CLRM – Assumption 2: Full Rank of Matrix X

Assumption 2 requires the matrix of explanatory variables $X$ to have full rank. This means that in case matrix $X$ is a $N \times K$ matrix the rank of matrix $X$ is $K$ . Namely, $Rank(X) = K$

Two conditions are necessary to ensure assumption 2. First, the number of observations ( $N$ ) cannot be smaller than the number of explanatory variables ( $K$ ) in the model. Formally, $N \geqslant K$ . Second, there cannot be an exact linear relationship between two explanatory variables. This means that it is impossible to include one variable twice or include a variable which is a linear combination of another variable as this would lead to exact multicollinearity. We have exact multicollinearity when our model includes two variables with an exact linear dependence. We cannot calculate the OLS estimator in the case perfect multicollinearity in our matrix X, because $X'X$ would be singular.