Updated for the 2026-2027 CFA® Level I curriculum.
A regression model can produce coefficients, test statistics, and predictions even when its underlying assumptions do not fit the data well. Residual analysis helps you check whether the model’s errors behave as expected.
For CFA Level I, you should know the four assumptions of simple linear regression, calculate and interpret a residual, and recognize common warning patterns in a residual plot.
Quick Answer
Simple linear regression assumes a linear relationship, constant error variance, independent errors, and normally distributed errors. A residual measures the difference between an observed value and the value predicted by the regression. Random scatter around zero generally supports the model, while curvature, funneling, repeated sequences, or unusual observations may indicate an assumption violation.
Key Takeaways About Linear Regression Assumptions
The CFA Level I framework uses four core assumptions: linearity, homoskedasticity, independence, and normality.
A residual is the observed value minus the value predicted by the regression model.
Positive residuals occur when observed values sit above the fitted regression line.
Negative residuals occur when observed values sit below the fitted regression line.
A well-behaved residual plot shows random scatter around zero with a reasonably stable vertical spread.
Curvature suggests that a straight-line model may not capture the relationship.
A funnel shape suggests heteroskedasticity, or changing error variance.
Runs or repeated sequences may suggest correlated errors.
An unusual residual should be investigated before the analyst changes or rejects the model.
What You Need to Know for CFA Level I
For CFA Level I, focus on:
Describing the four assumptions underlying simple linear regression.
Explaining the difference between an unobservable error term and an observable residual.
Calculating a residual from an observed and predicted value.
Identifying the variables shown on a residual plot.
Recognizing curvature, funneling, sequencing, and unusual observations.
Connecting each residual pattern with the assumption it may violate.
Explaining how violations can affect coefficient interpretation, standard errors, hypothesis tests, and predictions.
The official learning outcome asks candidates to explain the assumptions and use residuals and residual plots to identify possible violations.
What Are the Assumptions of Simple Linear Regression?
The CFA framework presents four assumptions behind the classic simple linear regression model.
Assumption | What It Means | Residual-Plot Warning |
|---|---|---|
Linearity | The dependent and independent variables have a linear relationship. | A curved or systematic pattern |
Homoskedasticity | The error term has the same variance across observations. | A funnel or changing vertical spread |
Independence | Error terms are uncorrelated across observations. | Runs, cycles, or repeated sequences |
Normality | Error terms follow a normal distribution. | Extreme observations or a clearly non-normal residual distribution |
Linearity
Linearity means the relationship between the dependent variable and independent variable can be represented by a straight-line model.
When the relationship curves, a linear regression tends to overpredict in some areas and underpredict in others. These alternating errors create a curved pattern in the residual plot.
Homoskedasticity
Homoskedasticity means the variance of the error term remains constant across observations.
A model may have small residuals at low fitted values and much larger residuals at high fitted values. This changing spread is called heteroskedasticity and often appears as a funnel or cone.
Under the other regression assumptions, ordinary least squares coefficient estimates may remain unbiased. However, heteroskedasticity can make the estimated standard errors unreliable, which affects t-tests, F-tests, confidence intervals, and prediction intervals.
Independence
Independence means one error term provides no information about another error term.
This assumption is especially relevant when observations are arranged over time. Residuals that remain positive for several periods, turn negative for several periods, or alternate in a regular cycle may indicate correlated errors.
Dependent errors can distort estimated standard errors and weaken the reliability of the model’s statistical inference.
Normality
Normality means the regression error terms follow a normal distribution.
This assumption supports the use of t-tests, F-tests, and interval estimates, particularly in smaller samples. It applies to the errors around the regression line, rather than requiring either the dependent variable or independent variable to be normally distributed.
Study note: Some general statistics resources list five or more regression assumptions because they separate conditions such as zero conditional mean from the four classic assumptions. For CFA Level I, use the four-assumption framework shown above.
What Is a Residual in Regression?
A residual measures the vertical distance between an observed data point and the estimated regression line.
