Testing multiple linear restrictions

In summary, the conversation discusses how to test the significance of one variable in a multiple regression model when the other variables are not significant. It is suggested to set up the null hypothesis as Beta1=1-Beta2, Beta3=0 or Beta1=1, Beta3=0, but the latter does not make sense as it does not take into account the effect of the other variables. Instead, the null hypothesis should reflect that the variable being tested is responsible for the variation in Y that is not explained by the other variables.
  • #1
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!
 
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  • #2
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?
 

1. What is the purpose of testing multiple linear restrictions?

The purpose of testing multiple linear restrictions is to determine whether a set of linear restrictions on the coefficients of a regression model can be rejected or not. This is important for understanding the relationships between the variables in the model and making valid inferences from the data.

2. How is the F-test used in testing multiple linear restrictions?

The F-test is commonly used in testing multiple linear restrictions to assess whether the restricted model (where the restrictions are imposed) is significantly different from the unrestricted model (where the restrictions are not imposed). A large F-statistic and a small p-value indicate that the restrictions can be rejected and the unrestricted model is a better fit for the data.

3. What is the difference between a joint test and a separate test in multiple linear restrictions?

In a joint test, all of the restrictions are tested simultaneously, while in a separate test, each restriction is tested individually. A joint test is more powerful in detecting overall differences between models, while separate tests may be more useful for understanding specific relationships between variables.

4. Can multiple linear restrictions be tested in non-linear models?

No, multiple linear restrictions can only be tested in linear models where the relationship between the dependent and independent variables is assumed to be linear. In non-linear models, different tests and techniques must be used to assess the significance of restrictions.

5. How can the results of testing multiple linear restrictions be interpreted?

If the restrictions are rejected, it means that the restricted model is not a good fit for the data and the unrestricted model should be used instead. If the restrictions are not rejected, it means that the restricted model is a reasonable fit for the data and can be used for making inferences and predictions.

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