Study guide
6.4 Multiple Regression
FRM Part I Financial Risk Manager Study Guide Exam
Track your progress as you read
Finished sections get a check mark.
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
6.4 Multiple Regression
Key Takeaways Explain This Topic AI
Multiple regression estimates Y = α + β₁X₁ + … + β_k X_k + ε; each β_j is a partial effect holding other included regressors fixed
Assumptions extend the simple-regression list: zero conditional mean given all X’s, homoskedasticity, no serial correlation, and no perfect multicollinearity among regressors
R² always rises (weakly) when a regressor is added; adjusted R² penalizes extra parameters and can fall when a variable adds little fit
Joint tests (F-tests) ask whether several coefficients are simultaneously zero (or equal specified values); insignificant individuals can still be jointly significant, and vice versa patterns matter
- TSS = ESS + RSS with an intercept; R² = ESS/TSS = 1 − RSS/TSS links fit to the variance decomposition
- Last updated: August 2026
- Multiple Regression
- Markets rarely offer a single driver. Multiple regression writes
- Y_i = α + β₁ X_{1i} + β₂ X_{2i} + … + β_k X_{ki} + ε_i
OLS again minimizes Σ ε̂_i², now over α and all β_j. Matrix formulas exist, but FRM focus is interpretation, assumptions, fit measures, and joint tests.
Multiple Versus Single-Regressor Assumptions
Core ideas carry over, with important extensions:
- Topic Simple regression Multiple regression
- Conditional mean E[ε given X] = 0 E[ε given all X's] = 0
- Homoskedasticity Var(ε given X) = σ² Var(ε given all X's) = σ²
- Dependence of errors Uncorrelated across observations Same
Variation in X X not constant No perfect multicollinearity : no X_j is an exact linear combination of the others (and constant)
BLUE OLS BLUE under GM OLS still BLUE under GM in the multiple-X class of linear unbiased estimators
Perfect multicollinearity makes OLS coefficients undefined (singular X′X)—e.g., including both a full set of country dummies and a constant without dropping one category, or including X and 2X. Imperfect multicollinearity (high correlation among X’s) leaves OLS defined but inflates SE(β̂_j), making individual t-tests weak even when the group of variables matters jointly.
Omitted variable bias becomes subtler: if you leave out Z that belongs in the CEF and Z correlates with included X_j, then β̂_j mixes the effect of X_j with the proxy channel through Z.
Coefficient Interpretation: Partial Effects
β̂_j estimates the change in E[Y] associated with a one-unit increase in X_j holding the other included regressors fixed (ceteris paribus within the model).
Example: Y = fund return, X₁ = market, X₂ = size factor.
β̂₁ is the market slope after controlling for size exposure—not the same number you get from regressing Y on market alone, unless size is uncorrelated with the market in the sample.
Worked contrast
Simple regression of Y on X₁ alone might give β̃₁ = 1.20.
Multiple regression of Y on X₁ and X₂ might give β̂₁ = 0.95, β̂₂ = 0.40.
The drop from 1.20 to 0.95 means part of the raw market association was shared with X₂. Neither coefficient is “wrong”; they answer different questions. FRM items often ask which interpretation matches which regression.
R² Versus Adjusted R²
As before:
- TSS = Σ (y_i − ȳ)²
- ESS = Σ (ŷ_i − ȳ)²
- RSS = Σ (y_i − ŷ_i)²
- TSS = ESS + RSS (with intercept)
- R² = ESS/TSS = 1 − RSS/TSS
Adding a regressor never increases RSS (OLS can always set the new coefficient to zero and recover the old fit). Therefore R² never falls when you add variables—it is a poor tool for choosing models if you only maximize R².
Adjusted R² penalizes complexity:
Adjusted R² = 1 − [(RSS/(n − k − 1)) / (TSS/(n − 1))]
where k is the number of slope parameters (excluding the intercept in the usual counting). Equivalently, compare residual variance estimates with and without the extra parameters. Adjusted R² can fall when the new variable reduces RSS too little to offset the lost degrees of freedom.
