Joint Linear Regression hypothesis test

Question

Suppose I have the following joint model:

$$ y_{f_i} = \beta_1 + \beta_2 t_i + \beta_3 s_i +\epsilon_i $$

$$ y_{m_i} = \beta_4 + \beta_5t_i + \beta_6 s_i +\epsilon_i $$

where $y_{f_i}$ corresponds to the female portion of the response variable, and $y_{m_i}$ corresponds to the male portion, and $t_i,s_i$ are some predictors.

I want to do a hypothesis test such as

$$H_0: \beta_2 = \beta_5 =0 \quad \text{and}\quad\beta_3 = \beta_6,\quad \text{against} \quad \text{not}\quad H_0$$

I would like to use the F test, but I don't understand how I would deal with the fact that we are using a joint model. Any help is appreciated.

Answer 1

The proper way to do it is by estimating $$ y_i = \beta_0 + \beta_1t_i + \beta_2D_i + \beta_3D_is_i + \beta_4s_i + \epsilon_i $$ where $D_i=1$ if $i$th patient is a male. This is your full model. The restricted model will be $$ y_i = \beta_0 + \beta_2D_i + \beta_3s_i + \epsilon. $$ Now you can take the $R^2$ of each model and compute the partial $F$-test.

Answer 2

In order to combine both regression equations, you can introduce a dummy variable $g_i$ into your data set. If the $i^\text{th}$ observation is belonging to a male person the dummy variable will be equal to $1$. The dummy variable will vanish if the $i^\text{th}$ observation is belonging to a female person.

The regression will be

$$y_i = \beta_1 + \alpha_1g_i + \beta_2t_i + \alpha_2g_it_i + \beta_3s_i+\alpha_3g_is_i+\varepsilon_i$$

The relationship $\beta_1+\alpha_1=\beta_4$, $\beta_2+\alpha_2=\beta_5$ and $\beta_3+\alpha_3=\beta_6$, can be derived if you consider $g_i=1$.

You wanted to test $\beta_2=\beta_5=0$ and $\beta_3=\beta_6$. Looking at the previous equations you should see that, we need to show the following conditions for the combined regression.

$$\beta_2=0 \quad \wedge \quad \alpha_2 = 0$$ $$\alpha_3=0$$

When I write that something needs to equal to $0$, I mean that the confidence interval must include zero.