Methodological Workshop 3:Methodological Workshop 3:

Fixed Effects Models and Multi-Fixed Effects Models and Multi-

Level ModelsLevel Models

Yu XieYu XieUniversity of MichiganUniversity of Michigan

What’s Common? What’s Common?

Both the fixed effects model and the multi-Both the fixed effects model and the multi-level model utilize clustered data.level model utilize clustered data.

Both the fixed effects model and the multi-Both the fixed effects model and the multi-level model are designed to handle cross-level model are designed to handle cross-context heterogeneity. context heterogeneity.

Different Objectives Different Objectives

Fixed effects model and multi-level model Fixed effects model and multi-level model are very different research designs:are very different research designs: Fixed effects model controls for (or absorbs) Fixed effects model controls for (or absorbs)

pre-treatment heterogeneity (type I pre-treatment heterogeneity (type I heterogeneity)heterogeneity)

Multi-level model models both forms of Multi-level model models both forms of heterogeneity across contexts. heterogeneity across contexts.

Application of Different PrinciplesApplication of Different Principles

The fixed effects model is essentially an The fixed effects model is essentially an application of the social grouping principle application of the social grouping principle (with a group being a cluster)(with a group being a cluster)

The multi-level model is essentially an The multi-level model is essentially an application of the social context principle. application of the social context principle.

Using Different AssumptionsUsing Different Assumptions

The fixed effects model assumes no type II The fixed effects model assumes no type II heterogeneity bias (often constant effects heterogeneity bias (often constant effects model), or additive effects of heterogeneity model), or additive effects of heterogeneity across contexts (i.e., clusters).across contexts (i.e., clusters).

The multi-level model relaxes homogeneity The multi-level model relaxes homogeneity assumption at the individual level but assumption at the individual level but assumes that both forms of heterogeneity assumes that both forms of heterogeneity are at the context level and can be modeled are at the context level and can be modeled adequately with contextual covariates. adequately with contextual covariates.

A General Lesson: Tradeoff between Data A General Lesson: Tradeoff between Data and Assumptionand Assumption

““When observed data are thin, it takes When observed data are thin, it takes strong assumptions to yield sharp results. strong assumptions to yield sharp results. There is no free information in statistics. There is no free information in statistics. Either you collect it, or you assume it.” Either you collect it, or you assume it.” (Xie 1996, (Xie 1996, AJSAJS).).

Fixed effects modelFixed effects model Sibling model as an exampleSibling model as an example

Family SES, environment are sharedFamily SES, environment are shared• YYi1 i1 == XXi1 i1 iii1i1

• YYi2 i2 == XXi2 i2 iii2i2

andandXX may be correlated. may be correlated. Take difference between the two eq.Take difference between the two eq.

• YYi2i2 - - YYi1i1== XXi2 i2 -- XXi1i1)) i2i2--i1i1))

• Resulting in a more robust equationResulting in a more robust equation Properties of the fixed effects approach:Properties of the fixed effects approach:

• All fixed-characteristics are controlledAll fixed-characteristics are controlled• It consumes a lot of informationIt consumes a lot of information• Unobserved heterogeneity (Type I) is controlled for at the Unobserved heterogeneity (Type I) is controlled for at the

group level (fixed effects)group level (fixed effects)

Example: Critique of Zhou and Hou (1999): Example: Critique of Zhou and Hou (1999): Positive Benefits of Send-Down?Positive Benefits of Send-Down?

““More interestingly, our findings also reveal some More interestingly, our findings also reveal some positive consequences of the send-down experience. positive consequences of the send-down experience. For instance, when compared with urban youth, a For instance, when compared with urban youth, a noticeably higher proportion of the send-down youth noticeably higher proportion of the send-down youth attained a college education after 1977. Partly as a attained a college education after 1977. Partly as a result of their educational attainment, these sent-down result of their educational attainment, these sent-down youth, especially those with shorter rural durations, were youth, especially those with shorter rural durations, were equally likely to enter favorable employment (type of equally likely to enter favorable employment (type of occupation and work organizations) in the urban labor occupation and work organizations) in the urban labor force, despite their relatively short force, despite their relatively short urbanurban labor force labor force experience.” (Zhou and Hou 1999: 32)experience.” (Zhou and Hou 1999: 32)

Speculated Reason for the Speculated Reason for the Beneficial EffectsBeneficial Effects

The unusual hardship faced by sent-down The unusual hardship faced by sent-down youth forced them to be more adaptive youth forced them to be more adaptive and thus acquire skills to survive. and thus acquire skills to survive.

In Our Recent Study (Xie, Yang, In Our Recent Study (Xie, Yang, and Greenman 2008)and Greenman 2008)

We analyze data from the survey of Family We analyze data from the survey of Family Life in Urban China that we conducted in Life in Urban China that we conducted in three large cities (Shanghai, Wuhan, and three large cities (Shanghai, Wuhan, and Xi’an) in 1999. Xi’an) in 1999.

We use some items designed for this We use some items designed for this study. study.