The simple linear regression model can be written as:
Where:
= observed value of the dependent variable for observation
= value of the independent variable for observation
= population intercept
= population slope coefficient
= unobservable population error term
After estimating the model, the residual is calculated as:
Where:
= residual for observation
= observed value of the dependent variable
= value predicted by the estimated regression model
For example, suppose a model predicts a return of 6%, while the observed return is 8%.
The residual is positive because the observed return sits above the fitted regression line.
If the observed return were 4%, the residual would be:
The negative residual would place the observation below the fitted line.
Error Term vs Residual
The error term and residual describe similar distances, but they refer to different regression lines.
The error term measures the distance from the true population regression line. That line is unknown, so the error cannot be directly observed.
The residual measures the distance from the estimated sample regression line. It can be calculated from the sample data.
Analysts therefore use residuals as observable estimates of the unobservable errors.
What Should a Residual Plot Look Like?
A residual plot usually places residuals on the vertical axis and fitted values on the horizontal axis. Some questions may place the independent variable on the horizontal axis instead.
A well-behaved residual plot should show:
Points scattered above and below zero.
No visible curve, cycle, or repeated sequence.
A reasonably stable vertical spread from left to right.
No single observation dominating the plot.
The desired pattern is random scatter rather than perfect visual symmetry. A residual plot provides diagnostic evidence, so a clean plot supports the assumptions without proving that every assumption holds.
In a simple regression with an intercept, the residuals sum to zero:
Where:
= number of observations
= residual for observation
This mathematical result does not guarantee that the residuals behave properly. They can sum to zero while still showing curvature, changing variance, or serial dependence.
How Do Residual Plots Reveal Assumption Violations?
Curved Residual Pattern
A curved residual pattern suggests that the linear form does not capture the relationship between the variables.
For example, residuals may begin below zero, rise above zero in the middle of the plot, and fall below zero again. The regression line is then making systematic errors rather than random ones.
An analyst may need to reconsider the model’s functional form or transform one of the variables.
Funnel-Shaped Residual Pattern
A funnel occurs when the residual spread expands or contracts as fitted values increase.
This pattern indicates heteroskedasticity. The error variance changes across observations instead of remaining constant.
The estimated regression line may still describe the average relationship reasonably well. However, unreliable standard errors make the reported hypothesis tests and intervals less dependable.
Serial or Sequenced Residual Pattern
A serial pattern appears when residuals form runs, cycles, or regular changes across ordered observations.
For example, several positive residuals followed by several negative residuals may indicate that the errors are correlated over time. This is especially relevant for financial and economic time-series data.
The model may be leaving out a time-related influence that continues from one period to another.
Unusual Residual
An observation that sits far above or below the other residuals deserves investigation.
Possible explanations include:
A data-entry or measurement error.
A genuine but unusual market event.
A structural change in the relationship.
An observation generated under different conditions.
The analyst should check the underlying data before removing the observation. The effect on the regression depends on where the observation lies and how strongly it influences the fitted line.
[INSERT regression-assumptions-residual-diagnostics.png HERE]
What Should an Analyst Do After Finding a Residual Pattern?
The next step depends on the pattern.
Curvature: Review the functional form and consider whether a variable transformation better represents the relationship.
Funneling: Treat the estimated standard errors and related hypothesis tests with caution.
Sequencing: Investigate whether the errors are correlated across time or whether an important time-related variable is missing.
Unusual residual: Confirm the observation is accurate and assess how strongly it affects the regression results.
Residual diagnostics identify possible problems rather than automatically determining the correction. CFA Level I mainly tests whether you can recognize the pattern, name the likely violation, and explain its effect on interpretation.
Worked Example: Interpreting Residual Diagnostics
An analyst regresses a company’s quarterly revenue growth on industry revenue growth using 40 observations. The analyst reviews four possible residual patterns.
Panel 1: Random Scatter
The residuals appear randomly above and below zero. Their vertical spread remains similar across the plot.
Interpretation: The plot does not show clear evidence of nonlinearity or changing variance. The residual behavior is broadly consistent with the model assumptions.
Panel 2: Curved Pattern
Residuals are negative at low fitted values, positive near the middle, and negative again at high fitted values.
