Multilevel Models 1 Sociology 229: Advanced Regression

Multilevel Models 1 Sociology 229: Advanced Regression
Copyright © 2010 by Evan Schofer Do not copy or distribute without permission

Announcements Assignment 4 Due Assignments 2 & 3 handed back.

Multilevel Data Often we wish to examine data that is “clustered” or “multilevel” in structure Classic example: Educational research Students are nested within classes Classes are nested within schools Schools are nested within districts or US states We often refer to these as “levels” Ex: If the study is individual/class/school… Level 1 = individual level Level 2 = classroom Level 3 = school Note: Some stats books/packages label differently!

Multilevel Data Students nested in class, school, and state
Variables at each level may affect student outcomes Class School California Oregon

Multilevel Data Simpler example: 2-level data Which can be shown as:
Class Which can be shown as: Class 1 S1 S2 S3 Class 2 Class 3 Level 2 Level 1

Multilevel Data We are often interested in effects of variables at multiple levels Ex: Predicting student test scores Individual level: grades, SES, gender, race, etc. Class level: Teacher qualifications, class size, track School: Private vs. public, resources State: Ed policies (funding, tests), budget And, it is useful to assess the relative importance of each level in predicting outcomes Should educational reforms target classrooms? Schools? Individual students? Which is most likely to have big consequences?

Multilevel Data Repeated measurement is also “multilevel” or “clustered” Measurement at over time (T1, T2, T3…) is nested within persons (or firms or countries) Level 1 is the measurement (at various points in time) Level 2 = the individual Person 1 T2 T1 T4 T3 T5 Person 2 Person 3 Person 4

Multilevel Data Examples of multilevel/clustered data:
Individuals from same family Ex: Religiosity People in same country (in a cross-national survey) Ex: Civic participation Firms from within the same industry Ex: Firm performance Individuals measured repeatedly Ex: Depression Workers within departments, firms, & industries Ex: Worker efficiency Can you think of others?

Aggregation If you want to focus on higher-level hypotheses (e.g., schools, not children), you can aggregate Make “school” the unit of analysis OLS regression analysis of school-level variables Individual-level variables (e.g., student achievement) can be included as school averages (aggregates) Ex: Model average school test score as a function of school resources and average student SES Problem: Approach destroys individual-level data Also: Loss of statistical power (Tabachnick & Fidel 2007) Also: Can’t draw individual-level interpretations: ecological fallacy.

Ecological Fallacy Issue: Data aggregation limits your ability to draw conclusions about level-1 units The “ecological fallacy” Robinson, W.S. (1950). "Ecological Correlations and the Behavior of Individuals". American Sociological Review 15: 351–357 Among US states, immigration rate correlates positively with average literacy Does this mean that immigrants tend to be more literate than US citizens? NO: You can’t assume an individual-level correlation! The correlation at individual level is actually negative But: immigrants settled in states with high levels of literacy – yielding a correlation in aggregate statistics.

OLS Approaches Another option: Just use OLS regression Problems:
Allows you to focus on lower-level units No need for aggregation Ex: Just analyze individuals as the unit of analysis, ignoring clustering among schools Include independent variables measured at the individual-level and other levels Problems: 1. Violates OLS assumptions (see below) 2. OLS can’t take full advantage of richness of multilevel data Ex: Complex variation in intercepts, slopes across groups.

Multilevel Data: Problems
Issue: Multilevel data often results in violation of OLS regression assumption OLS requires an independent random sample… Students from the same class (or school) are not independent… and may have correlated error If you don’t control for sources of correlated error, models tend to underestimate standard errors This leads to false rejection of H0 “Type I Error” -- Too many asterisks in table This is a serious issue, as we always want to err in the direction of conservatism

Why might nested data have correlated error? Example: Student performance on a test Students in a given classroom may share & experience common (unobserved) characteristics Ex: Maybe the classroom is too dark, causing all students to perform poorly on tests If all those students score poorly, they fall below the regression line… and have negative error But OLS regression requires that error be “random” Within-class error should be random, not consistently negative Other sources of within-class (or school) error An especially good teacher; poor school funding Other ideas?

Sources of correlated error within groups Ex: Cross-national study of homelessness People in welfare states have a common unobserved characteristic: access to generous benefits Ex: Study of worker efficiency in workgroups Group members may influence each other (peer pressure) leading to group commonalities.

