- CFA Exams
- 2023 Level I
- Topic 1. Quantitative Methods
- Learning Module 2. Multiple Regression
- Subject 7. Testing Whether All Population Regression Coefficients are Equal to Zero

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##### Subject 7. Testing Whether All Population Regression Coefficients are Equal to Zero PDF Download

The t-test can be used to test a hypothesis about any single coefficient in the regression model, but testing a hypothesis about βH H

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

Therefore, F = MSR/MSE = 287.3521 / 2.985 = 96.26.F = [SSR/K] / [SSE/(n - K - 1)] = [(n - K - 1) R

_{1}and then testing a hypothesis about β_{2}is not the same as performing a joint test about β_{1}and β_{2}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

_{0}: β

_{1}= β

_{2}= ... = β

_{k}= 0

against the alternative hypothesis

_{1}: At least one of the coefficients β

_{1}, β

_{2}, ..., β

_{k}is nonzero.

Sequentially performing a series of t tests, each at the 5% level of significance, on each of the coefficients β

_{1}, β_{2}, ..., 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 β_{1}and β_{2}, each at the 5% level of significance level, the joint level of significance will not be 0.05. If the probability of rejecting β_{1}= 0 is 0.05, and the probability of rejecting β_{1}= 0 is also 0.05, the joint probability of rejecting both β_{1}= 0 and β_{2}= 0 is less than 0.05 and depends on the correlation between the sample estimators b_{1}and b_{2}(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 H

_{0}: β_{1}= β_{2}= 0 against the alternative hypothesis H_{1}: β_{1}and β_{2}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 H_{0}in favor of H_{1}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 H

_{0}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 H

_{0}: β_{1}= β_{2}= ... = β_{k}= 0 in favor of H_{1}. Rejection of H_{0}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 R^{2}, 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 R

^{2}according to the following equation:

^{2}] / [K(1 - R

^{2})]

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 R

^{2}; consequently, the F-statistic does not provide much additional information about the quality of a model that cannot be learned from inspection of R^{2}alone.

**Learning Outcome Statements**

CFA® 2023 Level I Curriculum, Volume 1, Module 2

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**User Contributed Comments**
1

User |
Comment |
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nmech1984 |
so whatever we learned so far was BS, because F-statistic does not provide more info than the simple R2. perfect... |

I am using your study notes and I know of at least 5 other friends of mine who used it and passed the exam last Dec. Keep up your great work!