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FRM Part I · FRM Exam Part I

Linear Regression for FRM Part I: Chapter Guide

Linear regression models a dependent variable as a linear function of one or more explanatory variables. Ordinary least squares (OLS) picks the coefficients that minimise the sum of squared residuals. To solve exam questions, compute the estimates, test them with t or F statistics, read R-squared, and check the assumptions.

What this chapter covers

This chapter teaches you how to fit a straight-line relationship to data and judge whether it can be trusted. You start with simple regression and the OLS formulas for slope and intercept. Then you study the assumptions behind OLS, the standard errors of the estimates, and how to test and build confidence intervals for coefficients.

The second half extends the model. You learn goodness of fit (ESS, TSS, SSR, R-squared), multiple regression, joint tests with the F-statistic, and the problems that break a model: heteroskedasticity, multicollinearity and misspecification. You finish with dummy variables and how to interpret coefficients.

This chapter sits inside Quantitative Analysis and leans on probability, distributions and hypothesis testing. It feeds later chapters on time series, volatility modelling, and risk models in the Valuation and Risk Models topic. Factor models, beta estimation and hedge ratios all use the same regression logic, so the ideas return in several places.

Regression is a core tool in the Quantitative Analysis topic, and its questions are usually computational but short. Many can be solved in a minute if you know the formulas and the decision rule. The same skills (t-tests, standard errors, interpreting output) help you in other chapters, so time spent here pays back more than once. Candidates who only memorise definitions tend to lose marks on questions that give a regression output table and ask you to interpret it.

Linear Regression: topics in the order to study them

  1. 1Simple Linear Regression Model and OLSEverything else builds on the model, the residual and the OLS slope and intercept formulas.
  2. 2OLS Assumptions and Properties of EstimatorsYou need to know why OLS works (unbiased, consistent, BLUE) before you test anything.
  3. 3Hypothesis Testing and Confidence Intervals for CoefficientsOnce you have estimates and standard errors, you learn to test and bound them with t-statistics.
  4. 4Goodness of Fit: R-squared, ESS, TSS, SSRAfter testing single coefficients, you measure how well the whole model explains the data.
  5. 5Multiple Regression and Joint Hypothesis TestsThis extends the simple model to several variables and adds the F-test and adjusted R-squared.
  6. 6Heteroskedasticity, Multicollinearity and Model MisspecificationOnly after you know the full model can you see how violated assumptions distort results.
  7. 7Dummy Variables and Regression InterpretationThis closes the chapter by applying the model to categorical data and tests your reading of outputs.

How to prepare Linear Regression

Treat this chapter as a set of formulas plus a set of decision rules. Practise both with numbers, not just reading.

  1. Learn the simple regression formulas: slope = Cov(X, Y) ÷ Var(X), and intercept = mean of Y − slope × mean of X. Work two or three small data sets by hand.
  2. Write the assumptions in your own words and note what breaks when each fails: bias, wrong standard errors, or both.
  3. Drill the t-test: t = (estimate − hypothesised value) ÷ standard error. Practise the decision against the critical value and the confidence interval form: estimate ± critical value × standard error.
  4. Memorise the fit identities: TSS = ESS + SSR and R² = ESS ÷ TSS = 1 − SSR ÷ TSS. In simple regression, R² equals the squared correlation.
  5. Practise the F-statistic and adjusted R² with multiple regression output tables. Be clear on degrees of freedom: n − k − 1 for k explanatory variables.
  6. For each problem (heteroskedasticity, multicollinearity, misspecification), learn the symptom, the consequence and the usual remedy.
  7. Finish with timed mixed sets of output-table questions. Aim to answer each in about two minutes.

Common mistakes in Linear Regression

  • Mixing up ESS and SSR when computing R-squared

    Fix: Define R² as the explained share of total variation. Check each question's own definitions before substituting numbers.

  • Using the wrong degrees of freedom

    Fix: Count the estimated parameters. For k explanatory variables plus an intercept, use n − k − 1 for the t-test.

  • Saying heteroskedasticity biases the coefficients

    Fix: Remember it leaves OLS estimates unbiased but makes the usual standard errors unreliable, so test results are invalid.

  • Reading a significant F-test as meaning every variable is significant

    Fix: The F-test only says at least one slope differs from zero. Check each t-statistic separately.

  • Misinterpreting a dummy variable coefficient

    Fix: Read it as the average shift relative to the omitted base group, holding other variables constant.

  • Assuming a high R-squared means the model is correct

    Fix: A high R² can come from misspecification or spurious relationships. Always check the assumptions and the economic sense.

Last-day revision: Linear Regression

  • 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.

Linear Regression practice questions

Linear Regression in other exams

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

Linear Regression: frequently asked questions

How much of the FRM Part I exam is linear regression?

GARP publishes topic-level coverage in its annual Study Guide, and the curriculum changes each year. Check the current guide for the Quantitative Analysis share. Treat regression as a core chapter either way.

Do I need to derive the OLS formulas for the exam?

No. You need to apply them and understand what they mean. Know the slope and intercept formulas, and know why squared residuals are minimised.

Can I use a financial calculator for regression questions?

Most questions give you the summary numbers or an output table, so basic arithmetic is enough. A calculator with a square root and statistics functions helps with standard errors and correlations. Practise with the model approved for the exam.

What is the difference between R-squared and adjusted R-squared?

R² never falls when you add a variable. Adjusted R² adjusts for the number of variables and can fall if the new variable adds little. Use adjusted R² to compare models with different numbers of regressors.

How do I decide if a coefficient is statistically significant?

Compute the t-statistic and compare it with the critical value for your significance level and degrees of freedom. If its absolute value is larger, reject the null that the coefficient is zero. The same decision follows if zero lies outside the confidence interval.