Topic 4: The Frisch-Waugh-Lovell Theorum

Question 1

Suppose a variable z is constructed from x, where:

z_i = x_i - mean(x)

What are the properties of this variable?

Answer 1

This variable can be described as centered. It is orthogonal to ι (a column vector of ones.)

This can be seen below.

Question 2

How can a variable be centered with projections?

Answer 2

Question 3

What is the result of multiplying the projection matrix of X by the projection matrix of one of it’s columns?

Answer 3

P₁P_X=P_XP₁=P₁

The final projection is to the smallest space.

Question 4

When a set of regressors is orthogonal to another, what happens to the estimates β when one is dropped out of the regression.

Answer 4

The estimate of β for thre regressor group that remains is unchanged - because they were orthogonal, neither group of variables added any information to the other.

Question 5

How can orthogonal regressors be constructed?

Answer 5

By multiplying one regressor group by the complementary projection matrix of another.

i.e.

X*₂=M₁X₂

Question 6

What is the result of modelling

y = X₁α₁ +(X₂+X₁A)α₂ + u

Where A is an arbitrary k x k matrix.

What does this imply for the regression

y = X₁α₁ +M₁X₂α₂ + u

Answer 6

The regression would simplify to

y = X₁(α₁+Aα₂)+X₂α₂

Thus when estimated, α₂ would be unchanged, and the estimate of the X₁ parameter would be modified as given.

As M₁ is made up entirely of X₁, this implies such a regression would not change the estimate of X₂, only X₁ and the residuals.

Question 7

What is the difference between the following regressions?

Answer 7

B₁ != α₁

u!= v!= w

But B₂ = α₂ = γ₂

Question 8

How can the regression

y = M₁X₂α₂ + v

Be altered to have identical residuals to the original (full) regression?

Answer 8

By using the regression:

M₁y = M₁X₂α₂ + u

The diagram below illustrates.

Question 9

Prove that

β₂ = (X₂^TM₁X₂)^-1X₂^TM₁y

(from the FWL theorem)

is identical to the the Method of Moments estimator.

Answer 9

Question 10

Consider the model:

y = Sβ₁ + Xβ₂ + u

Where S is a n x 4 matrix of seasonal indicators. What is the significance of the following regression?

M_Sy = M_SXβ + u

Answer 10

This regression is run on seasonally adjusted data.

M_Sy is the seasonally adjusted result, and

MsX is the seasonally adjusted data.

Note that this is a very trivial seasonal adjustment - more work is always done in practice, and it is not the case that seasonalizing data ‘does not matter’ for statistics.

Question 11

For a model

y = Xβ₁ + Tβ₂

Where T_T = t. X₁ = col(1)

What is the purpose and method of centering the regressor T?

Answer 11

This can be achieved by premultiplying T by M_i

• The complementary projection to a vector of ones.

This makes it orthogonal to the constant and makes the constant the expected value of the regressant over the time period, rather then the expected starting value.

Question 12

What is R²?

Show the formula for the uncentered version R_u²

Answer 12

A measure of the goodness of fit, also known as the coefficient of determination.

Calculated by:

Where θ is the angle between y and S(X).

Question 13

What is the formula for the centered R², R_c²?

Question 14

What is the meaning of Trace(X) or Tr(X)?

Answer 14

Takes the sum of the diagonal elements of the matrix X

Question 15

What are the properties of the trace function?

Answer 15

• Cyclic Permutation: Tr(ABC)=Tr(BCA)=Tr(CAB)
• Tr(A + B) = Tr(A) + Tr(B)
• The trace and expectation operations can be interchanged. i.e. E(Tr(X))=Tr(E(X))

Question 16

What is the result of Tr(P_X) and Tr(M_X)?

Answer 16

Tr(Px) = Tr(I_k) = k

(The firstequality comes from cyclic permutation of the P formula).

Tr(M_X) = n-k