Further inference in the multiple linear regression model

数学
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Joint hypothesis testing

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  • Simple null hypothesis → involves a restriction on one sign (<,=,>) only

  • Joint null hypothesis → involves two or more restrictions at the same time

Testing the effect of advertising: The F-test

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restricted model :

  • An unrestricted model is the “full” regression specification, where you estimate all parameters freely without imposing any restrictions. For example, if your regression is

    then the unrestricted model estimates all at once.

  • A restricted model is the version of the model after you impose the null hypothesis restrictions. For instance, if

    then the restricted model reduces to

    because the restrictions eliminate and from the regression.

Testing the overall significance of the model

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t-test and F-test

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More general F-tests

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  • The F-test is the standard way to test any set of linear restrictions, not just “all equal to zero.”

  • Here the restriction links two parameters ( and together. That means you cannot just look at a single t-statistic — the test requires accounting for covariance between estimators.

  • The F-test compares restricted model vs. unrestricted model fit:

    • Unrestricted model: estimate freely.

    • Restricted model: force

What “restricted model” really means

  • The restricted model is any regression model you get after imposing the null hypothesis.

  • If the null says , then yes: the restricted model is simply dropping that variable.

  • But in general, the null may say something else, e.g.

    That is not a zero restriction, but it’s still a restriction on parameters.

So: restricted model ≠ only β=0. Instead, it means “the model under the null hypothesis.”

Why substitution is needed

When the null hypothesis involves a linear combination (like

  • You cannot just delete regressors, because neither nor is zero.

  • Instead, the restriction ties them together, so one becomes dependent on the other.

  • To build the restricted model, you must substitute that relationship back into the regression equation.

That’s why in your KFC example, they replaced with .

Substitution is exactly the way to re-express the restricted model so it still looks like a valid regression: