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

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 $\sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 }$ 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 $R ^ { 2 }$ 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 $\hat { \beta } _ { 1 }$ ?