Indicators, odds and influential observations

Learning goals

• Calculate the logistic default probability.
• Explain why log-odds relative to the baseline sector applies to this case.
• Identify the case-specific assumption or failure mode before taking action.

Draft · approximately 45 minutes, excluding further revision.

Shared case and decision context

A default model has log-odds z=−2+0.5L+0.8D. Leverage L=2 and sector indicator D=1. One highly leveraged observation dominates the coefficient; the team proposes deleting it without investigation.

Economic or professional mechanism

Logistic regression transforms a linear index into a probability between zero and one. A coefficient changes log-odds, not probability by a constant number of percentage points. A binary indicator shifts the fitted index relative to an explicitly omitted baseline; including every category plus an intercept causes exact collinearity.

Apply, challenge and communicate

High leverage concerns unusual regressors; influence combines leverage with effects on the fitted model. Verify the record and compare results with and without it, documenting why any exclusion is defensible. An economically real stress observation should not be erased solely because it makes model performance less attractive.

Log odds, odds and probabilities

A logistic model links a linear score z to probability p=1/(1+exp(−z)); log[p/(1−p)]=z. A coefficient b changes log odds by b for a one-unit input increase, holding other inputs fixed; the odds multiply by exp(b). The probability change depends on the starting probability, so b is not a fixed percentage-point probability change.

In an original credit-screening example z=−2+0.8x, x=1 implies z=−1.2 and p≈23.15%. Raising x to 2 gives z=−0.4 and p≈40.13%, a 16.98 percentage-point increase; odds multiply by exp(0.8)≈2.226. These are illustrative fitted associations, not a lending rule or verified default probabilities. Calibration, class imbalance, false positives and the decision threshold must be evaluated on time-separated data.

Formula and assumptions

p=1/(1+exp(−z)); odds=p/(1−p)

Case analysis with explicit assumptions

1. A default model has log-odds z=−2+0.5L+0.8D. Leverage L=2 and sector indicator D=1. One highly leveraged observation dominates the coefficient; the team proposes deleting it without investigation.
2. Method: p=1/(1+exp(−z)); odds=p/(1−p)
3. Case result: 45.02%. High leverage concerns unusual regressors; influence combines leverage with effects on the fitted model. Verify the record and compare results with and without it, documenting why any exclusion is defensible. An economically real stress observation should not be erased solely because it makes model performance less attractive.

Errors to catch

• Do not replace the stated assumptions with an unstated shortcut. Reconcile units, timing and the claim being valued or duty being assessed.
• Automatic causal identification is not a defensible conclusion for these case facts.

Original knowledge check

Answer every question before checking. Review explanations after submitting; retry starts a new attempt.

Sources and coverage status

These original lessons are drafts. Public outlines are used to check scope; the official learning-outcome text and proprietary question banks are not reproduced here. Follow the source for exam-year requirements and updates.

• Reference · www.cfainstitute.org
• Reference · www.itl.nist.gov