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Canvas Example MCQ — Q92

• A) approximately 18.2%
• B) approximately 31.0%
• C) approximately 42.6%
• D) approximately 57.4%
• E) None of A-D is close; write the correct answer.

z = -7 + .55(12) - .35(2) + .10(8) = -0.30. P = 1/(1+e^0.30) ≈ .4256 = 42.56%.

Rule: For Logit, insert z into the logistic CDF.

Canvas Example MCQ — Q93

Given: Logit model for choosing an adjustable-rate mortgage. At FIXRATE=12, MARGIN=2, NETWORTH=3 the index is z = −0.80. MARGIN coefficient = −0.35. Logit: P = 1/(1+e^(−z)); marginal effect = b·P·(1−P).

• A) approximately -35.0 percentage points
• B) approximately -12.8 percentage points
• C) approximately -7.5 percentage points
• D) approximately +7.5 percentage points
• E) None of A-D is close; write the correct answer.

z=-0.80, P≈0.3100. ME = (-.35)(.31)(.69) = -.0749 ≈ -7.49 pp.

Rule: Logit ME = β x P(1-P).

Canvas Example MCQ — Q94

• A) approximately -7.5 percentage points
• B) approximately -12.8 percentage points
• C) approximately -18.2 percentage points
• D) approximately +12.8 percentage points
• E) None of A-D is close; write the correct answer.

At MARGIN=2, z=-.80 -> P≈.3100. At MARGIN=4, z=-1.50 -> P≈.1824. Change = .1824-.3100 = -.1276 ≈ -12.8 pp.

Rule: Discrete change = Pnew - Pold, not local ME x 2.

Old Exam — Dec 2021 re-exam, Q30

Given: Average marginal effect = calculate each observation's marginal effect, then average them.

• A) Taking the marginal effect at the mean of the sample.
• B) Mean marginal effects are always 0.5 in a binary choice model.
• C) Calculate the marginal effects for each observation; then take the mean.
• D) Take the mean of the marginal effects at the 5th and 95th percentile.
• E) None of the above (A, B, C, D) is correct.

AME = (1/n) sum_i ME_i: compute the marginal effect for every observation, then average -> C.

Rule: Phi(.) gives probability; phi(.) enters Probit marginal effects. Raw Probit coefficients are not percentage-point effects.

Computer Lab 6 — Exercise 4, LPM Task 1

• A) +5.59 percentage points (0.5592×0.10)
• B) +0.5592 percentage points
• C) +55.92 percentage points
• D) +17.74 percentage points

ΔP = 0.5592 × 0.10 = 0.05592, i.e., about +5.59 percentage points, since LPM coefficients give direct probability (percentage-point) effects per unit of X.

Rule: LPM coefficients are directly interpretable as changes in probability (in proportion, i.e., ×100 for percentage points) per unit change in X.

Computer Lab 6 — Exercise 4, LPM Task 1b

• A) Holding PIRAT fixed, being Black is associated with a 17.74 percentage-point higher predicted denial probability (not a 17.74% relative increase)
• B) Being Black increases the denial probability by 17.74% in relative terms
• C) Being Black decreases denial probability by 17.74 percentage points
• D) The coefficient has no probability interpretation in an LPM

In an LPM, 0.1774 is a direct effect on the probability scale: +17.74 percentage points, holding PIRAT constant — this is a percentage-point difference, not a relative (%) increase.

Rule: Percentage points (an absolute difference in probability) differ from percent (a relative change); LPM coefficients are naturally in percentage points.

Computer Lab 6 — Exercise 4, LPM Task 2

• A) Because Model 2 has no interaction between PIRAT and BLACK, so the PIRAT slope is forced to be the same for both races
• B) Because BLACK has no effect on denial probability at all
• C) Because LPM models always produce identical slopes for every subgroup by mathematical necessity
• D) Coincidence; the two groups just happened to have the same PIRAT effect in this example

Without a PIRAT×BLACK interaction term, the coefficient on PIRAT (0.5592) applies uniformly across all values of BLACK, forcing an identical slope for both groups; adding an interaction term would allow the slope to differ by race.

