xtlogit - Stata

Methods and formulas
xtlogit — Fixed-effects, random-effects, and population-averaged logit models 15
xtlogit reports the population-averaged results obtained using xtgee, family(binomial)
link(logit) to obtain estimates. The fixed-effects results are obtained using clogit. See [XT] xtgee
and [R] clogit for details on the methods and formulas.
If we assume a normal distribution, N(0, σ 2 ν), for the random effects ν i ,
Pr(y i1 , . . . , y ini |x i1 , . . . , x ini ) =
∫ ∞
−∞
e −ν2 i /2σ2 ν
√
2πσν
}
∏
F (y it , x it β + ν i ) dν i
{
ni
t=1
where
⎧
1
⎪⎨
1 + exp(−z)
F (y, z) =
1 ⎪⎩
1 + exp(z)
if y ≠ 0
otherwise
The panel-level likelihood l i is given by
l i =
∫ ∞
−∞
e −ν2 i /2σ2 ν
√
2πσν
{
ni
}
∏
F (y it , x it β + ν i ) dν i
t=1
≡
∫ ∞
−∞
g(y it , x it , ν i )dν i
This integral can be approximated with M-point Gauss–Hermite quadrature
This is equivalent to
∫ ∞
−∞
∫ ∞
−∞
e −x2 h(x)dx ≈
f(x)dx ≈
M∑
wmh(a ∗ ∗ m)
m=1
M∑
wm ∗ exp { (a ∗ m) 2} f(a ∗ m)
m=1
where the wm ∗ denote the quadrature weights and the a∗ m denote the quadrature abscissas. The log
likelihood, L, is the sum of the logs of the panel-level likelihoods l i .
The default approximation of the log likelihood is by adaptive Gauss–Hermite quadrature, which
approximates the panel-level likelihood with
l i ≈ √ 2̂σ i
M ∑
m=1
w ∗ m exp { (a ∗ m) 2} g(y it , x it , √ 2̂σ i a ∗ m + ̂µ i )
where ̂σ i and ̂µ i are the adaptive parameters for panel i. Therefore, with the definition of g(y it , x it , ν i ),
the total log likelihood is approximated by