This document provides an introduction to panel data analysis and regression models for panel data. It defines panel data as longitudinal data collected on the same units (like individuals, firms, countries) over multiple time periods. Panel data allow researchers to study changes over time and estimate causal effects. The document outlines common panel data structures, reasons for using panel data analysis, and basic estimation techniques like fixed effects and random effects models to account for unobserved heterogeneity across units. It also discusses assumptions and limitations of different panel data models.
Ragui Assaad- University of Minnesota
Caroline Krafft- ST. Catherine University
ERF Training on Applied Micro-Econometrics and Public Policy Evaluation
Cairo, Egypt July 25-27, 2016
www.erf.org.eg
Ragui Assaad- University of Minnesota
Caroline Krafft- ST. Catherine University
ERF Training on Applied Micro-Econometrics and Public Policy Evaluation
Cairo, Egypt July 25-27, 2016
www.erf.org.eg
1) To understand the underlying structure of Time Series represented by sequence of observations by breaking it down to its components.
2) To fit a mathematical model and proceed to forecast the future.
Binary outcome models are widely used in many real world application. We can used Probit and Logit models to analysis this type of data. Specially, dose response data can be analyze using these two models.
Identification problem in simultaneous equations modelGarimaGupta229
In this presentation, identification problem is explained with the example of Supply and Demand equilibrium and why identification problem arises. In addition, the rank and order conditions are also introduced.
For further explanation, checkout the youtube link:
https://youtu.be/PyU_RJJspfE
Brief notes on heteroscedasticity, very helpful for those who are bigners to econometrics. i thought this course to the students of BS economics, these notes include all the necessary proofs.
1) To understand the underlying structure of Time Series represented by sequence of observations by breaking it down to its components.
2) To fit a mathematical model and proceed to forecast the future.
Binary outcome models are widely used in many real world application. We can used Probit and Logit models to analysis this type of data. Specially, dose response data can be analyze using these two models.
Identification problem in simultaneous equations modelGarimaGupta229
In this presentation, identification problem is explained with the example of Supply and Demand equilibrium and why identification problem arises. In addition, the rank and order conditions are also introduced.
For further explanation, checkout the youtube link:
https://youtu.be/PyU_RJJspfE
Brief notes on heteroscedasticity, very helpful for those who are bigners to econometrics. i thought this course to the students of BS economics, these notes include all the necessary proofs.
Survival analysis is an important method for analysis time to event data for biomedical and reliability applications. It is often done with semiparametric methods e.g. the Cox proportional hazards model. In this presentation I discuss an alternative parametric approach to survival analysis that can overcome some of the limitations of the Cox model and provide additional flexibility to the modeler. This approach may also be justified from a Bayesian perspective and the connection is shown as well. Simulations and case studies that illustrate the flexibility of the GAM approach for survival analysis and its equivalent performance to existing methods for survival data are discussed in the text.
The material presented herein are based on two publications:
1) Argyropoulos C, Unruh ML. Analysis of time to event outcomes in randomized controlled trials by generalized additive models. PLoS One. 2015 Apr 23;10(4):e0123784. doi: 10.1371/journal.pone.0123784. PMID: 25906075; PMCID: PMC4408032.
2)Bologa CG, Pankratz VS, Unruh ML, Roumelioti ME, Shah V, Shaffi SK, Arzhan S, Cook J, Argyropoulos C. High performance implementation of the hierarchical likelihood for generalized linear mixed models: an application to estimate the potassium reference range in massive electronic health records datasets. BMC Med Res Methodol. 2021 Jul 24;21(1):151. doi: 10.1186/s12874-021-01318-6. PMID: 34303362; PMCID: PMC8310602.
Episode 18 : Research Methodology ( Part 8 )
Approach to de-synthesizing data, informational, and/or factual elements to answer research questions
Method of putting together facts and figures
to solve research problem
Systematic process of utilizing data to address research questions
Breaking down research issues through utilizing controlled data and factual information
SAJJAD KHUDHUR ABBAS
Chemical Engineering , Al-Muthanna University, Iraq
Oil & Gas Safety and Health Professional – OSHACADEMY
Trainer of Trainers (TOT) - Canadian Center of Human
Development
Episode 12 : Research Methodology ( Part 2 )
Approach to de-synthesizing data, informational, and/or factual elements to answer research questions
Method of putting together facts and figures
to solve research problem
Systematic process of utilizing data to address research questions
Breaking down research issues through utilizing controlled data and factual information
SAJJAD KHUDHUR ABBAS
Chemical Engineering , Al-Muthanna University, Iraq
Oil & Gas Safety and Health Professional – OSHACADEMY
Trainer of Trainers (TOT) - Canadian Center of Human
Development
FIRE ADMIN UNIT 1 .orct121320#ffffff#fa951a#FFFFFF#e7b3513VERSON.docxAKHIL969626
FIRE ADMIN UNIT 1 .orct121320#ffffff#fa951a#FFFFFF#e7b3513VERSON2.2MAYOR/CITY COUNSELxNO#66b66cCITY MANAGER1zNO#CD6A80FIRE CHIEF2zNO#504DCDOPERATIONS ASSISTANT CHIEF3zNO#FF8C00ADMINISTRATIVE ASSISTANT CHIEF3zNO#8E388ECHIEF OF PREVENTION5zNO#00ae00CHIEF OF TRAINING5zNO#ff6e01CONFIDENTIAL AMINISTRSTIVE ASSISTANT3x8#935c24ADMINISTRATIVE ASSISTANT4x9#388E8EADMINISTRATIVE ASSISTANT5y10#5483a2BATTALION CHIEF (1 PER SHIFT4zNO#B0171FDISTRICT CHIEF (3 PER SHIFT)11zNO#912CEECAPTAIN (18 PER SHIFT)12zNO#0000EELIEUTANENT (18 PER SHIFT)13zNO#00868BDRIVER/OPERATOR (18 PER SHIFT)14zNO#698B22FIREFIGHTER-1 (18 PER SHIFT)15zNO#FFA500RESCUE SPECIALIST II (10 PER SHIFT)12zNO#7171C6RESCUE SPECIALIST I (10 PER SHIFT)17zNO#418cf0SENIOR FIRE INVESTIGATOR6zNO#00BFFFSENIOR FIRE SAFETY EDUCATOR6zNO#4682B4SENIOR FIRE INSPECTOR6zNO#FF8C00FIRE INVESTIGATOR II19zNO#0000EEFIRE INVESTIGATOR I22zNO#6E7B8BFIRE SAFETY EDUCATOR II20zNO#FF6103FIRE SAFETY EDUCATOR I24zNO#FFE4E1FIRE INSPECTOR II21zNO#808000FIRE INSPECTOR I (2)26zNO#9BCD9BSENIOR TRAINING OFFICER7zNO#87CEFATRAINING OFFICER II (2)28zNO#D02090TRAINING OFFICER I (3)29zNO#308014MAINTENANCE SUPERVISOR/MASTER MECHANIC5zNO#9ACD32ADMINISTRATIVE ASSISTANT31y32#418cf0MAINTENANCE TECHNICIAN II31zNO#CD6A80MAINTENANCE TECHNICIAN (2)33zNO#504DCDzNO#FF8C00yNO#8E388ExNO#00ae00zNO#ff6e01xNO#935c24yNO#388E8ExNO#5483a2zNO#B0171FxNO#912CEExNO#00ae00yNO#00868ByNO#698B22xNO#FFA500yNO#7171C6zNO#418cf0xNO#00BFFFyNO#4682B4xNO#FF8C00yNO#0000EExNO#6E7B8BxNO#FF6103zNO#FFE4E1xNO#808000yNO#9BCD9ByNO#87CEFAxNO#D02090xNO#308014yNO#9ACD32zNO#418cf0yNO#CD6A80xNO#504DCDyNO#FF8C00xNO#8E388ExNO#00ae00yNO#ff6e01zNO#935c24xNO#388E8EyNO#5483a2xNO#B0171FxNO#912CEEyNO#00ae00yNO#00868BxNO#698B22zNO#FFA500zNO#7171C6yNO#6E7B8BxNO#00BFFFyNO#FFE4E1zNO#FF8C00yNO#0000EEyNO#6E7B8BxNO#FF6103yNO#FFE4E1zNO#808000yNO#9BCD9BxNO#87CEFAyNO#D02090xNO#308014xNO#9ACD32yNO#418cf0xNO#CD6A80zNO#504DCDzNO#FF8C00yNO#8E388ExNO#00ae00yNO#ff6e01zNO#935c24yNO#388E8EyNO#5483a2xNO#B0171FyNO#912CEEzNO#00ae1eyNO#00868BxNO#698B22yNO#FFA500xNO#7171C6
Business Decision Making Project Part 2
Jared Linscombe
QNT/275
Dr. Davisson
September 12, 2016
Descriptive Statistics
Descriptive statistics are statistics that describe or summarize features of collected data. Descriptive statistics simply present quantitative information in a manner that can be easily managed. The large amount of data is reduced into a simple summary and therefore the whole process of describing the data is less laborious.
For example, finding the mean helps to summarize a lot of individual information into a way that is quickly understood. The samples are likely to produce different independent variables that affect the sales of Elite Technologies Limited. For this reason, we opt to use bivariate analysis in the describing the statistics. Bivariate analysis of the descriptive statistics that is derived from the data will help in drawing relationships between different variables.
For a more accurate representa ...
Explore our comprehensive data analysis project presentation on predicting product ad campaign performance. Learn how data-driven insights can optimize your marketing strategies and enhance campaign effectiveness. Perfect for professionals and students looking to understand the power of data analysis in advertising. for more details visit: https://bostoninstituteofanalytics.org/data-science-and-artificial-intelligence/
Techniques to optimize the pagerank algorithm usually fall in two categories. One is to try reducing the work per iteration, and the other is to try reducing the number of iterations. These goals are often at odds with one another. Skipping computation on vertices which have already converged has the potential to save iteration time. Skipping in-identical vertices, with the same in-links, helps reduce duplicate computations and thus could help reduce iteration time. Road networks often have chains which can be short-circuited before pagerank computation to improve performance. Final ranks of chain nodes can be easily calculated. This could reduce both the iteration time, and the number of iterations. If a graph has no dangling nodes, pagerank of each strongly connected component can be computed in topological order. This could help reduce the iteration time, no. of iterations, and also enable multi-iteration concurrency in pagerank computation. The combination of all of the above methods is the STICD algorithm. [sticd] For dynamic graphs, unchanged components whose ranks are unaffected can be skipped altogether.
StarCompliance is a leading firm specializing in the recovery of stolen cryptocurrency. Our comprehensive services are designed to assist individuals and organizations in navigating the complex process of fraud reporting, investigation, and fund recovery. We combine cutting-edge technology with expert legal support to provide a robust solution for victims of crypto theft.
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Chatty Kathy - UNC Bootcamp Final Project Presentation - Final Version - 5.23...John Andrews
SlideShare Description for "Chatty Kathy - UNC Bootcamp Final Project Presentation"
Title: Chatty Kathy: Enhancing Physical Activity Among Older Adults
Description:
Discover how Chatty Kathy, an innovative project developed at the UNC Bootcamp, aims to tackle the challenge of low physical activity among older adults. Our AI-driven solution uses peer interaction to boost and sustain exercise levels, significantly improving health outcomes. This presentation covers our problem statement, the rationale behind Chatty Kathy, synthetic data and persona creation, model performance metrics, a visual demonstration of the project, and potential future developments. Join us for an insightful Q&A session to explore the potential of this groundbreaking project.
Project Team: Jay Requarth, Jana Avery, John Andrews, Dr. Dick Davis II, Nee Buntoum, Nam Yeongjin & Mat Nicholas
1. Panel Data Analysis October 2011
Introduction to Regression Models for Panel Data Analysis
Indiana University
Workshop in Methods
October 7, 2011
Professor Patricia A. McManus
2. WIM Panel Data Analysis October 2011| Page 1
What are Panel Data?
Panel data are a type of longitudinal data, or data collected at different
points in time. Three main types of longitudinal data:
Time series data. Many observations (large t) on as few as one unit (small
N). Examples: stock price trends, aggregate national statistics.
Pooled cross sections. Two or more independent samples of many units
(large N) drawn from the same population at different time periods:
o General Social Surveys
o US Decennial Census extracts
o Current Population Surveys*
Panel data. Two or more observations (small t) on many units (large N).
o Panel surveys of households and individuals (PSID, NLSY, ANES)
o Data on organizations and firms at different time points
o Aggregated regional data over time
This workshop is a basic introduction to the analysis of panel data. In
particular, I will cover the linear error components model.
3. WIM Panel Data Analysis October 2011| Page 2
Why Analyze Panel Data?
We are interested in describing change over time
o social change, e.g. changing attitudes, behaviors, social relationships
o individual growth or development, e.g. life-course studies, child
development, career trajectories, school achievement
o occurrence (or non-occurrence) of events
We want superior estimates trends in social phenomena
o Panel models can be used to inform policy – e.g. health, obesity
o Multiple observations on each unit can provide superior estimates as
compared to cross-sectional models of association
We want to estimate causal models
o Policy evaluation
o Estimation of treatment effects
4. WIM Panel Data Analysis October 2011| Page 3
What kind of data are required for panel analysis?
Basic panel methods require at least two “waves” of measurement.
Consider student GPAs and job hours during two semesters of college.
One way to organize the panel data is to create a single record for each
combination of unit and time period:
StudentID Semester Female HSGPA GPA JobHrs
17 5 0 2.8 3.0 0
17 6 0 2.8 2.1 20
23 5 1 2.5 2.2 10
23 6 1 2.5 2.5 10
Notice that the data include:
o A time-invariant unique identifier for each unit (StudentID)
o A time-varying outcome (GPA)
o An indicator for time (Semester).
Panel datasets can include other time-varying or time-invariant variables
5. WIM Panel Data Analysis October 2011| Page 4
An alternative way to structure the data is to keep all the measures
related to each student in a single record. This is sometimes called
“wide” format.
