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I am confusing with the wording here. I was reading a book on linear regression. "The primary concern for linear models is that they display linearity in the parameters. Therefore, when we refer to a linear regression model we generally assume that the equation is linear in the parameters; it may or may not be linear in the variables"

Can anyone explain that last sentence to me? As it seems a bit odd to say so.. Like: y=a+b*x^2 In my understanding, this is linear in parameters, not linear in the variables, so that makes it a linear model? It feels like it should be the other way round, can someone please explain? Thanks.

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Lets take an example $$ y = \sum_{k=0}^n a_k x^k $$ where the first few terms are $$ y = a_0 + a_1 x + a_2 x^2 +\cdots a_nx^n $$ we can re-label things as $v_k = x^k$ we have $$ y = a_0 + a_1 v_1 + a_2 v_2 + \cdots + a_nv_n $$ where we regress on the different variables $v_k$'s or linear in parameters. A nonlinear function which is not linear in parameters are given by $$ y = \frac{x}{a_0+a_1x} $$ or $$ y = \sqrt{a_0 + a_1 x+a_2x^2} $$ etc..

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