I am experiencing some cognitive dissonance about what 'linear in the parameters' means. For example, here and here.
For example, my understanding is $y_i = \beta_0 + \beta_1\beta_2x_1 + \exp(\beta_3)(x_2)^2 + \epsilon$ is not linear in the parameters, because it has two parameter variables multiplied together (namely ${\beta_1, \beta_2}$).
If $\beta_1$ (say) was replaced with $\gamma_1$, a constant, it would be.
Appreciate if someone could clarify this point.
Your example model can be reëxpressed to be linear in the parameters $\alpha=\beta_1\beta_2$ & $\zeta=\exp\beta_3$:
$$g(\operatorname{E} Y) = \beta_0 + \alpha x_1 + \zeta x_2^2$$
(Clearly $\beta_1$ & $\beta_2$ aren't separately estimable; a non-linear model wouldn't help there. And note that $\hat\zeta$ must be constrained to be positive.) Some models can't be so reëxpressed:
$$g(\operatorname{E} Y) = \beta_0 + \beta_1 x_1 + x_2^{\beta_2}$$
Some can be though it's not obvious at first: https://stats.stackexchange.com/a/60504/17230.
There's a very thorough discussion of different meanings of "linear" at How to tell the difference between linear and non-linear regression models?.
Linear in the parameters means that you can write your prediction as
$$\beta_0+\sum_{j=1}^px_{ij}\beta_j $$
For some definition of $x_{ij} $. But these x's need not be linear functions of your data. For example, ploynomial fitting of a time series has $x_{ij}=t_i^j $ where $t_i $ is the time associated with data point $i $. The prediction is a non linear function of time, but it is linear in the betas.
UPDATE
In response to the comment, the answer is "sort of". If $\beta_2$ was constant, then the predictor is linear in $\beta_0,\beta_1,\exp (\beta_3) $. It is not linear in $\beta_3$, but a transformation of $\beta_3$. In terms of least squares estimates it doesn't make much difference here.
I think it's better for you to understand the three components of the GLM. Esp, you need understand how link function is defined.
You can refer to the page 7 in the slides below. 'linear in the parameters' is true after being transformed by the link function.
