g., Test score for unobserved
ability
Interest is in δ(xit,ci)=E[yit|xit,ci]/xit
Since ci is unobserved, we seek APE = Ec[δ(xit,ci)]
Proxy has two characteristics

Ignorable in the model: E[yit|xit,zi,ci] = E[yit|xit,ci]

‘Explains’ ci in that E[ci|zi,xit] = E[ci|zi]. In the presence of zi, xit does
not further ‘explain ci.’
Then, Ec[δ(xit,ci)] = Ez{E[yit|xit,zi]/xit}
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Proof: See Wooldridge, pp. 23-24.
Loose ends:
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Where do you get the proxy?
What is E[yit|xit,zi]? Use the linear projection and hope for the best.
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Estimating the Sampling Variance of b
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s2(X ́X)-1?
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Correlation across observations
Heteroscedasticity
A “robust” covariance matrix
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Robust estimation (in general)
The White estimator
A Robust estimator for OLS.
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A ‘Cluster’ Estimator
yit =xitβ+(ci +εit )
=xitβ+vit , Cov[vit , vis ]  0
Pseudo-log likelihood that produces OLS as the estimator
Ti
logL*=Ni=1 (-1/2)Σ t=1
(logσ 2 +log2π+v it2 /σ 2 
Ti
ˆ
The solution for 2 will always be [Ni=1Σ t=1
v it2 ] / Ni=1 Ti ,
so concentrate on β. The solution will be b=(X X )-1 X y
Ti
logL*/β = Ni=1 Σ t=1
x it v it /σ 2   Ni=1gi  g.
Ti
 2logL*/ββ = -Ni=1Σ t=1
x it x it /σ 2  (1 / σ 2 ) X X = H and = E[H]
Var[b] = (-H-1 )Var[g](-H-1 )
Var[g] is usually H, but not here because of correlation across
observations. Approximate Var[g] with Ni=1gigi.
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Cluster Estimator (cont.)
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Cornwell and Rupert Data
Cornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 Years
Variables in the file are
EXP
WKS
OCC
IND
SOUTH
SMSA
MS
FEM
UNION
ED
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
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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Application: Cornell and Rupert
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Bootstrapping
Some assumptions that underlie it - the sampling mechanism
Method:
1. Estimate using full sample: --> b
2. Repeat R times:
Draw n observations from the n, with replacement
Estimate  with b(r).
3. Estimate variance with
V = (1/R)r [b(r) - b][b(r) - b]’
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Bootstrap Application
matr;bboot=init(7,21,0.)$
Store results here
name;x=one,occ,…,exp$
Define X
regr;lhs=lwage;rhs=x$
Compute b
calc;i=0$
Counter
Proc
Define procedure
regr;lhs=lwage;rhs=x;quietly$
… Regression
matr;{i=i+1};bboot(*,i)=b$...
Store b(r)
Endproc
Ends procedure
exec;n=20;bootstrap=b$
20 bootstrap reps
matr;list;bboot' $
Display results
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Results of Bootstrap Procedure
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Bootstrap Replications
Full sample result
Bootstrapped
sample results
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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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Bootstrapping
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Naïve bootstrap: Why is it naïve?
Cases when it fails
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Time series
“Clustered data”
Order statistics
Parameters on the edge of the parameter space
Alternatives
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Block bootstrap
“Wild” bootstrap (injects extra randomness)
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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 + uit
Note: Time invariant variables become zero
Time trend becomes the constant term
Time dummy variables become (0,...,1,-1,0,0...)
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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  22

  2
 i,3  i,2   
Var 
 0

 
 i,T  i,T 1   0
 i
i

2
22
2
0
2
2
0 

 (Toeplitz form)
2


22 
GLS is unpleasantly complicated. In order to
compute a first step estimator of σε2 we would use
fixed effects. We should just stop there. Or, use OLS
in first differences and use Newey-West with one lag.
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Two Periods
With two periods and strict exogeneity,
y it = y i2 -y i,1 = 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 classical 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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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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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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http://dera.ioe.ac.uk/14610/1/oft1416.pdf
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Outcome is the fees charged.
Activity is collusion on fees.
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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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Treatment (Intervention)
Effect = 1 +
2 if SS school
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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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D-in-D Model: Natural Experiment
With two periods and strict exogeneity,
y it = 0  1Di 2  2 Tt  3 Tt Dit  it
Di2 = 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 classical 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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D-i-D

Card and Krueger: “Minimum Wages and Employment: A Case
Study of the Fast Food Industry in New Jersey and Pennsylvania,”
AER, 84(4), 1994, 772-793.

Pennsylvania vs. New Jersey

1991, NJ raises minimum wage


Compare change in employment PA after the change to change in
employment in NJ after the change.
Differences cancel out other things specific to the state that would
explain change in employment.
[Part 3: Common Effects ] 50/57
A Tale of Two Cities




A sharp change in policy can constitute a natural
experiment
The Mariel boatlift from Cuba to Miami (MaySeptember, 1980) increased the Miami labor force by
7%. Did it reduce wages or employment of nonimmigrants?
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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Difference in Differences
i  individual, T = 0 for no immigration, T=1 for immigration
(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
Unemployment 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.
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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.
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Implications for Differences

Neither city exposed to migration



Both cities exposed to migration



E[Yi,0|M,81] - E[Yi,0|M,79] = [β81 + γM ] – [β79 + γM] ( Miami)
E[Yi,0|L,81] - E[Yi,0|L,79] = [β81 + γL ] – [β79 + γL] (LA)
E[Yi,1|M,81] - E[Yi,1|M,79] = [β81 + γM ] – [β79 + γM] + δ (Miami)
E[Yi,1|L,81] - E[Yi,1|L,79] = [β81 + γL ] – [β79 + γL] + δ (LA)
One city (Miami) exposed to migration: The difference
in differences is.


Miami change - Los Angeles change
{E[Yi,1|M,81] - E[Yi,1|M,79]} – {E[Yi,0|L,81] - E[Yi,0|L,79]}
= δ (Miami)
[Part 3: Common Effects ] 54/57
The Tale
1979
1980
1981
1982
1983
1984
1985
In 79, Miami unemployment is 2.0% lower
In 80, Miami unemployment is 7.1% lower
From 79 to 80, Miami gets
5.1% better
In 81, Miami unemployment is 3.0% lower
In 82, Miami unemployment is 3.3% higher
From 81 to 82, Miami gets
6.3% worse
[Part 3: Common Effects ] 55/57
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
[Part 3: Common Effects ] 56/57
[Part 3: Common Effects ] 57/57
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





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
[Part 3: Common Effects ] 58/57
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)
[Part 3: Common Effects ] 59/57
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








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 3: Common Effects ] 60/57
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
[Part 3: Common Effects ] 61/57
Optimal treatment.
Conventional wisdom
and assumption of
policy.
Study finding
Note the implication of
the study for the
location of the optimal
point for the treatment.
Largest marginal benefit
moves from the left tail
to the center.
Finding