Simple Linear Regression at CFA Level 2: Reading the Output, Not Just the Formula

Regression at Level 2 is presented as output rather than as a calculation. You are given an ANOVA table and a coefficients table, and asked to interpret them. Candidates who memorised formulas without learning where each number sits in the output lose marks they had the knowledge to earn.

The model

Y_i = b₀ + b₁X_i + ε_i

• Y — the dependent variable, the thing being explained
• X — the independent variable, the explanatory factor
• b₀ — the intercept: the predicted value of Y when X is zero
• b₁ — the slope: the change in Y for a one-unit change in X
• ε — the error term, capturing everything the model does not explain

Coefficients are estimated by ordinary least squares, which chooses the line minimising the sum of squared vertical distances between the observed points and the fitted line. Squaring, rather than taking absolute values, is why large residuals dominate the fit.

The variation decomposition

Everything in the ANOVA table follows from one identity:

SST = SSR + SSE

• SST — total sum of squares. Total variation in Y around its own mean.
• SSR — regression sum of squares. The portion the model explains.
• SSE — sum of squared errors. The portion left unexplained.

The most common confusion in this topic is the naming: SSR is explained variation and SSE is unexplained variation. Both begin with S and both sound like they might refer to error. Fix this before anything else, because every ratio built on top depends on it.

The four numbers to extract

1. Coefficient of determination (R-squared)

R² = SSR / SST

The proportion of variation in Y explained by X. An R² of 0.64 means 64% of the variation in the dependent variable is explained by the model.

In simple linear regression only, R² equals the square of the correlation coefficient. So an R² of 0.64 implies a correlation of ±0.80 — with the sign taken from the slope. This shortcut does not extend to multiple regression.

2. Standard error of estimate

SEE = √(SSE / (n − 2)) = √MSE

The typical size of a residual, expressed in the units of Y. A lower SEE means observations sit closer to the fitted line.

The n − 2 denominator reflects two estimated parameters, the intercept and the slope. This same n − 2 recurs as the degrees of freedom for the slope test.

3. The t-test on the slope

This is the question that appears most often. The usual null is that the slope is zero — that X has no explanatory power.

t = (b₁ − B₁) / s_b1, with n − 2 degrees of freedom

Where B₁ is the hypothesised value, usually zero, and s_b1 is the standard error of the coefficient, given in the output.

Compare against the critical value, or use the reported p-value directly: reject the null when the p-value falls below your significance level. The same hypothesis testing framework from Level 1 applies unchanged.

Watch the hypothesised value. Questions sometimes test whether the slope differs from 1 rather than from 0 — common when the regression is of one return series on another and the question concerns beta. Subtracting the wrong B₁ gives a wrong t-statistic from correct inputs.

4. The F-test

F = MSR / MSE, with 1 and n − 2 degrees of freedom

It tests whether the model as a whole has explanatory power. In simple regression there is only one independent variable, so the F-test and the t-test on the slope test the same proposition — and F = t². If the reported t-statistic is 2.5, F is 6.25.

That identity is worth knowing: it lets you check one figure against the other, and questions occasionally supply one and ask for the other.

A worked reading

Output: n = 42, SST = 850, SSR = 578, slope 1.35 with standard error 0.28.

SSE = 850 − 578 = 272
R² = 578 / 850 = 0.68 — 68% of variation explained; correlation ≈ +0.82
MSE = 272 / 40 = 6.8, so SEE = √6.8 = 2.61
t on the slope = 1.35 / 0.28 = 4.82, with 40 degrees of freedom — comfortably significant
F = 4.82² = 23.2

Every figure came from four inputs and the SST identity.

The assumptions

Four assumptions underpin the model, and violations are examined:

• Linearity — the relationship between X and Y is linear in the parameters
• Homoskedasticity — the variance of the errors is constant across observations
• Independence — errors are uncorrelated with each other
• Normality — errors are normally distributed

Two violations matter most in practice. Heteroskedasticity — error variance changing systematically, often as X grows — leaves coefficients unbiased but makes standard errors unreliable, so t-statistics cannot be trusted. Serial correlation — errors correlated across observations, common in time series — has the same consequence and typically makes results look more significant than they are.

The pattern to remember: these violations do not usually bias the coefficient estimates. They corrupt the inference. A question asking what heteroskedasticity does to a regression is asking about standard errors, not about the slope.

Prediction intervals

A predicted value from a regression is a point estimate, and questions often ask for an interval around it. The interval widens as the value of X moves away from the mean of X, because the fitted line is estimated with least certainty at its extremes.

This is the statistical case against extrapolating far beyond the observed data: the model may be well fitted within its range and close to worthless outside it.

Exam checklist

• SSR is explained variation; SSE is unexplained
• R² = SSR / SST, and in simple regression equals the squared correlation
• SEE = √MSE, with n − 2 in the denominator
• Slope t-test uses n − 2 degrees of freedom — check the hypothesised value
• In simple regression, F = t²
• Heteroskedasticity and serial correlation corrupt standard errors, not the coefficients
• Prediction intervals widen away from the mean of X