Hypothesis testing on multiple regression coefficients In the simple linear regression setting, in order to determine
whether there is a relationship between the response and the predictor we
can simply check whether $\beta_1=0$. In multiple regression, it is to check $\beta_1 = \beta_2 = \cdots = \beta_p=0$ for hypothesis testing. 
Why do we set these coefficients to zero? What is the motivation/physical meaning for this? 
 A: Here's the full model:
$$ y_i = \beta_0 + \beta_1 x_{i,1} + \cdots + \beta_p x_{i,p} + \epsilon_i$$
where all the $\epsilon_i$s are independent and identically distributed as $\mathcal{N}(0,\sigma^2)$. If you set all the $\beta$s to zero, it simplifies the model to 
$$ y_i = \beta_0 + \epsilon_i$$
which means that your data is just normally distributed, or
$$Y_i \sim \mathcal{N}(\beta_0, \sigma^2)$$
and none of the $x$s can help you predict it.
A: Primarily null hypothesis might be that adding these P variables does not reduce residual variation, meaning does not explain variation in the response variable. 
But this null vector hypothesis is not only one, often parameter restrictions are put only to the subset of variables. 
One can have model A which is larger than model B, having more variables. Null hypothesis might be that model B is adequate for the explaining variation in the response variable and then it is natural that you have zero restrictions on the remaining P-S variables which are in model A.
