Hypothesis tests of a single coefficient in a multiple regression, using t-tests, are identical to those in simple regression. However, testing a hypothesis about β₁ and then testing a hypothesis about β₂ is not the same as performing a joint test about β₁ and β₂ simultaneously. To test whether our model explains a significant portion of the variation in Y, we need to perform a joint test of the regression coefficients. In the general regression model, we may want to test the null hypothesis

H₀: β₁ = β₂ = ... = β_k = 0

against the alternative hypothesis

H₁: At least one of the coefficients β₁, β₂, ..., β_k is nonzero.

Sequentially performing a series of t tests, each at the 5% level of significance, on each of the coefficients β₁, β₂, ..., β_k, is not the same as jointly testing the null hypothesis that all K coefficients are 0. For example, when we perform two separate t tests on the coefficients β₁ and β₂, each at the 5% level of significance level, the joint level of significance will not be 0.05. If the probability of rejecting β₁ = 0 is 0.05, and the probability of rejecting β₁ = 0 is also 0.05, the joint probability of rejecting both β₁ = 0 and β₂ = 0 is less than 0.05 and depends on the correlation between the sample estimators b₁ and b₂ (if the estimators were independent, the joint probability of rejecting both hypothesis would be 0.05 x 0.05 = 0.025).

The correct method for testing all the coefficients jointly is based on an F statistic, which follows the F distribution. To perform the test, calculate the F statistic:

where MSR is the mean regression sum of squares and MSE is the mean squared error.

Continue with the example we have used for previous sections, we want to use a 5% level of significance and test the joint null hypothesis H₀: β₁ = β₂ = 0 against the alternative hypothesis H₁: β₁ and β₂ are not both 0.

From the ANOVA table above, we see that SSE is 20.8958, thus we obtain MSE = 20.8958 / 7 = 2.985.

As RSS is 574.7042, we obtain MSR = 574.7042 / 2 = 287.3521.

Therefore, F = MSR/MSE = 287.3521 / 2.985 = 96.26.

The large value of F indicates that the null hypothesis is false. For α = 0.05, the critical value of F having K = 2 and (n - K - 1) = 7 degrees of freedom is F_{0.05, 2, 7} = 4.74. We reject H₀ in favor of H₁ the F statistic of 96.26 far exceeds the critical value.

In fact, the F-statistic can be obtained directly from an ANOVA table (the above is just a part of a complete ANOVA table).

The value "Significance F" represents the prob-value associated with F = 96.26 and is shown to be 0.0000. This provides extremely strong evidence that H₀ is false.

The example represents a typical regression model using business data. When we build a model to explain some business data, it's almost always the case that we will reject H₀: β₁ = β₂ = ... = β_k = 0 in favor of H₁. Rejection of H₀ means that, as a group, the explanatory variables do help explain the values of the dependent variable Y, but this result alone does not imply that we have a good model or that the estimated equation will yield good predictions of Y. Nor does it mean that the estimated equation provides a good fit to the data. In general, a good model explains most of the variation in Y and satisfies the statistical assumptions. In practice, most statisticians rely more on R², s_e, and the individual t tests than on the overall F test when assessing the quality of some regression model.

The F statistic is related to the value of R² according to the following equation:

F = [SSR/K] / [SSE/(n - K - 1)] = [(n - K - 1) R² ] / [K(1 - R²)]

A small value of F strongly indicates that our model is inadequate, but, in general, a small value of F will be accompanied by a small value of R²; consequently, the F-statistic does not provide much additional information about the quality of a model that cannot be learned from inspection of R² alone.