Linear regression expression of a non-linear model $Y=\frac{x_1x_2}{β_0+β_1x_1+β_2x_2}$
It was written on some slide of my econometrics class that such a model could be expressed in the form of a linear model, but I am struggling to derive it by myself.
Here is what I have attempted so far:
$\frac{1}{Y} = \frac{\beta_0+\beta_1x_1+\beta_2x_2}{x_1x_2}$
$\frac{1}{Y} = \frac{\beta_0}{x_1x_2}+\frac{\beta_1x_1}{x_1x_2}+\frac{\beta_2x_2}{x_1x_2}$
$\frac{1}{Y} = \frac{\beta_0}{x_1x_2}+\frac{\beta_1}{x_2}+\frac{\beta_2}{x_1}$
$Y^*=0 + \beta_0X_1^*+\beta_1X_2^*+\beta_2X_3^*$
where $X_1^*=\frac{1}{x_1x_2}$, $X_2^*=\frac{1}{x_2}$ and $X_3^*=\frac{1}{x_1}$

You had it before you removed all of your work.
$$
\dfrac{1}{Y}=\beta_0\dfrac{1}{x_1x_2}+\beta_1\dfrac{1}{x_2}+\beta_2\dfrac{1}{x_1}
$$
The $\dfrac{1}{x_1}$, $\dfrac{1}{x_2}$, and $\dfrac{1}{x_1x_2}$ are then nonlinear basis functions of the original variables.
Yes, the inversion lacks an intercept, but linear models don’t require intercepts. It’s often advisable to include an intercept in your model, but an intercept-free linear model remains a linear model.
