CFA Linear Regression Notes: Slope, Intercept and R-Squared [Level 1]

CFA linear regression is the Level 1 tool that fits a straight line to two variables, so a candidate can describe the relationship, test whether it is real, and predict one variable from the other. In fact, the model has one dependent variable, one independent variable, a slope, an intercept, and an error term. Overall, this note covers the equation, R-squared, the slope test, and ANOVA, with two worked examples. For the full Level 1 Quantitative Methods set, start at the CFA notes hub.

Regression sits in Quantitative Methods, which CFA Institute weights at 6 to 9 percent in 2026 and 11 to 14 percent from February 2027. For scale, with 180 questions, that 2026 range works out to about 11 to 16 Quantitative Methods questions, as an estimate. In addition, the August 2026 Level 1 pass rate was 43 percent, from 28,145 candidates.

What is CFA linear regression?

CFA linear regression fits a line to two variables so one can predict the other. The slope shows the change in the outcome for a one unit change in the predictor, R-squared shows variation explained, and a test on the slope shows whether the link is real.

The CFA Institute 2026 reading, Simple Linear Regression, covers regression with a single independent variable. After that, from February 2027, it continues as module 10, Applications of Simple Linear Regression in Finance.

The dependent variable, Y, is the outcome the model explains or predicts. The independent variable, X, does the explaining: it predicts Y. In finance, Y could be a stock return and X the market return. Above all, the direction matters: swapping X and Y gives a different line. A cross-sectional regression compares entities at one time; a time-series regression uses different periods for one entity. Next, least squares fits the line by minimizing the sum of squared residuals. Here, a residual is the vertical distance between an observed Y value and the fitted line. Finally, residuals sum to zero, and the line passes through the mean point.

CFA linear regression equation: slope and intercept

The model is Y = b0 + b1X + error. In the estimated sample version, Y-hat = b0-hat + b1-hat times X. Here b0-hat is the estimated intercept, b1-hat is the estimated slope, and the error term is the difference between the observed Y and the value expected from the true relationship.

First, the estimated slope is the covariance of X and Y over the variance of X. In sum form, b1-hat equals the sum of (X minus X-bar) times (Y minus Y-bar), divided by the sum of (X minus X-bar) squared. As a result, the slope is the change in Y for a one unit change in X. The estimated intercept is b0-hat = Y-bar minus b1-hat times X-bar, the Y value when X equals zero.

Worked example 1: slope, intercept, prediction, and residual

For example, use five observations, market return X and stock return Y, both in percent: X is 1, 2, 3, 4, 5 and Y is 2, 3, 5, 4, 6. Here, the means are X-bar 3 and Y-bar 4. The numerator sum is 9, and the denominator sum is 10.

Step 1, slope: b1-hat = 9 divided by 10 = 0.90. Step 2, intercept: b0-hat = 4 minus 0.90 times 3 = 4 minus 2.70 = 1.30. The fitted line is Y-hat = 1.30 + 0.90X. In plain words, the slope means the stock return rises by 0.90 percent for each one percent rise in the market return. Similarly, the intercept means the predicted stock return is 1.30 percent when the market return is zero.

Then, step 3, prediction: at X = 6, Y-hat = 1.30 + 0.90 times 6 = 6.70 percent. After that, step 4, residual: at X = 3, actual Y is 5 and predicted Y is 4.00, so the residual is 5 minus 4.00 = +1.00 percent. Overall, the five predicted values are 2.20, 3.10, 4.00, 4.90, and 5.80, and the five residuals are minus 0.20, minus 0.10, +1.00, minus 0.90, and +0.20, which sum to zero. In practice, residual plots check the model: random scatter supports the line, a curve points at a linearity problem, and a fan shape points at changing variance. Also, a prediction interval for a new observation is wider than a confidence interval for the mean.

CFA linear regression: R-squared, the standard error of estimate, and ANOVA

Total variation in Y is SST. It splits into SSR, the variation explained by the line, and SSE, the variation left unexplained: SST = SSR + SSE. In short, R-squared is SSR divided by SST, which also equals 1 minus SSE divided by SST. It shows the share of Y variation that X explains, and it runs from 0 to 1. For instance, in simple regression, R-squared equals the square of the correlation coefficient, r. In addition, the slope equals r times the standard deviation of Y divided by the standard deviation of X, and the slope and the correlation always have the same sign.

