1. Panel and Time Series Cross Section Models
Sociology 229: Advanced Regression
Copyright © 2010 by Evan Schofer
Do not copy or distribute without permission

2. Announcements
Assignment 5 Due
Assignment 6 handed out

3. Panel Data
Panel data typically refers to a particular type of multilevel data
Measurement at over time (T1, T2, T3…) is nested within persons (or firms or countries)
Each time is referred to as a “wave”
Person 1 T2 T1 T4 T3 T5
Person 2
Person 3
Person 4

4. Panel Data Panel data involves combining:
Information about multiple cases
A “cross-sectional” component
Information about cases over time
A ‘longitudinal’ or ‘time series’ component
Panel datasets are described in terms of:
N, the number of individual cases
T, the number of waves
If N is “large” compared to T, the dataset is called “cross-sectionally dominant”
If T is “large” compared to N, the dataset is called “time-series dominant”
“Time-series Cross-section” / TSCS data: small set of units (usually 10-30), moderate T

5. Panel Terminology Issue: Panel means 2 things:
1. Panel is an umbrella term for all data with time-series and cross-section components
2. Panel refers specifically to datasets with large N, small T (cross-sectionally dominant)
Example: a survey of 1000 people at 3 points in time
Another distinction: Balanced vs unbalanced
Panel data are said to be “balanced” if information on each person is available for all T
If data is missing for some cases at some points in time, the data are “unbalanced”
This is common for many datasets on countries or firms.

6. Panel Data Example ID Wave Grade Math Test Family income Class Size
Gender 1 5 67
80K 18 2 7 78
82K 22 3 9 85
88K 26 34
27K 41
23K 33

7. Benefits of Panel Data
1. Pooling (either cases or across time) provides richer information
More observations = better
2. Panel data is longitudinal
You can follow individual cases over time
Allows us to study dynamic processes
Baltagi: “Dynamics of adjustment”
Provides opportunities to better tease out causal relationships
3. Panel models allow us to control for individual heterogeneity.

8. Benefits of Panel Data
4. Panel data allows investigation of issues that are obscured in cross-sectional data
Example: women’s participation in the labor force
Suppose a cross-sectional dataset shows that 50% of women are in the labor force
What’s going on?
Are 50% of women pursuing work and 50% staying at home?
Are 100% of women seeking employment, but many experiencing unemployment at any given time?
Or something in between?
Without longitudinal information, we can’t develop a clear picture.

9. Problems with Panel Data
1. Violates OLS independence assumption
Clustering by cases
Clustering by time
Other sources? Ex: spatial correlation
2. For small N large T (TSCS): Poolability
Appropriate to combine very different cases?
3. Serial correlation
Temporally adjacent cases may have correlated error
4. Non-stationarity (for “larger T” data)
5. Other issues: heteroskedasticity, etc…

10. Panel Data Strategies Traditional strategy: Fixed vs. random effects
Use Hausman test to choose
More recent strategies:
1. Distinction between tools for “panel” vs “TSCS” data
Different problems, different solutions
2. Many more options
New solutions to existing problems
3. More attention to “dynamic” models
Models that include lagged values (lags of Y)
Issue: overall, not much consensus about what constitutes “best practices”.

11. The Econometric Tradition
Main focus of econometric studies:
Large N, small T panels
Concern with “the omitted variables problem”
Unobserved heterogeneity
“Individual” heterogeneity
Unobserved effects
Example: Individual wages
Often modeled as a function of education, experience...
But, individuals differ in ways that are difficult to control
Strategy: Use a model that “gets rid of” individual variability
Such as “fixed effects”…

12. The Econometric Tradition
Basic “static” panel model w/ unobserved effects:
Subscripts i, t refer to cases & time periods
bX = covariates
mi = unobserved unit-specific error
nit = idiosyncratic error
Look familiar? Basically the same as the random & fixed effects models discussed previously…
Other texts use a or u or zeta instead of “mu”
Other texts use e for error, instead of “nu”
Issue: whether to treat mi as “fixed” versus “random”
But, there are other choices, as well…

13. Unobserved Effects Basic “static” panel model w/ unobserved effects:
Issue: mi is the problem
The “unobserved” time-invariant features of an individual that may cause their wages to be especially high or low across time
Ex: Too much TV as a child; or lots of parental pressure to make $
Or genes or IQ or whatever…
Common strategies:
1. Wipe it out: first differences
2. Build it into the model: fixed effects
Or random effects, if we think mi is not correlated with Xs
3. Control for it in some other way – such as with a lagged DV
Which already might reflect the impact of mi.

