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Explain what a simple linear regression's ANOVA table and F-statistic tell you

CFA® Level I · Quantitative Methods · Applications of Simple Linear Regression in Finance · QM.19.b

The idea

The ANOVA table splits the total variation in the dependent variable, SST, into the part the regression explains, SSR, and the unexplained part, SSE. R-squared = SSR / SST is the share explained. Mean squares divide each sum by its degrees of freedom: MSR = SSR / 1 in a simple regression and MSE = SSE / (n - 2). The standard error of estimate is the square root of MSE. The F-statistic, MSR / MSE, tests whether the slope is zero; if it exceeds the critical value the null is rejected. With one independent variable, F equals the square of the slope's t-statistic, so the two tests always agree.

A practice question

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

A simple linear regression on 22 observations has a regression sum of squares of 60 and a sum of squared errors of 140. The F-statistic is closest to:

  1. 0.43Why not: Divided the sums of squares directly, without converting them to mean squares.
  2. 8.57 — correct
  3. 0.12Why not: Inverted the ratio, putting the mean squared error over the mean square regression.

MSR = 60 / 1 = 60 and MSE = 140 / (22 - 2) = 7. F = 60 / 7 = 8.57.

A harder one

A simple linear regression uses 32 observations and has a sum of squared errors of 180. The standard error of estimate is closest to:

  1. 2.45 — correct
  2. 6.00Why not: Stopped at the mean squared error and did not take its square root.
  3. 2.37Why not: Divided by n = 32 rather than by the n - 2 = 30 error degrees of freedom.

MSE = 180 / (32 - 2) = 6.00. The standard error of estimate is sqrt(6.00) = 2.45.

The hardest tier

A regression of 40 monthly excess returns on the market's has a total sum of squares of 250 and an R-squared of 0.36. The F-statistic is closest to:

  1. 21.94Why not: Used n - 1 = 39 error degrees of freedom instead of n - 2 = 38.
  2. 13.68Why not: Divided by the total sum of squares over 38 instead of by the error sum of squares.
  3. 21.38 — correct

SSR = 0.36 x 250 = 90 and SSE = 160. MSE = 160 / 38 = 4.2105, so F = 90 / 4.2105 = 21.38.

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