9.2 Motivating Example

To work through the model we will use data from the article “Empowering Women? Gender Quotas and Women’s Political Careers” by Yann Kerevel, in The Journal of Politics here

Women’s representation in executive office continues to lag behind that of their female peers in legislatures, and several studies find women face an executive glass ceiling. One way to shatter this executive glass ceiling is through the adoption of legislative gender quotas. However, scholars know little about how legislative quotas affect women’s access to executive office. Previous research has been unable to determine whether once elected through quotas, women advance to executive office at similar rates to men. Using data on the future career paths of nearly 2,000 Mexican legislators, I find women face a glass ceiling in winning nomination to executive office. Using career data before and after quota implementation, and exploiting lax enforcement in the district component of the mixed electoral system, I find quotas have done little to increase the advancement of women into executive office, although they have increased opportunities for women in other legislative positions.

On pg. 1169, Kerevel writes, “If a glass ceiling exists, women may find it harder to advance to executive office compared to men. Female legislators may still be able to develop successful political careers in similar roles, such as seeking reelection, winning nomination to other legislative offices, or receiving some type of political appointment. However, women may be less likely than men to secure executive nominations to elected positions or appointments to important cabinet posts. The introduction of gender quotas is unlikely to shatter this glass ceiling given the numerous ways women are marginalized once elected and the general lack of executive quotas.”

The first hypothesis following this claim is:

  • H1. Glass ceiling- Women legislators will be nominated to future executive office at lower rates than men.

We will focus on this first hypothesis, but if you are interested, the author also discusses specific hypotheses related to the potential negative and positive effects of gender quotas in the paper.

Data

The data include information on the future career moves of nearly 2000 Mexican federal deputies who served between 1997-2009.

  • Outcome: genderimmedambcat3, first future career move
    • ballot access for state legislature/city council, for Senate, for mayor/governor; appointment to cabinet or party leadership; bureaucratic appointment; other office
  • Explanatory variables: female, party_cat, leg_exp_dum, exec_exp_dum, leadership

Let’s load the data and take a look at the outcome variable.

library(foreign)
ker <- read.dta("https://github.com/ktmccabe/teachingdata/blob/main/kerevel.dta?raw=true")

table(ker$genderimmedambcat3)

                       other deputy/regidor ballot access 
                         680                          241 
        senate ballot access      mayor/gov ballot access 
                         130                          243 
        cabinet/party leader                     bur appt 
                         137                          329

The outcome categories focus on nominations to these offices (i.e., ballot access), as the author finds that gender has less to do with electoral success.

9.2.1 Assumption and Considerations

Let’s focus on the author’s research question. How could we evaluate this relationship? The UCLA site has a good discussion of these tradeoffs.

  • One option would be to collapse this to a more simple problem of multiple separate binary logit models comparing two outcome categories at a time.
    • One possible issue is there is no constraint such that multiple pairwise logits won’t generate probabilities that when summed together exceed 1. Another is that each pairwise comparison might involve a slightly different sample. In contrast, the multinomial logit uses information from all \(N\) observations when jointly estimating the likelihood.
  • A second option would be to collapse the outcome categories down to two levels only (E.g., 1=Mayor/governor or Bureaucrat vs. 0=Otherwise)
    • Just like we said collapsing a scale in ordinal models loses information, the same thing would happen here. This only makes sense to do in a case where a collapseed scale is theoretically interesting.
  • Assess the scale– maybe the categories actually are ordered!
    • Think about whether the categories actually could be considered ordered or even on an interval scale. If that is the case, then you might be able to return to a linear or ordinal model.

Instead, if we want to keep the nominal categories as they are, and we believe they are not ordered, we can move forward with the multinomial model.

  • If we do so, we will want to make sure we have enough data in each outcome category to feel comfortable estimating the predicted probability of being in that category. We also want to avoid situations where there is no variation in our independent variables for a given outcome category (e.g., maybe only men were nominated for mayor/governor)
    • If we run into those situations, we might need more data or think about alternative modelling strategies.

9.2.2 Key Assumption: Independence of Irrelevant Alternatives

The IIA assumption requires that our probability of choosing a particular outcome over another (A over B) does not depend on the choice set, in particular, the presence or absence of some choice C.

Classic case: bus example

  • Suppose you have the transportation choice set below, where the numbers are the probabilities of choosing a particular form of transportation.
  • Choice set: {train, red bus, blue bus} = {.5, .25, .25}
    • Choice set: {train, red bus} = {.5, .5}–Violates IIA
    • Choice set: {train, red bus} = {2/3, 1/3}–Does not violate IIA

The is because we assume the independence of the error terms \(\epsilon_i\) across choices. In a multinomial probit model, this can be relaxed. However, multinomial probit is very computationally intensive, often not used.