When is multilevel data NOT a problem? Answer: If you can successfully control for potential sources of correlated error Add a control to OLS model for: classroom, school, and state characteristics that would be sources of correlated error in each group Ex: Teacher quality, class size, budget, etc… But: We often can’t identify or measure all relevant sources of correlated error Thus, we need to abandon simple OLS regression and try other approaches.

Robust Standard Errors
Strategy #1: Improve our estimates of the standard errors Option 1: Robust Standard Errors reg y x1 x2 x3, vce(robust) The Huber / White / “Sandwich” estimator An alternative method of computing standard errors that is robust to a variety of assumption violations Provides accurate estimates in presence of heteroskedasticity Also, robust to model misspecification Note: Freedman’s criticism: What good are accurate SEs if coefficients are biased due to poor specification? Doesn’t fix the clustered error problem…

Robust Cluster Standard Errors
Option 2: “Robust cluster” standard errors An extension of robust SEs to address clustering reg y x1 x2 x3, vce(cluster groupid) Note: Cluster implies robust (vs. regular SEs) It is easy to adapt robust standard errors to address clustering in data; See: Result: SE estimates typically increase, which is appropriate because non-independent cases aren’t providing as much information compared to a sample of independent cases.

Dummy Variables Another solution to correlated error within groups/clusters: Add dummy variables Include a dummy variable for each Level-2 group, to explicitly model variance in means A simple version of a “fixed effects” model (see below) Ex: Student achievement; data from 3 classes Level 1: students; Level 2: classroom Create dummy variables for each class Include all but one dummy variable in the model Or include all dummies and suppress the intercept

Dummy Variables What is the consequence of adding group dummy variables? A separate intercept is estimated for each group Correlated error is absorbed into intercept Groups won’t systematically fall above or below the regression line In fact, all “between group” variation (not just error) is absorbed into the intercept Thus, other variables are really just looking at within group effects This can be good or bad, depending on your goals.

Dummy Variables Note: You can create a set of dummy variables in stata as follows: xi i.classid – creates dummy variables for each unique value of the variable “classid” Creates variables named _Iclassid_1, _Iclassid2, etc These dummies can be added to the analysis by specifying the variable: _Iclassid* Ex: reg y x1 x2 x3 _Iclassid*, nocons “nocons” removes the constant, allowing you to use a full set of dummies. Alternately, you could drop one dummy.

Dummy Variables Benefits of the dummy variable approach Weaknesses
It is simple Just estimate a different intercept for each group sometimes the dummy interpretations can be of interest Weaknesses Cumbersome if you have many groups Uses up lots of degrees of freedom (not parsimonious) Makes it hard to look at other kinds of group dummies Non-varying group variables = collinear with dummies Can be problematic if your main interest is to study effects of variables across groups Dummies purge that variation… focus on within-group variation If there isn’t much within group variation, there isn’t much to analyze Related point: fixed effects can amplify noise (e.g., in panel data).

Dummy Variables Note: Dummy variables are a simple example of a “fixed effects” model (FEM) Effect of each group is modeled as a “fixed effect” rather than a random variable Also can be thought of as the “within-group” estimator Looks purely at variation within groups Stata can do a Fixed Effects Model without the effort of using all the dummy variables Simply request the “fixed effects” estimator in xtreg.

Fixed Effects Model (FEM)
For i cases within j groups Therefore aj is a separate intercept for each group It is equivalent to solely at within-group variation: X-bar-sub-j is mean of X for group j, etc Model is “within group” because all variables are centered around mean of each group.

ANOVA: A Digression Suppose you wish to model variable Y for j groups (clusters) Ex: Wages for different racial groups Definitions: The grand mean is the mean of all groups Y-bar The group mean is the mean of a particular sub-group of the population Y-bar-sub-j

ANOVA: Concepts & Definitions
Y is the dependent variable We are looking to see if Y depends upon the particular group a person is in The effect of a group is the difference between a group’s mean & the grand mean Effect is denoted by alpha (a) If Y-bar = $8.75, YGroup 1 = $8.90, then aGroup 1= $0.15 Effect of being in group j is: It is like a deviation, but for a group.

