FRM: Paired-return regression

Explanations, applications and practice for paired-return regression.

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  Raw and excess return basis record

  Download the original constant risk-free adjustment alongside raw observations.

  Templates

• Benchmark beta and correlation are different

  Separate response sensitivity from standardized association.

  Concepts
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  Return-fit centered-sum and residual worksheet

  Download the original centroid, cross-product and residual reconciliation.

  Templates

• Paired-return centered-sum bridge

  Reconcile matched observations, cross-products, coefficients and residuals.

  Worked examples

• Fit a return line from centered sums

  Use the benchmark variation and cross-product to calculate slope and intercept.

  Lessons

• Paired-return slope and intercept practice

  Test centered sums, estimator divisors and the return basis.

  Practice

• Communicating a benchmark return fit

  Report the proxy, sample and return basis before treating a slope as a risk identity.

  Careers

• Constant asset returns and undefined R-squared

  Keep an exact constant fit separate from a defined explained-variation ratio.

  Concepts

• Benchmark-fit data and diagnostic practice

  Test pairing, nonlinear patterns and the limits of a fit score.

  Practice

• Inspect scatter and residual views of a return fit

  Use the observations and deviations to assess what one summary score leaves out.

  Lessons

• Matching covariance and variance divisors for beta

  Use a consistent moment estimator in the slope ratio.

  Worked examples

• Constant risk-free adjustment of a return fit

  Recalculate the intercept after shifting both return series by the same fixed rate.

  Worked examples

• Raw-return intercept and excess-return basis

  State the risk-free adjustment before interpreting an intercept as CAPM-relative performance.

  Lessons

• Return-fit quality and statistical inference

  Separate a descriptive line from evidence about population coefficients or predictions.

  Comparisons

• Leave one observation out of a return fit

  Use a sensitivity comparison without treating deletion as automatic data correction.

  Worked examples

• Choosing the return basis for a benchmark fit

  Identify matching dates, total-return treatment and raw versus excess returns before entering data.

  Tool guides

• Return-fit intercept from the centroid

  Set the line’s intercept after calculating its slope.

  Formulas

• Interpreting a paired-return beta fit

  Keep coefficient units, benchmark identity and the estimation window attached to the outputs.

  Tool guides

• Explanatory leverage in a straight-line fit

  Inspect benchmark distance from its sample mean without calling every distant point an error.

  Formulas

• Zero linear slope with a curved return pattern

  Inspect what a slope and R² omit when the supplied relationship is nonlinear.

  Worked examples

• Forcing a return line through the origin

  Keep an intercept model separate from a zero-intercept fitting restriction.

  Common mistakes
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  Paired-return input record

  Download a record retaining period keys, benchmark identity and return definitions.

  Templates

• Prepare matched returns for a benchmark fit

  Align periods, definitions and units before calculating a descriptive slope.

  Lessons

• Beta relative to a chosen benchmark

  Keep the explanatory return series attached to its estimated slope.

  Concepts

• R-squared for an intercept return fit

  Compare residual squared deviations with total response variation.

  Formulas

• Treating high R-squared as outperformance

  Keep an in-sample fit score separate from a risk-adjusted performance conclusion.

  Common mistakes

• Calling residual standard error the error of beta

  Distinguish response-scale dispersion from coefficient uncertainty.

  Common mistakes

• Fitted residual and model disturbance

  Distinguish an observable fitted deviation from an unobserved population-model error.

  Concepts

• Return-fit residual and score practice

  Test signed deviations, fit ratios and constant-response boundaries.

  Practice

• Reversing a return regression is not inverting beta

  Keep the response direction and residual criterion attached to the slope.

  Worked examples
• 

  Return-fit review record

  Download a structured record for data questions, residual patterns and sensitivity comparisons.

  Templates

• Paired-return fit review checklist

  Check paired identity, units, coefficient direction and residual interpretation.

  Revision

• Paired-return data to fit interpretation sequence

  Build a compatible record, reconcile coefficients and inspect what the residuals retain.

  Study plans

• Shifted return pairing changes a slope

  Show how misalignment alters covariance while leaving separate series summaries unchanged.

  Worked examples

• Dropping correlation sign when reading R-squared

  Keep the signed association beside its squared fit ratio.

  Common mistakes

• Benchmark return OLS slope

  Divide centered cross-product by the explanatory return’s centered squared sum.

  Formulas

• Residual standard error of a return line

  Use residual degrees of freedom after fitting slope and intercept.

  Formulas
• 

  Paired-return beta and intercept calculator

  Fit a line to compatible asset and benchmark period returns and inspect its residuals.

  Tools

• Tracking error and regression residual dispersion

  Keep active returns against a unit benchmark separate from deviations around an estimated line.

  Comparisons

• Return-unit scaling and a fitted slope

  Check whether both sides of a regression use the same unit representation.

  Worked examples