StudentID Female HSGPA GPA5 JobHrs5 GPA6 JobHrs6
17 0 2.8 3.0 0 2.1 20
23 1 2.5 2.2 10 2.5 10
o Why are there two variables for GPA and JobHrs ?
o Why is there only one variable for gender and high school GPA?
o Where is the indicator for time?
6. WIM Panel Data Analysis October 2011| Page 5
Estimation Techniques for Panel Models
We can write a simple panel equation predicting GPA from hours worked:
0
it it T it H it J it
GPA TERM HSGPA JOB v
General Linear Model is the foundation of linear panel model estimation
o Ordinary Least Squares (OLS)
o Weighted least squares (WLS)
o Generalized least squares (GLS)
Least-squares estimation of panel models typically entails three steps:
(a) Data transformation or first-stage estimation
(b) Estimation of the parameters using Ordinary Least Squares
(c) Estimation of the variance-covariance matrix of the estimates (VCE)
Parameter estimates are sometimes refined using iteratively reweighted
least squares (IRLS), a maximum likelihood estimator.
7. WIM Panel Data Analysis October 2011| Page 6
Basic Questions for the Panel Analyst
What’s the story you want to tell?
Is this a descriptive analysis? Less worry, fewer controls are usually better.
Is this an attempt at causal analysis using observational data? Careful
specification AND theory is essential.
How does time matter?
Some analyses, e.g. difference-in-difference analysis associates time with
an event (before and after)
Some analyses may be interested in growth trajectories.
Panel analysis may be appropriate even if time is irrelevant. Panel models
using cross-sectional data collected at fixed periods of time generally use
dummy variables for each time period in a two-way specification with
fixed-effects for time.
Are the data up to the demands of the analysis?
Panel analysis is data-intensive. Are two waves enough?
Can you perform the necessary specification tests?
How will you address panel attrition?
8. WIM Panel Data Analysis October 2011| Page 7
Review of the Classical Linear Regression Model
0 1 1 2 2 ...
i i i ki k i
y x x x u , i=1,2,3,…N
Where we assume that the linear model is correct and:
Covariates are Exogenous: 1 2
| , ,.., 0
i i i ki
E u x x x
Uncorrelated errors: , 0
i j
Cov u u
Homoskedastic errors: 2
1 2
| , ,...,
i i i i ki
Var u Var y x x x
If assumptions do not hold, OLS estimates are BIASED and/or INEFFICIENT
Biased - Expected value of parameter estimate is different from true.
o Consistency. If an estimator is unbiased, or if the bias shrinks as the
sample size increases, we say it is CONSISTENT
Inefficient - (Informally) Estimator is less accurate as sample size
increases than an alternative estimator.
o Estimators that take full advantage of information more efficient
9. WIM Panel Data Analysis October 2011| Page 8
OLS Bias Due to Endogeneity
Omitted Variable Bias
o Intervening variables, selectivity
Measurement Error in the Covariates
Simultaneity Bias
o Feedback loops
o Omitted variables
Conventional regression-based strategies to address endogeneity bias
Instrumental Variables estimation
Structural Equations Models
Propensity score estimation
Fixed effects panel models
10. WIM Panel Data Analysis October 2011| Page 9
OLS Inefficiency due to Correlated Errors
Many data structures are susceptible to error correlation:
Hierarchical data sample multiple individuals from each unit, e.g.
household members, employees in firms, multiple pupils from each school.
Multistage probability samples often incorporate cluster-based sampling
designs with errors that may be correlated within clusters.
Repeated observations data often show within-unit error correlation.
Time series data often have errors that are serially correlated, that is,
correlated over time.
Panel data have errors that can be correlated within unit (e.g. individuals),
within period.
Conventional regression-based strategies to address correlated errors
Cluster-consistent covariance matrix estimator to adjust standard errors.
Generalized Least Squares instead of OLS to exploit correlation structure.
11. WIM Panel Data Analysis October 2011| Page 10
Linear Panel Data Model (LPM)
Suppose the data are on each cross-section unit over T time periods:
, 1 , 1 1 , 1
, 2 , 2 2 , 1
, , ,
'
'
:::
'
i t i t t i t
i t i t t i t
i T i T T i T
y u
y u
y u
x
x
x
, t=1,2,…,T
We can express this concisely using i
y to represent the vector of individual
outcomes for person i across all time periods:
i i i
y X u , where '
, 1 , 2
, ,...,
i i t i t iT
y y y
y
For comparison, begin with two conventional OLS linear regression models,
one for each period. Note that the variables female highgpa (HS
GPA) is time-invariant.
