It Is Not Hard, but Tedious, to Derive the OLS

question 51

Essay

It is not hard, but tedious, to derive the OLS formulae for the slope coefficient in the
multiple regression case with two explanatory variables.The formula for the first
regression slope is $$\hat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } } { \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } x _ { i i } x _ { 2 i } \right) ^ { 2 } }$$
(small letters refer to deviations from means as in $z _ { i } = Z _ { i } - \bar { Z }$ ). Show that this formula reduces to the slope coefficient for the linear regression model
with one regressor if the sample correlation between the two explanatory variables is
zero.Given this result, what can you say about the effect of omitting the second
explanatory variable from the regression?

Describe the impact of neurocognitive disorders on individuals and their families.

Explain the diagnostic process and criteria for neurocognitive disorders.

Compare and contrast various neurocognitive disorders.

Discuss treatment and management options for neurocognitive disorders.

---

Definitions:

Exchange Theory

A social theory that views interactions and relationships as transactions where individuals seek to maximize rewards and minimize costs.

Decision-Making

The cognitive process of choosing a course of action among multiple alternatives, often involving considerations of risk and reward.

Coveted Rewards

Highly desired benefits or prizes that individuals or groups seek to obtain, often driving competition and motivation.

George Homans

An influential American sociologist known for his work in developing social exchange theory, which views social interactions in terms of cost-benefit analyses.