Books/notes recommendation for a rigorous explanation of Classical Linear Regression

Question

So far I have been following Econometrics text books which while intuitively clear, due to their lack of rigour and hence, poorly explained and maybe even contradictory notation, I am having a lot of trouble trying to understand the core math behind it which is unsurprising; they are books meant to teach you to put data into practice. What I am having trouble trying to understand is which ones are the vectors, variables and constants, conditional and unconditional expectation.

There is no upper limit to the rigour or prerequisites, they may use measure theory, group theory or quantum physics, the more the merrier, provided the concepts are either defined in the book or are concepts which can be easily found in some other pure math book. Also, they need not contain proofs.

They must contain explanation for the regression functions and equations of Simple and Multiple Linear Rigouression Model, rest like LS estimation, goodness of fit etc., I will understand on my own after the rigour is clear, they are not necessary. I would prefer if the simple linear regression chapters avoid matrix algebra and uses it later for the multiple case, though that is not necessary.

The recommendations need not be self contained or limited to one book.

(EDIT) I want the book to avoid matrix notation for SLRM only, and then introduce matrix notation. Also, the difference between PRF and SRF should be explained.

Examples of some poorly explained notation I can think of right now is I have read books defining $Y$ and $X$ as variables, and $Y_i$ and $X_i$ as the value of the variable at the $i$th observation where $Y_i$ is a random variable. Then, at least from what I believe $Y$ should not be defined as just a variable but a multivariate random variable. Yet in the same books, the unconditional $\mathbb E[Y]$ is defined as the sum of all possible $Y$ values times their probability i.e., the mean of the population, but the expectation of a vector as far as I know should be a vector of constants, not just a constant, I don't think it even be a scalar of all-one-vectors.

I also struggle with the assumptions of non stochasticity. Using same notation, is $X_i$ assumed to be constant values that $X$ a random variable takes, or is $X$ itself a non-stochastic vector which consists of elements $X_i$

Answer 1

You ask for clear definitions of concepts, coverage of linear models and their applications, and to avoid matrix algebra. These requirements suggest the following:

Michael J. Wichura, The Coordinate-Free Approach to Linear Models. 2009. Very geometric flavor.

Mervyn Stone, Coordinate-Free Multivariable Statistics. 1987. Very geometric. Requires substantial mathematical sophistication.

Ronald Christensen, Plane Answers to Complex Questions. 1996 (last updated 2020). Classic. Emphasizes vector space geometry. Wide array of applications.

Answer 2

Staying in econometrics land you could give Econometric Analysis of Cross Section and Panel Data by Wooldridge a shot. My grad school book and I found it sufficient. Alternatively, I have heard great things about Elements of Statistical Learning and have referenced it for several things although I am unsure of it's rigor for linear regression.