ANOVA is based on partitioning deviation We initially calculated deviation as the distance of a point from the grand mean: But, you can also think of deviation from a group mean (called “e”): Or, for any case i in group j:

The location of any case is determined by: The Grand Mean, m, common to all cases The group “effect” a, common to members The distance between a group and the grand mean “Between group” variation The within-group deviation (e): called “error” The distance from group mean to an case’s value

The ANOVA Model This is the basis for a formal model:
For any population with mean m Comprised of J subgroups, Nj in each group Each with a group effect a The location of any individual can be expressed as follows: Yij refers to the value of case i in group j eij refers to the “error” (i.e., deviation from group mean) for case i in group j

Sum of Squared Deviation
We are most interested in two parts of model The group effects: aj Deviation of the group from the grand mean Individual case error: eij Deviation of the individual from the group mean Each are deviations that can be summed up Remember, we square deviations when summing Otherwise, they add up to zero Remember variance is just squared deviation

The total deviation can partitioned into aj and eij components: That is, aj + eij = total deviation:

The total deviation can partitioned into aj and eij components: The total variance (SStotal) is made up of: aj : between group variance (SSbetween) eij : within group variance (SSwithin) SStotal = SSbetween + SSwithin

ANOVA & Fixed Effects Note that the ANOVA model is similar to the fixed effects model But FEM also includes a bX term to model linear trend ANOVA Fixed Effects Model In fact, if you don’t specify any X variables, they are pretty much the same

Within Group & Between Group Models
Group-effect dummy variables in regression model creates a specific estimate of group effects for all cases Bs & error are based on remaining “within group” variation We could do the opposite: ignore within-group variation and just look at differences between Stata’s xtreg command can do this, too This is essentially just modeling group means!

Fixed vs. Random Effects
Dummy variables produce a “fixed” estimate of the intercept for each group But, models don’t need to be based on fixed effects Example: The error term (ei) We could estimate a fixed value for all cases This would use up lots of degrees of freedom – even more than using group dummies In fact, we would use up ALL degrees of freedom Stata output would simply report back the raw data (expressed as deviations from the constant) Instead, we model e as a random variable We assume it is normal, with standard deviation sigma.

Random Effects A simple random intercept model
Notation from Rabe-Hesketh & Skrondal 2005, p. 4-5 Random Intercept Model Where b is the main intercept Zeta (z) is a random effect for each group Allowing each of j groups to have its own intercept Assumed to be independent & normally distributed Error (e) is the error term for each case Also assumed to be independent & normally distributed Note: Other texts refer to random intercepts as uj or nj.

Random Effects Issue: The dummy variable approach (ANOVA, FEM) treats group differences as a fixed effect Alternatively, we can treat it as a random effect Don’t estimate values for each case, but model it This requires making assumptions e.g., that group differences are normally distributed with a standard deviation that can be estimated from data.

Linear Random Intercepts Model
The random intercept idea can be applied to linear regression Often called a “random effects” model… Result is similar to FEM, BUT: FEM looks only at within group effects Aggregate models (“between effects”) looks across groups Random effects models is a hybrid: a weighted average of between & within group effects It exploits between & within information, and thus can be more efficient than FEM & aggregate models. IF distributional assumptions are correct.

Notes: Model can also be estimated with maximum likelihood estimation (MLE) Stata: xtreg y x1 x2 x3, i(groupid) mle Versus “re”, which specifies weighted least squares estimator Results tend to be similar But, MLE results include a formal test to see whether intercepts really vary across groups Significant p-value indicates that intercepts vary . xtreg supportenv age male dmar demp educ incomerel ses, i(country) mle Random-effects ML regression Number of obs = Group variable (i): country Number of groups = … MODEL RESULTS OMITTED … /sigma_u | /sigma_e | rho | Likelihood-ratio test of sigma_u=0: chibar2(01)= Prob>=chibar2 = 0.000

Choosing Models Which model is best?
There is much discussion (e.g, Halaby 2004) Fixed effects are most consistent under a wide range of circumstances Consistent: Estimates approach true parameter values as N grows very large But, they are less efficient than random effects In cases with low within-group variation (big between group variation) and small sample size, results can be very poor Random Effects = more efficient But, runs into problems if specification is poor Esp. if X variables correlate with random group effects Usually due to omitted variables.

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…

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

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…

Within & Between Effects
Issue: What is the relationship between within-group effects and between-group effects? FEM models within-group variation BEM models between group variation (aggregate) Usually they are similar Ex: Student skills & test performance Within any classroom, skilled students do best on tests Between classrooms, classes with more skilled students have higher mean test scores BUT…

But: Between and within effects can differ! Ex: Effects of wealth on attitudes toward welfare At the country level (between groups): Wealthier countries (high aggregate mean) tend to have pro-welfare attitudes (ex: Scandinavia) At the individual level (within group) Wealthier people are conservative, don’t support welfare Result: Wealth has opposite between vs within effects! Watch out for ecological fallacy!!! Issue: Such dynamics often result from omitted level-1 variables (omitted variable bias) Ex: If we control for individual “political conservatism”, effects may be consistent at both levels…

Within & Between Effects / Centering
Multilevel models & “centering” variables Grand mean centering: computing variables as deviations from overall mean Often done to X variables Has effect that baseline constant in model reflects mean of all cases Useful for interpretation Group mean centering: computing variables as deviation from group mean Useful for decomposing within vs. between effects Often in conjunction with aggregate group mean vars.

You can estimate BOTH within- and between-group effects in a single model Strategy: Split a variable (e.g., SES) into two new variables… 1. Group mean SES 2. Within-group deviation from mean SES Often called “group mean centering” Then, put both variables into a random effects model Model will estimate separate coefficients for between vs. within effects Ex: egen meanvar1 = mean(var1), by(groupid) egen withinvar1 = var1 – meanvar1 Include mean (aggregate) & within variable in model.

Example: Pro-environmental attitudes . xtreg supportenv meanage withinage male dmar demp educ incomerel ses, i(country) mle Random-effects ML regression Number of obs = Group variable (i): country Number of groups = Random effects u_i ~ Gaussian Obs per group: min = avg = max = LR chi2(8) = Log likelihood = Prob > chi = supportenv | Coef. Std. Err z P>|z| [95% Conf. Interval] meanage | withinage | male | dmar | demp | educ | incomerel | ses | _cons | Between & within effects are opposite. Older countries are MORE environmental, but older people are LESS Omitted variables? Wealthy European countries with strong green parties have older populations!

Generalizing: Random Coefficients
Linear random intercept model allows random variation in intercept (mean) for groups But, the same idea can be applied to other coefficients That is, slope coefficients can ALSO be random! Random Coefficient Model Which can be written as: Where zeta-1 is a random intercept component Zeta-2 is a random slope component.

Linear Random Coefficient Model
Rabe-Hesketh & Skrondal 2004, p. 63 Both intercepts and slopes vary randomly across j groups

Random Coefficients Summary
Some things to remember: Dummy variables allow fixed estimates of intercepts across groups Interactions allow fixed estimates of slopes across groups Random coefficients allow intercepts and/or slopes to have random variability The model does not directly estimate those effects Just as we don’t estimate coefficients of “e” for each case… BUT, random components can be predicted after you run a model Just as you can compute residuals – random error This allows you to examine some assumptions (normality).

STATA Notes: xtreg, xtmixed
xtreg – allows estimation of between, within (fixed), and random intercept models xtreg y x1 x2 x3, i(groupid) fe - fixed (within) model xtreg y x1 x2 x3, i(groupid) be - between model xtreg y x1 x2 x3, i(groupid) re - random intercept (GLS) xtreg y x1 x2 x3, i(groupid) mle - random intercept (MLE) xtmixed – allows random slopes & coefs “Mixed” models refer to models that have both fixed and random components xtmixed [depvar] [fixed equation] || [random eq], options Ex: xtmixed y x1 x2 x3 || groupid: x2 Random intercept is assumed. Random coef for X2 specified.

STATA Notes: xtreg, xtmixed
Random intercepts xtreg y x1 x2 x3, i(groupid) mle Is equivalent to xtmixed y x1 x2 x3 || groupid: , mle xtmixed assumes random intercept – even if no other random effects are specified after “groupid” But, we can add random coefficients for all Xs: xtmixed y x1 x2 x3 || groupid: x1 x2 x3 , mle cov(unstr) Useful to add: “cov(unstructured)” Stata default treats random terms (intercept, slope) as totally uncorrelated… not always reasonable “cov(unstr) relaxes constraints regarding covariance among random effects (See Rabe-Hesketh & Skrondal).

