FRM Part I · FRM Exam Part I

Linear Regression: formula sheet

Full chapter guide

Key formulas

Population regression model: Yᵢ = β₀ + β₁Xᵢ + εᵢ; β's are fixed unknown parameters. εᵢ is the unobservable error.
Sample regression function: Ŷᵢ = β̂₀ + β̂₁Xᵢ; Fitted values from estimated coefficients.
Residual: êᵢ = Yᵢ − Ŷᵢ; Residuals estimate the errors. OLS minimises Σêᵢ².
OLS slope: β̂₁ = Σ(Xᵢ − X̄)(Yᵢ − Ȳ) ÷ Σ(Xᵢ − X̄)² = Cov(X,Y) ÷ Var(X); Equals ρ × (σY ÷ σX), where ρ is the correlation. Use the same n or n−1 divisor in numerator and denominator.
OLS intercept: β̂₀ = Ȳ − β̂₁X̄; Ensures the line passes through (X̄, Ȳ).
Residual properties: Σêᵢ = 0 and ΣXᵢêᵢ = 0; Hold by construction when the model includes an intercept.
Standard error of regression: SER = √[SSR ÷ (n − 2)]; Divide by n − 2 because two coefficients are estimated.
Unbiasedness: E(β̂) = β; Holds at any sample size if zero conditional mean holds and the model is correctly specified.
Consistency: plim β̂ = β, i.e. β̂ → β in probability as n → ∞; Large-sample property. Needs the regressors to be uncorrelated with the error.
Zero conditional mean: E(ε | X) = 0; Violation causes bias and inconsistency.
Homoskedasticity and no serial correlation: Var(εi | X) = σ² and Cov(εi, εj | X) = 0 for i ≠ j; Needed for BLUE and valid usual standard errors.
Gauss-Markov theorem: Under the classical assumptions, Var(β̂OLS) ≤ Var(β̃) for any other linear unbiased β̃; Normality is not required.
Simple regression slope and its variance: β̂1 = Cov(X,Y) ÷ Var(X); Var(β̂1) = σ² ÷ Σ(Xi − X̄)²; More spread in X and more observations reduce variance.
Standard error of regression: s² = SSR ÷ (n − k − 1); Unbiased estimate of σ² with k slope coefficients.
t-statistic for a coefficient: t = (β̂ − β₀) ÷ SE(β̂); β₀ is the hypothesized value, often 0. Degrees of freedom = n − k − 1.
Confidence interval for a slope: β̂ ± t_c × SE(β̂); t_c is the two-sided critical value for the chosen confidence level, with n − k − 1 degrees of freedom.
Standard error of slope (simple regression): SE(β̂₁) = s ÷ √Σ(Xᵢ − X̄)², where s = √(SSR ÷ (n − 2)); s is the standard error of the regression. A larger spread in X gives a smaller SE.
Decision rule (two-sided): Reject H0 if |t| > t_c, or if p-value < α; Both rules give the same decision.
Confidence interval and test link: Reject H0: β = β₀ at level α if β₀ lies outside the (1 − α) confidence interval; Applies to two-sided tests.
Decomposition of variation: TSS = ESS + SSR; Holds for OLS with an intercept. TSS = Σ(Y − Ȳ)², ESS = Σ(Ŷ − Ȳ)², SSR = Σ(Y − Ŷ)².
R-squared: R² = ESS ÷ TSS = 1 − SSR ÷ TSS; Share of variation in Y explained by the regression.
Simple regression link to correlation: R² = ρ², so |ρ| = √R²; Only with one regressor. The sign of the correlation is the sign of the slope.
Standard error of regression: SER = √(SSR ÷ (n − k − 1)); k = number of slope coefficients. In simple regression, k = 1, so the divisor is n − 2.
Adjusted R-squared: Adjusted R² = 1 − [SSR ÷ (n − k − 1)] ÷ [TSS ÷ (n − 1)] = 1 − (1 − R²) × (n − 1) ÷ (n − k − 1); Is less than or equal to R² for k ≥ 1. Can be negative.
F-statistic from sums of squares: F = (ESS ÷ k) ÷ (SSR ÷ (n − k − 1)); Tests that all slope coefficients are jointly zero.
Multiple regression model: Y = β0 + β1X1 + β2X2 + ... + βkXk + ε; k is the number of slope coefficients, excluding the intercept. Each βj is a partial effect.
t-statistic for one coefficient: t = (β̂j − β_H0) ÷ SE(β̂j), df = n − k − 1; Usually β_H0 = 0. Two-tailed test unless the alternative is directional.
F-statistic (restricted vs unrestricted, SSR form): F = [(SSR_r − SSR_u) ÷ q] ÷ [SSR_u ÷ (n − k − 1)]; q is the number of restrictions. k is the number of slopes in the unrestricted model. Degrees of freedom are q and n − k − 1.
F-statistic (R² form): F = [(R²_u − R²_r) ÷ q] ÷ [(1 − R²_u) ÷ (n − k − 1)]; Valid only when both models have the same dependent variable.
Overall F-statistic: F = (ESS ÷ k) ÷ (SSR ÷ (n − k − 1)) = (R² ÷ k) ÷ ((1 − R²) ÷ (n − k − 1)); Tests H0: all slope coefficients equal zero. Here q = k and the restricted model has only an intercept.
Adjusted R²: Adjusted R² = 1 − (1 − R²) × (n − 1) ÷ (n − k − 1); Unlike R², it can fall when you add a variable that adds little.
Link between t and F: F = t² when q = 1; A joint test with one restriction is the same as a two-tailed t-test.
Sum of squares identity: TSS = ESS + SSR; R² = ESS ÷ TSS; SSR here is the sum of squared residuals. Check which label the question uses.
Variance inflation factor: VIF_j = 1 ÷ (1 − R²_j); R²_j is the R² from regressing regressor j on all the other regressors. A common rule of thumb flags VIF above 10, but it is a guideline, not a strict rule.
Omitted variable bias (one included, one omitted regressor): Bias in b1 ≈ β2 × [Cov(X1, X2) ÷ Var(X1)]; β2 is the true effect of the omitted variable. Bias direction is the sign of β2 times the sign of the correlation between X1 and X2. No bias if the correlation is zero.
Breusch-Pagan test statistic: n × R² from regressing squared residuals on the regressors ~ χ² with k degrees of freedom; Null hypothesis: homoskedasticity. A large statistic rejects the null.
Summary of effects: Heteroskedasticity: coefficients unbiased, standard errors wrong. Multicollinearity: coefficients unbiased, standard errors large. Omitted variable: coefficients biased; Use this as a quick matching table in the exam.
Dummy regression: Y = b0 + b1·D + e; b0 = mean of Y for D = 0 (base group); b1 = difference in mean Y, D = 1 minus D = 0.
Dummy trap rule: Number of dummies = k − 1 (with an intercept); Using k dummies plus an intercept gives perfect multicollinearity.
Interaction model: Y = b0 + b1·X + b2·D + b3·(D·X) + e; Slope on X: b1 if D = 0, b1 + b3 if D = 1. Intercept: b0 if D = 0, b0 + b2 if D = 1.
Linear model: ΔY = b1 × ΔX; Units of b1 are units of Y per unit of X.
Log-linear model: ln Y = b0 + b1·X; %ΔY ≈ 100 × b1 × ΔX; Approximation is good for small b1. Exact change is e^(b1·ΔX) − 1.
Linear-log model: Y = b0 + b1·ln X; ΔY ≈ b1 × (%ΔX ÷ 100); A 1% rise in X changes Y by about b1/100 units.
Log-log model: ln Y = b0 + b1·ln X; %ΔY ≈ b1 × %ΔX; b1 is an elasticity.
t-statistic for a coefficient: t = (b̂ − hypothesized value) ÷ SE(b̂); Usually tests whether the coefficient is zero. About |t| > 1.96 is significant at 5% for large samples.

Quick revision

OLS minimises the sum of squared residuals.
Slope = Cov(X, Y) ÷ Var(X); intercept = Ȳ − slope × X̄.
The fitted line always passes through the point (X̄, Ȳ).
t = (estimate − hypothesised value) ÷ standard error.
Confidence interval = estimate ± critical value × standard error.
TSS = ESS + SSR; R² = ESS ÷ TSS.
Simple regression: R² is the square of the correlation between X and Y.
Adjusted R² penalises extra variables and can fall when a useless one is added.
The F-test checks joint hypotheses, such as all slopes equal to zero.
Heteroskedasticity: coefficients stay unbiased but standard errors are wrong; use robust standard errors.
Multicollinearity: high R² with insignificant t-statistics; estimates become imprecise.
A dummy coefficient is the difference in the mean outcome versus the omitted category; avoid the dummy variable trap.

Common mistakes

Swapping X and Y in the slope formula, using Cov ÷ Var(Y). Fix: Always divide by the variance of the regressor X. The slope has units of Y per unit of X.
Confusing errors with residuals. Fix: Errors εᵢ belong to the PRF and are unobservable. Residuals êᵢ belong to the SRF and are computed from data.
Saying heteroskedasticity makes OLS coefficients biased. Fix: Heteroskedasticity leaves β̂ unbiased and consistent. It loses efficiency and makes the usual standard errors invalid.
Thinking normal errors are required for BLUE. Fix: Gauss-Markov does not use normality. Normality only gives exact small-sample t and F distributions.
Using n − 1 or n as degrees of freedom Fix: For regression use n − k − 1. In simple regression this is n − 2.
Dividing by the estimate instead of its standard error, or forgetting to subtract a nonzero hypothesized value Fix: Always write (β̂ − β₀) ÷ SE. Read the null value from the question.
Dividing SSR by n instead of n − k − 1 when computing SER. Fix: The model estimates k + 1 coefficients, so the degrees of freedom are n − k − 1. In simple regression that is n − 2.
Mixing up ESS and RSS abbreviations. Fix: Work from the definition. Explained means fitted values versus the mean. Residual means actual versus fitted.
Testing a joint hypothesis with several separate t-tests. Fix: Use the F-test whenever H0 involves more than one restriction. Individual t-tests ignore the correlation between the estimates.
Using n − k as the denominator degrees of freedom. Fix: Always use n − k − 1, where k counts slope coefficients only.

Exam tips

Slope questions usually give either Cov and Var, or correlation and standard deviations. Pick the matching formula at once.
Know that the OLS line passes through (X̄, Ȳ). It lets you find the intercept or a missing mean quickly.
Distinguish PRF and SRF, and errors and residuals. Conceptual questions test this wording.
For assumption questions, link each violation to its effect: omitted correlation with X biases estimates, while heteroskedasticity affects standard errors.
Check the sign and rough size of your answer against the correlation before moving on.
Memorize the consequence table: bias comes from correlation between X and the error; inefficiency comes from heteroskedasticity or serial correlation.
When an option says normality is required for BLUE, it is almost always a distractor.
Read for the words small sample versus large sample to separate unbiased from consistent.