You are reading the 2026 curriculum (exams up to November 2026). Sitting in February 2027 or later? Go to the 2027 Quantitative Methods guide.
The four regression assumptions and what residual plots reveal
The model assumes a linear relation, constant error variance (homoskedasticity), independent errors and normally distributed errors. Plot the residuals: a curve means non-linearity, a fan means heteroskedasticity, runs or waves over time mean dependent errors, and a lopsided histogram or an extreme point means non-normality.
Before this: The regression line: slope, intercept and least squares
On the exam
- Explain the assumptions underlying the simple linear regression model, and describe how residuals and residual plots indicate if these assumptions may have been violated
Explain the assumptions of simple linear regression and describe how residuals and residual plots reveal violations. Expect a described plot and a question on which assumption is broken. A fan is heteroskedasticity, not non-linearity, and the normality assumption is about the errors.
How deeply to learn each part
- Know the idea
- Understand it in general. It frames other topics, but you will not calculate with it.
- Use it
- Understand it well enough to solve a question with it. Recognize when it applies.
- Memorize
- Learn it by heart. The exam will not give it to you.
You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.
You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.
Why do the assumptions matter?
Least squares will draw a line through any cloud of points. The tests and prediction intervals in the next lessons are only reliable if four assumptions hold. You cannot see the errors themselves, but you can see their sample estimates, the residuals. A residual plot shows the residuals against X (or against time). A healthy plot looks like random scatter in a level band around zero. Any shape is a warning.
What are the four assumptions?
| Assumption | Meaning | Residual plot when broken | Example |
|---|---|---|---|
| linearity | Y is a linear function of X in the parameters | U-shape or arch | Revenue that compounds over time, fitted with a straight line |
| homoskedasticity | The error variance is the same for every observation | Fan or funnel, or a step change between two regimes | Return spreads that widened after a rate-policy change |
| independence of errors | Errors are uncorrelated with each other | Patterns over time: runs of positives then negatives, seasonal waves | Quarterly sales with a holiday-quarter spike |
| normality of errors | The errors are normally distributed | Skewed or fat-tailed residual histogram, extreme points | One data-entry error ten times larger than the rest |
- Linearity Memorize Y is a linear function of X in the parameters
- Broken: curve or arch in the residuals fix: transform the variables
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- Homoskedasticity Memorize error variance is the same for every observation
- Broken (heteroskedasticity): fan, funnel or step change fix: split the sample or transform
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- Independence of errors Memorize one error tells you nothing about the next
- Broken (serial correlation): runs, waves, seasonal pattern common in time series
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- Normality of errors Memorize matters most in small samples
- Broken: skewed or fat-tailed histogram, outliers check for data errors first
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Two more points complete the list. The independent variable is assumed not to be random (or at least uncorrelated with the error), and its values must vary, or there is nothing to learn a slope from.
Try this 0 of 3
- Diagnose all five data sets correctly.
- Load “One bad entry” and drag the ringed point down until R² is above 0.90.
- Load “Curved” and compare its R² with the well-behaved set.
Well-behaved data: least-squares line Ŷ = 3.61 + 0.729X, R² 0.931. No assumption looks broken: the residuals are random noise in a level band around zero, with no curve, fan or rhythm. Residuals should look like random noise in a level band; any shape is a warning.
Data and the fitted line
- Least-squares line
Residuals against X
A steady relation with random noise. Look at the residual plot, then pick the assumption that looks broken.
Show working
Least-squares line: Ŷ = 3.61 + 0.729X (n = 16). R² = SSR ÷ SST = 180.46 ÷ 193.76 = 0.931.
Residuals Y − Ŷ in X order: 1.56, −0.86, −0.29, 0.28, −1.05, −0.08, −0.01, −1.94, 1.34, 0.91, −0.42, 0.15, 1.02, 0.19, 0.26, −1.06.
Standard error of estimate SEE = √(SSE ÷ (n − 2)) = √(13.30 ÷ 14) = 0.975.
Read the residual plot, not R²: a curve means non-linearity, a fan means heteroskedasticity, a rhythm over time means dependent errors, and a few huge misses mean non-normal errors.
What does each violation look like?
Non-linearity. If the true relation curves, a straight line is too high in the middle and too low at the ends (or the reverse). The residuals trace a U or an arch. A company whose revenue grows 15% a year looks like a smooth upward bend, so a straight line leaves a bow in the residuals.
Heteroskedasticity. The spread of the residuals changes with X, which is heteroskedasticity . A fan opening to the right means larger X goes with larger errors. The word describes unequal variance, so the opposite, constant variance, is homoskedasticity. The same signal appears when two regime periods are pooled: the residuals are small in the early years and large in the later years because the relationship itself changed. Estimating each regime separately fits better than one pooled line.
Dependent errors. Plot the residuals in time order. Long runs of positives followed by negatives, or a repeating seasonal pattern, show that today’s error predicts tomorrow’s. This is serial correlation . It is the typical problem of regressions on time series, and it makes the usual standard errors unreliable.
Non-normal errors. A histogram of the residuals that is lopsided or has fat tails points to non-normality. Normality matters most in small samples, because the t and F tests lean on it. In large samples the central limit theorem rescues the coefficient tests.
How do outliers distort a regression?
An outlier is an observation far from the rest. Least squares squares each miss, so the line tilts toward the outlier. It can inflate R² (one point far up and to the right makes the cloud look like a strong line) or flatten a real slope.
Exam tip When a question describes a shape, name the shape first (curve, fan, wave, lopsided), then match it to the assumption. Never pick “linearity” for a fan.
Go deeper Regime changes: when one line is really two
Suppose a bond-yield relationship held before a central bank changed policy and held differently after. Pooling both periods gives a single line that fits neither regime and residuals with a visible step in spread. The sample comes from more than one distribution. The remedy is to estimate separate regressions for each regime, or to model the change explicitly, which needs more than one independent variable (multiple regression, covered at Level II).
Go deeper What can you do about a violation?
At this level, three responses are enough. Transform the variables, for example with logs, to straighten a curve or steady a widening spread (lesson 6). Split the sample where the relation changed. Or add more explanatory variables, which takes you into multiple regression. Do not simply drop a point because it hurts the fit: check first whether it is a data error.
Memorize this
MemorizeAssumption, tell-tale, example
| Assumption | Residual plot when broken | Typical cause |
|---|---|---|
| Linearity | U-shape or arch | compounding series fitted with a straight line |
| Homoskedasticity | fan or funnel; step change in spread | spread widens as X grows; two regimes pooled |
| Independence | runs of + then −, seasonal waves | time series with trends or seasonality |
| Normality | skewed or fat-tailed histogram, outliers | a data-entry error, extreme events |
Memory hook Four assumptions, four shapes: curve, fan, wave, lopsided. Good residuals look like static on a TV: a level band with no shape.
Know it cold The assumptions are about the errors and the model (linear in the parameters), not about X being normal. A high R² does not prove them.
- Linearity Memorize Y is a linear function of X in the parameters
- Broken: curve or arch in the residuals fix: transform the variables
-
- Homoskedasticity Memorize error variance is the same for every observation
- Broken (heteroskedasticity): fan, funnel or step change fix: split the sample or transform
-
- Independence of errors Memorize one error tells you nothing about the next
- Broken (serial correlation): runs, waves, seasonal pattern common in time series
-
- Normality of errors Memorize matters most in small samples
- Broken: skewed or fat-tailed histogram, outliers check for data errors first
-
How it is asked
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Residuals fan out
A residual plot shows residuals spreading wider as X increases. Which assumption?
Homoskedasticity is violated (heteroskedasticity). Trap: linearity.
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Pattern over time
Quarterly sales residuals are positive every fourth quarter and negative otherwise.
Errors are not independent (serial correlation, a seasonal pattern).
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Who must be normal?
The normality assumption applies to which variable?
The error term (residuals), not X and not the levels of Y.
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A good fit hiding a problem
R² is 0.92 but one extreme point drives it; without it R² is 0.31.
Outliers can create a misleadingly good fit. High R² does not confirm the assumptions.
Exam traps
- Calling a fan shape non-linearity: a fan is heteroskedasticity, a curve is non-linearity.
- Thinking the independent variable must be normally distributed. The assumption is about the errors.
- Assuming a high R² proves the assumptions hold. One outlier can inflate R² and tilt the line.
- Forgetting that pooling two regimes into one regression can make the residual spread jump between periods.
Checkpoint
4 exam-style questions. About 90 seconds each.
1A residual plot shows residuals that spread wider as X increases. This most likely indicates a violation of:
2Residuals from a regression on quarterly sales are positive every fourth quarter and negative in the others. The assumption most likely violated is:
3The normality assumption in simple linear regression applies to the:
4A regression has R² = 0.92, but removing one extreme observation drops R² to 0.31. This best illustrates that:
Key takeaways
- Four assumptions: linearity, homoskedasticity, independent errors, normal errors.
- Read the residual plot: curve, fan, wave or lopsided histogram each point to one assumption.
- Outliers can create a misleadingly good fit; R² alone never validates a model.
Questions about this lesson?
Where this comes back
- Goodness of fit and testing the regression coefficients Simple Linear Regression
- Simple Linear Regression Quantitative Methods
Questions candidates ask
What are the assumptions of simple linear regression?
The relationship is linear in the parameters, the error variance is constant (homoskedasticity), the errors are independent of each other, and the errors are normally distributed. The independent variable is also assumed not to be random, or at least uncorrelated with the errors, and to take varying values.
What does heteroskedasticity look like on a residual plot?
A fan or funnel: residuals are tightly bunched at one end of the X range and spread out at the other. A step change in spread between two periods, such as before and after a policy shift, is also heteroskedasticity.
How do outliers affect a regression?
Least squares squares the misses, so an extreme point pulls the line toward it. It can change the slope and intercept sharply and can inflate or deflate R². Check whether it is a data error or a genuine observation before deciding what to do.
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