The four regression assumptions and what residual plots reveal
Simple linear regression assumes linearity between the variables, homoskedasticity (constant residual variance), independence (uncorrelated residuals, so no autocorrelation) and normally distributed residuals. Residual plots expose breaches: a curve means non-linearity, a fan or a jump in spread heteroskedasticity, waves over time autocorrelation, a lopsided histogram non-normality.
Before this: The regression line: slope, intercept and least squares
On the exam
- Describe and compare the assumptions of simple linear regression, identify violations through analyzing residuals, evaluate the estimated model’s goodness-of-fit and regression coefficients, and results of ANOVA estimates
Describe and compare the assumptions of simple linear regression and identify violations from the residuals. Expect a described residual plot and the assumption it breaks, the definition of homoskedasticity, and where normality applies (the residuals, not X or Y). A fan is heteroskedasticity, not non-linearity.
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 between the variables | Y is linear in the parameters (X may be transformed, lesson 6) | U-shape or arch | Revenue that compounds over time, fitted with a straight line |
| Homoskedasticity of the residuals | the residual variance is the same for every observation | Fan or funnel, or a jump in spread between two regimes | Return spreads that widened after a rate-policy change |
| Independence of the observations | the observations are independent, so the residuals are uncorrelated | Patterns over time: runs of positives then negatives, seasonal waves | Quarterly sales with a holiday-quarter spike |
| Normality of the residuals | the residuals (not X or Y) are normally distributed | Skewed or fat-tailed residual histogram | One data-entry error ten times larger than the rest |
- Linearity between the variables Memorize Y is a linear function of X in the parameters
- Broken: curve or arch in the residuals fix: transform the variables (lesson 6)
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- Homoskedasticity of the residuals Memorize residual variance is the same for every observation
- Broken (heteroskedasticity): fan, funnel or a jump in spread often two regimes pooled
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- Independence of the observations Memorize one residual tells you nothing about the next
- Broken (autocorrelation): runs, waves, cycles over time common in time series
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- Normality of the residuals Memorize matters most in small samples
- Broken: skewed or fat-tailed residual histogram large samples relax it
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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 the estimate s_e = √(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 autocorrelation (independence broken), and a few huge misses mean non-normal residuals (check for a data error first).
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 changed. Fitting each regime separately often gives two different slopes, a sign that the relationship itself changed.
Dependent residuals. Plot the residuals in time order. Long runs of positives followed by negatives, or a repeating cycle, show that one residual predicts the next. This is autocorrelation , and it breaks the independence assumption. It is the typical problem of time-series regressions and it makes the estimates and their standard errors unreliable, so t-tests and intervals can mislead. Time-series models and regressions on changes rather than levels are the usual remedies; both go beyond this module.
Non-normal residuals. The normality assumption is about the residuals, not about X or Y. 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.
What about outliers?
An outlier is an observation far from the rest. Because least squares squares each miss, one outlier can tilt the line. Check first whether it is a data error before deciding what to do with it.
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, beyond this module).
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. For autocorrelation, the usual remedies are time-series models or regressions on changes rather than levels. Do not simply drop a point because it hurts the fit: check first whether it is a data error.
Memorize this
MemorizeAssumption, tell-tale, cause
| Assumption | Residual plot when broken | Typical cause |
|---|---|---|
| Linearity | U-shape or arch | a curved relation fitted with a straight line |
| Homoskedasticity | fan, funnel or a jump in spread | spread grows with X; two regimes pooled |
| Independence | runs or cycles over time (autocorrelation) | time series with trends or cycles |
| Normality | skewed or fat-tailed residual histogram | extreme events, data errors |
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 Normality is about the residuals, not X or Y, and large samples relax it (central limit theorem). A high R² (the share of the variation in Y the model explains) proves nothing about the assumptions.
- Linearity between the variables Memorize Y is a linear function of X in the parameters
- Broken: curve or arch in the residuals fix: transform the variables (lesson 6)
-
- Homoskedasticity of the residuals Memorize residual variance is the same for every observation
- Broken (heteroskedasticity): fan, funnel or a jump in spread often two regimes pooled
-
- Independence of the observations Memorize one residual tells you nothing about the next
- Broken (autocorrelation): runs, waves, cycles over time common in time series
-
- Normality of the residuals Memorize matters most in small samples
- Broken: skewed or fat-tailed residual histogram large samples relax it
-
Quick example
Residual variance about 4 in the first half of the sample and about 4 in the second: homoskedastic. About 1, then about 9: heteroskedastic, often two regimes pooled into one regression.
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.
Residuals are not independent: autocorrelation, a seasonal pattern.
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Who must be normal?
The normality assumption applies to which variable?
The residuals (the error term), not X and not the levels of Y.
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Large sample, skewed residuals
n = 400 monthly observations; the residual histogram is mildly skewed. Are the t-tests useless?
No: with a large sample the central limit theorem lets you relax normality. It matters in small samples.
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 residuals.
- Assuming a high R² proves the assumptions hold.
- Forgetting that pooling two regimes into one regression can make the residual spread jump between periods.
- Calling a cycle in the residuals over time heteroskedasticity. A pattern in the signs over time is autocorrelation (independence broken); heteroskedasticity is about the size of the spread.
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 of a company's quarterly sales growth on GDP growth shows long runs of positive residuals followed by long runs of negative ones. The most likely consequence is that the:
Key takeaways
- Four assumptions: linearity, homoskedasticity, independence (no autocorrelation), normality of the residuals.
- Read the residual plot: a curve, a fan, a cycle over time or a lopsided histogram each points to one assumption.
- Normality concerns the residuals and matters most in small samples; check an outlier for a data error first.
Quick example
Residual variance about 4 in the first half of the sample and about 4 in the second: homoskedastic. About 1, then about 9: heteroskedastic, often two regimes pooled into one regression.
All Quantitative Methods formulas, every lesson
Practice exam-style questions on Simple Linear Regression in Finance
Questions about this lesson?
Where this comes back
- Goodness of fit and testing the regression coefficients Simple Linear Regression in Finance
- Simple Linear Regression in Finance Quantitative Methods
Questions candidates ask
What are the assumptions of simple linear regression?
Four: the relationship is linear in the parameters, the residual variance is constant (homoskedasticity), the observations are independent so the residuals are uncorrelated (no autocorrelation), and the residuals are normally distributed. Check each with residual plots.
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. Check whether it is a data error or a genuine observation before deciding what to do.
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