FRM Exam Part I · Regression with Multiple Explanatory Variables

R-squared and Adjusted R-squared in Multiple Regression

Updated 11 October 2026 · Fact-checked

R-squared is the share of total variation in the dependent variable explained by the regression: R² = ESS ÷ TSS = 1 − SSR ÷ TSS. It never falls when you add a variable. Adjusted R² = 1 − (1 − R²) × (n − 1) ÷ (n − k − 1) penalises extra regressors, so it can fall.

Understand R-squared and Adjusted R-squared

A regression tries to explain why the dependent variable moves. The total sum of squares (TSS) measures all the variation of Y around its mean. The regression splits it into two parts: the explained sum of squares (ESS), which the model captures, and the sum of squared residuals (SSR), which it leaves unexplained. So TSS = ESS + SSR.

R-squared is the fraction explained: R² = ESS ÷ TSS, or equivalently 1 − SSR ÷ TSS. It lies between 0 and 1 for a regression with an intercept. An R² of 0.40 means the model explains 40% of the variation in Y in the sample.

The problem is that R² never decreases when you add an explanatory variable, even a useless one. Adding a variable can only reduce SSR or leave it unchanged, because OLS can always set the new coefficient to zero. So a model with more variables looks better even when the extra variables add nothing real. This is a form of overfitting.

Adjusted R² fixes this by penalising the number of regressors. It divides SSR and TSS by their degrees of freedom, n − k − 1 and n − 1, where k is the number of slope variables. A new variable raises adjusted R² only if it improves the fit by more than chance would. If it does not, adjusted R² falls.

Use adjusted R² to compare models with different numbers of regressors and the same dependent variable. Neither measure proves the model is correct, unbiased or free of omitted variable bias. Also, adjusted R² is always less than or equal to R² (when k ≥ 1), and it can be negative.

Key formulas to remember

Sum of squares decomposition: TSS = ESS + SSR; TSS = Σ(Yᵢ − Ȳ)², ESS = Σ(Ŷᵢ − Ȳ)², SSR = Σ(Yᵢ − Ŷᵢ)². Holds for OLS with an intercept.
R-squared: R² = ESS ÷ TSS = 1 − SSR ÷ TSS; Share of variation in Y explained by the model.
Adjusted R-squared: Adjusted R² = 1 − [SSR ÷ (n − k − 1)] ÷ [TSS ÷ (n − 1)]; n = observations, k = number of explanatory variables (excluding the intercept).
Adjusted R-squared from R-squared: Adjusted R² = 1 − (1 − R²) × (n − 1) ÷ (n − k − 1); The form you will use most often in the exam.
Standard error of regression: SER = √[SSR ÷ (n − k − 1)]; Uses the same degrees of freedom as adjusted R².
Link to F-statistic: F = (ESS ÷ k) ÷ [SSR ÷ (n − k − 1)]; Tests whether all slope coefficients are jointly zero.

How to solve R-squared and Adjusted R-squared questions

Use this routine for any question on R², adjusted R² or the sums of squares.

1Write down what you are given: n, k, and any of TSS, ESS, SSR, R².
2Fill the gaps with TSS = ESS + SSR. Find the missing sum of squares first.
3Compute R² = ESS ÷ TSS or 1 − SSR ÷ TSS.
4Find the degrees of freedom: n − 1 for total and n − k − 1 for residuals. Check that k counts only slopes, not the intercept.
5Apply Adjusted R² = 1 − (1 − R²) × (n − 1) ÷ (n − k − 1).
6Sanity check: adjusted R² must be below R² when k ≥ 1.
7For comparison questions, check that the models use the same dependent variable and sample, then pick the higher adjusted R².

Quickest way: Shortcut: work with the unexplained fraction

When to use it: Use when the question gives R² and asks for adjusted R², or the reverse, under time pressure.

Compute 1 − R², the unexplained fraction.
Multiply it by (n − 1) ÷ (n − k − 1), a ratio always above 1.
Subtract the result from 1.
If options are close, remember adjusted R² is lower than R², which removes any option above R².
If the question asks whether adding a variable helps, compare adjusted R² before and after. Do not use R², which can only rise.

