Updated for the 2026-2027 CFA® Level I curriculum.
A scatterplot can support many possible regression lines, but ordinary least squares gives you a consistent rule for choosing one. It selects the line with the smallest total squared residuals. For CFA Level I, you should understand what a residual represents, why residuals are squared, and how the criterion determines the estimated intercept and slope.
Quick Answer
The least squares criterion selects the regression line that minimizes the sum of squared residuals. A residual is the difference between an observed value and the value predicted by the line. Squaring prevents positive and negative residuals from canceling and gives larger prediction errors more influence over the fitted line.
Key Takeaways: Least Squares Criterion
A residual equals the observed value minus the predicted value.
A point above the fitted line has a positive residual, while a point below it has a negative residual.
The least squares method adds the squared residuals across all observations.
Ordinary least squares, or OLS, chooses the intercept and slope that produce the smallest possible squared-residual total.
Squaring prevents residuals with opposite signs from canceling.
Squaring also gives large misses more weight than small misses.
When an OLS regression includes an intercept, its residuals sum to zero.
The fitted line also passes through the point defined by the sample means of X and Y.
A distant observation can affect the fitted line considerably because its residual is squared.
What You Need to Know for CFA Level I
For this topic, focus on the logic behind the least squares criterion rather than the mathematical derivation of the coefficient estimates. You should be able to calculate a residual, square it, and compare candidate regression lines using their total squared residuals.
You should also understand why the raw sum of residuals is unsuitable as the fitting criterion. Positive and negative misses can offset each other, allowing a poorly fitted line to produce a small or even zero total. Squaring each residual before adding them prevents that cancellation.
Two properties of the OLS line are also worth remembering. When the model includes an estimated intercept, the fitted residuals sum to zero, and the regression line passes through the sample mean point .
What Is the Least Squares Criterion?
The least squares criterion is the rule used to select the estimated intercept and slope in a regression model. It chooses the coefficient values that minimize the total squared vertical distance between the observed values and the fitted regression line.
Ordinary least squares is the estimation method that applies this rule. The intercept determines where the line crosses the Y-axis, while the slope determines how much the predicted value of Y changes when X increases by one unit.
OLS estimates both coefficients together. Changing either coefficient changes the predicted values and residuals for every observation in the sample.
What Is a Residual?
A residual measures how far the regression line misses an observed value. For each observation, subtract the predicted value of the dependent variable from its actual value.
A positive residual means the observed value sits above the fitted line. A negative residual means it sits below the line. A residual of zero means the predicted and observed values are equal.
Residuals measure vertical differences because simple linear regression predicts Y from a given value of X.
Why Are Residuals Squared?
Squaring residuals prevents positive and negative values from canceling. For example, residuals of and sum to zero, even though both observations are far from the fitted line. Once squared, they contribute to the objective.
Squaring also gives larger errors more influence. A residual of contributes , while a residual of contributes . Doubling the size of the residual therefore quadruples its contribution to the total.
This weighting helps the fitted line avoid large prediction errors. It also makes OLS sensitive to observations that sit far from the rest of the data.
How Does Ordinary Least Squares Choose the Regression Line?
Ordinary least squares treats regression as a minimization problem. It considers the possible intercept and slope combinations and selects the pair that produces the smallest sum of squared residuals.
Where:
= residual for observation
= observed value of the dependent variable
= predicted value of the dependent variable
= estimated intercept
= estimated slope
= observed value of the independent variable
= number of observations
The expression is the residual written using the estimated regression equation. Since the intercept and slope affect every predicted value, changing either coefficient changes the full set of residuals.
The OLS estimates are the values of and that leave no other possible line with a smaller squared-residual total.
What Properties Does the OLS Fitted Line Have?
Two useful properties apply when a simple linear regression includes an estimated intercept.
First, the residuals from the OLS-fitted line sum to zero.
The positive and negative residuals balance exactly. This makes the residual sum a useful calculation check, but it does not measure how closely the line fits the observations.
Second, the fitted regression line passes through the sample mean point .
Substituting the sample mean of X into the fitted equation therefore produces the sample mean of Y.
Both properties depend on including an intercept in the model. A regression forced through the origin does not necessarily have residuals that sum to zero or a fitted line that passes through
Worked Example: Comparing Two Candidate Lines
A junior analyst has four observations of a company’s annual revenue growth (, in percent) and number of new product launches ().
X | Y |
|---|---|
1 | 3 |
2 | 4 |
3 | 6 |
4 | 7 |
She compares two candidate regression lines.
Line A:
Line B:
The sample means are and . Both candidate lines pass through this mean point, and the residuals for each line sum to zero. Comparing the raw residual totals would therefore give the analyst no basis for choosing between them.
Line A residuals
X | Y | Predicted | Residual | Squared Residual |
|---|---|---|---|---|
1 | 3 | 2.9 | 0.1 | 0.01 |
2 | 4 | 4.3 | -0.3 | 0.09 |
3 | 6 | 5.7 | 0.3 | 0.09 |
4 | 7 | 7.1 | -0.1 | 0.01 |
Sum | 0.0 | 0.20 |
Line B residuals
X | Y | Predicted | Residual | Squared Residual |
|---|---|---|---|---|
1 | 3 | 3.2 | -0.2 | 0.04 |
2 | 4 | 4.4 | -0.4 | 0.16 |
3 | 6 | 5.6 | 0.4 | 0.16 |
4 | 7 | 6.8 | 0.2 | 0.04 |
Sum | 0.0 | 0.40 |
Line A has a sum of squared residuals of 0.20, compared with 0.40 for Line B. The least squares criterion therefore prefers Line A.
For this sample, Line A is the OLS regression line. No other combination of intercept and slope can produce a squared-residual total below 0.20.
The example also shows why the residual sum cannot measure fit. Both lines have residuals that sum to zero, but Line A stays closer to the observations overall.
Common Exam Traps
Minimizing the wrong quantity. Ordinary least squares minimizes the sum of squared residuals, not the sum of the residuals.
Reversing the residual formula. A residual is observed minus predicted. Reversing the order changes its sign.
Judging fit from the residual sum. Positive and negative residuals can cancel, so a zero total does not prove that a line fits the data well.
Confusing OLS with least absolute deviations. OLS squares each residual. An absolute-deviation method uses the absolute value instead and may select a different fitted line.
Ignoring the weight placed on large residuals. A large residual contributes disproportionately to the objective because it is squared.
Applying OLS properties without checking for an intercept. Residuals sum to zero and the fitted line passes through when the model includes an estimated intercept.
Practice Question
An analyst fits two candidate lines to the same four-observation sample and records the residuals from each.
Observation | Line 1 Residual | Line 2 Residual |
|---|---|---|
1 | -1.2 | -0.9 |
2 | 0.8 | 0.3 |
3 | 0.6 | 1.1 |
4 | -0.2 | -0.5 |
Which line better satisfies the least squares criterion?
Line 1, because its largest residual appears in only one observation.
Line 2, because its sum of squared residuals is smaller.
Neither, because both sets of residuals sum to zero.
Solution
Correct Answer: B.
Square the residuals for each line and add the results.
Line 2 has the smaller sum of squared residuals, so it provides the better fit under the least squares criterion.
Option A. focuses on one residual rather than the total across all observations. The criterion requires every squared residual to be included.
Option C. identifies a property shared by the two candidate lines, but the raw residual sum does not measure fit. The squared totals are what distinguish the lines.
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FAQs About Least Squares Criterion
What does the least squares criterion minimize?
The least squares criterion minimizes the sum of squared residuals. Each residual is the difference between an observed value and the value predicted by the regression line.
The estimated intercept and slope that produce the smallest total are the ordinary least squares estimates.
Why are regression residuals squared?
Residuals are squared to prevent positive and negative values from canceling. Squaring also places more weight on large prediction errors than on small ones.
For example, a residual of contributes four times as much to the total as a residual of , because and .
What is the difference between a residual and an error term?
An error term represents the difference between an observed value and the true population regression line. The true population relationship is unknown, so its error terms cannot be observed directly.
A residual is calculated from the estimated sample regression line. Analysts use residuals to assess how well the fitted model describes the sample data.
Is ordinary least squares sensitive to outliers?
Yes. Squaring means a large residual contributes considerably more to the objective than a small residual. A distant observation can therefore pull the fitted regression line toward itself.
This sensitivity is one reason analysts review residual plots and unusual observations when evaluating a regression model.