Study guide
6.3 Linear Regression
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
6.3 Linear Regression
Key Takeaways Explain This Topic AI
OLS for Y = α + βX + ε chooses α̂ and β̂ to minimize the sum of squared residuals; β̂ = Cov̂(X,Y) / Var̂(X) in the single-regressor case
Classical assumptions (linearity, exogeneity/zero conditional mean, homoskedasticity, no serial correlation, and often normality for exact t tests) underwrite BLUE and standard inference
Under the classical model, β̂ is unbiased with sampling variance σ² / Σ(x_i − x̄)²; estimated SE uses residual variance s²
Coefficient t-stats, p-values, and confidence intervals are three views of the same evidence against H₀: β = β₀
- In simple regression with an intercept, R² equals the square of the sample correlation between X and Y
- Last updated: August 2026
Linear Regression
Simple linear regression models a dependent variable as a straight-line function of one regressor plus an error:
Y_i = α + β X_i + ε_i
In risk applications, Y might be a stock’s excess return and X the market excess return (CAPM-style), or Y a loss rate and X a macro factor. FRM Part I expects you to estimate, interpret, and test β.
OLS With a Single Regressor
Ordinary least squares (OLS) picks α̂ and β̂ to minimize Σ ε̂_i² where ε̂_i = Y_i − α̂ − β̂ X_i.
Closed forms:
- β̂ = Σ (x_i − x̄)(y_i − ȳ) / Σ (x_i − x̄)² = Cov̂(X, Y) / Var̂(X)
- α̂ = ȳ − β̂ x̄
Fitted values: ŷ_i = α̂ + β̂ x_i. Residuals are orthogonal to the constant and to X in the sample (normal equations).
- Worked example
- X (factor): 1, 2, 3, 4
- Y (return): 2, 2, 4, 6
- x̄ = 2.5, ȳ = 3.5
- Σ(x − x̄)(y − ȳ) = (−1.5)(−1.5) + (−0.5)(−1.5) + (0.5)(0.5) + (1.5)(2.5) = 2.25 + 0.75 + 0.25 + 3.75 = 7
- Σ(x − x̄)² = 2.25 + 0.25 + 0.25 + 2.25 = 5
- β̂ = 7/5 = 1.4
- α̂ = 3.5 − 1.4 × 2.5 = 0
So Ŷ = 0 + 1.4 X. At X = 3, ŷ = 4.2; residual = 4 − 4.2 = −0.2.
Classical Assumptions (Simple Regression)
A standard teaching set (Gauss–Markov plus extras for exact inference):
Linearity in parameters : true CEF is α + βX (or model is otherwise correctly specified as linear in α, β).
Strict exogeneity / zero conditional mean : E[ε | X] = 0 (no omitted confounding correlated with X; regressor variation is “as good as exogenous” for the effect of interest).
Homoskedasticity : Var(ε | X) = σ² constant.
No serial correlation (in cross-section: independent errors; in time series: uncorrelated errors across i).
No perfect collinearity : Var(X) > 0 (X is not constant).
Often for exact normal t/F tests : ε | X ~ Normal(0, σ²). Asymptotically, CLT delivers approximate normality of β̂ without normal errors if other conditions hold and n is large.
Under 1–5, OLS is BLUE . Normality (6) justifies exact small-sample t tests; without it, use large-sample (robust) reasoning.
- Assumption failure Typical symptom
- E[ε given X] ≠ 0 Biased/inconsistent β̂ (endogeneity, omitted variables)
- Heteroskedasticity OLS still unbiased under exogeneity, but usual SEs wrong
- Serial correlation Usual SEs wrong; efficiency loss
- Nonlinearity Linear fit misrepresents marginal effects
- Sampling Distribution of the Slope
Under classical assumptions with fixed regressors (or conditioning on X):
- E[β̂] = β
- Var(β̂) = σ² / Σ(x_i − x̄)²
- More spread-out X → smaller variance of β̂ (stronger experimental design). Estimate σ² with
- s² = Σ ε̂_i² / (n − 2)
- (two parameters estimated: α and β). Then
- SE(β̂) = s / √[Σ(x_i − x̄)²]
t = (β̂ − β₀) / SE(β̂) ~ t_{n−2} under H₀ when errors are normal.