14. Unobserved Effects: First Differences
First differences (2 waves of data):
Which can be expressed generally as:
Result: mi goes away!
Differencing allows us to estimate coefficients, purged of unit-specific (time-invariant) unobserved effects

15. Unobserved Effects: Fixed Effects
Fixed Effects – start with the same basic model:
A second approach: the “within” transformation
Center everything around unit-specific mean (‘time demeaning’)
Which wipes out mu:
Equivalent to putting in dummy variables for every case
Estimating m as a “fixed” effect
In stata: xtreg wage educ, fe

16. Fixed Effects vs. First Differencing
FE and FD are very similar approaches
In fact, for 2 wave datasets, results are identical
For 3+ waves, results can differ
Which is better for large N, small T?
FE is more efficient if no serial correlation
FD is better if lots of serial correlation
Ex: concerns about nonstationarity
Which is better for small N, larger T (TSCS)?
More likely issues with nonstationarity & spurious regression problem… FE = problematic, FD = better
But, FD can turn a I(1) process into weakly dependent. Fixed!

17. Fixed Effects vs. First Differencing
General remarks (Wooldridge 2009):
Both FE and FD are biased if variables aren’t strictly exogenous
X variables should be uncorrelated with present, past, & future error…
Including inclusion of lagged dependent variable
BUT: bias in FE declines with large T
Bias in FD does not…
Wooldridge 2009, Ch 14 (p. 587): “Generally, it is difficult to choose between FE and FD when they give substantially different results. It makes sense to report both sets of results and try to determine why they differ.”

18. Fixed Effects vs. First Differencing
More remarks
1. FE and FD cannot estimate effects of variables that do not change over time
And can have problems if variables change rarely…
2. Both FE and FD are not as efficient as models that include “between case” variability
A trade off between efficiency and potential bias (if unobserved effects are correlated with Xs)
3. Both FE and FD are very sensitive to measurement error
Within-case variability is often small… may be swamped by error/noise.

19. “Two-way” Error Components
Unobserved effects may occur over time
Example: common effect of year (e.g., a recession) on wages
A “crossed” multilevel model
What Baltagi (2008) calls 2-way error components
Versus basic FE/RE, which has one eror component
Strategies:
Use dummies for time in combination with FE/FD
Specify a “crossed” multilevel model in Stata
See Rabe-Hesketh and Skrondal
Basically, create a 3-level model, nesting groups underneath.

20. Random Effects
Random effects: Additional efficiency at the cost of additional assumptions
Key assumption: unit-specific unobserved effect is not correlated with X variables
If assumptions aren’t met, results are biased
Omitted X variables often induce correlation between other X variables and the unobserved effect
If main purpose of panel analysis is to avoid unobserved effects, fixed effects is a safer choice
In stata:
xtreg wage educ, re – for GLS estimator
xtreg wage educ, mle – for ML estimator

21. Random Effects The random effects model
Then, instead of the “within” transformation, we take out part of within-case variation:
GLS Estimator: “Quasi time-demeaned data”
Where:

22. Random Effects
Random effects model is a hybrid: An intermediate model between OLS and FE
If T is large, it becomes more like FE
If the unobserved effect has small variance (isn’t important), RE becomes more like OLS
If unobserved effect is big (cases hugely differ), results will be more like FE
GLS random effects estimator is for large N
Its properties for small N large T aren’t well studied
Beck and Katz advise against it
And, if you must use random effects, they recommend ML estimator

23. Random Effects When to use random effects?
1. If you are confident that unit unobserved effect isn’t correlated with Xs
Either for theoretical reasons…
Or, because you have lots of good controls
Ex: lots of regional dummies for countries…
2. Your main focus is time-constant variables
FE isn’t an option. Hopefully #1 applies.
3. When your focus is between case variability
And/or there is hardly any within-case variability (#1!)
4. When a Hausman test indicates that they yield similar results