Common mistakes in R-squared and Adjusted R-squared

Using k = number of coefficients including the intercept.
Some textbooks define k as all parameters, so n − k is used instead.
Fix: In the FRM formula, k is the number of slope variables and the denominator is n − k − 1. Count only explanatory variables.
Believing a higher R² means a better model after adding variables.
R² looks like a score, and it always rises or stays equal when regressors are added.
Fix: Compare models with different numbers of regressors using adjusted R². A higher R² alone is not evidence of a better model.
Confusing ESS and SSR, since some sources use RSS for different things.
Abbreviations differ across textbooks and software.
Fix: Use the FRM convention: ESS is explained, SSR is residual. R² = ESS ÷ TSS. If a question defines terms, follow its definitions.
Thinking adjusted R² cannot be negative or can exceed R².
Students assume it is bounded between 0 and 1 like R².
Fix: Adjusted R² is at most R² when k ≥ 1 and can be negative when R² is very low relative to k and n.
Treating a high R² as proof the model is correct or that coefficients are unbiased.
Goodness of fit is mixed up with validity.
Fix: R² measures in-sample fit only. It says nothing about omitted variables, bias, causality or the significance of individual coefficients.
Comparing R² across models with different dependent variables, such as Y and ln(Y).
Both models report an R² and they look comparable.
Fix: The R² values measure variation in different quantities. Compare only models with the same dependent variable.

Worked examples

Example 1

A regression of monthly fund returns on three factors uses 60 observations. TSS = 480 and SSR = 120. Calculate R² and adjusted R².

Show the solution

ESS = TSS − SSR = 480 − 120 = 360.
R² = ESS ÷ TSS = 360 ÷ 480 = 0.75.
n = 60, k = 3, so n − 1 = 59 and n − k − 1 = 56.
Adjusted R² = 1 − (1 − 0.75) × 59 ÷ 56 = 1 − 0.25 × 1.05357.
0.25 × 1.05357 = 0.26339, so adjusted R² = 0.7366.

Answer: R² = 0.75 and adjusted R² ≈ 0.7366 (about 0.737).

Example 2

An analyst has 30 observations. Model A has 2 regressors with R² = 0.600. Model B adds 3 more regressors (5 in total) and has R² = 0.620. Which model has the higher adjusted R²?

Show the solution

Model A: n − 1 = 29, n − k − 1 = 30 − 2 − 1 = 27. Ratio = 29 ÷ 27 = 1.07407.
Adjusted R² (A) = 1 − 0.400 × 1.07407 = 1 − 0.42963 = 0.5704.
Model B: n − k − 1 = 30 − 5 − 1 = 24. Ratio = 29 ÷ 24 = 1.20833.
Adjusted R² (B) = 1 − 0.380 × 1.20833 = 1 − 0.45917 = 0.5408.
0.5704 > 0.5408, so Model A is better on this measure.

Answer: Model A has the higher adjusted R² (about 0.570 versus 0.541). The three extra variables raised R² slightly but did not justify the lost degrees of freedom.

Exam tips

Read k carefully. Questions often state 'including the intercept' or give the number of coefficients, so check before using n − k − 1.
If a question adds a variable and asks what happens to R², the answer is 'does not decrease'. For adjusted R² the answer is 'may rise or fall'.
Do the sum-of-squares algebra first. Many questions give ESS and SSR, or TSS and R², and hide the figure you need.
Use the calculator memory for the ratio (n − 1) ÷ (n − k − 1) to avoid rounding errors, then check against the options.
Watch for conceptual options that overstate R²: 'proves causality', 'guarantees no bias', or 'valid for comparing different dependent variables' are wrong.

Practice questions from Regression with Multiple Explanatory Variables

R-squared and Adjusted R-squared in other exams

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

CFA Level IIGoodness of Fit: R-squared and Adjusted R-squared

R-squared and Adjusted R-squared: frequently asked questions

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

R-squared is the proportion of variation in Y explained by the model and never falls when you add variables. Adjusted R-squared corrects for the number of regressors using degrees of freedom, so it rises only if a new variable improves the fit enough. Use adjusted R-squared to compare models of different size.

How do I calculate R-squared from ESS, TSS and SSR?

Use R² = ESS ÷ TSS, or 1 − SSR ÷ TSS. If you have only two of the three sums of squares, find the third from TSS = ESS + SSR. The result is a number between 0 and 1.

Can adjusted R-squared be negative?

Yes. If R² is small and the model has many regressors relative to the sample size, the penalty can push adjusted R² below zero. It is also never above R² when there is at least one slope variable.

Does a high R-squared mean the regression is good?

Not necessarily. R-squared measures in-sample fit only. It does not show that coefficients are unbiased, that important variables are not omitted, or that the model will forecast well.