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FRM Part I · FRM Exam Part I

Regression with Multiple Explanatory Variables for FRM Part 1

Multiple regression explains one dependent variable using two or more explanatory variables, estimated by ordinary least squares (OLS). To solve questions, read the coefficients, test them with t-statistics, test groups of them with the F-statistic, judge fit with adjusted R², and check for multicollinearity, heteroskedasticity and omitted variable bias.

What this chapter covers

This chapter extends simple regression to models with several explanatory variables: Y = β0 + β1X1 + β2X2 + ... + βkXk + ε. Each slope coefficient measures the change in Y for a one-unit change in that variable, holding the others constant. You learn how OLS estimates the coefficients, what assumptions make those estimates reliable, and how to test and interpret the results.

The chapter has two halves. The first half is mechanical: t-tests, confidence intervals, the F-statistic, R² and adjusted R². The second half is diagnostic: what goes wrong when variables are highly correlated, when error variance is not constant, when a relevant variable is left out, or when categorical information enters through dummy variables.

It builds directly on probability, hypothesis testing and simple linear regression earlier in Quantitative Analysis. It also feeds later chapters. Time series, volatility models, factor models and risk estimation in Valuation and Risk Models all rely on regression logic. Exam questions often give you a regression output table and ask you to read it, so the skill is interpretation as much as calculation.

Regression is core to the Quantitative Analysis topic, and FRM Part I is 100 multiple-choice questions in 4 hours, so quick and accurate reading of output tables saves time. Many questions are short and formula-based: compute a t-statistic, compare it to a critical value, or adjust R². Others are conceptual, asking what a violated assumption does to estimates or standard errors. Once you know the patterns, these are some of the most predictable marks in the quantitative section. The ideas also reappear in later chapters, so time invested here pays off twice.

Regression with Multiple Explanatory Variables: topics in the order to study them

  1. 1Multiple Regression Model and OLS AssumptionsEverything else depends on knowing what the model says, what OLS does and which assumptions make the results valid.
  2. 2Hypothesis Testing and Confidence Intervals for CoefficientsThis applies the t-statistic to one coefficient at a time, the most common calculation in the chapter.
  3. 3Joint Hypothesis Tests and the F-StatisticIt extends single-coefficient tests to several restrictions at once, so you need the t-test logic first.
  4. 4R-squared and Adjusted R-squaredFit measures are easier once you understand sums of squares, which also appear in the F-statistic.
  5. 5MulticollinearityThis is the first diagnostic problem, and it shows up in the output as high R² with weak individual t-statistics.
  6. 6Heteroskedasticity and Omitted Variable BiasThese are further assumption failures; you can now judge their effect on coefficients and standard errors.
  7. 7Dummy Variables and Model SpecificationThis closes the chapter by combining interpretation and design: how to build a sensible model and read its coefficients.

How to prepare Regression with Multiple Explanatory Variables

Aim for fluency with regression output tables. Calculation steps are short; the marks are lost on interpretation and on mixing up similar concepts.

  1. Write the model and the OLS assumptions from memory, then note what each violation causes: biased coefficients, wrong standard errors, or both.
  2. Practise the t-statistic: t = (estimated coefficient − hypothesized value) ÷ standard error. Compare it with the critical value and build confidence intervals as coefficient ± critical value × standard error.
  3. Learn the F-statistic in both forms: for the overall model, F = (ESS ÷ k) ÷ (SSR ÷ (n − k − 1)), and for restrictions, compare restricted and unrestricted models. Know the degrees of freedom.
  4. Compute adjusted R² with: adjusted R² = 1 − [(n − 1) ÷ (n − k − 1)] × (1 − R²). Check by hand that it can fall when you add a useless variable.
  5. Make a one-page table of multicollinearity, heteroskedasticity and omitted variable bias: symptom, effect, detection and remedy.
  6. Do timed practice questions using regression output tables. Use a calculator for the arithmetic, but first set up the formula on paper.
  7. Revise dummy variables by interpreting coefficients: the intercept shift for the group and the base category left out to avoid the dummy variable trap.

Common mistakes in Regression with Multiple Explanatory Variables

  • Treating a significant F-test as proof that every coefficient is significant.

    Fix: Remember that F tests the joint null that the coefficients are all zero. One or more could still have insignificant t-statistics.

  • Using n − k instead of n − k − 1 for degrees of freedom.

    Fix: Count the intercept every time. With k explanatory variables, the residual degrees of freedom are n − k − 1.

  • Saying multicollinearity biases the coefficients.

    Fix: State that high multicollinearity leaves OLS unbiased but inflates standard errors, which weakens t-statistics.

  • Choosing a model because it has the highest R².

    Fix: Compare models using adjusted R² and economic sense, and remember neither measure proves the model is correctly specified.

  • Mixing up the effects of heteroskedasticity and omitted variable bias.

    Fix: Heteroskedasticity affects standard errors and inference. Omitted variable bias affects the coefficients themselves.

  • Misreading dummy variable coefficients or including a dummy for every category.

    Fix: Interpret each dummy coefficient as the difference from the base group, and use one fewer dummy than categories when there is an intercept.

Last-day revision: Regression with Multiple Explanatory Variables

  • Slope coefficient = change in Y for a one-unit change in that X, holding the other variables constant.
  • OLS chooses coefficients that minimize the sum of squared residuals.
  • t = (estimate − hypothesized value) ÷ standard error; degrees of freedom = n − k − 1.
  • Confidence interval = estimate ± critical value × standard error.
  • The F-test checks whether several coefficients are jointly zero; a single t-test cannot do this.
  • Overall F = (ESS ÷ k) ÷ (SSR ÷ (n − k − 1)).
  • R² never falls when you add a variable; adjusted R² can fall.
  • Adjusted R² = 1 − [(n − 1) ÷ (n − k − 1)] × (1 − R²).
  • Perfect multicollinearity makes OLS impossible; high multicollinearity inflates standard errors but coefficients stay unbiased.
  • Classic sign of multicollinearity: high R² and significant F-test, but insignificant individual t-statistics.
  • Heteroskedasticity leaves coefficients unbiased but makes the usual standard errors unreliable; robust standard errors are a common fix.
  • Omitted variable bias arises when the left-out variable affects Y and is correlated with an included X.
  • With a categorical variable of m categories, use m − 1 dummies and leave one as the base.

Regression with Multiple Explanatory Variables practice questions

Regression with Multiple Explanatory Variables in other exams

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

Regression with Multiple Explanatory Variables: frequently asked questions

How much of FRM Part I is regression?

GARP does not make a per-chapter mark count public that you can rely on. Regression sits within Quantitative Analysis, one of four topics in the 100-question exam, and its ideas return in later chapters. Treat it as a core area.

Do I need to calculate regression coefficients by hand?

Usually no. You are more likely to be given output or summary figures and asked to compute test statistics, intervals or adjusted R², or to interpret the results. Know the formulas well enough to apply them quickly.

What is the difference between R² and adjusted R²?

R² measures the share of variation in Y explained by the model and never falls when variables are added. Adjusted R² penalizes extra variables through the degrees of freedom, so it can fall if a new variable adds little. Use it to compare models with different numbers of variables.

How do I tell multicollinearity from heteroskedasticity?

Multicollinearity concerns correlation among the explanatory variables and shows up as large standard errors despite a good overall fit. Heteroskedasticity concerns non-constant variance of the errors, often seen in residuals that spread out as a variable grows. They are different problems with different fixes.