Questions: Ordered Choice Models: Ordered Logit and Probit

Question 1

A researcher wants to model customer satisfaction ratings (1=very dissatisfied, 2=dissatisfied, 3=neutral, 4=satisfied, 5=very satisfied) as a function of price and service quality. Why is OLS inappropriate here?

• A. OLS cannot handle more than two outcome categories
• B. OLS treats the categories as having equal spacing, which imposes false precision on ordinal data
• C. OLS always predicts values outside the 1–5 range for this type of data
• D. Satisfaction data always violates the OLS normality assumption

Answer: (B)

The core problem is equal spacing: OLS treats a one-unit move from 3 to 4 as identical in magnitude to a move from 1 to 2. For ordinal categories, there is no guarantee these gaps are equal in the underlying construct. Ordered probit/logit avoids this by estimating where threshold parameters fall on a latent continuous scale, letting the data determine the spacing. OLS can handle multiple categories technically; the problem is interpretive, not mechanical.

Question 2

In an ordered logit model of bond credit ratings (AAA, AA, A, BBB, ...), a firm's leverage ratio has a positive coefficient. A ratings analyst concludes: 'Higher leverage increases the probability of every higher-quality rating.' This interpretation is:

• A. Correct — a positive coefficient shifts probability toward all higher categories
• B. Incorrect — a positive coefficient on leverage would shift probability toward lower-quality (worse) ratings
• C. Incorrect — the coefficient's sign cannot be interpreted without computing marginal effects
• D. Correct only if the proportional odds assumption holds

Answer: (B)

The analyst has the direction right but the logic wrong in a subtle way. A positive coefficient on leverage shifts the latent index y* upward — but whether that moves probability toward higher or lower rating categories depends on how ratings are coded. More importantly, a coefficient shift does NOT increase probability for all categories simultaneously: middle categories can actually lose probability while the tails gain. The marginal effect of a variable on the probability of any specific category can be positive, negative, or non-monotone. The analyst should compute marginal effects for each category, not rely on the coefficient's sign alone.

Question 3

The proportional odds assumption in ordered logit requires that the effect of each predictor on the latent index is the same regardless of which threshold is being crossed.

• T. True
• F. False

Answer: True

This is exactly the parallel regression (proportional odds) assumption: the β coefficients are constant across all thresholds. Intuitively, a one-unit increase in x shifts the latent propensity by β regardless of whether you're comparing 'category 1 vs. 2+' or 'categories 1–4 vs. 5.' Only the intercepts (threshold values μ) differ across comparisons. When this assumption fails — when the effect of a predictor changes depending on which transition you're examining — you need a generalized ordered logit with threshold-specific slopes.

Question 4

In ordered logit, a variable with a positive coefficient generally increases the probability of the highest outcome category.

• T. True
• F. False

Answer: False

This is only guaranteed for the extreme top category under specific distributional conditions. For middle categories, the marginal effect is non-monotone: a positive shift in the latent index can *decrease* the probability of an intermediate category while increasing the probabilities of the top and bottom extremes simultaneously, or it can increase the probability of high categories while decreasing low ones. The direction depends on where probability mass is concentrated relative to the threshold locations. Always compute category-specific marginal effects.

Question 5

Why can a positive coefficient in an ordered probit model actually decrease the probability of some categories, and what should be reported instead of just the coefficient?

Model answer: A positive coefficient shifts the entire latent distribution upward, moving probability mass from low categories toward high categories. Middle categories can lose probability from both ends — the lower threshold steals from them below while the upper threshold steals from above. The net effect on any specific middle category depends on the shape of the distribution and threshold locations. Instead of just reporting β, researchers should compute and report marginal effects: the change in the predicted probability of each outcome category for a one-unit change in the predictor.

This non-monotonicity is one of the most commonly misunderstood aspects of ordered choice models. It is analogous to the logic in multinomial models where adding a covariate can increase probability for distant categories while reducing it for adjacent ones. The coefficient β answers 'which direction does the latent propensity shift?' but marginal effects answer the policy-relevant question: 'by how much does the probability of each outcome change?'