Methodological Workshop 3:
Fixed Effects Models and
Multi-Level Models
Yu Xie
University of Michigan
What’s Common?
 Both
the fixed effects model and the multilevel model utilize clustered data.
 Both the fixed effects model and the multilevel model are designed to handle crosscontext heterogeneity.
Different Objectives
 Fixed
effects model and multi-level model
are very different research designs:


Fixed effects model controls for (or absorbs)
pre-treatment heterogeneity (type I
heterogeneity)
Multi-level model models both forms of
heterogeneity across contexts.
Application of Different Principles
 The
fixed effects model is essentially an
application of the social grouping principle
(with a group being a cluster)
 The multi-level model is essentially an
application of the social context principle.
Using Different Assumptions
 The
fixed effects model assumes no type II
heterogeneity bias (often constant effects
model), or additive effects of heterogeneity
across contexts (i.e., clusters).
 The multi-level model relaxes homogeneity
assumption at the individual level but
assumes that both forms of heterogeneity
are at the context level and can be modeled
adequately with contextual covariates.
A General Lesson: Tradeoff between Data
and Assumption
 “When
observed data are thin, it takes
strong assumptions to yield sharp results.
There is no free information in statistics.
Either you collect it, or you assume it.”
(Xie 1996, AJS).
Fixed effects model

Sibling model as an example

Family SES, environment are shared
• Yi1 = b0 + b1Xi1 + ai + ei1
• Yi2 = b0 + b1Xi2 + ai + ei2


a and X may be correlated.
Take difference between the two eq.
• Yi2 - Yi1= b1 (Xi2 - Xi1) + (ei2 - ei1)
• Resulting in a more robust equation

Properties of the fixed effects approach:
• All fixed-characteristics are controlled
• It consumes a lot of information
• Unobserved heterogeneity (Type I) is controlled for at the
group level (fixed effects)
Example: Critique of Zhou and Hou (1999):
Positive Benefits of Send-Down?

“More interestingly, our findings also reveal some
positive consequences of the send-down experience.
For instance, when compared with urban youth, a
noticeably higher proportion of the send-down youth
attained a college education after 1977. Partly as a
result of their educational attainment, these sent-down
youth, especially those with shorter rural durations, were
equally likely to enter favorable employment (type of
occupation and work organizations) in the urban labor
force, despite their relatively short urban labor force
experience.” (Zhou and Hou 1999: 32)
Speculated Reason for the
Beneficial Effects
 The
unusual hardship faced by sent-down
youth forced them to be more adaptive
and thus acquire skills to survive.
In Our Recent Study (Xie, Yang,
and Greenman 2008)
 We
analyze data from the survey of Family
Life in Urban China that we conducted in
three large cities (Shanghai, Wuhan, and
Xi’an) in 1999.
 We use some items designed for this study.
Statistical Analyses

(1) We present the differences in six
socioeconomic indicators between respondents
who experienced send-down with those who did
not experience send-down.
 (2) We present results from a fixed-effects model
capitalizing on the sibling structure in our data.
 (3) We examine educational attainment closely
as a time-varying covariate and its endogenous
role in affecting early returns of sent-down youth.
Table 1: Descriptive Differences between Respondents with Send-Down
Experience and Respondents without Send-Down Experience
Not
Sent Down
Sent down
Sent Down
Duration <6
10.9
11.9
15.2
*
11
10.8
11.3
**
Annual Salary (yuan)
5,318
4,983
4,567
Total Annual Income (yuan)
8,468
8,680
5.3
SEI
N
College Education (%)
Years of Schooling
Cadre (%)
Notes: *p<.1, **p<.05, ***p<.01
Sent Down
Duration
6+
3
***
9.4
***
6,083
***
7,976
10,542
***
6.3
6.6
5.3
42.5
42
42.5
40.6
651
481
349
132
***
After We Control for Covariates (Table 2)
 There
are no differences in salary or
income.
 Short-term sent-down youth still have
higher levels of education than the other
two groups (non-sent-down and long-term
sent-down).
Potential Sources of Bias
 Some
sent-down youth did not return to
cities or did not return to the same cities.
 There can be unobserved family-level
characteristics associated with both senddown and outcomes.
 We use a fixed effects model based on
sibling pairs to address both problems.
Table 3 : Unadjusted Differences by Send-Down Experience
Using Sibling Pairs
Not Sent
down
Sent Down
d
College Education (%)
11.4
11.7
-0.3
Years of Schooling
10.9
10.8
0.1
8.9
5.4
3.5
SEI
43.7
44.5
-0.7
N
344
344
Cadre (%)
Notes: *p<.1, **p<.05, ***p<.01
What’s Going On?
 If
there are no effects of send-down (from
the fixed effects model), why do we
observe differences in education between
short-term sent-down youth and long-term
sent-down youth?
 The answer largely lies in “pre-treatment”
differences.
Table 4: Unadjusted Differences by Duration
Duration <6 Duration > 6
53
13.6
***
Years of Schooling at Send-Down
10.5
9.2
***
Years of Schooling at Return
10.7
9.3
***
College Enrollment in Year of Return (%)
13.2
1.5
***
College Education (%)
15.2
3
***
Truncated Sample
11.9
2.3
***
11.3
9.5
***
11.1
9.4
***
HS Graduate at Send Down (%)
Current Years of Schooling
Truncated Sample
N
Notes: *p<.1, **p<.05, ***p<.01
349
132
Conclusion
 Did
send-down experience benefit youth?
-- No.
 Our analyses of the new data show that
the send-down experience did not benefit
the youth who were affected.
 Differences in social outcomes between
those who experienced send-down and
those who did not are either non-existent
or spurious due to other social processes.
Accounting for Heterogeneous Responses with
Social Context Principle
 Possible
with nested data, assuming that
patterns of relationships are homogeneous
(or following a distribution) within social
contexts (by time or space).
 dk is allowed to vary across k (k=1,…K),
social context, but is homogeneous within k,
conditional on X.
Multi-level Model (MLM)




Yik = ak + dkDik + b’Xik + eik
ak = l+fzk+mk
dk = g+szk+nk
Other names: hierarchical linear models, randomcoefficient models, growth-curve models, and mixed
models.
Units of analysis at a lower level are nested within higherlevel units of analysis
Examples:


Students within schools
Observations over time within persons (growth curve)
Problems without MLM





If we ignore higher-level units of analysis => we
cannot account for context (individualistic approach)
If we ignore individual-level observation and rely on
higher-level units of analysis, we may commit
ecological fallacy (aggregated data approach)
Without explicit modeling, sampling errors at
second level may be large =>unreliable slopes
Homoscedasticity and no serial correlation
assumptions of OLS are violated (an efficiency
problem).
No distinction between parameter variability and
sampling variability.
Advantages of MLM
 Cross-level
comparisons
 Controls for differences across higher
levels
Example: Xie and Hannum (1996)
T  log Y  b 0 + b1 X 1 + b 2 X 2 + b 3 X 22 + b 4 X 4 + b 5 X 5 + b 6 X 1 X 5 + 


(1)
Where

Y = earnings,

X1 = years of schooling,

X2 = years of work experience,

X4 = a dummy variable denoting membership in the Communist
Party of China (1 = party member),

X5 a dummy variable denoting gender (1 = female).
Note two interactions.
Consider regional heterogeneity

For the ith person in kth city:
log ( yik   b 0 k + b1k x1ik + b 2 k x2ik + b 3 x22ik + b 4 k x4ik + b 5 k x5ik + b 6 x1ik x5ik + ik .


Instead of using fixed effects for the intercept b0k, and full
interactions for slope parameters, Xie and Hannum
modeled these parameters in a multilevel model.
Let z be a city-level covariate that measures the degree
of economic reform. Let us assume that individual-level
parameters depend on z in the following linear
regressions:
Cross-City Model (“meta analysis”)
b0k  a 0 + l0 zk + m0k
b1k  a1 + l1 zk + m1k
b 2 k  a 2 + l2 zk + m2 k
b3  a 3
b 4 k  a 4 + l4 zk + m4 k
b5k  a5 + l5 zk + m5k
b6  a 6
Combining the two levels =>
log ( yik   a 0 + a1 x1ik + a 2 x2ik + a 3 x22ik + a 4 x4ik + a 5 x5ik + a 6 x1ik x5ik
+ l0 zk + l1 x1ik z k +l2 x2ik zk + l4 x4ik zk + l5 x5ik zk
+ (m0 k + m1k x1ik + m2 k x2ik + m4 k x4ik + m5k x5ik + ik 
We can see that the city-level covariate z interacts
with most of the individual-level predictors.
Special Cases
 Special
case 1: If all the coefficients of the
city-level covariate (z) are zero, we have
what is called “random coefficient model”
 Special case 2: If all the coefficients of the
city-level covariate (z) are zero and there
are no random coefficients in all slope
coefficients (except the intercept), we have
what is called “variance component model”.
[See Table 3.]
Summary: Four ways to conceptualize
variability in parameters
Specification Complete
Random
homogeneity variation
Regression
Fixed
Degree of
Freedom
1
1+Pk
K
Parsimony
(DF for
Model)
High
Low
Accuracy
(like R2)
Low
High
2
where Pk is the number of predictors at the 2nd level, and K is the number of
units at the second level.
References



Xie, Yu. 1996. “Review of Identification Problems in the
Social Sciences by Charles Manski.” American Journal
of Sociology 101:1131-1133.
Xie, Yu and Emily Hannum. 1996. “Regional Variation
in Earnings Inequality in Reform-Era Urban China.”
American Journal of Sociology 101:950-992.
Xie, Yu, Yang Jiang, and Emily, Greenman. 2008. “Did
Send-Down Experience Benefit Youth? A Reevaluation
of the Social Consequences of Forced Urban-Rural
Migration during China’s Cultural Revolution.” Social
Science Research 37: 686-700.