Study guide
7.1 Regression Diagnostics
FRM Part I Financial Risk Manager Study Guide Exam
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1 Introduction & Exam Overview
1.1 Current Exam Facts 1.2 Study Strategy & Domain Weights
2 Foundations: Risk Building Blocks & Governance
2.1 Building Blocks of Risk Management 2.2 How Firms Manage Financial Risk 2.3 Governance of Risk Management 2.4 Credit Risk Transfer Mechanisms
3 Foundations: Portfolio Models, Data & ERM
3.1 MPT and the CAPM 3.2 APT and Multifactor Models 3.3 Risk Data Aggregation & Reporting 3.4 Enterprise Risk Management
4 Foundations: Disasters, Crisis & Ethics
4.1 Learning from Financial Disasters 4.2 Anatomy of the Great Financial Crisis 4.3 GARP Code of Conduct
5 Quantitative Analysis: Probability Foundations
5.1 Fundamentals of Probability 5.2 Random Variables 5.3 Common Univariate Distributions 5.4 Multivariate Random Variables
6 Quantitative Analysis: Inference & Regression
6.1 Sample Moments 6.2 Hypothesis Testing 6.3 Linear Regression 6.4 Multiple Regression
7 Quantitative Analysis: Diagnostics & Time Series
7.1 Regression Diagnostics 7.2 Stationary Time Series 7.3 Non-Stationary Time Series 7.4 Returns, Volatility & Correlation
8 Quantitative Analysis: Simulation & Machine Learning
8.1 Simulation and Bootstrapping 8.2 Machine-Learning Methods 8.3 Machine Learning and Prediction
9 Financial Markets: Institutions & Derivatives Intro
9.1 Banks 9.2 Insurance Companies and Pension Plans 9.3 Fund Management 9.4 Introduction to Derivatives
10 Financial Markets: Clearing, Futures & Hedging
10.1 Exchanges and OTC Markets 10.2 Central Clearing 10.3 Futures Markets 10.4 Using Futures for Hedging
11 Financial Markets: FX, Forwards, Commodities & Options
11.1 Foreign Exchange Markets 11.2 Pricing Financial Forwards and Futures 11.3 Commodity Forwards and Futures 11.4 Options Markets
12 Financial Markets: Option Properties, Strategies & Rates
12.1 Properties of Options 12.2 Trading Strategies 12.3 Exotic Options 12.4 Properties of Interest Rates
13 Financial Markets: Bonds, MBS, Rate Futures & Swaps
13.1 Corporate Bonds 13.2 Mortgages and Mortgage-Backed Securities 13.3 Interest Rate Futures 13.4 Swaps
14 Valuation & Risk Models: VaR, Volatility & Ratings
14.1 Measures of Financial Risk 14.2 Calculating and Applying VaR 14.3 Measuring and Monitoring Volatility 14.4 External and Internal Credit Ratings
15 Valuation & Risk Models: Credit, OpRisk & Stress
15.1 Country Risk 15.2 Measuring Credit Risk 15.3 Operational Risk 15.4 Stress Testing
16 Valuation & Risk Models: Fixed-Income Valuation
16.1 Pricing, Discounting & Arbitrage 16.2 Interest Rates 16.3 Bond Yields and Return Calculations 16.4 Duration, Convexity & DV01
17 Valuation & Risk Models: Term Structure, Options & Greeks
17.1 Non-Parallel Term Structure Shifts 17.2 Binomial Trees 17.3 The Black-Scholes-Merton Model 17.4 Option Sensitivity Measures: The Greeks
7.1 Regression Diagnostics
Key Takeaways Explain This Topic AI
Heteroskedasticity leaves OLS coefficients unbiased under classical mean-zero assumptions but invalidates usual SEs; use White/Breusch–Pagan tests and robust or GLS-style fixes
Multicollinearity inflates coefficient SEs when regressors are highly correlated; perfect collinearity makes OLS undefined because X′X is singular
Omitting a relevant correlated regressor biases included coefficients; adding irrelevant variables raises variance without fixing that omitted-variable bias
Bias–variance trade-offs and information criteria (AIC/BIC) guide model selection; residual plots and outlier diagnostics reveal functional-form and leverage problems
OLS is BLUE only under Gauss–Markov conditions (linearity, mean-zero errors, homoskedasticity, uncorrelated errors, fixed or exogenous regressors in the classical statement)
Last updated: August 2026
Regression Diagnostics
After fitting a linear model, FRM candidates must ask whether the Gauss–Markov world still holds. Regression diagnostics detect heteroskedasticity, collinearity, misspecification, and influential points so that coefficient estimates and standard errors are interpreted correctly.
Heteroskedasticity: Problem, Tests, and Fixes
Homoskedasticity means Var(εᵢ | X) = σ² for all i. Heteroskedasticity means the conditional error variance changes with regressors or with the level of the fitted value—common in cross-section risk data (large banks more volatile than small ones) and in some return regressions.
Under classical assumptions with E[ε|X] = 0, OLS slope estimates remain unbiased and consistent even if variances differ. What breaks is the usual variance formula Var(β̂) = σ²(X′X)⁻¹. Reported t- and F-statistics become unreliable: often too optimistic when large residuals coincide with high-leverage x-values.
- Approach What it does FRM takeaway
- Residual vs fitted plot Visual fan shape First screen
- Breusch–Pagan / Cook–Weisberg Regress ê² on X (or fitted ŷ) Formal test of variance linked to levels
- White test Regress ê² on levels, squares, cross-products of X Broader alternative; eats degrees of freedom
- White / HC robust SEs Consistent SE without modeling σᵢ² Default reporting fix in many risk papers
- WLS / feasible GLS Weight by 1/σ̂ᵢ² Efficiency gain if variance model is good
- Worked sketch: White-style intuition
Suppose ê² rises with firm size S. A Breusch–Pagan regression ê² = a + b S + u with a large t-stat on b rejects constant variance. Reporting OLS with ordinary SEs is then unsafe; use heteroskedasticity-consistent (HC) standard errors or model the variance and use weighted least squares.
