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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.

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    $\begingroup$ 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. $\endgroup$ – user158565 Oct 8 '18 at 1:46
  • $\begingroup$ Thanks for pointing that out. I have edited my post. $\endgroup$ – Remy Oct 8 '18 at 1:48
  • $\begingroup$ What definition of "linear" are you using? $\endgroup$ – whuber Oct 8 '18 at 15:24
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My understanding is that linear (or not) refers to "linear in the coefficients", such that varying any parameter will linearly vary that part of the equation. For example, if you double A or B in the equation "Y = A * X + B * sin(X)" you double that part of the equation, and this is not true for "Y = A * X + sin(B * X)" because doubling B does not double the sin(B * X) part of the equation. If the equation is "Y = log(A + X)" this also is not "linear in the coefficients" and is called a non-linear equation.

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