Conditional exposures in a return model

Learning goals

• Predict y for the stated M and S.
• Explain why conditional sensitivity to M with S held fixed 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

An analyst fits monthly excess return in percentage points: y=0.20+1.10M−0.30S. Next month M=2 and S=1. The sample has 120 months. Residual spread grows with fitted returns and the analyst calls the market coefficient a causal effect.

Economic or professional mechanism

A slope measures the fitted change in y for a one-unit change in that regressor, holding the other included regressors fixed. It is conditional association; confounding, selection and reverse causality can prevent a causal interpretation. Keep the dependent variable, regressors and coefficients in consistent percentage-point units.

Apply, challenge and communicate

A fan-shaped residual plot raises a heteroskedasticity concern. Examine economic specification and influential observations, then use an appropriate inference procedure; an attractive coefficient is not evidence that the assumptions hold. Out-of-sample stability matters before the equation informs portfolio positions.

Joint restrictions versus one coefficient

An individual t test asks whether one slope equals a stated value, conditional on the other variables. A joint F test compares a restricted model with an unrestricted model on the same dependent variable and observations. For q restrictions, F=[(SSE_R−SSE_U)/q]/[SSE_U/(n−k−1)], where k is the number of unrestricted slopes. The usual finite-sample F reference distribution requires the classical normal, homoskedastic error setup; use a suitable robust joint test when those assumptions fail.

Original exercise: n=80, k=3, q=2, restricted SSE 120 and unrestricted SSE 96 give F=(24/2)/(96/76)=9.50. Compare with the relevant F(2,76) critical value or p-value at the chosen significance level. This calculation does not itself identify which restriction fails, demonstrate economic importance or establish causality. A large joint F can coexist with weak individual t statistics when regressors overlap strongly.

Formula and assumptions

yhat=b0+b1*M+b2*S

Case analysis with explicit assumptions

1. An analyst fits monthly excess return in percentage points: y=0.20+1.10M−0.30S. Next month M=2 and S=1. The sample has 120 months. Residual spread grows with fitted returns and the analyst calls the market coefficient a causal effect.
2. Method: yhat=b0+b1*M+b2*S
3. Case result: 2.10 percentage points. A fan-shaped residual plot raises a heteroskedasticity concern. Examine economic specification and influential observations, then use an appropriate inference procedure; an attractive coefficient is not evidence that the assumptions hold. Out-of-sample stability matters before the equation informs portfolio positions.

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.
• Interpret the intercept as a risk-free promise 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