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Outline the different ways to visualise residuals and the strengths of each relative to the others.

FRM® Part I · Quantitative Analysis · Regression Diagnostics · FRM1.QA.9.f

The idea

Residual plots reveal violated assumptions that summary statistics hide, and each plot suits a different problem. A scatter of residuals against fitted values shows non-linearity (a curve or U shape) and heteroskedasticity (a funnel), and in a multiple regression it summarises all the regressors in one picture. Residuals against each individual regressor locate which variable causes a problem. Residuals plotted in time order, or their autocorrelation function, show serial correlation as runs of same-sign values. A histogram gives a quick look at shape but depends on the bin width. A QQ plot of standardised residuals against normal quantiles avoids that choice and exposes tail behaviour. Points bending away from the line at both ends mean fat tails. Standardising residuals also makes outliers easier to spot. Match the plot to the assumption you are checking.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which plot is best suited to assess whether regression residuals follow a normal distribution, in particular in the tails?

  1. A scatter of the residuals against the fitted values.Why not: Shows non-linearity and changing variance, but gives no direct comparison with a normal distribution.
  2. A plot of the residuals against the order of the observations.Why not: Shows serial dependence and shifts over time; it does not compare the distribution with a normal one.
  3. A QQ plot of standardised residuals against normal quantiles. - correct
  4. A scatter of the dependent variable against one regressor.Why not: Gives the raw relationship between two variables and does not describe the shape of the residual distribution.

A QQ plot compares the sample quantiles with those of a normal distribution; departures at either end reveal fat or thin tails. A histogram can also show shape but depends on the bin width and is poor in the tails.

A harder one

In a plot of residuals against fitted values, the residuals show a clear U-shaped pattern, with positive values at both low and high fitted values and negative values in the middle. What does this suggest?

  1. The errors are heteroskedastic, with a variance that grows with the fitted value.Why not: Heteroskedasticity appears as a widening or narrowing spread, not as a systematic curve in the residual mean.
  2. The relationship is non-linear, so a squared term or another transformation may be missing. - correct
  3. The errors are serially correlated and the data should be sorted by time.Why not: A pattern in the mean against fitted values points to misspecified functional form; serial correlation would be seen in a time-ordered plot.
  4. The residuals are normally distributed, since they are symmetric around zero overall.Why not: A systematic curve means residuals are not random noise; the symmetry of the overall sum is guaranteed by OLS and says nothing about normality.

Residuals should scatter randomly around zero. A systematic curve means the conditional mean is mis-specified, usually because a non-linear term such as the square of a regressor has been omitted.

The hardest tier

In a QQ plot of standardised regression residuals (normal quantiles on the horizontal axis, sample quantiles on the vertical axis), the points lie close to the line in the middle but curve above the line at the right end and below it at the left end. What does this indicate?

  1. The residuals have thinner tails than a normal distribution.Why not: Thin tails would curve the other way, with the sample extremes falling inside the normal quantiles.
  2. The residuals are skewed to the right but have normal tails.Why not: Right skew would bend the points at one end only and in the same direction for the whole plot, rather than at both ends.
  3. The residuals have fatter tails than a normal distribution. - correct
  4. The residuals are normally distributed, and the curvature is a sign of heteroskedasticity.Why not: Curvature at both ends is a distributional departure from normality; heteroskedasticity is seen in residual-versus-fitted plots.

Sample quantiles more extreme than the normal quantiles at both ends mean more probability in the tails: fat tails (excess kurtosis). That matters because exact t and F tests rely on normality in small samples, and loss distributions are often fat-tailed.

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