This (trivial) question is about the assumptions of linear regression.
[1] For example, I have a matrix X of size n*p consisting of p explanatory variables and n units, and vector y consisting of n dependent units. I want to perform linear regression on X, y. There are some common assumptions for the model such as linear relationship between $x_1, x_2, ..., x_p$ and y, uncorrelation of error terms,.... Let's name these set of assumptions $\Sigma^1$.
[2] However, let's say the result turns out to be not good (could be due to multicolinearity), so I remove one column from the matrix X to get the new matrix Z, and I perform linear regression again on Z and y. There are some common assumptions for the model such as linear relationship between $z_1, z_2, ..., z_{p-1}$ and y, uncorrelation of error terms,.... Let's name these set of assumptions $\Sigma^2$.
[3] It seems that the model is still not really good. I decide to perform PCA on X to get P, and I perform linear regression on P, y. There are some common assumptions for the model such as linear relationship between $p_1, p_2, ..., _p$ and y, uncorrelation of error terms,.... Let's name these set of assumptions $\Sigma^3$.
Every time I am not happy with my model. I modify the model by removing redundant columns or transforming the data, and I apply the model again. My question is that, for every time I do it, is it true that I have to start with a new set of assumptions (e.g. $\Sigma^2$) and completely forget assumptions that associated with previous models that you design (e.g. $\Sigma^1$)?