24. Hausman Specification Test
Hausman Specification Test: A tool to help evaluate fit of fixed vs. random effects
Logic: Both fixed & random effects models are consistent if models are properly specified
However, some model violations cause random effects models to be inconsistent
Ex: if X variables are correlated to random error
In short: Models should give the same results… If not, random effects may be biased
If results are similar, use the most efficient model: random effects
If results diverge, odds are that the random effects model is biased. In that case use fixed effects…

25. Hausman Specification Test
Strategy: Estimate both fixed & random effects models
Save the estimates each time
Finally invoke Hausman test
Ex:
xtreg var1 var2 var3, i(groupid) fe
estimates store fixed
xtreg var1 var2 var3, i(groupid) re
estimates store random
hausman fixed random

26. Hausman Specification Test
Example: Environmental attitudes fe vs re
. hausman fixed random
---- Coefficients ----
| (b) (B) (b-B) sqrt(diag(V_b-V_B))
| fixed random Difference S.E.
age |
male |
dmar |
demp |
educ |
incomerel |
ses |
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(7) = (b-B)'[(V_b-V_B)^(-1)](b-B) =
Prob>chi2 =
Direct comparison of coefficients…
Non-significant p-value indicates that models yield similar results…

27. Hausman Specification Test
Issues with Hausman tests (Wooldridge 2009)
1. Fail to reject means either:
FE and RE are similar… you’re good!
FE estimates are VERY imprecise
Large differences from RE are nevertheless insignificant
That can happen if the data are awful/noisy. Watch out!
2. Watch for difference between “statistical significance” and “practical significance”
With a huge sample, the Hausman test may “fail” even though RE is nearly the same as FE
If differences are tiny, you could argue that RE is appropriate.

28. Allison’s Hybrid Approach
Allison (2009) suggests a ‘hybrid’ approach that provides benefits of FE and RE
Also discussed in Gelman & Hill
Builds on idea of decomposing X vars into mean, deviation
1. Compute case-specific mean variables
egen meanvar1 = mean(var1), by(groupid)
2. Transform X variables into deviations
Subtract case-specific mean
egen withinvar1 = var1 – meanvar1
3. Do not transform the dependent variable Y
4. Include both X deviation & X mean variables
5. Estimate with a RE model

29. Allison’s Hybrid Approach
Benefits of hybrid approach:
1. Effects of “X-deviation” variables are equivalent to results from a fixed effects model
All time-constant factors are controlled
2. Allows inclusion of time-constant X variables
3. You can build a general multilevel model
Random slope coefficients; more than 2 level models…
4. You can directly test FE vs RE
No Hausman test needed
X-mean and X-deviation coefficients should be equal
Conduct a Wald test for equality of coefficients
Also differing X-mean & X-deviation coefs are informative.

30. SEM Approaches Both FE and RE can be estimated using SEM Benefits:
See Allison 2009, Bollen & Brand forthcoming (SF)
Software: LISREL, EQS, AMOS, Mplus
Some limited models in Stata using GLLAMM
Benefits:
FE and RE are nested… can directly compare with fit statistics (BIC, IFI, RMSEA) – HUGE advantage
Allows tremendous flexibility
Lagged DVs
Can relax assumptions that covariate effects or variances are constant across waves of data
Flexibility in correlation between unobserved effect and Xs
And more!

31. SEM Approaches RE in SEM (from Allison 2009)
Unobserved effect correlated with Y across waves
(Assumes no correlation with X)
RE in SEM (from Allison 2009)

32. SEM Approaches FE in SEM (from Allison 2009)
Unobserved effect correlated with Y AND X at all waves

33. Serial Correlation
Issue: What about correlated error in nearby waves of data?
So far we’ve focused on correlated error due to unobserved effect (within each case)
Strategies:
Use a model that accounts for serial correlation
STATA: xtregar – FE and RE with “AR(1) disturbance”
Many other options… xtgee, etc.
Develop a “dynamic model”
Actually model the patterns of correlation across Y
Include the lagged dependent variable (Y)… or many lags!
This causes bias in FE, RE – so other models needed.

34. Dynamic Panel Models What if we have a dynamic process?
Examples from Baltagi 2008:
Cigarette consumption (in US states) – lots of inertia
Democracy (in countries)
We might consider a model like:
Y from the prior period is included as an independent variable
Issue: FE, RE estimators are biased
Time-demeaned (or quasi-demeaned) lag y correlated with error
FE is biased for small T. Gets better as T gets bigger (like 30)
RE also biased.

35. Dynamic Panel Models One solution: Use FD and instrumental variables
Strategy: If there’s a problem between error and lag Y, let’s find a way to calculate a NEW version of lag y that doesn’t pose a problem
Idea: Further lags of Y aren’t an issue in a FD model.
Use them as “instrumental variables” as a proxy for lag Y
Arellano Bond GMM estimator
A FD estimator
Lag of levels as an instrument for differenced Y
Arellano-Bover/Blundell-Bond “System GMM” estimator
Expand on this by using lags of differences and levels as instruments
Generalized Method of Moments (GMM) estimation.

36. Dynamic Panel Models Stata: xtabond – basic Arellano-Bond GMM model
xtdpdsys – System GMM estimator
xtdpd – A flexible command to build system GMM models
Lots of control over lag structure
Can run non-dynamic models
Key assumptions / issues
Serial correlation of differenced errors limited to 1 lag
No overidentifying restrictions (“Sargan test”)
How many instruments?
Criticisms:
Angrist & Pichke 2009: assumptions not always plausible
Allison 2009
Bollen and Brand, forth: Hard to compare models.

37. Dynamic Panel Models General remarks:
It is important to think carefully about dynamic processes…
How long does it take things to unfold?
What lags does it make sense to include?
With huge datasets, we can just throw lots in
With smaller datasets, it is important to think things through.

38. Methods: IV panel models
Traditional instrumental variable panel estimator:
bX = exogenous covariates
gZ = endogenous covariates (may be related to nit)
mi = unobserved unit-specific error
nit = idiosyncratic error
Treat mi as random, fixed, or use differencing to wipe it out
Use contemporaneous or lagged X and (appropriate) lags of Z as instruments in two-stage estimation of yit.
Works if lag Z is plausibly exogenous.

39. TSCS Data Time Series Cross Section Data
Example: economic variables for industrialized countries
Often countries
Often ~30-40 years of data
Beck (2001)
No specific minimum, but be suspicious of T<10
Large N isn’t required (though not harmful)

40. TSCS Data: OLS PCSE Beck & Katz 2001
“Old” view: Use FGLS to deal with heteroskedasticity & correlated errors
Problem: This underestimates standard errors
New view: Use OLS regression
With “panel corrected” standard errors
To address panel heteroskedasticity
With FE to deal with unit heterogeneity
With lagged dependent variable in the model
To address serial correlation.

41. TSCS Data: Dynamics Beck & Katz 2009 examine dynamic models
OLS PCSE with lagged Y and FE
Still appropriate
Better than some IV estimators
But, didn’t compare to System GMM.
Plumper, Troeger, Manow (2005)
FE isn’t theoretically justified and absorbs theoretically important variance
Lagged Y absorbs theoretically important temporal variation
Theory must guide model choices…

42. TSCS Data: Nonstationary Data
Issue: Analysis of longitudinal (time-series) data is going through big changes
Realization that strongly trending data cause problems
Random walk / unit root processes / integrated of order 1 / non-stationary data
Converse: stationary data, integrated of order zero
The “spurious regression” problem
Strategies:
Tests for “unit root” in time series & panel data
Differencing as a solution
A reason to try FD models.

43. Panel Data Remarks 1. Panel data strategies are taught as “fixes”
How do I “fix” unobserved effects?
How do I “fix” dynamics/serial correlation?
But, the fixes really change what you are modeling
A FE (within) model is a very different look at your data, compared to OLS
Goal: learn the “fixes”… but get past that… start to think about interpretation
2. Much strife in literature
People arguing over what “fix” is best
Don’t be afraid of criticism… but expect that people will weigh in with different views

44. Panel Data Remarks 3. MOST IMPORTANT THING: Try a wide range of models
If your findings are robust, you’re golden
If not, differences will help you figure out what is going on…
Either way, you don’t get “surprised” when your results go away after following the suggestion of a reviewer!

45. Reading Discussion
Schofer, Evan and Wesley Longhofer. “The Structural Sources of Associational Life.” Working Paper.