# 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 $$X_3^*=\frac{1}{x_1}$$

• Welcome to Cross Validated! I like your middle equation. Why do you have a problem with lacking an intercept?
– Dave
Oct 21, 2022 at 2:06
• Thank you for your answer. It's a good question that you raise lol. I felt like there should be one but I could run the model without intercept... I have edited my post and corrected for the re-inversion mistake. What do you think ? Oct 21, 2022 at 10:28
• Try it only with y=x_1/beta0, and see what happens, maybe try a few values. Oct 22, 2022 at 21:18

$$\dfrac{1}{Y}=\beta_0\dfrac{1}{x_1x_2}+\beta_1\dfrac{1}{x_2}+\beta_2\dfrac{1}{x_1}$$
The $$\dfrac{1}{x_1}$$, $$\dfrac{1}{x_2}$$, and $$\dfrac{1}{x_1x_2}$$ are then nonlinear basis functions of the original variables.