Worked example: SE and t
Continue the toy fit Ŷ = 1.4X, n = 4 (tiny—for method only). Fitted values are 1.4, 2.8, 4.2, 5.6, so residuals are +0.6, −0.8, −0.2, +0.4 and Σ ε̂_i² = 0.36 + 0.64 + 0.04 + 0.16 = 1.2. Then s² = 1.2/(4 − 2) = 0.6, s ≈ 0.775.
Σ(x − x̄)² = 5, SE(β̂) = 0.775/√5 ≈ 0.346.
Test H₀: β = 0: t = 1.4/0.346 ≈ 4.04. Even with n − 2 = 2 df, this is large—but never trust inference from n = 4 in practice; FRM wants the mechanics.
Realistic sketch: n = 60, β̂ = 1.10, SE = 0.20.
t = 1.10/0.20 = 5.5 → decisively reject β = 0 at standard levels.
Coefficient Tests, p-Values, and Confidence Intervals
For H₀: β = β₀ (often β₀ = 0):
Compute t = (β̂ − β₀)/SE(β̂).
Two-sided p-value from t_{n−2} (or normal approximation for large n).
(1 − α) CI: β̂ ± t_{α/2, n−2} · SE(β̂).
Trinity of equivalence (two-sided): reject H₀: β = β₀ at level α ⟺ p ≤ α ⟺ β₀ outside the (1 − α) CI.
Worked CI
β̂ = 0.85, SE = 0.25, n large so use z ≈ 1.96 for 95%.
CI: 0.85 ± 1.96 × 0.25 = 0.85 ± 0.49 = [0.36, 1.34].
H₀: β = 0 rejected; H₀: β = 1 not rejected (1 is inside the CI). Same story from t for β = 1: (0.85 − 1)/0.25 = −0.6.
R² and Correlation
Total sum of squares TSS = Σ (y_i − ȳ)² Explained sum of squares ESS = Σ (ŷ_i − ȳ)² Residual sum of squares RSS = Σ (y_i − ŷ_i)²
TSS = ESS + RSS (with intercept).
R² = ESS/TSS = 1 − RSS/TSS
In simple linear regression with an intercept:
R² = r_{X,Y}²
where r_{X,Y} is the sample correlation between X and Y. If r = −0.6, R² = 0.36: 36% of sample variation in Y is linearly associated with X. Sign of dependence lives in β̂ (and r), not in R².
High R² is not the same as a causal story, nor does it guarantee good forecasting out of sample. Low R² is common in daily return regressions even when β is estimated precisely (because residual variance is large).
- Metric Meaning
- β̂ Estimated slope; units of Y per unit X
- SE(β̂) Sampling noise of the slope
- t / p / CI Evidence on H₀ about β
- R² Fraction of Y variation fit in-sample by X
- r Signed linear association; r² = R² in simple regression
- Interpreting β in Risk Contexts
If Y is asset excess return and X is market excess return, β̂ ≈ 1.2 means the asset moved about 1.2% for a 1% market move on average in the sample , under the linear model. That is a statistical association from OLS—not automatic proof of equilibrium CAPM pricing. Still, testing H₀: β = 1 (market-like) versus H₀: β = 0 (no linear market link) uses exactly the t/CI tools above.
Common Traps
Confusing statistical significance of β with economic size of β.
Using R² to “validate” a trading signal without out-of-sample tests.
Ignoring that heteroskedasticity/autocorrelation invalidate textbook SEs.
Forgetting df = n − 2 in simple regression.
Claiming R² = r when r is negative—R² = r² always in the simple intercept case.
Simple OLS is the backbone for multiple regression next: same least-squares idea, richer interpretation and joint tests when several X’s enter together.