# Multiple linear regression, categorical predictors

I'm trying to understand how multiple linear regression with only categorical predictors works. In lectures we were told that, given a data set $(c_1, y_1), (c_2, y_2), ..., (c_n, y_n)$, $c_i \in \{1, 2, ..., K\}$. We can define a model, $j = 1, 2, ..., K$ such that:

$x_{i, j} = 1\ \ if\ \ c_i = j$

$x_{i, j} = 0\ \ otherwise$

The lecturer told us that the two models

$Y_{i} = \beta_0 + \beta_2x_{i, 2} + \beta_3x_{i, 3} + \dots + \beta_K x_{i, K} + \epsilon_i\ \ \ \ \ \ \ \ \ (1)$

and

$Y_i = \alpha_1x_{i, 1} + \alpha_2x_{i, 2} + \dots + \alpha_Kx_{i, K} + \epsilon_i\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$

are equivalent. Here I assume the usual assumptions on $\epsilon$ are made, i.e. $\epsilon_i\ i.i.d.\ N(0, 1)$. If I understand this model correctly, each row in the design matrix will only have two 1s (one from the column corresponding to the intercept parameter, and one from another column where $c_i = j$). ̃Intuitively, the fact that they are equivalent makes sense, as both systems have $n$ equations and $K$ unknowns (since one parameter is "eliminated"). I also get that when it comes to linear regression with categorical predictors you choose one variable as a reference level. However, how does one prove this equivalence mathematically? Is it possible to express $\beta$ in terms of $\alpha$? I figure this has to do with proving that the linear systems are equivalent, and hence one can be written as a linear combination as the other.

Furthermore, how can one interpret the predictors in these models? I get that, since $Y_i$ is a qualitative response, the parameters say something about how much each category contribute to the response. I understand that this will depend on which variable is the reference level. Additionally, how does this interpretation change if we do not eliminate a predictor and include all from, say $\beta_0$ to $\beta_K$?

I understand this is a lengthy question, and I am truly grateful for any answers to this even if every question is not answered.

As for the second question, if $Y_i$ is qualitative you probably should not be fitting an ordinary linear model, but rather a logistic or generalized linear model.