CFA Level II Exam · Basics of Multiple Regression and Underlying Assumptions

Multiple Linear Regression Model and Coefficient Interpretation

Updated 6 October 2026 · Fact-checked

A multiple regression model explains a dependent variable using two or more independent variables: Y = b0 + b1X1 + b2X2 + ... + bkXk + ε. Ordinary least squares picks coefficients that minimise squared residuals. Each partial slope is the expected change in Y for a one-unit rise in that variable, holding the other variables constant.

Understand Multiple Linear Regression Model and Coefficients

A simple regression uses one independent variable to explain a dependent variable. A multiple regression uses two or more. In practice, an outcome such as a stock's return rarely depends on one thing. Multiple regression lets you measure several drivers together.

The model is Y = b0 + b1X1 + b2X2 + ... + bkXk + ε. Y is the dependent variable. The X variables are the independent variables. b0 is the intercept. b1 to bk are the partial slope coefficients. ε is the error term, the part of Y the model does not explain.

The coefficients are estimated by ordinary least squares (OLS). OLS chooses the intercept and slopes that make the sum of squared residuals as small as possible. A residual is the actual Y minus the fitted Y. You will not run OLS by hand on the exam. You will be given the output and asked to read it.

The key idea is the word partial. A partial slope measures the effect of one variable with all the other independent variables held constant. That is why a coefficient on X1 in a multiple regression usually differs from its slope in a simple regression of Y on X1 alone. The simple slope also picks up the effect of any related variable that was left out.

The intercept is the expected value of Y when every independent variable equals zero. It is often not meaningful in practice, because zero may be outside the range of the data. Do not give it an economic story unless the vignette does.

Key formulas to remember

Multiple regression model: Yi = b0 + b1X1i + b2X2i + ... + bkXki + εi; i indexes the observation. k is the number of independent variables.
Predicted (fitted) value: Ŷ = b̂0 + b̂1X1 + b̂2X2 + ... + b̂kXk; Use the estimated coefficients and drop the error term, which is expected to be zero.
Residual: ei = Yi − Ŷi; OLS minimises Σei², the sum of squared residuals.
Interpretation of a partial slope: bj = expected change in Y for a one-unit change in Xj, other X variables held constant; Always state the units of Y and Xj.
Degrees of freedom of the regression: Residual df = n − k − 1; n is observations and k is independent variables. The extra 1 is for the intercept.

How to solve Multiple Linear Regression Model and Coefficients questions

Use this method for any question that asks you to read or use multiple regression output.

1Identify Y and each X from the vignette. Note their units, such as percent, basis points or millions.
2Write the estimated equation with the coefficient values from the output table.
3For interpretation, state the slope as the expected change in Y per one-unit change in that X, holding the other variables constant.
4For prediction, substitute the given X values into the equation. Check that the units match the way the coefficients were estimated.
5Add the intercept last, and keep signs carefully when a coefficient is negative.
6If asked about a change, multiply the coefficient by the change in X. The intercept and other variables drop out.
7Compare your answer with the options and check that the size and sign are sensible.

Quickest way: Plug in, or multiply the change

When to use it: Use when the question asks for a predicted value or the effect of a specified change in one variable.

Underline the X values or the change in X given in the vignette.
For a level, compute b0 + b1X1 + b2X2 and so on. For a change, compute only bj × ΔXj.
Check units, for example percent versus decimal, before choosing the option.
Ignore the standard errors and t-statistics unless the question asks about significance.

Common mistakes in Multiple Linear Regression Model and Coefficients

Interpreting a slope without 'holding other variables constant'.
Students carry over the simple regression wording.
Fix: Always add that the other independent variables are held constant. This is what makes the slope partial.
Assuming the multiple regression slope equals the simple regression slope.
The same variable appears in both models.
Fix: The slopes differ when X variables are correlated with each other. Use the slope from the model in the question.
Giving the intercept an economic meaning when X = 0 is unrealistic.
The textbook definition is memorised mechanically.
Fix: Define it as the expected Y when all X equal zero, and treat it as meaningful only if zero is plausible within the data.
Using the wrong degrees of freedom.
Students forget the intercept uses one degree of freedom.
Fix: Residual df = n − k − 1. With 60 observations and 3 variables, df = 56.
Mixing units, such as entering 5% as 5 when the model uses decimals.
Vignettes often show data in percent.
Fix: Check how the variables are defined in the exhibit and enter values in the same units.
Reading a large coefficient as a more important variable.
Coefficient size depends on the units of X.
Fix: Judge importance with significance tests, not raw size. Variables in different units cannot be compared by slope alone.

