FRM Exam Part I · Linear Regression

Goodness of Fit: R-squared, ESS, TSS and SSR

Updated 11 October 2026 · Fact-checked

Goodness of fit shows how much of the variation in Y a regression explains. Total sum of squares (TSS) splits into explained (ESS) and residual (SSR): TSS = ESS + SSR. R² = ESS ÷ TSS = 1 − SSR ÷ TSS. Adjusted R² penalises extra regressors. SER = √(SSR ÷ (n − k − 1)).

Understand Goodness of Fit: R-squared, ESS, TSS, SSR

A regression tries to explain why Y moves around its mean. The total variation in Y is measured by the total sum of squares (TSS): Σ(Yᵢ − Ȳ)². This is the starting point for every goodness-of-fit measure.

The fitted line splits that variation in two. The explained sum of squares (ESS) is Σ(Ŷᵢ − Ȳ)², the variation captured by the model. The sum of squared residuals (SSR) is Σ(Yᵢ − Ŷᵢ)², the variation left over. With an intercept in the model, TSS = ESS + SSR. Watch the labels: some texts call the residual part RSS and the explained part RegSS or SSE. Read the definition in the question, not the abbreviation.

R² is the share of total variation explained: ESS ÷ TSS. It lies between 0 and 1 when an intercept is included. In simple regression with one explanatory variable, R² equals the square of the sample correlation between X and Y. It does not say that the model is correct or that the slope is significant.

Adding a regressor never lowers R², even if the variable is useless. Adjusted R² fixes this by correcting for degrees of freedom, so it can fall when a new variable adds little. It can also be negative. The standard error of regression (SER) measures the typical size of residuals in the units of Y. Smaller is better, and it is useful for building prediction intervals.

Key formulas to remember

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.

How to solve Goodness of Fit: R-squared, ESS, TSS, SSR questions

Use this routine for any goodness-of-fit question. First decide which sums of squares you are given and which measure is asked.

1Write down n and k (number of slope coefficients, not counting the intercept).
2Identify what is given: TSS, ESS, SSR, R², correlation or SER. Check the labels used for explained and residual parts.
3Fill the gap with TSS = ESS + SSR if one of the three is missing.
4Compute R² = ESS ÷ TSS or 1 − SSR ÷ TSS.
5For adjusted R², use 1 − (1 − R²)(n − 1) ÷ (n − k − 1).
6For SER, divide SSR by n − k − 1 and take the square root. Do not use n alone.
7In simple regression, convert between R² and correlation by squaring or taking the square root, then set the sign from the slope.
8Sanity check: R² between 0 and 1, adjusted R² ≤ R², SER in the units of Y.

Quickest way: Three-number shortcut

When to use it: When the question gives two of TSS, ESS, SSR, or an R², and asks for another measure.

Get R² first. Everything else follows from it.
If you have R² and TSS, then SSR = TSS × (1 − R²).
Adjusted R² only needs R², n and k.
For SER you need SSR and the degrees of freedom n − k − 1.
Eliminate options where adjusted R² exceeds R² or where R² is outside 0 to 1.

Common mistakes in Goodness of Fit: R-squared, ESS, TSS, SSR

Dividing SSR by n instead of n − k − 1 when computing SER.
It looks like the usual variance formula.
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.
Texts differ: RSS can mean residual or regression sum of squares.
Fix: Work from the definition. Explained means fitted values versus the mean. Residual means actual versus fitted.
Believing a higher R² always means a better model.
R² rises mechanically when variables are added.
Fix: Compare models with adjusted R², and also consider significance, theory and overfitting.
Using √R² as the correlation without checking the sign.
The square root only gives the magnitude.
Fix: Give the correlation the same sign as the slope coefficient. This works only in simple regression.
Using k = number of coefficients including the intercept in the adjusted R² formula.
Confusion between parameters and regressors.
Fix: In n − k − 1, k counts only the slope coefficients.
Treating R² as a measure of causality or of correct specification.
A high fit feels like proof.
Fix: R² only describes in-sample fit. A spurious regression can show a very high R².

Worked examples

Example 1

A regression of a fund's monthly excess returns on the market's excess returns uses n = 62 observations. TSS = 480 and SSR = 120. Compute R² and the standard error of regression.

Show the solution

ESS = TSS − SSR = 480 − 120 = 360.
R² = ESS ÷ TSS = 360 ÷ 480 = 0.75.
This is simple regression, so k = 1 and degrees of freedom = 62 − 1 − 1 = 60.
SER = √(SSR ÷ 60) = √(120 ÷ 60) = √2 = 1.4142.

Answer: R² = 0.75 and SER ≈ 1.414 (in units of the returns, here percentage points if returns are in percent).

Example 2

A multiple regression with n = 40 observations and k = 4 regressors has R² = 0.60. Compute adjusted R². If a fifth regressor is added and R² rises to 0.61, does adjusted R² rise?

Show the solution

Four regressors: degrees of freedom = 40 − 4 − 1 = 35.
Adjusted R² = 1 − (1 − 0.60) × (39 ÷ 35) = 1 − 0.40 × 1.114286 = 1 − 0.445714 = 0.5543.
Five regressors: degrees of freedom = 40 − 5 − 1 = 34.
Adjusted R² = 1 − (1 − 0.61) × (39 ÷ 34) = 1 − 0.39 × 1.147059 = 1 − 0.447353 = 0.5526.
0.5526 is below 0.5543, so adjusted R² falls.

Answer: Adjusted R² with four regressors is about 0.554. After adding the fifth regressor it falls to about 0.553, so the extra variable does not improve the model on this measure.

Exam tips

Questions often hand you two of TSS, ESS, SSR. Fill in the third first, then compute.
Check how the question defines ESS and RSS before using any formula.
Expect a conceptual item: adding a regressor never lowers R² but can lower adjusted R².
In simple regression, a quick conversion between R² and correlation is a common trap. Check the sign.
Carry at least four decimals through adjusted R² calculations, because answer options can be close.

Practice questions from Linear Regression

Goodness of Fit: R-squared, ESS, TSS, SSR: frequently asked questions

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

R² is the share of variation in Y explained by the model and never falls when you add a regressor. Adjusted R² corrects for degrees of freedom, so it rises only if the new variable improves the fit enough to justify its cost. It is the better measure for comparing models with different numbers of regressors.

How is R-squared related to correlation in simple regression?

With one explanatory variable and an intercept, R² equals the square of the sample correlation between X and Y. So a correlation of −0.8 gives R² = 0.64. This link does not hold in multiple regression.

How do you calculate the standard error of regression?

Divide SSR by n − k − 1 and take the square root. In simple regression the divisor is n − 2. The result is in the same units as Y and estimates the typical residual size.

Can adjusted R-squared be negative?

Yes. If the model explains very little and has many regressors, the correction can push adjusted R² below zero. R² itself stays between 0 and 1 when an intercept is included.