12. WIM Panel Data Analysis October 2011| Page 11
OLS Results for each term:
Term 5 GPA Term 6 GPA
Estimate SE t-stat Estimate SE t-stat
Intercept 3.02 0.17 17.8 3.02 0.17 18.3
jobhrs -0.182 0.05 -4.0 -0.174 0.05 -3.6
female 0.108 0.04 2.5 0.145 0.05 3.2
highgpa -0.004 0.04 -0.1 0.003 0.04 0.1
Pooled OLS Results for both terms:
Term 5&6 GPA Term 5&6 GPA (Clustered SE)
Estimate SE t-stat Estimate SE t-stat
Intercept 2.97 0.17 25.1 2.97 0.17 17.2
jobhrs -0.178 0.05 -5.4 -0.178 0.05 -5.8
female 0.125 0.04 4.1 0.125 0.04 3.0
highgpa -0.0001 0.03 -0.01 0.0001 0.03 -0.0004
term6 0.095 0.016 6.1 0.095 0.016 6.1
13. WIM Panel Data Analysis October 2011| Page 12
Linear Unobserved Effects Panel Data Model
Motivation: Unobserved heterogeneity
Suppose we have a model with an unobserved, time-constant variable c:
0 1 1 2 2 ... k k
y x x x c u
Where u is uncorrelated with all explanatory variables in x.
Because c is unobserved it is absorbed into the error term, so
we can write the model as follows:
0 1 1 2 2 ... k k
y x x x v
v c u
The error term v consists of two components, an “idiosyncratic”
component u and an “unobserved heterogeneity” component c .
14. WIM Panel Data Analysis October 2011| Page 13
OLS Estimation of the Error Components Model
If the unobserved heterogeneity i
c is correlated with one or more of the
explanatory variables, OLS parameter estimates are biased and
inconsistent.
If the unobserved heterogeneity c is uncorrelated with the explanatory
variables in i
x , OLS is unbiased even in a single cross-section.
If we have more than one observation on any unit, the errors will be
correlated and OLS estimates will be inefficient
1 1 1
2 2 2
,1 0 1 1 2 2 ,1
,2 0 1 1 2 2 ,2
,1 ,1
,2 ,2
,1 ,2
...
...
, ) 0
i i i
i i i
i k k i
i k k i
i i i
i i i
i i
y x x x v
y x x x v
v c u
v c u
cov(v v
15. WIM Panel Data Analysis October 2011| Page 14
Unobserved Heterogeneity in Panel Data
Suppose the data are on each cross-section unit over T time periods.
This is an unobserved effects model (UEM), also called the error
components model. We can write the model for each time period:
1 1
2 2
i i i i
i i i i
iT iT i iT
y c u
y c u
y c u
1
2
x
x
x
,
Where there are T observations on outcome y for person i,
it
x is a vector of explanatory variables measured at time t,
i
c is unobserved in all periods but constant over time
it
u is a time-varying idiosyncratic error
Define it i it
v c u as the composite error.
16. WIM Panel Data Analysis October 2011| Page 15
Consistent estimation of the Error Components Model with Pooled OLS
If we assume no contemporaneous correlation of the errors and the
explanatory variables, pooled OLS estimation is consistent:
'
( )
it it
E u
x 0 and '
( )
it i
E c
x 0, t=1,2,…,T
Efficient estimation of the Error Components Model with Pooled OLS
Even if estimation is consistent, pooled OLS may not be efficient.
One strategy is to combine pooled OLS with cluster-consistent standard
errors.
Panel GLS methods may be preferred.
In the next sections, we consider the dominant approaches to estimation
of the error components panel model: fixed effects and random effects.
18. WIM Panel Data Analysis October 2011| Page 17
Just a few panel data examples:
Propper and Van Reenen (2010)
Effect of regulation of nursing pay on hospital quality
Data: 209 NHS Hospitals in the UK 1997-2005
Western, Bruce (2002)
Effect of Incarceration on wages and income inequality
Data: NLSY
Cherlin, Chase-Lansdale and McRae (1998)
Effect of parental divorce on mental health over life-course
Data: British Cohort Study
Jacobs and Carmichael (2002)
Determinants of Death Penalty in US states
Data: US Census 1970, 1980, 1990 + other sources
Baum and Lake (2003)
Effect of Democracy on Human Capital and Economic Growth
Data: Aggregate data on 128 countries over 30 years
19. WIM Panel Data Analysis October 2011| Page 18
Fixed Effects Methods for Panel Data
Suppose the unobserved effect i
c is correlated with the covariates.
Example: Motherhood wage penalty
We observe that mothers earn less than other women, cet par.
ˆ 0.08
OLS
KIDS in a log wage model suggests that each additional
child reduces mothers’ hourly wages by about 8%
But if women who are less oriented towards work are also more likely to
have more children, omitting “work orientations” from the model will bias
the coefficient on children.
Fixed-effects methods transform the model to remove i
c
ˆ 0.03
FE
KIDS FE estimates a persistent but much smaller penalty.
