Week 2 Flashcards
(27 cards)
What is autocorrelation? Verbal and mathematical
Relationship between variable and its lag
E(εiεj) = σij, i!=j
E(εε’) = Ω = matrix of all different σij -> symmetric and positive definite
Also possible heteroskedasticity
Can see in plot of residual against residual lag - correlated
What are the properties of OLS under autocorrelation?
- Unbiased
- Consistent
- Inefficient
- Incorrect SE
What estimators should we use if autocorrelation?
Newey-West Estimator Covariance Matrix
Where do the weights in the Newey-West estimator come from?
The Kernel function
What are the properties of Newey-West SE?
HAC (Heteroskedastic and autocorrelation consistent)
What is the idea of the Cochrane-Orcutt procedure?
Errors are from an autoregressive model of order 1
You have the normal regression, the lag regression, and the regression of error term with lag
Find value of y(i) - py(i-1)
What are the alternatives to Cochrane-Orcutt procedure?
NLS for y(i) = py(i-1) + B1(1-p) + B2(xi-px(i-1)) + n(i)
In Eviews, AR(1)
What is the idea of GLS?
Transform data s.t. the conditions for efficient OLS hold
What is the Choleski decomposition?
PP’ = Ω
Transformed data: y* = P^(-1)y ; X* = P^(-1)X ; ε* = P^(-1)ε
Properties of GLS disturbances
Homoskedastic and no autocorrelation
=> OLS for transformed model efficient estimator for β
Compare GLS to Cochrane-Orcutt
- In GLS 1st observation is included
- In GLS scaling factor 1/σ(n)
GLS estimator + expected value and variance
b = (X'Ω^(-1)X)^(-1)X'Ω^(-1)y E(b) = β Var(b) = (X'Ω^(-1)X)^(-1)
Why do we need feasible GLS?
In practice often Ω unknown and have to estimate it
What are the steps of FGLS?
1) Estimate the covariance matrix
a) Apply OLS in y=Xβ + ε -> b consistent
b) Estimate Ω using residuals e = y - Xb : Ω^ = ee’
2) Apply OLS on the transformed data
a) Use Ω^ to determine P^
b) Transform data with P^^(-1) : y=P^^(-1)y and X=P^^(-1)X
c) Estimate β with OLS in the model for the transformed data: y* = Xβ + ε -> b(FLGS)
What is the null hypothesis of the autocorrelation tests?
No autocorrelation
What is the equation for the autocorrelation of residuals?
r(k) = Σe(i)e(i-k)/Σe(i)^2 : first sum is from i=k+1 to n; second sum i=1 to n
Durbin-Watson test
DW = Σ(i=2->n) (e(i) - e(i-1))^2/Σe(i)^2 ≈ 2(1 - r(1))
0 (r(1) = 1 => perfect correlation)
4 (r(1) = -1 => perfect negative correlation)
H0: Value should be around 2
What are the disadvantages of the Durbin-Watson test?
- Distribution under H0 depends on the properties of regressors
- Not applicable when lagged dependent variables are included as regressors
Box-Pierce Test
H0: No autocorrelation
BP = nΣ(k=1 -> p) r(k)^2 ≈ χ2(p)
Ljung-Box Test
LB = nΣ(k=1 -> p) (n+2)/(n-k) r(k)^2 ≈ χ2(p)
What type of test is a Breusch-Godfrey test?
Lagrange Multiplier (LM) test
Which is the procedure for the Breusch-Godfrey test?
1) OLS on y(i) = x(i)’β + ε(i)
2) Run auxiliary regression
3) Under H0 (no autocorrelation) have nR^2 ≈ χ2(p)
What is the main difference between the Box-Pierce and Ljung-Box test?
The Box-Pierce test is an approximated version of the Ljung-Box test
What happens to the significance of the parameter of the independent variable with NW SE?
The significance may change
