2

I'm dealing with an exercise that tells me that the natural relationship between $Y$ and two predictors $x_1$ and $x_2$ can be modelled as follows:

$Y = e^{B_0+x_1B_1+x_2B_2}$/$1+e^{B_0+x_1B_1+x_2B_2}$

Then, the exercise asks me to propose a transformation that would make this model linear, so that it can easiliy be estimated with data. What I have done so far is:

I've taken natural logs on both sides, which gives $logY = B_0+x_1B_1+x_2B_2 - log[1+e^{B_0+x_1B_1+x_2B_2}]$. I was thinking of using the fact that $log(1+r)=r$ for $r$ small enough, but then I would get $logY = B_0+x_1B_1+x_2B_2 - e^{B_0+x_1B_1+x_2B_2}$. After I rewrite it as $logY = B_0+x_1B_1+x_2B_2 - e^{B_0}e^{x_1B_1}e^{x_2B_2}$, it doesn't seem to have gotten much better. Could someone help me out?

jamesstealth
  • 31

1 Answers1

1

Think of

$$ Y = \frac{\Phi}{1-\Phi}\Rightarrow \Phi = \frac{Y}{Y+1} $$

and then

$$ B_0 +x_1B_1+x_2B_2 = \ln\left(\frac{Y}{Y+1} \right) $$

Cesareo
  • 33,252