In the Case of Perfect Multicollinearity, OLS Is Unable to Calculate

question 64

Essay

In the case of perfect multicollinearity, OLS is unable to calculate the coefficients for the explanatory variables, because it is impossible to change one variable while holding all other variables constant. To see why this is the case, consider the coefficient for the first explanatory variable in the case of a multiple regression model with two explanatory variables: $\hat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } } { \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } \right) ^ { 2 } }$ (small letters refer to deviations from means as in $z _ { i } = Z _ { i } - \bar { Z }$ ).
Divide each of the four terms by $\sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 }$ to derive an expression in terms of regression coefficients from the simple (one explanatory variable)regression model. In case of perfect multicollinearity, what would be R² from the regression of $X _ { 1 i }$ on $X _ { 2 i }$ ? As a result, what would be the value of the denominator in the above expression for $\beta _ { 1 }$ ?

Distinguish between types of stresses and their sources, including daily hassles and major life events.

Recognize different types of conflicts and the stress associated with decision-making.

Identify strategies for managing stress and the role of positive events (uplifts).

Understand the definition and examples of social categories and social groups, including primary groups.

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Definitions:

Cost Flow Assumptions

An accounting method that determines the cost of goods sold and ending inventory based on the presumed flow of inventory costs.

Cost-to-Retail Ratio

A method used in retail to calculate the cost of goods sold based on the ratio of the cost of goods available for sale to the retail price of the goods.

Retail Inventory Method

An accounting method used by retailers to estimate inventory levels by incorporating the cost to retail price ratio.

Sales Discounts

Reductions in the price of goods or services offered to customers, typically as an incentive to encourage prompt payment.