• Linear in Variables, Linear in Parameters: y=β0​+β1​x+ϵ
• Nonlinear in Variables, Linear in Parameters: y=β0​+β1​x+β2​x2+ϵ
• A nonlinear equation in variables involves at least one independent variable raised to a power other than one or involved in a product with another variable, leading to a non-linear relationship between the dependent variable and the independent variables.
• Nonlinear in Parameters: y=α0​+α1​eβx+ϵ
• - Nonlinear in Parameters: An equation is nonlinear in parameters if any parameter is raised to a power other than one, appears in a product with another parameter, or is involved in a nonlinear function
• - Linear in Parameters: An equation is linear in parameters if the parameters \(β0​,β1​,…,βn​) appear linearly, meaning that each parameter is multiplied by a variable (or is constant) and is not raised to any power other than one, nor involved in a product with another parameter.

A. Linear in Variables and linear in parameters-LV+LP. That means you have a linear (line) relation that describe the relation between variables. Quoting Mona Alariqi , you have y=β0​+β1​x+ϵ or y=a0+a1x1+a2x2+ ...anxn+u. See in this sense Keynes consumption model: Cons=a0+a1Income. Please see for example: https://pure.uvt.nl/ws/files/522521/113.pdf

B. Linear in Variables non-linear in parameters-LV+NLP. This means the econometric model (last form) has a linear functional form in variable x and y, but the coefficients have a more complex expression. An example is when you want to predict the productivity(w) of a region based on the odd-ratio of recovery(OR) after a shock. Therefore you have w=c0+c1OR+u. ODD ratio is already exp(b) from a previous model. Therefore w=c0+c1*exp(b)+u.

C. Non-linear in variables but linear in parameters-NLV+LP.

The connection between the dependent variable and possible causes is non-linear. For example,in econonomics, we have Kuznet curves, Cobb-Douglas model, Phillips curves. Some of the NLV+LP can be liniearized....

D. NLV+NLP . It is a combination of previous. I see a fast example the CES function. Also we may combine a Kuznets curve or Phillips curve with some exp(beta*x).https://estima.com/ecourse/samples/BayesSampleChapter.pdf

https://www.sciencedirect.com/topics/earth-and-planetary-sciences/kuznets-curve#:~:text=The%20Kuznets%20Curve%20is%20a,Kuznets%20in%201955.

A linear equation is of the form where each term involves variables raised to the first power, and variables are not multiplied together. For example, ( 3x + 4y = 12 ) is linear.

A nonlinear equation includes terms where variables are raised to powers other than one, or variables are multiplied together.

In general, parameters are simply nubers (ineger, rational, real). They can, in principle, be the result of of a second equation, but if one looks at one equation, I would say, that parameters are not variables and therefore neither linearly nor non-lineary dependent.

A simple example for a linear equation may be a consumptions function like

C=c0+c1Y with Y as disposible income.

An example for a non-linear relation is the Cobb-Douglas production function with production Q=cKaLb (I mention it although I think it is nonsense). For estimating a and b, mostly equations like this are logged to get a linear equation lnQ=lnc+a*lnK+b*lnL or - as dln X is near %X (as long as percentage changes are not too far from zero) - %Q=a*%K+b*%L. Many non-linear functions can (within limits) be transformed into linear ones. Therefore, the question of linear of non-linear is often a definitional one.