Rule: Without an interaction term, a regression imposes the same slope for all subgroups defined by the dummy variable.

Computer Lab 6 — Exercise 4, LPM weaknesses

• A) Predicted probabilities can fall outside [0,1], and errors are heteroskedastic by construction, so robust standard errors are generally preferred
• B) LPM cannot estimate coefficients at all for binary outcomes
• C) LPM coefficients cannot be interpreted in percentage-point terms
• D) LPM always has a lower R² than Probit or Logit

Because LPM is linear, predicted probabilities are not constrained to [0,1], and the binary nature of Y produces heteroskedastic errors, making robust standard errors advisable, even though the coefficients remain directly interpretable in probability terms.

Rule: Key LPM limitations: possible out-of-range probabilities and heteroskedastic errors; use robust SEs.

Computer Lab 6 — Exercise 4, Probit vs LPM nonlinearity

• A) Because Probit is nonlinear: the marginal/discrete effect of PIRAT depends on the predicted probability level itself, which differs by BLACK status even without an explicit interaction term
• B) Because the Probit model actually includes an interaction term between PIRAT and BLACK
• C) Because Probit coefficients differ by group automatically
• D) This is a calculation error; Probit should give identical effects just like LPM

Even without an interaction term, Probit's nonlinear Φ(·) link function makes the implied probability effect of any regressor depend on the overall linear index (and thus on BLACK, which shifts the index), unlike LPM where effects are mechanically constant across groups.

Rule: Nonlinear models (Probit/Logit) naturally produce covariate-dependent marginal effects even without explicit interaction terms, unlike the linear LPM.

Computer Lab 6 — Exercise 4, Logit vs Probit marginal effect

• A) Logit's marginal effect uses the logistic density P(1−P), not the normal PDF φ(·), which belongs to Probit
• B) Logit and Probit always use the exact same marginal effect formula
• C) Logit does not have a marginal effect concept
• D) P(1-P) and φ(Xβ) always give numerically identical results

Logit's link function is the logistic CDF Λ(z), whose derivative is the logistic density P(1-P); the normal PDF φ(·) is specific to Probit's standard normal CDF link, so using φ for Logit's marginal effect is a template/model-mismatch error.

Rule: Match the marginal-effect density to the model: Probit uses φ(Xβ); Logit uses P(1−P).

Computer Lab 6 — Exercise 4, Logit calc

• A) ≈27.28%
• B) ≈72.72%
• C) ≈49.51%
• D) ≈9.05%

Λ(-0.9805) = 1/(1+e^0.9805) = 1/(1+2.6654) ≈ 1/3.6654 ≈ 0.2728 = 27.28%, matching the lab's reported value.

Rule: Logit predicted probability: Λ(z) = 1/(1+e^(-z)), computed directly from the linear index z.

Computer Lab 6 — Exercise 4, Probit vs Logit comparison

• A) Compare predicted probabilities or marginal effects across models rather than comparing raw coefficients directly, since the two models' coefficient scales differ
• B) Since the coefficients differ so much, one of the two models must be wrong
• C) The Logit model is always exactly twice the Probit coefficient
• D) Raw coefficients from Probit and Logit can be directly compared and interpreted the same way

Probit and Logit use different link functions (normal vs. logistic CDF) with different inherent scales, so their raw coefficients are not directly comparable; however, their implied predicted probabilities and marginal effects typically tell a similar story, as seen here.

Rule: Compare nonlinear-model results via predicted probabilities or marginal effects, not raw coefficient magnitudes, across different link functions.

Formula Battle — method recognition (built from the Formula Book)

• A) β·P(1 − P)
• B) β·φ(z)
• C) β·P
• D) P(1 − P)

Logistic slope = P(1 − P). Correct method: β·P(1 − P).

Rule: ME = β·P(1 − P)