# Systematic way to determine if a model is linear or nonlinear? [duplicate]

Determine whether the following models are linear, intrinsically linear, or nonlinear (disregard the error structure):

$$y=\beta_0+\beta_1 x_1 +\beta_2 x_2^{\beta_3}+\epsilon$$

$$y=\beta_1 + \left(\frac{\beta_2}{\beta_1}\right)x+\epsilon$$

$$y=\beta_1+\beta_2 e^{\beta_3 x}+\epsilon$$

I'm having trouble finding a systematic way to determine whether a model is linear, intrinsically linear, or nonlinear.

For $$y=\beta_0+\beta_1 x_1 +\beta_2 x_2^{\beta_3}$$ I have that the best you can do is to have

$$log(y-\beta_0)=log(\beta_1 x_1 +\beta_2 x_2^{\beta_3})$$

but there is no way to separate $$x_1$$ and $$x_2$$ so this model is nonlinear. Is this valid reasoning for this being nonlinear though?

For $$y=\beta_1 + \left(\frac{\beta_2}{\beta_1}\right)x$$ I have that $$y$$ can be expressed as

$$y=\theta_1+\theta_2 x$$

which is linear in the transformed parameters $$\theta_1$$ and $$\theta_2$$ so this model is intrinsically linear.

For $$y=\beta_1+\beta_2 e^{\beta_3 x}$$ since $$\beta_1$$ is just constant, we have

$$log(y-\beta_1)=log(\beta_2)+\beta_3 x$$

so this model is intrinsically linear.

• Three of them are mathematical equations, and none of them is statistical model. One example of statistical model: $Y = X\beta + \epsilon$, where $\epsilon$ follows normal distribution with mean zero. – user158565 Oct 8 '18 at 1:46
• Thanks for pointing that out. I have edited my post. – Remy Oct 8 '18 at 1:48
• What definition of "linear" are you using? – whuber Oct 8 '18 at 15:24