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Further inference in the multiple linear regression model

news 2025/9/23 9:29:53

Joint hypothesis testing

Simple null hypothesis → involves a restriction on one sign (<,=,>) only $H_0: \beta_2 = 0).$
Joint null hypothesis → involves two or more restrictions at the same time $β4=0).(e.g.H_0: \beta_2 = 0, \; \beta_3 = 0, \; \beta_4 = 0).$

restricted model : $βi=0\beta_{i}=0$

An unrestricted model is the “full” regression specification, where you estimate all parameters freely without imposing any restrictions. For example, if your regression is
$\beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + u$

then the unrestricted model estimates $β 0, β 1, β 2, β 3$ all at once.
A restricted model is the version of the model after you impose the null hypothesis restrictions. For instance, if
$H0:β2=β3=0H_0: \beta_2 = \beta_3 = 0$

then the restricted model reduces to
$\beta_0 + \beta_1 x_1 + u$

because the restrictions eliminate $x_2$ and $x_3$ from the regression.

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 ( $β3\beta_3$ and $β_{4})$ 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 $β3,β4\beta_3, \beta_4$ freely.
- Restricted model: force $β_{3}=1−3.8$

The restricted model is any regression model you get after imposing the null hypothesis.
If the null says $βj=0\beta_j = 0$ , then yes: the restricted model is simply dropping that variable.
But in general, the null may say something else, e.g.
$β3+3.8β4=1\beta_3 + 3.8\beta_4 = 1$
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.”

When the null hypothesis involves a linear combination (like $β3+3.8β4=1:\beta_3 + 3.8\beta_4 = 1:$

You cannot just delete regressors, because neither $β3\beta_3$ nor $β4\beta_4$ 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.