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.
