5- Heteroskedasticity

Question 1

What is the result of OLS under heteroskedasticity?

Answer 1

The result is still β but OLS is no longer efficient

Question 2

What are the 4 main consequences of Heteroskedasticity?

Answer 2

1.T-statistics using the standard error are not valid
2.Regression estimates can’t be used for confidence intervals or inferences
3.t and F statistics no longer reliable for hypothesis testing
4.Rejection of the null hypothesis too often

Question 3

What is the informal way of detecting Heteroskedasticity?

Answer 3

Plotting the residuals from the regression against the estimated dependent variable to see if the spread of residuals seems to depend on the variable

Question 4

What is the formal way of detecting Heteroskedasticity?

Answer 4

Regressing the squared residuals (u^2) on predicted values (X̂) or explanatory variables

Question 5

What differs the White test from Breusch-Pagan?

Answer 5

The White test residual regression is for all pairs of independent variables too

Question 6

What are the 5 steps of the Breusch-Pagan test?

Answer 6

1.Estimate the model and obtain the residuals ^ui
2.Regress the squared residual on all independent variables
3.Formulate null hypothesis all coefficients =0
4.Compute LM=nR²
5.Check LM table, if LM>χ² reject null

Question 7

What is the point of General Least Squares (GLS)?

Answer 7

Transform the observation matrix [y X] so that the variance in the transformed model is I

Question 8

How can you transform a regression function to homoskedastic when Ω is known?

Answer 8

Divide all the variables by σi, because you’re dividing each observation by something proportional to the error standard deviation for the observation

Question 9

What is the P matrix?

Answer 9

Matrix of 1/σi along principal diagonal and zeros elsewhere used to transform functions such that Py=y*

Question 10

What is the Cholesky root?

Answer 10

Ω⁻¹ = P’P

Question 11

How do you find the inverse of a diagonal nxn matrix?

Answer 11

Just take the inverse of the principal diagonal elements

Question 12

What is the transpose of a diagonal nxn matrix?

Answer 12

Itself i.e. P=P’

Question 13

In the transformed model y, what is the expected value of u?

Answer 13

E(u*) = PE(u) = 0

Question 14

In the transformed model y, what is the variance of u?

Answer 14

Var(u*) = Var(Pu) = I

Question 15

How do you find ^βGLS?

Answer 15

Same process as OLS but with * values then sub in equivalent P values i.e. y*=Py

Question 16

How can you prove the ^βGLS of the transformed model is unbiased?

Answer 16

E(^βGLS) = β
Take the expectation of the function, sub in for y and expand out

Question 17

How do you find the variance of ^βGLS?

Answer 17

var(^βGLS)=(X’Ω⁻¹X)⁻¹
Sub in for y and expand out

Question 18

How can you show ^βOLS is less efficient than ^βGLS?

Answer 18

Derive both their variances and show OLS is greater

Question 19

When is GLS infeasible?

Answer 19

When Ω is unknown it has n(n+1)/2 elements and n observations so is impossible to estimate

Question 20

What is the Feasible GLS technique and how does it work?

Answer 20

When Ω is unknown we can create an estimated version ^Ω

Question 21

What are the 3 steps of Feasible GLS?

Answer 21

1.Estimate OLS to obtain residuals ûᵢ
2.Construct 2 groups of variance estimates
3.Proceed with GLS procedure using ^Ω

Question 22

How is Feasible GLS un/biased?

Answer 22

Feasible GLS is naturally biased, but it is consistent in that for large values of n it will converge to true value

Question 23

What are the 4 steps of GLS when you can’t split variance groups?

Answer 23

1.Get OLS residuals from original regression ûᵢ
2.Run auxiliary regression on squared residuals to get an estimate of γ
3.Use this to estimate ^V=V(γ)
4.Apply FGLS using V instead of omega

Question 24

What is gamma (γ) in the context of GLS?

Answer 24

Coefficient of known variable

Question 25

What are the 2 Least squares methods when heteroskedasticity is suspected but the variance matrix is unknown?

Answer 25

-Feasible GLS if variance matrix structure is known
-OLS using White standard errors if variance matrix structure is not known

Question 26

In GLS, what do you use instead of Ω if variance has a coefficient e.g. var(u)=σV

Answer 26

Use the coefficient instead of Ω

Question 27

What is the definition of a positive definite matrix?

Answer 27

A matrix is positive definite if we have vector b such that vector b’Ab≥0

Question 28

What is the result if you sum the variances of a homoscedastic error term from 1 to N?

Answer 28

Nσ²
Because variance for the error term of each observation is the same

Question 29

What is the formula for the variance of the error terms for a subset of observations (e.g. a certain country)?

Answer 29

Variance of the sum of each error term over the number of them (N)

Question 30

What does homoscedasticity imply for the variance coefficient?

Answer 30

It must be equal to I

Question 31

How can you show a model is unbiased?

Answer 31

When the expected value of beta is equal to the true value of beta

Question 32

What are the 3 characteristics of the GLS estimator?

Answer 32

Efficient, consistent, unbiased- regardless of whether data is heteroskedastic or auto-correlated

Question 33

Describe the 3 steps of GLS when the variance matrix is known

Answer 33

1. Divide each y and x observation by σᵢ
2. Run OLS on the transformed data
3. Calculate variance matrix using transformed variables

Question 34

What 2 things does a normal distribution of error terms tell us u~N(0,σ²I)?

Answer 34

-Error terms are homoskedastic
-Error terms are independent

Question 35

How can you use the rule x’Ax ~ χ²(r), to prove if xi is Σx² ~ χ²(n)?

Answer 35

Set A=I and note that normal random variables are independent if they are uncorrelated