case study · maximum likelihood · multinomial

An Executive Glass Ceiling

Women legislators are nominated to executive office at lower rates than men — by fitting a multinomial logit to the career paths of ~1,900 Mexican federal deputies, with the likelihood machinery written from scratch.

R / nnetPython · numpy · scipyMLE from scratchMarginal effectsDelta method

1,760

legislators modeled (listwise)

unordered career outcomes

exact

match to published Table A1 & log-likelihood

// the question

Women's representation in executive office lags behind their numbers in legislatures. Kerevel asks whether the bottleneck is nomination: when a deputy's term ends, what is their first future career move, and are women steered away from the executive track? The hypothesis is a glass ceiling —

H1. Women legislators are nominated to future executive office at lower rates than men.

The outcome, genderimmedambcat3, records that first move across six unordered destinations — they have no natural ranking, which rules out ordered models and motivates a multinomial approach.

Fig 1. The dependent variable: six nominal destinations. Other (no immediate move) is the modal outcome and the model's baseline category.

// the model

The multinomial logit generalizes binary logit to J categories, estimating one coefficient vector per outcome, each relative to a baseline category (here, other):

log( π_j / π_J ) = α_j + Xβ_j // β_J = 0 by design; probabilities across all j sum to 1

Predicted probabilities convert those log-odds back to an interpretable scale, with the baseline numerator fixed to exp(0) = 1.

the key assumption — IIA

Multinomial logit assumes independence of irrelevant alternatives: the odds of A over B don't depend on whether some option C is in the choice set (the red-bus / blue-bus problem). If IIA is suspect, the fallbacks are multinomial probit, or nested / mixed logit.

// what I built

Rather than calling a black-box solver, I implemented the estimator directly. The log-likelihood sums, over every legislator and category, the log-probability assigned to the destination they actually took:

ℓ(β) = Σ_i Σ_j 1(Y_i = C_j) · log π_ij

Coded the negative log-likelihood and its analytic gradient in numpy, with a log-sum-exp trick for numerical stability.
Optimized with scipy.optimize; recovered standard errors from the inverse observed information.

// result · table

The full model adds party (PRI baseline), legislative and executive experience, and chamber leadership. Each column is a destination contrasted with other; coefficients are log-odds, standard errors in parentheses.

Influence of gender on first future elected office — multinomial logit (baseline: *other*)
outcome vs. other	Woman	PAN	PRD	Leg. exp.	Exec. exp.	Leadership	Constant
State deputy / councilor	0.26 (0.19)	0.55^* (0.18)	0.00 (0.21)	-0.28⁺ (0.16)	-0.05 (0.17)	-0.33 (0.26)	-1.11^* (0.17)
Senator	0.70^* (0.24)	0.50^* (0.22)	-0.51⁺ (0.30)	0.44^* (0.20)	0.45^* (0.21)	0.71^* (0.24)	-2.42^* (0.23)
Mayor / Governor	-0.38⁺ (0.23)	0.76^* (0.18)	0.49^* (0.20)	-0.07 (0.16)	0.35^* (0.17)	-0.45⁺ (0.26)	-1.41^* (0.17)
Cabinet / Party leader	-0.04 (0.26)	-0.08 (0.24)	0.18 (0.23)	-0.06 (0.20)	0.37⁺ (0.21)	0.21 (0.27)	-1.78^* (0.21)
Bureaucracy	0.43^* (0.18)	1.32^* (0.16)	-0.09 (0.21)	-0.11 (0.14)	0.05 (0.16)	0.11 (0.21)	-1.34^* (0.16)

^* p < .05 ⁺ p < .10 · N = 1,760 · log-likelihood = −2755.61

Being a woman raises the log-odds of a senate run (0.70^*) and a bureaucratic appointment (0.43^*) but lowers the odds of the highest-value executive track, mayor / governor (−0.38⁺).

// result · quantities of interest

With one slope per covariate per category, the raw output is hard to read; the honest summary is predicted probabilities and marginal effects.

Fig 2. Predicted probability of each first move, women vs. men (bivariate model). Women are more likely to land in bureaucracy and senate runs; men are notably more likely to be nominated for mayor / governor.

Fig 3. Replication of Kerevel's Figure 1. Cyan intervals exclude zero: women are significantly more likely to head to bureaucracy (+5.5pp) and the senate (+4.8pp), and significantly less likely to be nominated for executive office — mayor / governor (−6.0pp).

takeaway

The glass ceiling is specific, not blanket. Women aren't shut out of all advancement — they over-route into legislative and bureaucratic tracks — but they are systematically diverted away from the single highest-office executive nomination. A binary "executive vs. not" split would have averaged the +senate and −governor effects into noise.

// method notes

MLE from first principles — likelihood, gradient, and information matrix built by hand, not delegated to a library.
Discrete-choice modeling — multinomial logit, baseline identification, and the IIA assumption with its alternatives.
Quantities of interest — predicted probabilities and average marginal effects with delta-method uncertainty.