Let's say I have an equation $Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + ... + \beta_k X_k $, where $\beta_i$ represents an estimated coefficient and $X_i$ represent independent variables. How can I test whether the function $\beta_1/\beta_2$ is significantly different from zero?
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1$\begingroup$ Do you mean to ask a question about coefficients rather than the independent variables? $\endgroup$– Glen_bCommented Dec 7, 2015 at 12:07
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1$\begingroup$ Have a look at package lmtest or car. $\endgroup$– Helix123Commented Dec 7, 2015 at 12:27
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$\begingroup$ For constructing confidence inertvals for ratio's see stats.stackexchange.com/questions/164738/…. Note that form a confidence interval you can derive a hypopthesis test. $\endgroup$– user83346Commented Jan 2, 2016 at 11:29
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1 Answer
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Note that -- as long as $\beta_2$ is not zero -- if $\beta_1=0$, then the ratio $\beta_1/\beta_2$ must logically be zero, so if you reject $\beta_1=0$ you can reject the ratio being $0$.
More generally, there's a nifty trick you can do with ratios (again keeping in mind that this only works as long as $\beta_2$ isn't exactly 0):
$H_0: \beta_1/\beta_2=k$
is equivalent to
$H_0: \beta_1=k\beta_2$
so you can test such a ratio simply by testing the linear restriction:
$H_0: \beta_1-k\beta_2=0$
