5- Heteroskedasticity Flashcards
What is the result of OLS under heteroskedasticity?
The result is still β but OLS is no longer efficient
What are the 4 main consequences of Heteroskedasticity?
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
What is the informal way of detecting Heteroskedasticity?
Plotting the residuals from the regression against the estimated dependent variable to see if the spread of residuals seems to depend on the variable
What is the formal way of detecting Heteroskedasticity?
Regressing the squared residuals (u^2) on predicted values (X̂) or explanatory variables
What differs the White test from Breusch-Pagan?
The White test residual regression is for all pairs of independent variables too
What are the 5 steps of the Breusch-Pagan test?
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
What is the point of General Least Squares (GLS)?
Transform the observation matrix [y X] so that the variance in the transformed model is I
How can you transform a regression function to homoskedastic when Ω is known?
Divide all the variables by σi, because you’re dividing each observation by something proportional to the error standard deviation for the observation
What is the P matrix?
Matrix of 1/σi along principal diagonal and zeros elsewhere used to transform functions such that Py=y*
What is the Cholesky root?
Ω⁻¹ = P’P
How do you find the inverse of a diagonal nxn matrix?
Just take the inverse of the principal diagonal elements
What is the transpose of a diagonal nxn matrix?
Itself i.e. P=P’
In the transformed model y, what is the expected value of u?
E(u*) = PE(u) = 0
In the transformed model y, what is the variance of u?
Var(u*) = Var(Pu) = I
How do you find ^βGLS?
Same process as OLS but with * values then sub in equivalent P values i.e. y*=Py
How can you prove the ^βGLS of the transformed model is unbiased?
E(^βGLS) = β
Take the expectation of the function, sub in for y and expand out
How do you find the variance of ^βGLS?
var(^βGLS)=(X’Ω⁻¹X)⁻¹
Sub in for y and expand out
How can you show ^βOLS is less efficient than ^βGLS?
Derive both their variances and show OLS is greater
When is GLS infeasible?
When Ω is unknown it has n(n+1)/2 elements and n observations so is impossible to estimate
What is the Feasible GLS technique and how does it work?
When Ω is unknown we can create an estimated version ^Ω
What are the 3 steps of Feasible GLS?
1.Estimate OLS to obtain residuals ûᵢ
2.Construct 2 groups of variance estimates
3.Proceed with GLS procedure using ^Ω
How is Feasible GLS un/biased?
Feasible GLS is naturally biased, but it is consistent in that for large values of n it will converge to true value
What are the 4 steps of GLS when you can’t split variance groups?
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
What is gamma (γ) in the context of GLS?
Coefficient of known variable
What are the 2 Least squares methods when heteroskedasticity is suspected but the variance matrix is unknown?
-Feasible GLS if variance matrix structure is known
-OLS using White standard errors if variance matrix structure is not known
In GLS, what do you use instead of Ω if variance has a coefficient e.g. var(u)=σV
Use the coefficient instead of Ω
What is the definition of a positive definite matrix?
A matrix is positive definite if we have vector b such that vector b’Ab≥0
What is the result if you sum the variances of a homoscedastic error term from 1 to N?
Nσ²
Because variance for the error term of each observation is the same
What is the formula for the variance of the error terms for a subset of observations (e.g. a certain country)?
Variance of the sum of each error term over the number of them (N)
What does homoscedasticity imply for the variance coefficient?
It must be equal to I
How can you show a model is unbiased?
When the expected value of beta is equal to the true value of beta
What are the 3 characteristics of the GLS estimator?
Efficient, consistent, unbiased- regardless of whether data is heteroskedastic or auto-correlated
Describe the 3 steps of GLS when the variance matrix is known
- Divide each y and x observation by σᵢ
- Run OLS on the transformed data
- Calculate variance matrix using transformed variables
What 2 things does a normal distribution of error terms tell us u~N(0,σ²I)?
-Error terms are homoskedastic
-Error terms are independent
How can you use the rule x’Ax ~ χ²(r), to prove if xi is Σx² ~ χ²(n)?
Set A=I and note that normal random variables are independent if they are uncorrelated
