What exactly does the ANOVA F-test tell us in a regression, and how do I interpret the F-statistic?

The ANOVA F-test in regression asks a single overarching question: Does the regression model, taken as a whole, explain a statistically significant portion of the variation in the dependent variable?

Formally, the null hypothesis is:

H₀: All slope coefficients equal zero (the model has no explanatory power)

H₁: At least one slope coefficient is non-zero

ANOVA Table Breakdown:

Where k = number of independent variables, n = number of observations.

Worked Example — Thornberry Capital Revenue Model

An analyst at Thornberry Capital regresses quarterly GDP growth on three predictors: consumer spending growth, industrial production index, and net exports (n = 48 quarters).

F-statistic = MSR / MSE = 47.53 / 1.895 = 25.08

The critical F-value at α = 0.05 with (3, 44) degrees of freedom is approximately 2.82. Since 25.08 >> 2.82, we reject H₀ and conclude the model has significant explanatory power.

R² Connection:

R² = RSS / TSS = 142.6 / 226.0 = 63.1%. The model explains about 63% of the variance in GDP growth.

Key Distinctions for the Exam:

• F-test = joint test of all slopes simultaneously
• t-test = test of one individual slope coefficient
• A model can pass the F-test while individual t-tests fail (multicollinearity) or vice versa (rare but possible with many weak predictors)

Exam Tip: If given an ANOVA table, always compute F = MSR/MSE, compare to the critical value, and state your conclusion. For simple regression (k = 1), F = t² — the two tests are equivalent.

Check our CFA Level I question bank for more regression practice problems.