- Measure Behavior when adding a weak regressor
- R² Stays same or rises
- Adjusted R² May fall
- Residual df Falls by 1
- Worked numerical sketch
n = 100, k = 2 slopes, RSS = 80, TSS = 200.
R² = 1 − 80/200 = 0.60.
Adjusted R² = 1 − (80/97)/(200/99) = 1 − (0.8247)/(2.0202) ≈ 1 − 0.408 ≈ 0.592.
Add a third regressor, RSS falls only to 79.5.
R² = 1 − 79.5/200 = 0.6025 (up).
Adjusted R² = 1 − (79.5/96)/(200/99) ≈ 1 − 0.8281/2.0202 ≈ 1 − 0.410 ≈ 0.590 (down). The variable added fit in-sample by R² but failed the adjusted-R² bar.
Joint Tests (F-Tests)
Individual t-tests ask about one restriction: H₀: β_j = 0. Joint tests ask about several restrictions at once, e.g.
H₀: β₂ = β₃ = 0
(the two extra factors jointly useless) while allowing β₁ free.
Standard F-statistic comparing restricted and unrestricted models:
F = [(RSS_R − RSS_U)/q] / [RSS_U/(n − k_U − 1)]
where q is the number of restrictions, RSS_R is residual SS under H₀, RSS_U under the bigger model. Under classical normal assumptions and H₀, F ~ F_{q, n−k_U−1}.
Large F → the restricted model’s worse fit is unlikely under H₀ → reject the joint null.
Worked example
Unrestricted model (k_U = 3 slopes): RSS_U = 120, n = 84, so df_U = 84 − 3 − 1 = 80.
Restricted (drop two regressors, q = 2): RSS_R = 150.
F = [(150 − 120)/2] / [120/80] = (15) / (1.5) = 10.
An F of 10 with (2, 80) df far exceeds typical 5% critical values near 3.1 → reject H₀: the two coefficients are jointly zero. Note: each separate t might be only marginally significant when multicollinearity is present, yet the F can still reject—or the opposite pattern can occur depending on correlation structure.
Overall regression F tests H₀: all slopes zero (β₁ = … = β_k = 0). It is the joint test that “the model has no linear explanatory power from the X’s.”
ESS / TSS / RSS Decomposition
Keep the accounting clear:
TSS : total variation of Y around ȳ.
ESS : variation of fitted Ŷ around ȳ (what the regressors jointly explain in-sample).
RSS : leftover variation of residuals.
R² = ESS/TSS. Reporting only R² without checking residual diagnostics, coefficient stability, and economic magnitude is incomplete risk practice—but the decomposition is the language of FRM exam questions on fit.
Worked decomposition
TSS = 500, RSS = 200 → ESS = 300, R² = 300/500 = 0.60.
If a competing model posts RSS = 250 on the same Y (same TSS), then ESS = 250, R² = 0.50—worse in-sample linear fit. Whether that model is “better” still depends on out-of-sample performance, theory, and adjusted R² / information criteria—but TSS/ESS/RSS arithmetic must add up.
Putting Multiple Regression to Work in Risk
A desk might regress P&L on market, rates, and FX factors. Partial β̂_j feed risk reporting and hedging ratios conditional on the other factors included . Joint F-tests decide whether a block of style factors belongs in the specification. Adjusted R² discourages kitchen-sink factor lists that inflate plain R². Multicollinearity warns that you may know “the factors matter together” without pinning down each β precisely—use joint tests and economically motivated restrictions rather than hunting every individual p-value.
Exam Checklist
Interpret β_j as a partial effect given the other included X’s.
State the multiple-regression versions of exogeneity, homo/serial assumptions, and no perfect collinearity.
Know R² vs adjusted R² incentives when variables are added.
Compute or interpret F from RSS_R, RSS_U, q, and residual df.
Tie R² to ESS/TSS/RSS with TSS = ESS + RSS.
Master these and you connect sample moments and hypothesis tests to the multivariate tools used throughout market risk, performance attribution, and empirical FRM applications.