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Work out the standard error of estimate and explain what it shows

CFA® Level I · Quantitative Methods · Applications of Simple Linear Regression in Finance · QM.8.c

The idea

The standard error of the estimate measures the typical distance between the observed values and the fitted line, in the units of the dependent variable. It is the square root of the mean squared residual, adjusted for the degrees of freedom used by the model. A smaller standard error means a tighter fit and narrower prediction intervals. It complements R-squared rather than repeating it: R-squared says what share of variation is explained, and the standard error says how large the remaining errors actually are in the units a decision would be made in.

Formulas

Standard error of the estimate

SEE = √(SSE ÷ (n − 2))

SSE
the sum of squared residuals
n
the number of observations

Simple regression, hence n − 2: one degree of freedom lost to the slope and one to the intercept. Lower SEE means a tighter fit.

A practice question

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

In what units is the standard error of the estimate expressed?

  1. It is unitless, like R-squaredWhy not: R-squared is a proportion and so has no units. The standard error is an average residual size, which is measured in whatever the dependent variable is.
  2. In the units of the independent variableWhy not: Residuals are vertical distances — observed minus fitted values of the *dependent* variable. The independent variable's units do not enter.
  3. In the units of the dependent variable — correct

It measures the typical size of a residual, so it shares the dependent variable's units. That is what makes it directly interpretable when judging whether a fit is good enough to use.

A harder one

Two regressions of the same dependent variable report identical R-squared but different standard errors of the estimate. How is that possible?

  1. R-squared and the standard error measure different things and need not move together — correct
  2. One of the two must have been calculated incorrectlyWhy not: Both can be right. A sample with more total variation can explain the same *share* of it while leaving larger residuals in absolute terms.
  3. The samples must have different dependent variablesWhy not: The question stipulates the same dependent variable. What differs is how much it varies in each sample.

R-squared is a proportion of variance explained; the standard error is an absolute magnitude. A sample with more total variation can explain the same share while leaving larger residuals.

The hardest tier

Why is the standard error of the estimate often more useful than R-squared when deciding whether to act on a forecast?

  1. It is always smaller and therefore more encouragingWhy not: They are not comparable — one is a proportion between 0 and 1, the other an amount in the variable's units. Neither is systematically smaller.
  2. It states the likely forecast error in the units the decision is made in — correct
  3. It is unaffected by outliersWhy not: It is built from squared residuals, so a single large outlier moves it substantially. Robustness is not the reason to prefer it.

A decision needs to know how wrong the forecast is likely to be in percentage points or currency, not what fraction of variance was explained. A high R-squared on a very noisy variable can still leave errors too large to trade on.

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