Compare 

- **model 1** which assumes all the coefficients for $x_1$, $x_2$, $x_3$, and $x_4$ are equal $$y = \beta_0 + \beta(x_1+x_2+x_3+x_4) + \epsilon$$  

versus

- **model 2**  which assumes all the coefficients for $x_1$, $x_2$, $x_3$, and $x_4$ are free (not equal) $$y = \beta_0+\beta(x_1+x_2+x_3+x_4) + \beta_2^\prime x_2 + \beta_3^\prime x_3 + \beta_4^\prime x_4 + \epsilon$$ or differently parameterized 
$$y = \beta_0+ \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_4 x_4 + \epsilon$$

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You can do this comparison, for instance, by means of 

 1. an  F-test comparing the sum of squared residuals, which is also called analysis of variance (ANOVA).

 2.  In your case you seem to want to do three t-tests $$\begin{array}{rcl} \beta_i^\prime &=& \beta_i-\beta \\ &=&  \beta_i-\beta_1 \\ &=& 0 \end{array}$$ for $i = 2,3,4$