In the Multiple Regression Model with Two Regressors, the Formula

question 26

Essay

In the multiple regression model with two regressors, the 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 $z _ { i } = Z _ { i } - \bar { Z }$ ).
An alternative way to derive the OLS estimator is given through the following three step procedure.
Step 1: regress Y on a constant and $X _ { 2 }$ , and calculate the residual (Res1).
Step 2: regress $X _ { 1 }$ on a constant and $X _ { 2 }$ , and calculate the residual (Res2).
Step 3: regress Res1 on a constant and Res2.
Prove that the slope of the regression in Step 3 is identical to the above formula.

Understand the general population trends and movements within specific geographic regions.

Recognize the environmental challenges and effects of urbanization, including pollution and resource consumption.

Identify factors influencing migration to urban areas and the growth of cities.

Describe the concept of ecocities and strategies for sustainable urban living.

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Definitions:

Capital Flight

The rapid movement of large sums of money or capital out of a country due to economic or political instability.

Financial Capital

The funds that corporations and individuals use to invest in and finance new business ventures and operations.

Real Capital

Physical assets that are used in the production of goods and services, such as machinery, buildings, and equipment.

Vicious Circle

A complex sequence of negative events or conditions that reinforce each other, leading to a deteriorating situation or a series of worsening cycles.