20. WIM Panel Data Analysis October 2011| Page 19
Caution: Fixed effects has some disadvantages
FE is not a panacea for all sources of endogeneity bias.
time-varying unobserved effects
time-varying measurement error
simultaneity or feedback loops
All time-constant effects are removed.
No estimation of effects of race, gender, birth order, etc.
Poor estimates if little variation (e.g. education in adulthood)
FE trades consistency for efficiency.
FE uses only within-unit change, ignores between-unit variation.
Parameter estimates may be imprecise, standard errors large.
Despite limitations, FE is an indispensable tool in the panel analyst’s
toolbox.
Fixed Effects Transformation - the “Within” Estimator
21. WIM Panel Data Analysis October 2011| Page 20
Suppose we have the UEM model:
'
it it i it
y c u
x , t=1,2,…,T
For each unit, average this equation over all time periods t:
'
i i i i
y c u
x
Subtract the within-unit average from each observation on that unit:
' '
it i it i i i it i
y y c c u u
x x , t=1,2,…,T
This is the fixed effects transformation. We can write it as:
'
it it it
y u
x ,
where 0
i i
c c and it it i
y y y , it it i
x x x , it it i
u u u
and it
x does not contain an intercept term.
22. WIM Panel Data Analysis October 2011| Page 21
The fixed-effects estimator, also called the within estimator, applies pooled
OLS to the transformed equation:
1 1
' ' ' '
1 1 1 1 1 1
ˆ
N N N T N T
FE i i i i it it it it
i i i t i t
y
X X X y x x x
Recall the student GPA Data:
StudentID Semester Female HSGPA GPA JobHrs
17 5 0 2.8 3.0 0
17 6 0 2.8 2.1 20
23 5 1 2.5 2.2 10
23 6 1 2.5 2.5 10
After applying the fixed-effects transform, the demeaned (mean-centered)
data:
StudentID Semester CFemale CHSGPA CGPA CJobHrs
17 -.5 0 0 .45 -10
17 .5 0 0 -.45 10
23 -.5 0 0 -.15 0
23 .5 0 0 .15 0
23. WIM Panel Data Analysis October 2011| Page 22
Fixed Effects Dummy Variables Regression
Up to now, we’ve treated the unobservables i
c as random variables:
'
it it i it
y c u
x
An alternative approach is to treat i
c as a fixed parameter for each unit. In
this case, we can use dummy variables regression to estimate i
c .
Step one: Create a dummy variable for each of sample unit i
Step two: Substitute the vector of N-1 dummies for i
c :
'
1 2 3
2 3 ...
it it N it
y d d dN u
x ,
(where the intercept 1 estimates the effect when 1
d =1)
Step three: Estimate the equation using pooled OLS.
The fixed effects dummy variables (FEDV) estimator produces precisely
the same coefficient vector and standard errors as the FE estimator.
24. WIM Panel Data Analysis October 2011| Page 23
*** Practical asides
One way or two? Sometimes you will see “one-way” or “two-way” FE.
One-way fixed effects error components model - only the unit effects are
conditioned out.
Two-way fixed effects error components model – both the unit effects and
period effects are conditioned out.
In the conventional FE model with large N and small T, it is a simple
matter to create dummy variables for each period, and most panel models
will include controls for period effects.
Statistical Software
Most statistical packages offer several alternatives for estimating the FEM.
STATA xtreg areg reg (with factor variables)
SAS proc panel proc glm (with “absorb” statement)
25. WIM Panel Data Analysis October 2011| Page 24
First Differencing Methods
The first difference (FD) model transforms the UEM model to remove the
unobserved effects i
c
FD is sometimes called a first-difference fixed effects model
Suppose we have the unobserved effects model (UEM):
'
it it i it
y c u
x ,
For each observation, subtract the previous within-unit observation:
' '
, 1 , 1 , 1
it i t it i t i i it i t
y y c c u u
x x
This is the first-difference transformation. We can write it as:
'
it it it
y u
x ,
where 0
i
c and it
x does not contain an intercept term.
27. WIM Panel Data Analysis October 2011| Page 26
Fixed Effects and First Differences in the Two-Period Case
FE (Within) Transform
StudentID Semester CFemale CHSGPA CGPA CJobHrs
17 -.5 0 0 .45 -10
17 .5 0 0 -.45 10
23 -.5 0 0 -.15 0
23 .5 0 0 .15 0
FD (Differenced) Transform:
StudentID DSemester dFemale dHSGPA dGPA dJobHrs
17 1 0 0 -.9 20
23 1 0 0 .3 0
Compare the transformed (FE) and differenced (FD) data. Each FD variable
is equal to the difference between the second-period FE demeaned
variable and the first-period demeaned variable.
This symmetry will always be present in the two-period panel model.