Worked examples

Example 1

An analyst regresses a fund's monthly excess return (%) on market excess return (%) and a size factor (%). Estimated equation: Ŷ = 0.20 + 0.95 X1 + 0.40 X2. In one month, X1 = 2.0 and X2 = −1.5. (1) Predict the fund's excess return. (2) Interpret the coefficient 0.40. (3) What is the expected change in Y if X1 rises by 1.5 and X2 is unchanged?

Show the solution

(1) Substitute: 0.20 + 0.95 × 2.0 + 0.40 × (−1.5).
0.95 × 2.0 = 1.90. 0.40 × (−1.5) = −0.60.
Sum: 0.20 + 1.90 − 0.60 = 1.50.
(2) The slope 0.40 means that a one-percentage-point rise in the size factor is associated with an expected 0.40 percentage-point rise in the fund's excess return, holding market excess return constant.
(3) Change = 0.95 × 1.5 = 1.425. The intercept and X2 do not matter because X2 is unchanged.

Answer: (1) 1.50%. (2) 0.40 percentage points per one-point rise in the size factor, holding the market factor constant. (3) An increase of 1.425 percentage points.

Example 2

A regression of a company's annual revenue growth (%) on GDP growth (%) and the change in an industry price index (%) uses 48 observations. The simple regression of revenue growth on GDP growth alone gives a slope of 1.80. The multiple regression gives: intercept 1.10, GDP slope 1.20, price index slope 0.50. (1) What are the residual degrees of freedom? (2) Why does the GDP slope differ? (3) Predict revenue growth when GDP growth is 3% and the price index change is 4%.

Show the solution

(1) k = 2 and n = 48. Residual df = 48 − 2 − 1 = 45.
(2) The simple slope of 1.80 also captures the effect of the price index to the extent that it moves with GDP. The multiple slope of 1.20 is partial: it holds the price index constant. The two differ because the independent variables are correlated.
(3) Ŷ = 1.10 + 1.20 × 3 + 0.50 × 4.
1.20 × 3 = 3.60. 0.50 × 4 = 2.00.
Sum: 1.10 + 3.60 + 2.00 = 6.70.

Answer: (1) 45. (2) The multiple regression slope holds the price index constant, while the simple slope absorbs part of its effect because the two variables are correlated. (3) 6.70%.

Exam tips

Level II gives you regression output inside a vignette. Find the coefficient table first, then read the question.
Interpretation questions test the phrase 'holding other variables constant'. Pick the option that includes it.
For prediction, watch the units of each variable and any negative values in the vignette.
When asked about a change in one variable, multiply only that slope by the change. Do not add the intercept.
If a question shifts to significance or fit, the same output also holds standard errors, t-statistics and R-squared. Read what is asked before using them.

Multiple Linear Regression Model and Coefficients in other exams

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

Multiple Linear Regression Model and Coefficients: frequently asked questions

What is the difference between simple and multiple regression?

Simple regression has one independent variable. Multiple regression has two or more. In multiple regression each slope is partial, meaning it measures the effect of one variable with the others held constant.

How do you interpret a partial slope coefficient?

It is the expected change in the dependent variable for a one-unit increase in that independent variable, holding all other independent variables constant. Include the units of both variables in your answer.

Why does a coefficient change when I add another variable?

If the new variable is correlated with the existing one and also affects Y, the old slope was partly capturing its effect. Adding it separates the two effects, so the slope changes.

What does OLS do in multiple regression?

Ordinary least squares chooses the intercept and slopes that minimise the sum of squared residuals. Residuals are the gaps between actual and fitted values of the dependent variable.