Diagnose the error before repairing inference

Learning goals

• Calculate the conventional t-ratio against a zero slope.
• Explain why omitted-variable bias from the missing sector exposure 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 monthly stock-return regression omits a relevant sector factor correlated with the market regressor. Residuals cluster in time, and x1 and x2 are highly correlated. The reported slope is 0.8 with conventional SE 0.2.

Economic or professional mechanism

Omitting a relevant correlated variable can bias slopes; a robust standard error does not repair that misspecification. Heteroskedasticity and serial correlation concern the error structure and can invalidate conventional inference. Multicollinearity inflates uncertainty of individual slopes and can make estimates unstable without necessarily biasing OLS under otherwise valid assumptions.

Apply, challenge and communicate

The numerical t-ratio is descriptive until the SE estimator is justified. Add the economically justified missing factor, inspect residual dependence and choose an estimator aligned with the data frequency. Do not delete a relevant factor merely to make a t-statistic larger.

Diagnose correlation before changing the estimator

With an intercept, VIF_j=1/(1−R_j²), where R_j² comes from regressing predictor j on the other predictors. If that auxiliary R² is 0.90, VIF is 10 and the coefficient standard error is √10 times the comparable orthogonal-design standard error, holding residual variance and that predictor’s centered sum of squares fixed. Multicollinearity does not by itself bias OLS under exogeneity; it weakens identification of separate conditional effects.

Robust standard errors change estimated uncertainty, not a misspecified conditional mean. Heteroskedasticity-robust errors do not automatically handle time dependence; HAC procedures require their own lag and dependence choices. Dropping an economically necessary correlated predictor can introduce omitted-variable bias. Record the decision objective: prediction may tolerate correlated inputs that make coefficient interpretation unstable.

Formula and assumptions

t=(b−b_null)/SE(b)

Case analysis with explicit assumptions

1. A monthly stock-return regression omits a relevant sector factor correlated with the market regressor. Residuals cluster in time, and x1 and x2 are highly correlated. The reported slope is 0.8 with conventional SE 0.2.
2. Method: t=(b−b_null)/SE(b)
3. Case result: 4.00. The numerical t-ratio is descriptive until the SE estimator is justified. Add the economically justified missing factor, inspect residual dependence and choose an estimator aligned with the data frequency. Do not delete a relevant factor merely to make a t-statistic larger.

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.
• Declare the t-ratio causal 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
• Reference · www.itl.nist.gov