testing multiple linear restrictions

[randomchef]

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. 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.

The hypothesis would be setup so that the null (H-0) is: Beta1=1, Beta3=0
H-1 would be: Beta 1 ≠ 1 and / or Beta3≠0.

Would it make sense for me to rearrange the regression thus:

Unrestricted model: Y - [Beta2*(Divert)] = Beta1*(Wealth) + Beta3*(Race)
Restricted model: Y - [Beta2*(Divert)] = Beta1*(Wealth)

There is a good chance that the variable "Divert" contributes so I want to nullify its affect and then figure out after it has been netted out whether Wealth is 100% responsible and Race is insignificant.

[startz]

No. If your null hypothesis is beta1=1 and beta3=0 then just test that. The regression will take care of properly accounting for Divert.

[randomchef]

EDIT: Oh I forgot to mention that "Wealth" is also out of 100 so it can't be more than 1 and it can't = 1 unless everything else is 0. (since if it =1 it is responsible for all the movement in Y which is also out of 100).

That's what I'm not sure about. I know that there is a good chance "Divert" contributes so that DadPrestige will not show up as 1 in this test. But it could show up as one if I re-paramitise (I think).

Maybe I could set beta1 = 1-Beta2 in my null hypothesis, would this be better?

[randomchef]

Here's why I don't really think testing for Beta1=1 is good...


If the true population regression is given by Y = β0 + .5*Wealth .5*Divert
and I arbitrarily set Wealth coefficient =1 then
Y - Wealth = β0 + h .5*Divert but this is not true

so what if, when constructing my restricted and unrestricted model, instead I tested for Beta1 = 1- Divert

Y - (1-.5)*Wealth= β0 + .5*Divert -->
Y - (.5)*Wealth= β0 + .5*Divert --> This is true.

[startz]

If you hypothesize that when wealth and divert both increase by some amount X, then y also increases by X, then your suggestion should work well. Note that in EViews you can do a Wald test, so you don't have to work out the restricted regression.

[randomchef]

Thanks startz...

I was thinking what about a situation where my variable "Divert" is now a dummy?

If I set the null as B1-Divert then this is not true unless the person is considered under part of the group "Divert". How would I account for this if I want to test whether B1 is responsible for all left over variation in Y?

If I set the null to just B1=0 then I'm testing, if under the base case, B1=1 which will make sense since then the dummy is disabled. But what about when someone is part of "Divert" there is now some contribution so that I cannot force the null to be B1=1 as this is false.

Should I perform then multiple tests, so that the first is using the null b1=1 and the second the null b1=1-divert... ?


An added complication would be multiple dummy categories...