Definition of Linear

Question

I'm not sure if this question is too broad, but here goes: What does the word "linear" mean in mathematical models? In my econometrics class, one of the Gauss Markov assumptions for running regressions, as I understand it, is that the model is linear in parameters. What would a model that is non-linear in parameters look like? As a side note, is a curve different from a line? If so, how?

Answer 1

Théophile gave the answer for linear or nonlinear functions.

If we consider regression, the model is linear if the derivative of the equation with respect to any of the parameters does not involve any parameter.

So, for the simplest case $y=a x$, the model is linear since $\frac{dy}{da}=x$ while for $y=a^2 x$ the model is nonlinear since $\frac{dy}{da}=2ax$.

For the regression part, that is all; the nonlinearity of the involved terms does not matter. For example, the model $$y = a\sin\left(x_1 + \frac{x_2}{x_3}\right) + b(x_1x_2x_3)^7$$ is linear with respect to $a,b$.