I have read some explanations about the properties of linear vs nonlinear models, but still I am sometimes not sure if a model on hand is a linear or a nonlinear one. For example, is the following model linear or nonlinear?
$$y_t=\beta_0 + \beta_1B(L;\theta)X_t+\varepsilon_t$$
With:
$$B(L;\theta)=\sum_{k=1}^{K}b(k;\theta)L^k$$
$$L^kX_t=X_{t-k}$$
Where $b(k;\theta)$ represents (a decaying) Exponential Almon Polynomial function of the form:
$$b(k;\theta)=\frac{\exp(\theta_1 k+\theta_2k^2)}{\sum_{k=1}^{K}\exp(\theta_1k+\theta_2k^2)}$$
In my view, my main equation (the first one) is linear with respect to $X_t$, because this term is just multiplied with a weight. But I would say the weighting function (the last equation) is nonlinear with respect to the parameters $\theta_1$ ans $\theta_2$.
Can someone explain to me if my main function is a linear or a nonlinear one and what does it mean for the estimation procedure – do I have to apply linear or nonlinear least s squares method?. Furthermore, what is the discernible feature by means of which I can definitely identify if a function is a nonlinear or linear one?
Best Answer
With the usual definitions of linear and nonlinear with regard to modelling, it's not linearity with respect to the predictors that's the critical aspect, but linearity with respect to the parameters. A nonlinear model is nonlinear because it's not linear in parameters.
For example, the first sentence here says:
By contrast, Generalized Linear Models generally have a nonlinear relationship between response and predictors, but the link-transformed mean response (the linear predictor, $\eta$) is linear in the parameters.
[By that definition, I believe your model is nonlinear in the $\theta$s, though if the $\theta$s are specified (known) then that nonlinearity isn't relevant to estimation. If they're being fitted, then the model is nonlinear.]