Exam trap: “heteroskedasticity biases OLS slopes.” Under mean-zero errors, slopes stay unbiased; inference is what fails unless you correct SEs or the variance model.
Multicollinearity Versus Perfect Collinearity
Multicollinearity means two or more regressors are highly linearly related. Coefficients remain unbiased if the model is otherwise correct, but Var(β̂ⱼ) inflates because (X′X) is nearly singular. Individual t-stats look weak even when the group of variables jointly matters (significant F, insignificant individuals).
Diagnostics: variance inflation factors (VIFⱼ = 1 / (1 − Rⱼ²) from regressing xⱼ on the other x’s); pairwise correlations; unstable signs when one correlated variable is dropped.
Perfect collinearity is the extreme case: one column of X is an exact linear combination of others (e.g., including both a dummy for every category and an intercept). Then X′X is singular and OLS cannot be computed —software drops a column. Perfect collinearity is a design error; multicollinearity is a precision problem.
- Concept X′X OLS estimate Typical symptom
- Mild multicollinearity Well-conditioned Unique, higher SE Wide CIs
- Severe multicollinearity Near-singular Unique but unstable Sign flips across samples
- Perfect collinearity Singular Undefined / not unique Software drops a variable
- Worked example: two correlated factors
Regress excess return on market and industry factors with sample correlation 0.95. Both may be “insignificant” individually while a joint test rejects zero. The model still estimates an overall fit; you simply cannot cleanly separate the two slopes. Solutions: drop or combine variables, use a principal component, or accept the joint story and stop over-interpreting single coefficients.
Omitted Variables Versus Irrelevant Variables
Omitted variable bias arises when a relevant regressor z that belongs in the true model is left out and Corr(x, z) ≠ 0. The OLS slope on the included x absorbs part of z’s effect. Direction of bias follows the product of (effect of z on y) and (correlation of z with x).
Irrelevant variables are regressors with true coefficient zero. Including them does not bias the other coefficients if the rest of the model is correct, but it increases the variance of the estimators (degrees of freedom and multicollinearity costs). So: omission can create bias; kitchen-sink inclusion mainly hurts precision.
- Misspecification Bias of included slopes Variance impact
- Omit relevant correlated z Generally biased May look “precise” but wrong
- Include irrelevant w Unbiased (if model otherwise OK) SEs typically larger
- Wrong functional form Often biased Residual patterns
- Worked bias direction
True model: credit spread = β₀ + β₁ Leverage + β₂ Volatility + ε, with β₂ > 0 and Leverage positively correlated with Volatility. If you omit Volatility, the OLS slope on Leverage is typically biased upward —leverage “steals” part of volatility’s positive effect.
Model Selection and the Bias–Variance Trade-Off
Richer models reduce bias (better approximation to E[y|X]) but raise variance (more parameters estimated from the same n). Underfitting leaves systematic residual structure; overfitting fits noise and forecasts poorly out of sample.
Practical tools:
Adjusted R² penalizes extra regressors relative to raw R².
AIC and BIC (BIC penalizes complexity more heavily) compare likelihood versus parameter count.
Cross-validation / out-of-sample MSE for predictive risk models.
FRM emphasis: statistical fit on the estimation sample is not the same as economic validity or stable forecast performance. A high in-sample R² with fragile coefficients is a red flag in risk factor models.
Residual Plots, Outliers, and Influence
Plot residuals êᵢ against fitted ŷᵢ and against each xⱼ:
Random scatter around zero → comfort with linearity and constant variance.
Curved pattern → missing nonlinear term or transformation.
Fan shape → heteroskedasticity.
Clump of large |êᵢ| → outliers or heavy tails.
An outlier is an unusual y given X; a leverage point has unusual X; an influential point moves β̂ a lot (Cook’s distance, DFBETAS). Always ask whether the point is a data error, a regime the model should include (crisis dummy), or a genuine rare event.
Worked residual check
n = 5 toy residuals after a line fit: −0.1, 0.2, −0.05, 0.1, 3.5. The last point dominates SSE. Refitting without it may change the slope sharply—report both fits and investigate the observation before “deleting inconvenient crises.”
When Is OLS BLUE?
BLUE = Best Linear Unbiased Estimator. Under the Gauss–Markov conditions—linear correctly specified conditional mean, E[ε|X] = 0 (or at least E[ε] = 0 and exogenous regressors in the classical finite-sample statement), spherical errors (homoskedastic and uncorrelated), and fixed regressors / appropriate sampling—the OLS estimator has the smallest variance among all linear unbiased estimators of β.
OLS is not BLUE if errors are heteroskedastic or serially correlated (GLS can dominate), if the linear mean is wrong, or if you leave the class of linear unbiased rules (nonlinear or biased estimators may win on MSE). Normality is not required for BLUE; normality underwrites exact t/F finite-sample distributions.
Diagnostic Workflow for FRM
Fit OLS; inspect residual plots and summary fit.
Test / robustify against heteroskedasticity; check serial correlation in time-series regressions (next sections).
Screen VIFs and perfect collinearity.
Ask what was omitted; avoid stuffing irrelevant controls without a reason.
Use information criteria or out-of-sample tests for competing specifications.
Document outliers and influence; justify any exclusions.
Master these checks and you will know when a published beta, hedge ratio, or factor loading is statistically trustworthy—and when only the point estimate (not the starred t-stat) deserves attention.