# Solved – How useful are linear hypotheses

In a linear model $$Y=Xbeta + varepsilon$$, one can easily test linear hypotheses of the form $$H_0: Cbeta = gamma,$$ where $$C$$ is a matrix and $$gamma$$ is a vector with dimension equal to the number of rows in $$C$$. Namely, one can derive a test statistic which has an F distribution under the null and go from there.

Theoretically, these tests are very interesting to me and seem quite flexible, as $$C$$ and $$gamma$$ can be anything.

However, I'm interested to know how useful these hypotheses are in practical applications and what are some interesting examples of these applications? (besides testing if a single coefficient is 0 or all coefficients of the model being zero, which is included in every `lm` call in R for example)

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These linear hypotheses on the coefficient vector have three main uses:

• Testing the existence of relationships: We can test the existence of relationships between some subset of the explanatory variables and the response variable. To do this, let $$mathbf{e}_mathcal{S}$$ denote the indicator vector for the subset $$mathcal{S}$$ and test the linear hypotheses:

$$H_0: mathbf{e}_mathcal{S} boldsymbol{beta} = 0 quad quad quad H_A: mathbf{e}_mathcal{S} boldsymbol{beta} neq 0.$$

• Testing a specified magnitude for the relationship: We can test the magnitude of a relationship between an explanatory variables and the response variable using some specified value of interest. This is often useful when a particular specified magnitude has some practical significance (e.g., it is often useful to test if the true coefficient is equal to one). To test $$beta_k = b$$ we use the linear hypotheses:

$$H_0: mathbf{e}_k boldsymbol{beta} = b quad quad quad H_A: mathbf{e}_k boldsymbol{beta} neq b.$$

• Testing the expected responses of new explanatory variables: We can test the values expected values of responses corresponding to a new set of explanatory variables. Taking new explanatory data $$boldsymbol{X}_text{new}$$ we get corresponding expected values $$mathbb{E}(boldsymbol{Y}_text{new}) = boldsymbol{X}_text{new} boldsymbol{beta}$$. This means that we can test the hypothesis $$mathbb{E}(boldsymbol{Y}_text{new}) = boldsymbol{y}$$ via the hypotheses:

$$H_0: boldsymbol{X}_text{new} boldsymbol{beta} = boldsymbol{y} quad quad quad H_A: boldsymbol{X}_text{new} boldsymbol{beta} neq boldsymbol{y}.$$

As you can see, the first use is to test for whether some of the coefficients are zero, which is a test of whether those explanatory variables are related to the response in the model. However, you can also undertake more general tests of a specific magnitude for the relationship. You can also use the linear test to test the expected response for new data.

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