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Linear fixed- and random-effects models


Stata fits fixed-effects (within), between-effects, and random-effects (mixed) models on balanced and unbalanced data. We use the notation

    y[i,t] = X[i,t]*b + u[i] + v[i,t]

That is, u[i] is the fixed or random effect and v[i,t] is the pure residual.

xtreg is Stata's feature for fitting fixed- and random-effects models.
xtreg, fe estimates the parameters of fixed-effects models:

. webuse nlswork (National Longitudinal Survey. Young Women 14-26 years of age in 1968) . xtset panel variable: idcode (unbalanced) time variable: year, 68 to 88, but with gaps delta: 1 unit . xtreg ln_w grade age c.age#c.age ttl_exp c.ttl_exp#c.ttl_exp tenure > c.tenure#c.tenure 2.race not_smsa south, fe note: grade omitted because of collinearity note: 2.race omitted because of collinearity Fixed-effects (within) regression Number of obs = 28091 Group variable: idcode Number of groups = 4697 R-sq: within = 0.1727 Obs per group: min = 1 between = 0.3505 avg = 6.0 overall = 0.2625 max = 15 F(8,23386) = 610.12 corr(u_i, Xb) = 0.1936 Prob > F = 0.0000
ln_wage Coef. Std. Err. t P>|t| [95% Conf. Interval]
grade 0 (omitted)
age .0359987 .0033864 10.63 0.000 .0293611 .0426362
 
c.age#c.age -.000723 .0000533 -13.58 0.000 -.0008274 -.0006186
 
ttl_exp .0334668 .0029653 11.29 0.000 .0276545 .039279
 
c.ttl_exp#c.ttl_exp .0002163 .0001277 1.69 0.090 -.0000341 .0004666
 
tenure .0357539 .0018487 19.34 0.000 .0321303 .0393775
 
c.tenure#c.tenure -.0019701 .000125 -15.76 0.000 -.0022151 -.0017251
 
race
black 0 (omitted)
not_smsa -.0890108 .0095316 -9.34 0.000 -.1076933 -.0703282
south -.0606309 .0109319 -5.55 0.000 -.0820582 -.0392036
_cons 1.03732 .0485546 21.36 0.000 .9421496 1.13249
sigma_u .35562203
sigma_e .29068923
rho .59946283 (fraction of variance due to u_i)
F test that all u_i=0: F(4696, 23386) = 6.65 Prob > F = 0.0000

We have used factor variables in the above example. The terms c.age#c.age, c.ttl_exp#c.ttl_exp, and c.tenure#c.tenure are just age-squared, total work experience-squared, and tenure-squared, respectively.

The syntax of all estimation commands is the same: the name of the dependent variable is followed by the names of the independent variables.

In this case, the dependent variable, ln_w (log of wage), was modeled as a function of a number of explanatory variables. Note that grade and black were omitted from the model because they do not vary within person.

Our dataset contains 28,091 “observations”, which are 4,697 people, each observed, on average, on 6.0 different years. An observation in our data is a person in a given year. The dataset contains variable idcode, which identifies the persons — the i index in x[i,t]. Before fitting the model, we typed xtset to show that we had previously told Stata the panel variable. Told once, Stata remembers.

To fit the corresponding random-effects model, we use the same command but change the fe option to re.

. xtreg ln_w grade age c.age#c.age ttl_exp c.ttl_exp#c.ttl_exp tenure > c.tenure#c.tenure 2.race not_smsa south, re Random-effects GLS regression Number of obs = 28091 Group variable: idcode Number of groups = 4697 R-sq: within = 0.1715 Obs per group: min = 1 between = 0.4784 avg = 6.0 overall = 0.3708 max = 15 Wald chi2(10) = 9244.74 corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
ln_wage Coef. Std. Err. z P>|z| [95% Conf. Interval]
grade .0646499 .0017812 36.30 0.000 .0611589 .0681409
age .0368059 .0031195 11.80 0.000 .0306918 .0429201
 
c.age#c.age -.0007133 .00005 -14.27 0.000 -.0008113 -.0006153
 
ttl_exp .0290208 .002422 11.98 0.000 .0242739 .0337678
 
c.ttl_exp#c.ttl_exp .0003049 .0001162 2.62 0.009 .000077 .0005327
 
tenure .0392519 .0017554 22.36 0.000 .0358113 .0426925
 
c.tenure#c.tenure -.0020035 .0001193 -16.80 0.000 -.0022373 -.0017697
 
race
black -.053053 .0099926 -5.31 0.000 -.0726381 -.0334679
not_smsa -.1308252 .0071751 -18.23 0.000 -.1448881 -.1167622
south -.0868922 .0073032 -11.90 0.000 -.1012062 -.0725781
_cons .2387207 .049469 4.83 0.000 .1417633 .3356781
sigma_u .25790526
sigma_e .29068923
rho .44045273 (fraction of variance due to u_i)

We can also perform the Hausman specification test, which compares the consistent fixed-effects model with the efficient random-effects model. To do that, we must first store the results from our random-effects model, refit the fixed-effects model to make those results current, and then perform the test.

. estimates store random_effects . quietly xtreg ln_w grade age c.age#c.age ttl_exp c.ttl_exp#c.ttl_exp tenure > c.tenure#c.tenure 2.race not_smsa south, fe . hausman . random_effects
Coefficients
  (b) (B) (b-B) sqrt(diag(V_b-V_B))
  . random_eff~s Difference S.E.
age .0359987 .0368059 -.0008073 .0013177
c.age#c.age -.000723 -.0007133 -9.68e-06 .0000184
ttl_exp .0334668 .0290208 .0044459 .001711
c.ttl_exp#~p .0002163 .0003049 -.0000886 .000053
tenure .0357539 .0392519 -.003498 .0005797
c.tenure#c~e -.0019701 -.0020035 .0000334 .0000373
not_smsa -.0890108 -.1308252 .0418144 .0062745
south -.0606309 -.0868922 .0262613 .0081345
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(8) = (b-B)'[(V_b-V_B)^(-1)](b-B) = 149.43 Prob>chi2 = 0.0000

In addition, Stata can perform the Breusch and Pagan Lagrange multiplier (LM) test for random effects and can calculate various predictions, including the random effect, based on the estimates.

Equally as important as its ability to fit statistical models with cross-sectional time-series data is Stata's ability to provide meaningful summary statistics.

xtsum reports means and standard deviations in a meaningful way:

. xtsum hours
Variable Mean Std. Dev. Min Max Observations
hours overall 36.55956 9.869623 1 168 N = 28467
between 7.846585 1 83.5 n = 4710
within 7.520712 -2.154726 130.0596 T-bar = 6.04395

The negative minimum for hours within is not a mistake; the within shows the variation of hours within person around the global mean 36.55956.

xttab does the same for one-way tabulations:

. xttab msp Overall Between Within
msp Freq. Percent Freq. Percent Percent
0 11324 39.71 3113 66.08 62.69
1 17194 60.29 3643 77.33 75.75
Total 28518 100.00 6756 143.41 69.73
(n = 4711)

msp is a variable that takes on the value 1 if the surveyed woman is married and the spouse is present in the household. Overall, some 60% of our person-year observations are msp. Taking women individually, 66% of the women are at some point msp, and 77% are not; thus some women are msp one year and not others. Taking women one at a time, if a woman is ever msp, 55% of her observations are msp observations. If a woman is ever not msp, 72% of her observations are not msp. (If marital status never varied in our data, the within percentages would all be 100.)

xttrans reports the transition matrix:

. xttrans msp
1 if
married, 1 if married, spouse
spouse present
present 0 1 Total
0 80.49 19.51 100.00
1 7.96 92.04 100.00
Total 37.11 62.89 100.00
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