Multilevel models for longitudinal data, and fixed effects regression provide two alternative methods for analyzing longitudinal data.
Briefly…
Here, level 2 units are countries. Level 1 is a country-year. Level 2 is the country.
Multilevel models use both within level 2 unit and between level 2 unit variation, and provide statistical control for observed variables that are included in the model.
Fixed effect regressions use only within level 2 unit 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.
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 country specific intercept, and $e_{it}$ is a measurement specific error term.
In the multilevel model, 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.
We are going to data from the World Bank World Development Indicators
. use "WorldBankData.dta", clear
.
. describe life_expectancy year Gini GDP undernourishment region
Variable Storage Display Value
name type format label Variable label
------------------------------------------------------------------------------------------
life_expectancy double %10.0g Life expectancy at birth, total (years)
year long %12.0g
Gini double %10.0g Gini index (World Bank estimate)
GDP double %10.0g GDP per capita (current US$)
undernourishm~t double %10.0g Prevalence of undernourishment (% of
population)
region str26 %-9s
. drop if region == "Aggregates" // drop rows of data representing Aggregates
(1,927 observations deleted)
.
. encode region, generate(regionNUMERIC) // numeric version of region
. histogram life_expectancy, scheme(michigan) title("Life Expectancy at Birth") fcolor(%50
> )
(bin=39, start=26.172, width=1.5191044)
mixed y x || id:)The model uses within and between level 2 unit variation. Estimates are provided for all variables. The model only controls for variables that are included in the model.
. mixed life_expectancy year Gini GDP undernourishment i.regionNUMERIC || country:
Performing EM optimization ...
Performing gradient-based optimization:
Iteration 0: log likelihood = -1941.3024
Iteration 1: log likelihood = -1941.3024
Computing standard errors ...
Mixed-effects ML regression Number of obs = 1,226
Group variable: country Number of groups = 140
Obs per group:
min = 1
avg = 8.8
max = 19
Wald chi2(10) = 3393.70
Log likelihood = -1941.3024 Prob > chi2 = 0.0000
-----------------------------------------------------------------------------------------
life_expectancy | Coefficient Std. err. z P>|z| [95% conf. interval]
------------------------+----------------------------------------------------------------
year | .2681652 .0070473 38.05 0.000 .2543527 .2819777
Gini | -.0051665 .012107 -0.43 0.670 -.0288958 .0185627
GDP | -2.70e-06 6.03e-06 -0.45 0.655 -.0000145 9.13e-06
undernourishment | -.0712303 .0099802 -7.14 0.000 -.0907911 -.0516696
|
regionNUMERIC |
Europe & Central Asia | 5.023854 1.293808 3.88 0.000 2.488037 7.559671
Latin America & Cari.. | 2.760078 1.51168 1.83 0.068 -.2027596 5.722916
Middle East & North .. | 2.318189 1.695885 1.37 0.172 -1.005684 5.642062
North America | 8.060808 3.494181 2.31 0.021 1.21234 14.90928
South Asia | -2.013143 2.365438 -0.85 0.395 -6.649317 2.62303
Sub-Saharan Africa | -12.27801 1.343098 -9.14 0.000 -14.91043 -9.645584
|
_cons | -467.0279 14.36493 -32.51 0.000 -495.1826 -438.8731
-----------------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects parameters | Estimate Std. err. [95% conf. interval]
-----------------------------+------------------------------------------------
country: Identity |
var(_cons) | 21.89217 2.712469 17.17211 27.90961
-----------------------------+------------------------------------------------
var(Residual) | .7722833 .0332602 .70977 .8403025
------------------------------------------------------------------------------
LR test vs. linear model: chibar2(01) = 2403.93 Prob >= chibar2 = 0.0000
.
. est store MLM
xtreg y x, i(id) fe)The model uses only within level 2 unit variation. Estimates are only provided for within level 2 unit change over time. The model controls for all time invariant variables whether observed or unobserved.
. xtreg life_expectancy year Gini GDP undernourishment i.regionNUMERIC, i(country) fe
note: 2.regionNUMERIC omitted because of collinearity.
note: 3.regionNUMERIC omitted because of collinearity.
note: 4.regionNUMERIC omitted because of collinearity.
note: 5.regionNUMERIC omitted because of collinearity.
note: 6.regionNUMERIC omitted because of collinearity.
note: 7.regionNUMERIC omitted because of collinearity.
Fixed-effects (within) regression Number of obs = 1,226
Group variable: country Number of groups = 140
R-squared: Obs per group:
Within = 0.7362 min = 1
Between = 0.2773 avg = 8.8
Overall = 0.1152 max = 19
F(4,1082) = 754.84
corr(u_i, Xb) = 0.1247 Prob > F = 0.0000
-----------------------------------------------------------------------------------------
life_expectancy | Coefficient Std. err. t P>|t| [95% conf. interval]
------------------------+----------------------------------------------------------------
year | .2765033 .0071426 38.71 0.000 .2624883 .2905183
Gini | -.0020466 .0122625 -0.17 0.867 -.0261077 .0220144
GDP | -.0000126 6.21e-06 -2.04 0.042 -.0000248 -4.63e-07
undernourishment | -.0597049 .0102307 -5.84 0.000 -.0797792 -.0396306
|
regionNUMERIC |
Europe & Central Asia | 0 (omitted)
Latin America & Cari.. | 0 (omitted)
Middle East & North .. | 0 (omitted)
North America | 0 (omitted)
South Asia | 0 (omitted)
Sub-Saharan Africa | 0 (omitted)
|
_cons | -480.9276 14.48041 -33.21 0.000 -509.3405 -452.5148
------------------------+----------------------------------------------------------------
sigma_u | 8.6615053
sigma_e | .87884005
rho | .98980975 (fraction of variance due to u_i)
-----------------------------------------------------------------------------------------
F test that all u_i=0: F(139, 1082) = 196.61 Prob > F = 0.0000
.
. est store FE
estimates table)NB that the omitted category for region is “East Asia & Pacific”.
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 countries.
The multilevel model uses both within and between country variation; the fixed effects model uses only within country variation.
The fixed effects model is unable to provide information on time invariant characteristics of countries 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).
. est table MLM FE, star equations(1) b(%9.3f) stats(N r2_a)
--------------------------------------------
Variable | MLM FE
-------------+------------------------------
#1 |
year | 0.268*** 0.277***
Gini | -0.005 -0.002
GDP | -0.000 -0.000*
undernouri~t | -0.071*** -0.060***
|
regionNUME~C |
Europe & .. | 5.024*** (omitted)
Latin Ame.. | 2.760 (omitted)
Middle Ea.. | 2.318 (omitted)
North Ame.. | 8.061* (omitted)
South Asia | -2.013 (omitted)
Sub-Sahar.. | -12.278*** (omitted)
|
_cons | -467.028*** -480.928***
-------------+------------------------------
lns1_1_1 |
_cons | 1.543***
-------------+------------------------------
lnsig_e |
_cons | -0.129***
-------------+------------------------------
Statistics |
N | 1226 1226
r2_a | 0.701
--------------------------------------------
Legend: * p<0.05; ** p<0.01; *** p<0.001