# Geometric interpretation of the maximum likelihood estimation

I was reading the book The Identification Problem In Econometrics by Franklin M. Fisher, and was confused by the part that he demonstrates the identification by visualizing the likelihood function.

The problem could be simplified as:

For a regression $Y=a+Xb+u$, where $u \sim i.i.d. N(0,\sigma^2I)$, $a$ and $b$ are the parameters. Suppose $Y$ has a coefficient $c$ which equals to unity. Then the likelihood function in the space of $c, a,b$ would have a ridge along the ray corresponding to the vector of true parameters and its scalar multiples. When considering only the place given by $c=1$, the likelihood function would have a unique maximum at the point where the ray intersected that plane.

My questions are:

1. How should one understand and reason about the ridge and the ray mentioned in the demonstration.
2. Since the ray are the true parameters and scalars, why is the ray not on the plane given by $c=1$ since the true value of parameter $c$ is 1.

Out of context this passage is a bit vague but here is how I interpreted it.

Suppose I wanted to perform a linear regression on $cY$. I would write $cY = a' +Xb' + u$ where $u \sim N(0, c^2 \sigma^2)$. If $Y=a_0+Xb_0$ are the true parameters then clearly $cY = c a_0 + Xcb_0$ are the true parameters of $cY$.

For fixed $c$ the likelihood function for this regression on $cY$ has a unique maximum at the point $a'=ca_0$ and $b' = cb_0$. Thus, for general $c$ the ray of scalar multiplies of the true parameter forms the ridge of the likelihood function as a function of three variables. Now take $c=1$ to intersect with the $c=1$ plane.