GEC (Katz-family) regression with high-dimensional unit fixed effects

Fits gec() with a full set of unit fixed effects, concentrating (profiling) out the unit intercepts by a one-dimensional inner maximization per unit, so the outer optimizer handles only the covariate coefficients and the free dispersion parameter delta. This is the cpb_fe() concentration generalized to the whole Katz family: the panel can be under-, equi-, or overdispersed and the direction is estimated, not presumed. For delta < 1 the Katz support is finite, but the likelihood is continuous across an integer ceiling (the entering support point's mass grows from zero) and only kinks there, so each unit's objective is unimodal in its intercept and is maximized by golden section on a bracket around the unit's Poisson intercept, expanded while the maximum sits at an edge; a maximum at a kink is tracked in the analytic gradient of the concentrated likelihood, which the outer BFGS uses. The feasibility floor is exact: every count must lie in 0..ceiling(mu/(1-delta)). Runtime: about one and a half times that of cpb_fe() on the same panel (the Katz recursion runs the whole support): on the positive counts of the bundled peacekeeping panel, 2,602 rows in 146 units, gec_fe() takes about two and a half minutes and cpb_fe() about one and a half, and the two return the same fit.

Usage

gec_fe(
  formula,
  data,
  fe,
  se = c("none", "bootstrap"),
  B = 500,
  cluster = NULL,
  offset = NULL,
  weights = NULL,
  cores = 1L,
  max.support = NULL,
  inner_it = 30L,
  maxit = 3000L,
  reltol = 1e-07,
  bias_correct = c("none", "jackknife"),
  time = NULL
)

Arguments

formula: A model formula for the covariates only (no unit factor, no intercept; the fixed effects absorb it).
data: A data frame.
fe: Name of the column holding the unit identifier.
se: Inference for the covariate coefficients: "none" (default) or "bootstrap", a pairs/cluster bootstrap over units.
B: Bootstrap resamples when se = "bootstrap".
cluster: Cluster for the bootstrap; NULL (default) resamples the fixed-effects units. See cpb_fe().
offset: Optional offset on the log-mean scale (an exposure): a numeric vector or the name of a column in data.
weights: Optional frequency weights (a numeric vector or a column name); see cpb().
cores: Worker processes for the bootstrap; see cpb().
max.support: Guard on the evaluated support; NULL (default) sets max(500, 10 * max(y)). See gec().
inner_it: Golden-section iterations for each unit's inner maximization.
maxit, reltol: Outer optimizer controls.
bias_correct: "none" (default) or "jackknife", the split-panel jackknife correction for the 1/T incidental-parameters bias (see Details).
time: Name of the column of data that orders each unit's rows in time; required by bias_correct = "jackknife" (see cpb_fe()).

Value

An object of class c("gec_fe", "gec").

Details

Limitation: unlike cpb_fe(), gec_fe() has no truncated argument – a concentrated zero-truncated GEC is not currently implemented. For a zero-truncated GEC with fixed effects, enter the unit factor as dummies in gec()'s formula (feasible for moderate unit counts).

Short panels: delta and the fixed effects carry the incidental-parameters bias of nonlinear fixed-effects estimation, of order 1/T; treat delta cautiously below about T = 30. The covariate coefficients carry a smaller bias of the same order. bias_correct = "jackknife" removes the leading 1/T term by the split-panel jackknife exactly as in cpb_fe(): refit on each unit's temporal halves and report 2 * full - mean(halves), with the unit effects and fitted values re-concentrated at the corrected parameters and logLik/AIC kept at the maximum-likelihood fit (uncorrected estimates in $uncorrected). The same time-homogeneity validity gate applies: on panels where the two halves do not estimate a common parameter the correction is refused with a warning and the maximum-likelihood fit is returned (see cpb_fe(), Details). As there, the covariate coefficients are corrected too, and with se = "bootstrap" each replicate is corrected the same way, so the standard errors include the correction's own variance.

Examples

# \donttest{
set.seed(5)
d <- do.call(rbind, lapply(1:30, function(i) {
  x <- rnorm(10); N <- pmax(round(exp(1 + rnorm(1, 0, 0.4) + 0.3 * x) / 0.5), 1)
  data.frame(unit = i, x = x, y = rbinom(10, N, 0.5))
}))
gec_fe(y ~ x, data = d, fe = "unit")   # delta ~ 0.5: underdispersed
#> GEC (Katz family) regression with 30 unit fixed effects (concentrated likelihood)
#> Coefficients:
#>      x 
#> 0.2897 
#> 
#> dispersion delta (Katz; Var/Mean on an unbounded support) = 0.488  [underdispersed (point estimate)],  n = 300
#> Note: delta is subject to incidental-parameters bias for short panels; see ?gec_fe.
# }