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I was running a t test over two regression betas with the assumption of equal variance. I know that if the condition of homoscedasticity do not hold then there are chances to have type 2 error but what if in my regression I do not have the intercept? (The models without errors do not require the error mean to be zero). I am very confused, I may have mixing two different concepts so, please tell me the associated readings to get it more clear.

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You are generally better off asking statistics questions at cross validated, simply because there are more statisticians there. Also, you will get a better answer if you supply some more detail of what you want to test exactly, and what program you use if you want hints on where to read up on these things

I assume that you want to test whether two coefficients that you found in a linear regression are equal or different from another, where your regression model has no intercept. That is an example of a linear restriction. These kind of restrictions are generally tested with F-tests. You can read up on them a.o. here (which implements them in R).

How to implement them depends on your stats program.

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