STATA Notes: GLLAMM Note: xtmixed can do a lot… but GLLAMM can do even more! “General linear & latent mixed models” Must be downloaded into stata. Type “search gllamm” and follow instructions to install… GLLAMM can do a wide range of mixed & latent-variable models Multilevel models; Some kinds of latent class models; Confirmatory factor analysis; Some kinds of Structural Equation Models with latent variables… and others… Documentation available via Stata help And, in the Rabe-Hesketh & Skrondal text.

Random intercepts: xtmixed
Example: Pro-environmental attitudes . xtmixed supportenv age male dmar demp educ incomerel ses || country: , mle Mixed-effects ML regression Number of obs = Group variable: country Number of groups = Obs per group: min = avg = max = Wald chi2(7) = Log likelihood = Prob > chi = supportenv | Coef. Std. Err z P>|z| [95% Conf. Interval] age | male | dmar | demp | educ | incomerel | ses | _cons | [remainder of output cut off] Note: xtmixed yields identical results to xtreg , mle

Random Coefficients: xtmixed
Ex: Pro-environmental attitudes (cont’d) . xtmixed supportenv age male dmar demp educ incomerel ses || country: educ, mle [output omitted] supportenv | Coef. Std. Err z P>|z| [95% Conf. Interval] age | male | dmar | demp | educ | incomerel | ses | _cons | Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] country: Independent | sd(educ) | sd(_cons) | sd(Residual) | LR test vs. linear regression: chi2(2) = Prob > chi2 = Educ (slope) varies, too! Here, we have allowed the slope of educ to vary randomly across countries

What if the random intercept or slope coefficients aren’t significantly different from zero? Answer: that means there isn’t much random variability in the slope/intercept Conclusion: You don’t need to specify that random parameter Also: Models include a LRtest to compare with a simple OLS model (no random effects) If models don’t differ (Chi-square is not significant) stick with a simpler model.

What are random coefficients doing? Let’s look at results from a simplified model Only random slope & intercept for education Model fits a different slope & intercept for each group!

Random Coefficients Why bother with random coefficients?
1. A solution for clustering (non-independence) Usually people just use random intercepts, but slopes may be an issue also 2. You can create a better-fitting model If slopes & intercepts vary, a random coefficient model may fit better Assuming distributional assumptions are met Model fit compared to OLS can be tested…. 3. Better predictions Attention to group-specific random effects can yield better predictions (e.g., slopes) for each group Rather than just looking at “average” slope for all groups.

Random Coefficients 4. Multilevel models explicitly put attention on levels of causality Higher level / “contextual” effects versus individual / unit-level effects A technology for separating out between/within NOTE: this can be done w/out random effects But it goes hand-in-hand with clustered data… Note: Be sure you have enough level-2 units! Ex: Models of individual environmental attitudes Adding level-2 effects: Democracy, GDP, etc. Ex: Classrooms Is it student SES, or “contextual” class/school SES?

Multilevel Model Notation
So far, we have expressed random effects in a single equation: Random Coefficient Model However, it is common to separate levels: Level 1 equation Gamma = constant u = random effect Here, we specify a random component for level-1 constant & slope Intercept equation Slope Equation

Multilevel Model Notation
The “separate equation” formulation is no different from what we did before… But it is a vivid & clear way to present your models All random components are obvious because they are stated in separate equations NOTE: Some software (e.g., HLM) requires this Rules: 1. Specify an OLS model, just like normal 2. Consider which OLS coefficients should have a random component These could be the intercept or any X (slope) coefficient 3. Specify an additional formula for each random coefficient… adding random components when desired

Cross-Level Interactions
Does context (i.e., level-2) influence the effect of level-1 variables? Example: Effect of poverty on homelessness Does it interact with welfare state variables? Ex: Effect of gender on math test scores Is it different in coed vs. single-sex schools? Can you think of others?

Cross-level interactions
Idea: specify a level-2 variable that affects a level-1 slope Level 1 equation Intercept equation Slope equation with interaction Cross-level interaction: Level-2 variable Z affects slope (B2) of a level-1 X variable Coefficient g3 reflects size of interaction (effect on B2 per unit change in Z)

Cross-level Interactions
Cross-level interaction in single-equation form: Random Coefficient Model with cross-level interaction Stata strategy: manually compute cross-level interaction variables Ex: Poverty*WelfareState, Gender*SingleSexSchool Then, put interaction variable in the “fixed” model Interpretation: B3 coefficient indicates the impact of each unit change in Z on slope B2 If B3 is positive, increase in Z results in larger B2 slope.

Pro-environmental attitudes . xtmixed supportenv age male dmar demp educ income_dev inc_meanXeduc ses || country: income_mean , mle cov(unstr) Mixed-effects ML regression Number of obs = Group variable: country Number of groups = supportenv | Coef. Std. Err z P>|z| [95% Conf. Interval] age | male | dmar | demp | educ | income_dev | inc_meanXeduc| ses | _cons | Interaction between country mean income and individual-level education Interaction: inc_meanXeduc has a positive effect… The education slope is bigger in wealthy countries Note: main effects change. “educ” indicates slope when inc_mean = 0

Random part of output (cont’d from last slide) . xtmixed supportenv age male dmar demp educ income_dev inc_meanXeduc ses || country: income_mean , mle cov(unstr) Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] country: Unstructured | sd(income~n) | sd(_cons) | corr(income~n,_cons) | sd(Residual) | LR test vs. linear regression: chi2(3) = Prob > chi2 = Random components: Income_mean slope allowed to have random variation Interceps (“cons”) allowed to have random variation “cov(unstr)” allows for the possibility of correlation between random slopes & intercepts… generally a good idea.

Beyond 2-level models Sometimes data has 3 levels or more
Ex: School, classroom, individual Ex: Family, individual, time (repeated measures) Can be dealt with in xtmixed, GLLAMM, HLM Note: stata manual doesn’t count lowest level What we call 3-level is described as “2-level” in stata manuals xtmixed syntax: specify “fixed” equation and then random effects starting with “top” level xtmixed var1 var2 var3 || schoolid: var2 || classid:var3 Again, specify unstructured covariance: cov(unstr)

Beyond Linear Models Stata can specify multilevel models for dichotomous & count variables Random intercept models xtlogit – logistic regression – dichotomous xtpois – poisson regression – counts xtnbreg – negative binomial – counts xtgee – any family, link… w/random intercept Random intercept & coefficient models Plus, allows more than 2 levels… xtmelogit – mixed logit model xtmepoisson – mixed poisson model

Panel Data Panel data is a multilevel structure
Cases measured repeatedly over time Measurements are ‘nested’ within cases Person 1 T2 T1 T4 T3 T5 Person 2 Person 3 Person 4 Obviously, error is clustered within cases… but… Error may also be clustered by time Historical time events or life-course events may mean that cases aren’t independent Ex: All T1s and all T5s Ex: Models of economic growth… certain periods (e.g., Oil shocks of 1970s) affect all countries.

Panel Data Issue: panel data may involve clustering across cases & time Good news: Stata’s “xt” commands were made for this Allow specification of both ID and TIME clusters… Ex: xtreg var1 var2 var3, mle i(countryid) t(year) Note: You can also “mix and match” fixed and random effects Ex: You can use dummies (manually) to deal with time-clustering with a random effect for case ids

Panel Data: serial correlation
Panel data may have another problem: Sequential cases may have correlated error Ex: Adjacent years (1950 & 1951 or 2007 & 2008) may be very similar. Correlation denoted by “rho” (r) Called “autocorrelation” or “serial correlation” “Time-series” models are needed xtregar – xtreg, for cases in which the error-term is “first-order autoregressive” First order means the prior time influences the current Only adjacent time-points… assumes no effect of those prior Can be used to estimate FEM, BEM, or GLS model Use option “lbi” to test for autocorrelation (rho = 0?).

Panel Data: Choosing a Model
If clustering is mainly a nuisance: Adjust SEs: vce(cluster caseid) Or simple fixed or random effects Choice between fixed & random Fixed is “safer” – reviewers are less likely to complain If hausman test works, random = OK, too But, if cross-sectional variation is of interest, fixed can be a problem… In that case, use random effects… and hope the reviewers don’t give you grief.

Panel Data: Choosing a Model
If you have substantive interests in cross-level dynamics, mixed models are probably the way to go… Plus, you can create a better-fitting model Allows you to relax the assumption that slopes are the same across groups.

Multilevel Models 1 Sociology 229: Advanced Regression

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