Introductory Econometrics: A Modern Approach 6th Edition by Jeffrey M Wooldridge

Exercise 6

Consider the setup of the Frisch-Waugh Theorem.

(i) Using partitioned matrices, show that the first order conditions  can be written as

(ii) Multiply the first set of equations by  and subtract the result from the second set of equations to show that

where . Conclude that

(iii) Use part (ii) to show that

(iv) Use the fact that M₁X₁ = 0 to show that the residuals ü from the regression ÿ on  are identical to the residuals ü from the regression y on X₁, X₂. [Hint: By definition and the FW theorem,

Now you do the rest.]

Step 1 of 6

(i)

Consider that matrix X have n rows and k columns. Matrix X is partitioned as below:

Here,

The transpose of matrix X is written below:

Therefore,can be written as shown below:

Consider that vector have k rows and 1 column. Vector is partitioned as below:

Here,

Therefore,can be written as shown below:

The first order conditions is written below:

This is because: