Matrices in OLS II

The Technicals

In the book Quantitative Methods in Finance by Watsham and Parramore and in class, it tells us how in empirical data, the series of $x$'s or the matrix with $x$ variables are usually not square, thus not invertible. We just have to work on what was given which are  $\hat{x}$ and  $\hat{y}$.
That's why we have to transform $$\hat{b} = \hat{x}^{-1}\hat{y}$$ into
$$\hat{b} = (\hat{x}^T \hat{x})^{-1} \hat{x}^T \hat{y}$$
To transform it, identity is used and so is $\hat{x}^T$ as can be seen below.  This is to satisfy my curiosity about the identity.
$$\begin{array}{rl} b = & x^{-1}y & \\ = & x^{-1}I_n y \\ = & x^{-1}(x x^{-1})^T y \\ = & x^{-1}(x^{-1})^T x^T y \\ = & (x^T x)^{-1} (x^T y)  \end{array}$$
I looked for other ways to simplify the equation but so far, this is the easiest with the given data.

Application Found

• Regression Analysis (of course) - The measurement of change in one variable (y)  that is the result of changes in other variables (x) . Regression analysis is used frequently in identifying the variables that affect a certain stock's price.
• Hedging or Hedge Ratio (211-212 book)