Does the dependent variable Y always need to have a linear relationship to all covariates when building a linear regression model? If it doesn't meet this requirement, can I still fit the model? From my understanding, quadratic terms can be included (and still counted as a linear model) to account for the non-linear relationship. So is this requirement really necessary?
I'm a bit confused by your question? So here's my response, apologies if I have misinterpreted your question. There are no assumptions of linearity in the relationships between dependent variables and regressors (independent variables). This is what you are investigating? There can be linear, non-linear or NO relationship and you can still perform fit model. There are, assumptions regarding residuals (NID(0,variance), but not for the terms in the model. You can certainly fit non-linear terms when performing GLM.
The phrase "Linear Regression" actually refers to fitting a model that is linear in the coefficients.
The first model (equation) below is linear with respect to its coefficients and is appropriate for linear regression. The second model is not linear in its coefficients, since the exponential transcendental function is involved.
Typically, if I have a model that is non linear, I try to reparameterize the model to be linear in it coefficients. For example, for the second model, if I take the natural log of Y, and fit ln(Y) as a linear function of X. Otherwise I use JMP to fit a non-inear function.
I hope that helps. This is just an addendum to @statman's response.
Thank you @gzmorgan0 and @statman! I randomly came across a video and the person said that we need to make sure that a linear relationship between the predictor and the response exists before using the linear regression model. So I was very confused and thought I missed something. Thank you for clarifying my confusion.