As a result, the parameter estimates for the two-period panel model can
be obtained using FD or FE, with identical results. Not so if T>2 !
28. WIM Panel Data Analysis October 2011| Page 27
FE and FD Results for two terms:
Term 5&6 GPA (FE, N=400) Term 5&6 GPA (FD, N=200)
Estimate SE Estimate SE________
jobhrs -0.0640159 0.0223835 -0.0640159 0.0223835
term6 0.1133996 0.0125627 0.1133996 0.0125627
FE and FD Results for six terms:
Terms 1-6 GPA (FE, N=1200) Term 1-6 GPA (FD, N=1000)
Estimate SE Estimate SE________
jobhrs -0.1285521 0.0188415 -0.087316 0.0174187
term 0.1037983 0.0040011 0.1066726 0.0091661
29. WIM Panel Data Analysis October 2011| Page 28
Difference-in-Difference Model for Panel Data using FD
Suppose we have a treatment that affects some but not all units in the
population. The “difference-in-differences” estimator is the difference
between the change over time in the treatment group and the change over
time in the control:
,2 ,1 ,2 ,1
B B A A
DID y y y y
If we have panel data from a time period prior to treatment and a second
observation drawn after the treatment event, we can study treatment
effects using 2-period panel data FD and DID:
1 0 1 1 1
2 0 1 2 2
i PD i TREAT i i
i PD i TREAT i i
y PERIOD treatment v
y PERIOD treatment v
, and
Where 1
i
treatment indicates treatment, zero for all at time t=1
PERIOD is a dummy for the time period.
30. WIM Panel Data Analysis October 2011| Page 29
The first difference model for difference-in-difference:
i PD TREAT i i
y treatment v
Where the intercept is replaced by a period effect ( 1
PERIOD for all
units and the change in treatment is either 0 or 1.
Designate A as the control group (i.e. 0
i A
treatment )
Designate B as the treatment group (i.e. 1
i B
treatment )
i PD TREAT i i
y treatment v
Difference in differences estimator:
,2 ,1 ,2 ,1
ˆ ˆ ˆ ˆ
) (
ˆ
B B A A
PD TREAT i B PD TREAT i A
TREAT
DID y y y y
treatment treatment
e.g. Card & Krueger (2000) Minimum Wage increases & Employment
31. WIM Panel Data Analysis October 2011| Page 30
Choosing an Estimator: Fixed Effects vs. First Differences (FE vs FD)
If T=2, (two period model) the FE and FD are identical
If T>2 FE is more efficient than if there is no serial correlation of the
idiosyncratic errors.
If T>2 FD is more efficient if there is serial correlation.
If the unobserved error is not correlated with the covariates, neither the
FE nor the FD model is efficient.
32. WIM Panel Data Analysis October 2011| Page 31
Why not Just Use a Lagged Dependent Variable?
Source: David Johnson. Journal of Marriage and Family, Vol. 67, No. 4 (Nov., 2005), pp. 1061-1075
33. WIM Panel Data Analysis October 2011| Page 32
Random Effects Methods
If we can assume that the unobserved heterogeneity will not bias the
estimates:
Fixed effects methods are inefficient. They throw away information.
Pooled OLS is inefficient because it does not exploit the autocorrelation
in the composite error term.
Random effects methods use feasible GLS estimation (RE FGLS) to
exploit within-cluster correlation
Random effects estimation is more efficient than FE or OLS
The “random effects assumption” of no bias due to i
c is more stringent
1
( | ,..., ) ( ) 0
i i iT i
E c E c
x x
34. WIM Panel Data Analysis October 2011| Page 33
A Conventional FGLS Random Effects Estimator
Assume the errors are correlated within each unit
Assume the errors are uncorrelated across units
Assume the variance in the composite errors is equal to the sum of the
variances in the unobserved effect i
c and the idiosyncratic error i
u :
2 2 2
v u c
RE strategy: If 2 2 2
v u c , find estimators such that 2 2 2
ˆ ˆ ˆ
v u c
35. WIM Panel Data Analysis October 2011| Page 34
Practical Feature of Random Effects Estimation
Recall that the fixed effects “within” estimator essentially transforms
the data by centering each variable on the unit-specific mean.
OLS is then performed on the “fully demeaned” transformed data.
The random effects estimator essentially transforms the data by
“partially demeaning” each variable. Instead of subtracting the entire
unit-specific mean, only part of the mean is subtracted.
The demeaning factor is between 0 and 1, with the specific value
based on the variance components estimation.
