Testing multiple linear restrictions

[mrcleanhands]

If I have 4 variables in my multiple regression and I'm told to test whether one is significant, and 3 others are not what would I do with the one left over?

I thought it would be easy as I could just test whether for H0: variable 1>0, variable 2,3 =0

But what if I'm told to restrict variable 1 so that under H0 it must =1?

If you think about it variable cannot = 1 unless there is no contribution from variable 4 (which I'm told I must "take into account"). This is because my Y variable has a maximum value of 100 and my variable 1 also has a maximum value of 100.

By forcing variable=1 I'm saying that the whole of the change in Y is caused by variable 1. But we have already agreed that variable 4 is a contributer. So how do I "take it into account" when building a restricted and unrestricted model?


If I exclude variable 4 from both the restricted and unrestricted than variable 1 won't be equal to 1, and if I include it won't be = 1 either!

 

[mrcleanhands]

Allow me to restate this in better terms...

I have 3 variables in an oversimplified model. Wealth, Divert, Race.
So that Y = a function of Beta1*(Wealth) + Beta2*(Divert) + Beta3*(Race).

My Y variable ranges from 0-100. Beta1 also varies from 0-100. I'm trying to test whether Wealth is 100% responsible AND that race does not matter for the variation in Y, after taking into account the variable "Divert". I'm going to construct a restricted and unrestricted model and then conduct an F-test.Would it make sense for me to setup the null hypothesis thus:

(H-0) is: Beta1=1-Beta2, Beta3=0
H-1 would be: Beta 1 ≠ 1-Beta2 and / or Beta3≠0.I was told I could also use:
set null (H-0) as: Beta1=1, Beta3=0
H-1 would be: Beta 1 ≠ 1 and / or Beta3≠0.But this doesn't make any sense to me as if we want to account for "Divert" then it is silly to test for Beta1=1. Beta1 will not equal one if "Divert" contributes to the regression, it will only equal the variation in Y which is unexplained by "Divert" (1-Divert).Is my thinking correct?