10 Dec 2024 10:32:51
This example draws from the Stata documentation for the
xtregcommand.
Multilevel models for longitudinal data, and fixed effects regression provide two alternative methods for analyzing longitudinal data.
Briefly…
Multilevel models use both within person and between person variation, and provide statistical control for observed variables that are included in the model.
Fixed effect regressions use only within person variation. As a consequence, fixed effects regression is unable to provide parameter estimates for time invariant variables, even when they are included in the statistical model. Fixed effects regressions provide statistical controls for all time invariant variables, whether observed or unobserved.
use)We are going to use the sample NLS data on work from Stata Corporation.
. clear all
. use https://www.stata-press.com/data/r16/nlswork, clear (National Longitudinal Survey. Young Women 14-26 years of age in 1968)
describe). describe ln_w grade age race union south
Variable Storage Display Value
name type format label Variable label
───────────────────────────────────────────────────────────────────────────────────────────
ln_wage float %9.0g ln(wage/GNP deflator)
grade byte %8.0g current grade completed
age byte %8.0g age in current year
race byte %8.0g racelbl race
union byte %8.0g 1 if union
south byte %8.0g 1 if south
Both models estimate the following equation.
\[y_{it} = \beta_0 + \beta_1 x_{it} + u_{0i} + e_{it}\]
Here \(\beta_0\) is the intercept, \(\beta_1\) is a slope, \(u_{0i}\) is a person specific intercept, and \(e_{it}\) is a measurement specific error term.
In the multilevel model discussed below, the \(u_{0i}\) are considered to have a distribution, with a mean of 0 and a standard deviation \(\sigma_{u0}\). In the fixed effects regression model, the \(u_{0i}\) are considered to be fixed, and directly estimable, although in practice, estimates for each of the \(u_{0i}\) are usually not provided.
mixed y x || id:)The model uses within and between person variation. Estimates are provided for all variables. The model only controls for variables that are included in the model.
. mixed ln_w grade age i.race union south || idcode:
Performing EM optimization ...
Performing gradient-based optimization:
Iteration 0: Log likelihood = -5486.826
Iteration 1: Log likelihood = -5486.826
Computing standard errors ...
Mixed-effects ML regression Number of obs = 19,224
Group variable: idcode Number of groups = 4,148
Obs per group:
min = 1
avg = 4.6
max = 12
Wald chi2(6) = 3471.83
Log likelihood = -5486.826 Prob > chi2 = 0.0000
─────────────┬────────────────────────────────────────────────────────────────
ln_wage │ Coefficient Std. err. z P>|z| [95% conf. interval]
─────────────┼────────────────────────────────────────────────────────────────
grade │ .0781541 .0021992 35.54 0.000 .0738438 .0824644
age │ .0137491 .0003907 35.19 0.000 .0129833 .0145149
│
race │
black │ -.0405347 .0126091 -3.21 0.001 -.0652482 -.0158212
other │ .0404357 .0508123 0.80 0.426 -.0591545 .140026
│
union │ .1243977 .0065614 18.96 0.000 .1115375 .1372579
south │ -.1019453 .0090188 -11.30 0.000 -.1196219 -.0842687
_cons │ .3110752 .0314868 9.88 0.000 .2493622 .3727882
─────────────┴────────────────────────────────────────────────────────────────
─────────────────────────────┬────────────────────────────────────────────────
Random-effects parameters │ Estimate Std. err. [95% conf. interval]
─────────────────────────────┼────────────────────────────────────────────────
idcode: Identity │
var(_cons) │ .0998265 .0027427 .0945931 .1053494
─────────────────────────────┼────────────────────────────────────────────────
var(Residual) │ .0691308 .0007996 .0675813 .0707159
─────────────────────────────┴────────────────────────────────────────────────
LR test vs. linear model: chibar2(01) = 8473.10 Prob >= chibar2 = 0.0000
. est store MLM
xtreg y x, i(id) fe)The model uses only within person variation. Estimates are only provided for within person change over time. The model controls for all time invariant variables whether observed or unobserved.
. xtreg ln_w grade age i.race union south, i(idcode) fe
note: grade omitted because of collinearity.
note: 2.race omitted because of collinearity.
note: 3.race omitted because of collinearity.
Fixed-effects (within) regression Number of obs = 19,224
Group variable: idcode Number of groups = 4,148
R-squared: Obs per group:
Within = 0.0983 min = 1
Between = 0.0712 avg = 4.6
Overall = 0.0847 max = 12
F(3, 15073) = 547.57
corr(u_i, Xb) = 0.0599 Prob > F = 0.0000
─────────────┬────────────────────────────────────────────────────────────────
ln_wage │ Coefficient Std. err. t P>|t| [95% conf. interval]
─────────────┼────────────────────────────────────────────────────────────────
grade │ 0 (omitted)
age │ .0153807 .0004154 37.03 0.000 .0145665 .0161949
│
race │
black │ 0 (omitted)
other │ 0 (omitted)
│
union │ .1034851 .0070913 14.59 0.000 .0895853 .1173849
south │ -.0759973 .0135167 -5.62 0.000 -.1024917 -.0495029
_cons │ 1.279453 .0143464 89.18 0.000 1.251332 1.307573
─────────────┼────────────────────────────────────────────────────────────────
sigma_u │ .41784013
sigma_e │ .2618843
rho │ .71796552 (fraction of variance due to u_i)
─────────────┴────────────────────────────────────────────────────────────────
F test that all u_i=0: F(4147, 15073) = 9.60 Prob > F = 0.0000
. est store FE
estimates table)
The multilevel model controls for variables that are included in the model.
The fixed effects model controls for variables that are included in the model, as well as all time invariant characteristics of participants.
The multilevel model uses both within and between person variation; the fixed effects model uses only within person variation.
The fixed effects model is unable to provide information on time invariant characteristics of individuals even if they are included in the model.
Coefficients in the fixed effects model are generally smaller than coefficients in the multilevel model. (Often, though not in this example, coefficients that were significant in the multilevel model are not significant in the fixed effects model).
. etable, estimates(MLM FE) column(estimate) showstars showstarsnote
────────────────────────────────--─────────--
MLM FE
────────────────────────────────--─────────--
current grade completed 0.078 **
(0.002)
age in current year 0.014 ** 0.015 **
(0.000) (0.000)
race
black -0.041 **
(0.013)
other 0.040
(0.051)
1 if union 0.124 ** 0.103 **
(0.007) (0.007)
1 if south -0.102 ** -0.076 **
(0.009) (0.014)
Intercept 0.311 ** 1.279 **
(0.031) (0.014)
var(_cons) 0.100
(0.003)
var(e) 0.069
(0.001)
Number of observations 19224 19224
────────────────────────────────--─────────--
** p<.01, * p<.05