Random effects routines are standard in statistical software packages:
SAS: PROC GLM or PROC PANEL
STATA: xtreg
36. WIM Panel Data Analysis October 2011| Page 35
RE Results compared to pooled OLS Results for two terms:
RE Term 5&6 GPA OLS Term 5&6 GPA
Estimate SE z-stat Estimate SE t-stat
Intercept 2.81 0.16 18.0 2.97 0.17 17.2
jobhrs -0.108 0.02 -4.8 -0.178 0.05 -5.8
female 0.126 0.04 3.0 0.125 0.04 3.0
highgpa -0.001 0.03 -0.04 0.0001 0.03 -0.0004
term6 0.096 0.015 5.6 0.095 0.016 6.1
RE Results for six terms:
Terms 1-6 GPA (FE, N=400)
Estimate SE
Intercept 2.41 0.10 23.5
jobhrs -0.129 0.02 -7.0
female 0.086 0.03 2.8
highgpa -0.030 0.02 1.2
term 0.088 0.006 13.6
37. WIM Panel Data Analysis October 2011| Page 36
Random Effects or Fixed Effects - How to decide?
Hausman test for the Exogeneity of the Unobserved Error Component
If the unobserved effects are exogenous, the FE and RE are asymptotically
equivalent. This suggests the null hypothesis for the Hausman test:
0
ˆ ˆ
: RE FE
H ,
where ˆ
RE and ˆ
FE are coefficient vectors for the time-varying explanatory
variables, excluding the time variables.
If the null hypothesis is rejected, we conclude that RE is inconsistent, and the
FE model is preferred.
If the null hypothesis cannot be rejected, random effects is preferred because
it is a more efficient estimator.
38. WIM Panel Data Analysis October 2011| Page 37
Hausman Test in Stata:
. xtreg gpa job sex highgpa,fe
. estimates store fe
. xtreg gpa job sex highgpa,re
. estimates store re
. hausman fe re
---- Coefficients ----
| (b) (B) (b-B) sqrt(diag(V_b-V_B))
| fe re Difference S.E.
-------------+---------------------------------------------------------------
job | -.0748115 -.1232374 .048426 .0088051
-----------------------------------------------------------------------------
b = consistent under Ho and Ha; obtained from xtreg
B = inconsistent under Ha, efficient under Ho; obtained from xtreg
Test: Ho: difference in coefficients not systematic
chi2(1) = (b-B)'[(V_b-V_B)^(-1)](b-B)
= 30.25
Prob>chi2 = 0.0000
We reject the null and conclude the fixed effects estimator is appropriate.
39. WIM Panel Data Analysis October 2011| Page 38
Interpretation of Results from the Error Components Model
Since the UEM model is derived as a levels model, coefficients can be
interpreted much the same as interpretations of a conventional OLS
model, but there are nuances:
For example, suppose we estimate the relationship between marriage and
men’s wages, ˆ 0.05
MARRIED in every model.
Pooled OLS cross-section coefficients contain information about average
differences between units.
[ | ]
it it it i
E y c
x x
This is a population-averaged effect. On average, married men earn 5%
more than men who are not married.
This says nothing about the causal effect of marriage on men’s earnings.
40. WIM Panel Data Analysis October 2011| Page 39
RE/FE/FD estimate average effects within units.
If the unobserved effects are exogenous these are asymptotically
equivalent to the population averaged effect.
[ | , ]
it it i it
E y c
x x
This is sometimes called an average treatment effect. On average,
entering marriage increases men’s earnings by 5%.
RE coefficients represent average change within units, estimated from all
units whether they experience change or not.
FE and FD coefficients represent average changes within units, only for
units that did experience change
This is akin to a treatment effect among the treated. On average, men
who married increased their earnings by 5%.
41. WIM Panel Data Analysis October 2011| Page 40
Best Practices
Theorize the model
What exactly does this unobserved heterogeneity represent?
Why would you expect it to be correlated / uncorrelated with the
regressors?
Specification Testing for Panel Analysis - Interval/Continuous Outcomes
Before ruling out pooled OLS, test for appropriateness of panel methods
vs. pooled ordinary least square.
Optional: Obtain intraclass correlation coefficient (ICC) as indicator of the
extent of within-unit clustering. This is a descriptive statistic, not a test.
Specification tests for strict exogeneity
Test for serial correlation in the idiosyncratic errors
Hausman test for random effects vs. fixed effects
42. WIM Panel Data Analysis October 2011| Page 41
Extensions
FE Models with Time-Invariant Predictors
Interactions between time and covariate
Panel Models for Categorical Outcomes
Fixed effects logit and random effects logit for binary outcomes
Fixed and random effects Poisson models can be used for count outcomes.
Population averaged models can be estimated using General Estimation
Equations (GEE).
Dynamic panel models i.e. lagged dependent variable as a covariate:
0 , 1
it i t GPA it T it H it J it
GPA GPA TERM HSGPA JOB v
GLM models for instrumental variables (IV) estimation
Generalized Method of Moments (GMM) is used for some dynamic panel
models because it allows a flexible specification of the instruments