To prove this I started by using: vector of Beta estimates for category k = [Bk], matrix of estimates for category k = [Xk], Y estimates for category = [Yk].

1. Find betas for single category:

Calculating [Ba] = (Xa' * Xa)inv * (Xa' *Ya) for the

2. Find betas for matrix of combined categories for the general case.

In block diagonal form:

[X] becomes [Xa 0 ... 0]

becomes [0 Xb... 0]

becomes [0 0... Xk]

[Y] becomes [Ya]

[Yb] ...

[Yk]

and you do the same multiplication B = (X' * X)inverse * (X'*Y).

Multiplying this out I get (X' * X)inverse =

[1/Xa^2 0 ... 0]

[0 1/Xb^2... 0]

[0 0... 1/Xk^2]

and X' * Y:

[XaYa ..... 0]

[0 XbYb.... 0]

[0 0.. XkYk]

and eventually I get B = Ya * Xa'.

Now that I'm this far I am not sure how to prove that the singular case for Ba = Ya*Xa' without adding specific numbers. Does anyone know how to proceed? Any help would be appreciated.