Statistical AnalysesStatistical Analyses

(1) We present the differences in six (1) We present the differences in six socioeconomic indicators between respondents socioeconomic indicators between respondents who experienced send-down with those who did who experienced send-down with those who did not experience send-down. not experience send-down.

(2) We present results from a fixed-effects model (2) We present results from a fixed-effects model capitalizing on the sibling structure in our data. capitalizing on the sibling structure in our data.

(3) We examine educational attainment closely (3) We examine educational attainment closely as a time-varying covariate and its endogenous as a time-varying covariate and its endogenous role in affecting early returns of sent-down youth. role in affecting early returns of sent-down youth.

Table 1: Descriptive Differences between Respondents with Send-Down Table 1: Descriptive Differences between Respondents with Send-Down Experience and Respondents without Send-Down ExperienceExperience and Respondents without Send-Down Experience

Not

Sent Down

Sent Down

Sent Down Duration

6+Sent down Duration <6

College Education (%) 10.9 11.9 15.2 * 3 ***

Years of Schooling 11 10.8 11.3 ** 9.4 ***

Annual Salary (yuan) 5,318 4,983 4,567 *** 6,083 ***

Total Annual Income (yuan) 8,468 8,680 7,976 10,542 ***

Cadre (%) 5.3 6.3 6.6 5.3

SEI 42.5 42 42.5 40.6

N 651 481   349   132  

Notes: *p<.1, **p<.05, ***p<.01

After We Control for Covariates (Table 2)After We Control for Covariates (Table 2)

There are no differences in salary or There are no differences in salary or income.income.

Short-term sent-down youth still have Short-term sent-down youth still have higher levels of education than the other higher levels of education than the other two groups (non-sent-down and long-term two groups (non-sent-down and long-term sent-down).sent-down).

Potential Sources of BiasPotential Sources of Bias

Some sent-down youth did not return to Some sent-down youth did not return to cities or did not return to the same cities.cities or did not return to the same cities.

There can be unobserved family-level There can be unobserved family-level characteristics associated with both send-characteristics associated with both send-down and outcomes. down and outcomes.

We use a fixed effects model based on We use a fixed effects model based on sibling pairs to address both problems. sibling pairs to address both problems.

Table 3 : Unadjusted Differences by Send-Down Experience Table 3 : Unadjusted Differences by Send-Down Experience Using Sibling PairsUsing Sibling Pairs

 Not Sent

down Sent Down

College Education (%) 11.4 11.7 -0.3

Years of Schooling 10.9 10.8 0.1

Cadre (%) 8.9 5.4 3.5

SEI 43.7 44.5 -0.7

N 344 344  

Notes: *p<.1, **p<.05, ***p<.01

What’s Going On?What’s Going On?

If there are no effects of send-down (from If there are no effects of send-down (from the fixed effects model), why do we the fixed effects model), why do we observe differences in education between observe differences in education between short-term sent-down youth and long-term short-term sent-down youth and long-term sent-down youth? sent-down youth?

The answer largely lies in “pre-treatment” The answer largely lies in “pre-treatment” differences. differences.

Table 4: Unadjusted Differences by DurationTable 4: Unadjusted Differences by Duration

HS Graduate at Send Down (%) 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 ***

Current Years of Schooling 11.3 9.5 ***

Truncated Sample 11.1 9.4 ***

N 349 132

Duration <6 Duration > 6

Notes: *p<.1, **p<.05, ***p<.01

ConclusionConclusion

Did send-down experience benefit youth? Did send-down experience benefit youth? -- No. -- No.

Our analyses of the new data show that Our analyses of the new data show that the send-down experience did not benefit the send-down experience did not benefit the youth who were affected. the youth who were affected.

Differences in social outcomes between Differences in social outcomes between those who experienced send-down and those who experienced send-down and those who did not are either non-existent those who did not are either non-existent or spurious due to other social processes. or spurious due to other social processes.

Accounting for Heterogeneous Responses with Accounting for Heterogeneous Responses with Social Context PrincipleSocial Context Principle

Possible with nested data, assuming that Possible with nested data, assuming that patterns of relationships are homogeneous patterns of relationships are homogeneous (or following a distribution) within social (or following a distribution) within social contexts (by time or space).contexts (by time or space).

kk is allowed to vary across k (k=1,…K), is allowed to vary across k (k=1,…K),

social context, but is homogeneous within k, social context, but is homogeneous within k, conditional on X. conditional on X.

Multi-level Model (MLM)Multi-level Model (MLM)

YYikik = = kk + + kkDDikik + + ’X’Xikik + + ikik

kk = = ++zzkk++k k

kk = = ++zzkk++kk

Other names: hierarchical linear models, random-Other names: hierarchical linear models, random-coefficient models, growth-curve models, and mixed coefficient models, growth-curve models, and mixed models. models.