See the probability note for how correlation is defined. However, a high R-squared does not prove that X causes Y, and a low R-squared does not prove that X is unrelated to Y. Next, SEE, the standard error of estimate, is the square root of SSE over (n minus 2), the standard deviation of the residuals.

Worked example 2: R-squared, SEE, and the slope test

To continue, use the same five observations. Here, SST is 10.00, SSR is 8.10, and SSE is 1.90: 8.10 plus 1.90 equals 10.00. As a result, R-squared is 8.10 divided by 10.00 = 0.81. In other words, the line explains 81 percent of the variation in the stock return. Also, the correlation is 0.90, and 0.90 squared is 0.81.

First, SEE is the square root of 1.90 over (5 minus 2), the square root of 0.6333, equal to 0.80 percent. Next, the standard error of the slope is 0.80 over the square root of 10, about 0.25. The t-statistic for the slope is (0.90 minus 0) over the standard error of the slope, equal to 3.58, using the unrounded standard error of 0.2517.

Then, with n minus 2 = 3 degrees of freedom, the two-tailed critical t at the 5 percent level is 3.18. As a result, the calculated t of 3.58 is larger in absolute value, so reject the null hypothesis that the slope is zero.

Finally, ANOVA puts the same numbers in order: regression SSR 8.10 (df 1), mean square 8.10; error SSE 1.90 (df 3), mean square 0.6333; total SST 10.00 (df 4). The F-statistic is 8.10 over 0.6333 = 12.79. As a check, in simple regression, F equals the square of the slope t-statistic, and the square of the unrounded t-statistic (3.576) is 12.79.

The four assumptions in plain words

To start, simple linear regression rests on four assumptions about the error term: linearity, the true relationship is a straight line; homoskedasticity, the error variance is the same for all X; independence, the error terms are independent; and normality, the error terms follow a normal distribution. In short, candidates need normality for inference, not for the least squares estimates. By contrast, heteroskedasticity means variance is not constant: the slope stays unbiased, but its standard error is wrong, so the t-test misleads.

CFA linear regression hypothesis test on the slope

In the exam, the usual test asks whether the slope differs from zero. Start with the null, H0: b1 = 0. Next, the alternative is Ha: b1 is not equal to 0. Finally, the t-statistic is (b1-hat minus the hypothesized b1) over the standard error of b1-hat, with n minus 2 degrees of freedom. As a rule, reject the null at the 5 percent level, two-tailed, when the absolute t beats the critical value. In short, a rejection says the data support a linear link, not a causal one.

Exam traps

• Correlation is not causation. A strong link is not proof of cause.
• Do not extrapolate. Predict only inside the sample range of X.
• R-squared always rises with more variables. Adding a variable can never reduce it.
• Heteroskedasticity damages inference, not the slope. The slope stays unbiased, but standard errors fail.
• Outliers and leverage move the line. One extreme point can change the slope.
• Use n minus 2 degrees of freedom. The slope test uses n minus 2, not n minus 1.

One-page revision box

• Model: Y = b0 + b1X + error. Y is dependent, X is independent.
• Slope: b1-hat = Cov(X, Y) / Var(X). Change in Y for one unit of X.
• Intercept: b0-hat = Y-bar minus b1-hat times X-bar. Y when X is zero.
• Residual: e = Y minus Y-hat. Residuals sum to zero.
• R-squared: SSR / SST = 1 minus SSE / SST. Share of variation explained, 0 to 1.
• SEE: Square root of SSE / (n minus 2). Typical miss from the line.
• Slope test: t = (b1-hat minus b1) / SE(b1-hat), df = n minus 2. Reject H0: b1 = 0 when absolute t exceeds the critical value.

Finally, before the exam, work through both worked examples by hand, then change one Y value and watch the slope, R-squared, and t move. For time value of money and hypothesis testing, see the CFA quantitative methods note. For the full Level 1 set, return to the CFA notes hub.

Sources

Level I exam page, CFA Institute: https://www.cfainstitute.org/programs/cfa-program/candidate-resources/level-i-exam

Simple Linear Regression refresher reading, CFA Institute 2026 Curriculum: https://www.cfainstitute.org/insights/professional-learning/refresher-readings/2026/simple-linear-regression

August 2026 Level I results, CFA Institute press release: https://www.cfainstitute.org/about/press-room/2026/cfa-institute-reports-results-cfa-program-level-i-testing-august-2026