# Appropriate null hypothesis for a two-stage regression

#### Daoist

##### New Member
Hi – thanks everyone for your help in advance. I’m working on a two-stage regression, in which a coefficient from the first regression is used as an independent variable for the second regression. Both regressions turn out significant, but I have no idea what that proves, since the hypothesis tests are mis-specified.

1st stage (simplified): y = a + bx (where a and b are normal regression coefficients)
2nd stage: y = c + dzb (where c and d are normal regression coefficients, z is new data, and b is the coefficient from the 1st stage.)

The null hypotheses, as specified, are b = 0 and d = 0. However, I strongly suspect the following regression is significant: y = c + dz – which would lead to a tautological conclusion. It seems that the implicit hypothesis test is that b is actually adding any information to the regression. Perhaps the appropriate test would be b = k, where k is a constant – or perhaps that would just be a precondition, I’m not sure. If this were a more normal regression, an F test would be appropriate, but we can’t separate b from z here under the assumptions of that test.

I need to be able to make my comments in the most concise way possible – ideally with cites to back up my point. This isn’t going to look good for my boss, and in any case he can point to peer-reviewed studies that take this same problematic approach. Such are the standards in financial statistics. I'm pretty sure I haven't oversimplified this in any significant way.
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