# Hypothesis testing: two estimates included

Consider the following two regression models:
(a) $Y_i=\beta_0+\beta_1 X_i +u_i$
(b) $X_i=\gamma_0+\gamma_1 Y_i+\epsilon_i$
Assume that both models (a) and (b) are valid regressions.
Test $H_0: \gamma=\frac{1}{\beta}$.

In that case, first I have to estimate $\hat{\beta}$ from model (a) and estimate $\hat{\gamma}$ from model (b),
and then using that estimates test $H_0: \gamma=\frac{1}{\beta}$?
I wonder whether my approach makes sense or not...

• This appears to be a self-study question, please add the tag and detail your issues with the problem, it is not clear enough as written. – Xi'an Feb 21 '16 at 17:15
• Your description doesn't seem to present any actionable "approach" at all: could you explain how you are thinking of testing $H_0$ using $\hat\beta$ and $\hat\gamma$? I suspect this (intriguing) question intends to make you sit back and think about what a bivariate random variable $(X,Y)$ with $\gamma=1/\beta$ must look like, rather than asking you to apply rote statistical procedures. – whuber Feb 21 '16 at 17:38