Units of analysis at a lower level are nested within higher-Units of analysis at a lower level are nested within higher-level units of analysislevel units of analysis

Examples:Examples: Students within schoolsStudents within schools Observations over time within persons (growth curve)Observations over time within persons (growth curve)

Problems without MLMProblems without MLM If we ignore higher-level units of analysis => we If we ignore higher-level units of analysis => we

cannot account for context (individualistic cannot account for context (individualistic approach) approach)

If we ignore individual-level observation and rely on If we ignore individual-level observation and rely on higher-level units of analysis, we may commit higher-level units of analysis, we may commit ecological fallacy (aggregated data approach)ecological fallacy (aggregated data approach)

Without explicit modeling, sampling errors at Without explicit modeling, sampling errors at second level may be large =>unreliable slopessecond level may be large =>unreliable slopes

Homoscedasticity and no serial correlation Homoscedasticity and no serial correlation assumptions of OLS are violated (an efficiency assumptions of OLS are violated (an efficiency problem).problem).

No distinction between parameter variability and No distinction between parameter variability and sampling variability. sampling variability.

Advantages of MLMAdvantages of MLM

Cross-level comparisonsCross-level comparisons Controls for differences across higher Controls for differences across higher

levelslevels

Example: Xie and Hannum (1996)Example: Xie and Hannum (1996)

(1) (1) Where Where Y = earnings, Y = earnings, X1 = years of schooling, X1 = years of schooling, X2 = years of work experience, X2 = years of work experience, X4 = a dummy variable denoting membership in the Communist X4 = a dummy variable denoting membership in the Communist

Party of China (1 = party member), Party of China (1 = party member), X5 a dummy variable denoting gender (1 = female).X5 a dummy variable denoting gender (1 = female).

Note two interactions. Note two interactions.

516554422322110log XXXXXXXYT

Consider regional heterogeneityConsider regional heterogeneity

For the ith person in kth city: For the ith person in kth city:

Instead of using fixed effects for the intercept bInstead of using fixed effects for the intercept b0k0k, and full , and full interactions for slope parameters, Xie and Hannum interactions for slope parameters, Xie and Hannum modeled these parameters in a multilevel model. modeled these parameters in a multilevel model.

Let Let zz be a city-level covariate that measures the degree be a city-level covariate that measures the degree of economic reform. Let us assume that individual-level of economic reform. Let us assume that individual-level parameters depend on parameters depend on zz in the following linear in the following linear regressions: regressions:

.log 516554422322110 ikikikikkikkikikkikkkik xxxxxxxy

Cross-City Model (“meta analysis”)Cross-City Model (“meta analysis”)

kkk z 0000

kkk z 1111

kkk z 2222

33 kkk z 4444

kkk z 5555

66

Combining the two levels =>Combining the two levels =>

ikikikikikikikik xxxxxxxy 516554422322110log

kikkikkikkikk zxzxzxzxz 554422110

ikikkikkikkikkk xxxx 554422110

We can see that the city-level covariate z interacts with most of the individual-level predictors.

Special CasesSpecial Cases

Special case 1: If all the coefficients of the Special case 1: If all the coefficients of the city-level covariate (city-level covariate (zz) are zero, we have ) are zero, we have what is called “random coefficient model”what is called “random coefficient model”

Special case 2: If all the coefficients of the Special case 2: If all the coefficients of the city-level covariate (city-level covariate (zz) are zero ) are zero andand there there are no random coefficients in all slope are no random coefficients in all slope coefficients (except the intercept), we have coefficients (except the intercept), we have what is called “variance component what is called “variance component model”. [See Table 3.] model”. [See Table 3.]

Summary: Four ways to conceptualize Summary: Four ways to conceptualize variability in parametersvariability in parameters

Specification Specification Complete Complete homogeneity homogeneity

Random Random variation variation

Regression Regression Fixed Fixed

Degree of Degree of Freedom Freedom

1 1 2 2 1+Pk1+Pk K K

Parsimony Parsimony (DF for (DF for Model) Model)

High Low High Low

Accuracy Accuracy (like R2)(like R2)

Low High Low High

where Pk is the number of predictors at the 2nd level, and K is the number of units at the second level.

ReferencesReferences

Xie, Yu. 1996. “Review of Xie, Yu. 1996. “Review of Identification Problems in the Identification Problems in the Social SciencesSocial Sciences by Charles Manski.” by Charles Manski.” American Journal American Journal of Sociologyof Sociology 101:1131-1133. 101:1131-1133.

Xie, Yu and Emily Hannum. 1996. “Regional Variation Xie, Yu and Emily Hannum. 1996. “Regional Variation in Earnings Inequality in Reform-Era Urban China.” in Earnings Inequality in Reform-Era Urban China.” American Journal of SociologyAmerican Journal of Sociology 101:950-992. 101:950-992.

Xie, Yu, Yang Jiang, and Emily, Greenman. 2008. “Did Xie, Yu, Yang Jiang, and Emily, Greenman. 2008. “Did Send-Down Experience Benefit Youth? A Reevaluation Send-Down Experience Benefit Youth? A Reevaluation of the Social Consequences of Forced Urban-Rural of the Social Consequences of Forced Urban-Rural Migration during China’s Cultural Revolution.” Migration during China’s Cultural Revolution.” Social Social Science ResearchScience Research 37: 686-700. 37: 686-700.