Week 4 Flashcards
What is leverage?
The influence of y(j) on y(j)^ (fitted value)
What is the formula for leverage?
h(j) = X(j)’(X’X)^(-1)X(j)
How do you derive the hat matrix (H)? What is the formula for it?
y^ = Xb = X(X'X)^(-1)X'y = Hy H = X(X'X)^(-1)X'
How do you test whether an observation is an outlier?
To test whether the j-th observation is an outlier:
y(i) = x(i)’β + γD(j,i) + ε(i), where D(j,i) = 1 if i=j - basically identity matrix
H0: j-th observation fits general pattern of data, γ=0
Estimate γ with OLS
y = Xβ + D(j)γ + ε
FW Theorem: γ^ = (Dj’MDj)^(-1)DjMy ; M = I - H , My = e
= ej/(1 - hj)
γ^ ~ N(0, σ^2/(1-hj)
Studentized Residual
γ^ / (sj / sqrt(1 - hj)) = ej / (sj / sqrt(1 - hj)) = ej*
t with (n-k) dof
What test tests for normality?
Jarque-Bera
Jarque-Bera Test
H0: Normality
JB = [sqrt(n/24) (K-3)^2] + [sqrt(n/6) S]^2 ~ χ2(2)
What does the Chow break test do?
Tests the existence of groups in data (1 full regression + 2 small regressions)
Chow Break Test
H0: no groups
F = (S0 - (S1 + S2))/k / (S1 + S2)/(n1 + n2 - 2k)
What does the Chow forecast test do?
Lets the alternative completely “unspecified”
Chow Forecast Test
F = (S0 - S1) / n2 / S1/(n1 - k)
MSE
MSE = E[β^ - β)(β^ - β)’]
TMSE
TMSE = E[(Xβ^ - Xβ)’(Xβ^ - Xβ)]
AIC and SIC
AIC = -2logL + 2p
SIC = -2logL + plogn
p = # of estimated parameters in model
