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 $\beta _ { 1 }$ ?

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Definitions:

Demand Shifts

Movements of the demand curve to the left or right in a market diagram, indicating a change in the amount consumers are willing and able to purchase at various prices.

Equilibrium Price

The cost at which the amount of a product or service that consumers want to buy matches the amount available for sale, leading to a state of equilibrium in the market.

Supply Shifts

Supply shifts refer to changes in the supply curve caused by factors other than price, such as technology, production costs, and supplier expectations, leading to different quantities being supplied at the same price.

Surplus

The situation in which the quantity of a good or service supplied exceeds the quantity demanded, often leading to lower prices.