One of the assumptions of the Gauss-Markov Theorem is that the explanatory varia

Question

One of the assumptions of the Gauss-Markov Theorem is that the explanatory variable matrix has full column rank. Please answer the following questions: (a) Explain the concept of full column rank in terms of the explanatory variable matrix. (b) Express mathematically the least squares estimator for beta and explain the consequences for this estimator if the explanatory variable matrix does not have full column rank. (c) What is the relationship, if any, between the lack of full column rank and multicollinearity?

Explanation / Answer

The Gauss Markov Theorem solves the linear regression model, and under the conditions of exogeneity and homoskedasticity gives a solution for 1 and 2 as the Best Linear Unbiased Estimator (BLUE) meaning that they ( 1 and 2) have the smallest variances among the class of all linear unbiased estimators. 1 and 2 are calcuated using Ordinary Least Square method.

(a) To Solve the Gauss Markov Model, we use the ordinary Least Square Method, and that uses matrix solution. When we put the sample data of the explanatory variable in a matrix, the rank of this matrix must be non singular which means it should have full rank. If the matrix does not have full rank, or if matrix is singular, then the matrix will become non invertible and we will not get the solution of Gauss Markov model.

(b)

The general linear regression model is

yi= Xi+ei

where e is the error or the residuals.

y is the dependent variable and

X is the independent variable.

i=1,2,....,n

so residual is expressed as

ei = yi - Xi

Now the sum of the squared residuals is expressed in matrix vector form as

e'e =(y - X)'(y - X)

     =y'y - 'X'y - Xy' + 'X'X

      now transpose of a scalar is scalar , so -Xy' = (-Xy')' = -'X'y

      Therefore

e'e= y'y - 2'X'y + 'X'X

To ensure that residue is minumum, we will take the derivative of e'e and equate it to 0

therefore

derivative(e'e)/derivative()= -2X'y +2X'X =0

taking another derivate with respect to , we get 2X'X.

So if X has a full rank, meaning that it is invertible and its determinant is non zero, then we get a positive value for 2X'X which would ensure the value remains minimum.

(c) Lack of full rank indicates multicollinearity.

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