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)
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