FRM Exam Part I · Linear Regression

Simple Linear Regression Model and OLS for FRM Part I

Updated 11 October 2026 · Fact-checked

Simple linear regression models Y as a straight-line function of one variable X plus an error. OLS picks the slope and intercept that minimise the sum of squared residuals. Slope = Cov(X,Y) ÷ Var(X). Intercept = mean of Y minus slope times mean of X. The fitted line always passes through the sample means.

Understand Simple Linear Regression Model and OLS

Regression describes how one variable, the dependent variable Y, changes on average with another, the independent variable X. The population model is Yᵢ = β₀ + β₁Xᵢ + εᵢ. Here β₀ is the intercept, β₁ is the slope and εᵢ is the error term. The error captures everything that affects Y other than X.

The population regression function (PRF) is the true, unobservable relationship E(Y | X) = β₀ + β₁X. You never see it. You only see a sample of data. From that sample you estimate the sample regression function (SRF): Ŷᵢ = β̂₀ + β̂₁Xᵢ. The hats mean estimated values. The error εᵢ is unobservable. Its sample counterpart is the residual, êᵢ = Yᵢ − Ŷᵢ.

Ordinary least squares (OLS) chooses β̂₀ and β̂₁ to minimise the sum of squared residuals, Σêᵢ². Squaring stops positive and negative residuals cancelling and penalises big misses more. Solving the minimisation gives closed-form formulas for the slope and intercept.

The slope tells you the expected change in Y for a one-unit change in X. The intercept is the expected Y when X = 0. That may have no real meaning if X = 0 is outside your data. A useful feature of OLS: the residuals sum to zero, they are uncorrelated with X in the sample, and the line passes through (mean X, mean Y).

OLS has good properties under standard assumptions: the model is linear in parameters, the errors have zero conditional mean, the observations are i.i.d. (or errors are uncorrelated), large outliers are unlikely, and the variance of errors is constant (homoskedasticity). Under the first set, OLS is unbiased and consistent. With homoskedastic, uncorrelated errors it is also BLUE (Gauss-Markov). Normality of errors is needed only for exact small-sample t and F tests.

Key formulas to remember

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.

How to solve Simple Linear Regression Model and OLS questions

Use this routine for any question asking you to estimate, interpret or check a simple regression.

1Identify Y (dependent) and X (independent) and write the model Y = β₀ + β₁X + ε.
2Collect the summary inputs: means X̄ and Ȳ, plus Cov(X,Y) and Var(X), or the sums Σ(X−X̄)(Y−Ȳ) and Σ(X−X̄)².
3Compute the slope: β̂₁ = Cov(X,Y) ÷ Var(X), or ρ × σY ÷ σX.
4Compute the intercept: β̂₀ = Ȳ − β̂₁X̄.
5Write the fitted line and plug in X to get a prediction Ŷ. Residual = actual Y − Ŷ.
6Interpret the slope in units: Y changes by β̂₁ units for a one-unit rise in X.
7If asked about assumptions or properties, check zero conditional mean of errors, constant variance, no serial correlation, and linearity in parameters.

Quickest way: Slope from correlation and standard deviations

When to use it: When the question gives correlation and standard deviations instead of raw data.

Compute slope = correlation × (σY ÷ σX).
Compute intercept = Ȳ − slope × X̄.
Predict by plugging in X, and sanity-check that the sign of the slope matches the sign of the correlation.
Eliminate options with the wrong sign first, then compute only what you need.

Common mistakes in Simple Linear Regression Model and OLS

Swapping X and Y in the slope formula, using Cov ÷ Var(Y).
Students remember 'covariance over variance' but forget which variance.
Fix: Always divide by the variance of the regressor X. The slope has units of Y per unit of X.
Confusing errors with residuals.
Both are called 'noise' in casual use.
Fix: Errors εᵢ belong to the PRF and are unobservable. Residuals êᵢ belong to the SRF and are computed from data.
Mixing n and n−1 divisors for covariance and variance.
Different data sources use sample or population formulas.
Fix: Use the same divisor in both. If they match, the ratio is unaffected.
Forgetting the intercept formula needs the slope first.
Students try to compute both in parallel.
Fix: Find β̂₁, then β̂₀ = Ȳ − β̂₁X̄.
Saying OLS requires normally distributed errors to be unbiased.
Normality shows up in hypothesis tests and gets blended with estimation.
Fix: Unbiasedness needs zero conditional mean of errors. Normality matters only for exact small-sample inference.
Interpreting the intercept literally when X = 0 is outside the data.
The intercept looks like a meaningful number.
Fix: Treat it as a fitting constant unless X = 0 is plausible.

Worked examples

Example 1

A risk analyst regresses monthly returns of a fund (Y) on market returns (X). Data: X̄ = 1.0%, Ȳ = 1.4%, Cov(X,Y) = 6.0, Var(X) = 5.0 (both in %²). Find the intercept and slope, and predict the fund return if the market returns 2.0%.

Show the solution

Slope: β̂₁ = 6.0 ÷ 5.0 = 1.2.
Intercept: β̂₀ = 1.4 − 1.2 × 1.0 = 0.2%.
Fitted line: Ŷ = 0.2 + 1.2X.
Prediction at X = 2.0: Ŷ = 0.2 + 1.2 × 2.0 = 0.2 + 2.4 = 2.6%.

Answer: Slope = 1.2, intercept = 0.2%, predicted fund return = 2.6%.

Example 2

For a regression of Y on X, the correlation is 0.60, σX = 4, σY = 10, X̄ = 5 and Ȳ = 20. What are the OLS slope and intercept, and what is the residual for an observation with X = 7 and Y = 38?

Show the solution

Slope: β̂₁ = 0.60 × (10 ÷ 4) = 0.60 × 2.5 = 1.5.
Intercept: β̂₀ = 20 − 1.5 × 5 = 20 − 7.5 = 12.5.
Fitted value at X = 7: Ŷ = 12.5 + 1.5 × 7 = 12.5 + 10.5 = 23.
Residual: ê = 38 − 23 = 15.

Answer: Slope = 1.5, intercept = 12.5, residual = 15.

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.

Practice questions from Linear Regression

Simple Linear Regression Model and OLS in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Simple Linear Regression Model and OLS: frequently asked questions

What is the difference between the population and sample regression function?

The PRF is the true relationship E(Y | X) = β₀ + β₁X with unknown parameters. The SRF uses estimates β̂₀ and β̂₁ computed from your sample. Because samples vary, the SRF differs from the PRF and from sample to sample.

How do I calculate the slope and intercept in OLS?

Slope = Cov(X,Y) ÷ Var(X), which equals correlation × σY ÷ σX. Intercept = Ȳ − slope × X̄. Compute the slope first.

What are the OLS assumptions for FRM?

The model is linear in parameters, errors have zero conditional mean given X, observations are independent and identically distributed, and large outliers are unlikely. For OLS to be BLUE, errors also need constant variance and no serial correlation. Normality is needed only for exact small-sample tests.

Why does OLS minimise squared residuals rather than absolute residuals?

Squaring avoids cancellation of positive and negative residuals and gives simple closed-form solutions. It also weights large errors more heavily. Under the Gauss-Markov conditions the resulting estimator has the lowest variance among linear unbiased estimators.