The OLS Formula for the Slope Coefficients in the Multiple

question 9

Essay

The OLS formula for the slope coefficients in the multiple regression model become increasingly more complicated, using the "sums" expressions, as you add more regressors. For example, in the regression with a single explanatory variable, the formula is
$$\frac { \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) \left( Y _ { i } - \bar { X } \right) } { \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } }$$
whereas this formula for the slope of the first explanatory variable is
$$\hat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } } { \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } \right) ^ { 2 } }$$
(small letters refer to deviations from means as in $z _ { i } = Z _ { i } - \bar { Z }$ ) in the case of two explanatory variables.Give an intuitive explanations as to why this is
the case.

---

Definitions:

Interval Level

A measurement scale that allows for the comparison of differences in quantities, where equal differences between values represent equal differences in what is being measured, but does not have a true zero point.

Frequency (f)

The number of times a particular value or class of values occurs in a data set.

Ordinal Level

A level of measurement where values can be ordered or ranked, but the intervals between values are not necessarily consistent or known.

Cyber-Phishing

A fraudulent practice of sending emails purporting to be from reputable companies to induce individuals to reveal personal information.