Interpretation: The repeating curve suggests that the relationship is nonlinear. The straight-line model systematically overpredicts and underpredicts in different parts of the data.
Panel 3: Funnel Pattern
Residuals remain close to zero at low fitted values but become increasingly spread out as fitted values rise.
Interpretation: The changing spread suggests heteroskedasticity. The standard errors, test statistics, and intervals calculated from the model may be unreliable.
Panel 4: One Unusual Observation
Most residuals fall between −4 and +4 percentage points, while one residual equals +14 percentage points.
Interpretation: The observation requires a data-quality check. If it is valid, the analyst should determine how strongly it affects the estimated regression line before drawing a conclusion.
Common Exam Traps
Counting five assumptions when the question follows the CFA framework. CFA Level I presents four classic assumptions: linearity, homoskedasticity, independence, and normality.
Treating the error term and residual as identical. The error is measured from the unknown population line. The residual is measured from the estimated sample line.
Calling a funnel pattern nonlinearity. A funnel indicates changing variance. A curved pattern indicates a functional-form problem.
Assuming a high R-squared confirms the assumptions. R-squared measures explained variation. It does not show whether residuals satisfy the model assumptions.
Focusing only on whether residuals average zero. Residuals can average zero while still forming a problematic pattern.
Assuming normality applies to the variables. The assumption concerns the regression errors.
Removing an unusual observation immediately. The analyst should check the data and assess its effect before deciding how to handle it.
Saying heteroskedasticity automatically biases the coefficients. Under the remaining assumptions, the main concern is unreliable standard errors and inference.
Practice Question
An analyst estimates a simple linear regression of portfolio return on benchmark return using 50 monthly observations.
The residuals are tightly clustered around zero at low fitted values but spread progressively wider as the fitted values increase. The residual plot forms a clear cone, with no visible curvature.
Which regression assumption is most likely violated?
Linearity of the relationship
Constant variance of the error term
Independence of the error terms
Solution
Correct Answer: B
The cone-shaped residual plot shows that the vertical spread changes with the fitted values. This pattern indicates heteroskedasticity, which violates the assumption that the error term has constant variance.
The regression coefficients may remain unbiased if the other assumptions hold. However, the estimated standard errors become unreliable, affecting t-tests, F-tests, confidence intervals, and prediction intervals.
Option A. Linearity would usually be questioned when the residual plot shows a curve or another systematic shape. The question states that no curvature is present.
Option C. Independence would usually be questioned when ordered residuals show runs, cycles, or repeated sequences. The plot instead shows changing vertical spread.
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FAQs About Regression Assumptions and Residual Diagnostics
What Are the Main Assumptions of Simple Linear Regression?
The CFA Level I framework uses four assumptions: linearity, homoskedasticity, independence, and normality.
These assumptions describe the form of the relationship, the consistency of error variance, the relationship between errors across observations, and the distribution of the errors.
Why Do Some Sources List Five Assumptions of Linear Regression?
Different textbooks divide the underlying statistical conditions in different ways. Some list zero conditional mean or exogeneity as a separate assumption, resulting in five or more conditions.
For CFA Level I, candidates should follow the curriculum’s four-part framework unless the question provides a different set of assumptions.
What Does a Residual Plot Show?
A residual plot shows the differences between observed and predicted values across the regression sample.
Random scatter around zero with a stable vertical spread generally supports the model. Curvature, funneling, sequences, or unusual observations may indicate that an assumption has been violated.
What Does a Funnel-Shaped Residual Plot Indicate?
A funnel-shaped residual plot indicates heteroskedasticity, meaning the variance of the error term changes across observations.
The estimated coefficients may remain unbiased under the other assumptions, but the standard errors and statistical inference can become unreliable.
Does a High R-Squared Mean the Regression Assumptions Are Satisfied?
A high R-squared means the model explains a large percentage of the variation in the dependent variable.
A model can still have a high R-squared while its residuals show curvature, heteroskedasticity, dependence, or unusual observations. Model fit and assumption validity require separate evaluation.