Full rank assumption in the linear regression model explanation:

I need some help understanding the full-rank assumption. My book, Econometric Analysis by Greene, presents the following example:

Suppose that a cross-section model specifies that consumption, $C$, relates to income as follows: $$C = \beta_1 + \beta_2X_{non-labor\ income} + \beta_3X_{salary} + \beta_4X_{total\ income} + \epsilon$$ where $X_{total\ income}$is exactly equal to $X_{salary}$ plus $X_{non-labor\ income}$. Clearly, there is an exact linear dependency in the model. Now let $\beta_2' = \beta_2 + a, \beta_3' = \beta_3 + a$, and $\beta_4' = \beta_4 - a$, where a is any number. Then the exact same value appears on the right-hand side of $C$ if we substitute $\beta_2', \beta_3', \beta_4'$ for $\beta_2, \beta_3$, and $\beta_4$. Obviously, there is no way to estimate the parameters of this model.

What I do not understand of the above explanation are the second-to-last and last sentences. What do they mean? What are they trying to convey??

• I only really got linear regression when I learned the linear algebra behind it. Basically, when the columns of your matrix X aren't independent, then the matrix $$X^{T}* X$$can't be inverted. In other words, you put that matrix formula in your calculator and you get an error--not even a bad answer, you get no answer. – Steve S Sep 30 '15 at 6:00
• your new $\beta' s$ are only old $\beta s$ plus a constant $a$ what are that for? It is not even a random coefficient regression. – Deep North Sep 30 '15 at 6:06
• Hey, I just tried to clean up the LaTeX a little--hopefully it will be a little clearer with the names written as subscripts... – Steve S Sep 30 '15 at 6:38
• Important: The problem has $\beta_4 - a$! (and not $\beta_4 + a$)--does that clear things up?? – Steve S Sep 30 '15 at 6:56
• To see what the author is getting at, try substituting in $(\beta' - a)$ for each $\beta$-value (and $\beta' + a$ for $\beta_{total\ income}$). Then, replace the term $a*X_{total\ income}$ with $a*(X_{non-labor\ income} + X_{salary})$ and notice how when you simplify the equation all the "a" terms cancel out. – Steve S Sep 30 '15 at 6:56

There is an error. It should be $\beta_4'=\beta_4-a$. If we substitute these new $\beta$s into the regression equation we get: