How do you analyze residuals?

Residual analysis checks the errors left by a regression model to see whether the model is capturing the important structure in the data.

A well-specified regression model usually produces residuals that look like random noise rather than showing clear patterns.

Start with residuals vs fitted values

Plot:

Residuals → vertical axis
Predicted values → horizontal axis

Ideally, the residuals should be scattered roughly randomly around zero.

What patterns can reveal

• Curved pattern: the relationship may not be adequately linear.
• Funnel shape: the residual variance may change with the prediction, suggesting heteroscedasticity.
• Clusters or bands: the model may be missing an important grouping variable or structure.
• Large isolated residuals: some observations may be unusual or poorly predicted.

Check whether residuals are centered around zero

Residuals should not be systematically positive or negative across a region of the data.

If they are, the model may be consistently underpredicting or overpredicting in that region.

Examine residuals against individual predictors

Plotting residuals against important input features can reveal relationships that the fitted model did not capture.

For example, a curved residual pattern against a feature may suggest that a transformation or nonlinear term is needed.

Check residual spread

The vertical spread of residuals should be reasonably similar across fitted values.

If the spread becomes much wider or narrower, the constant-variance assumption may be violated.

Check dependence for ordered or time-based data

For time-ordered observations, inspect residuals in chronological order.

Long runs of positive or negative residuals or repeating patterns can indicate autocorrelation or missing time-related structure.

Examine the residual distribution when inference matters

A histogram or Q-Q plot can help assess whether residuals are approximately normally distributed.

Normality is mainly important for classical confidence intervals and hypothesis tests, especially with smaller samples.

Investigate unusual observations

Very large residuals may indicate:

• data-quality problems
• unusual cases
• missing variables
• model misspecification

Large residuals should be investigated rather than automatically removed.

Residual analysis is diagnostic, not a single pass/fail test

No real dataset will produce perfectly random residuals.

The goal is to look for systematic patterns that suggest the model can be improved.

In short, analyze residuals by checking whether they are randomly scattered around zero with roughly constant spread and no systematic pattern. Curves, funnels, dependence, or extreme residuals can reveal problems with the regression model.