Week 2 - Serial Correlation Flashcards
What is serial correlation and what assumption does it go against?
- Serial correlation is the presence of some from of linear dependence over time for some series z
- This goes against assumption 3, which is Cov(et, et-j) = 0 for j does not equal, which is the idea that there is no serial correlation
What is TheAutoCorrelationFunction (ACF)
- The ACF is a pictorial representation of linear dependency over time which is measured in the form of the correlation between Z and Z t-k for different values of K
What is TheAutoCorrelation Function (ACF) equal to?
- Corr(Zt,Zt-k) = Gamma K / Gamma o = Pk, where Po=1, and Cov(Zt,Zt-k)=Gamma k, and V(Zt)=Gamma o
What are the 4 different types of models we look at the ACF for?
- White Noise model
- Autoregressive model (AR)
- Moving average model (MA)
- Autoregressive moving average models (ARMA)
What is Zt equal to for a white noise model?
Zt = et
For a white noise model what is the E(Zt), V(Zt) , Cov(Zt,Zt-1), and Pj equal to?
- E(Zt)=0
- V(Zt)= Theta Squared
- Cov(Zt,Zt-1)=0
- Pj=0 for J not equal to 0
For a white noise model what does the ACF look like when plotted?
See notes
What is Zt equal to in an AR(1) model?
Zt = Phi Zt-1 + et
What is E(Zt) V(Zt) and the Cov (Zt, Zt-1) equal to in an AR(1) model?
- E(Zt)=0, See notes for derivation
- V(Zt)= Theta squared / 1 + Theta Squared, for all values of T, See notes for derivation
- Cov(Zt,Zt-1) = Phi^h, see notes for derivation
What does the ACF look like for AR(1) model when plotted?
See notes zig zag up and down
What can you conclude about the ACF for AR(1)
- The ACF indicates we’ve got the condition of stationarity implying the correlation between 2 points gets closer to zero as the points are further away.
What is Zt equal to for an AR(2) model?
- Zt = Phi 1 Zt-1 + Phi 2 Zt-2 + et
For an AR(2) model what is E(Zt) V(Zt) and Cov(Zt, Zt-1) equal to generally?
- E(Zt)=0
- V(Zt)= E(Zt)^2= Gamma 0, generally Gamma j(the variance of Zt) = Phi 1 Gamma j-1 + Phi 2 Gamma j-2
What does the ACF look like for an AR(2) model?
See notes
What is Z equal to in a moving average MA(1) model?
- Zt = Theta et-1 + et
In a MA(1) model, what are E(Zt) , V(Zt) and Cov(Zt,Zt-1), and the Cov(Zt,Zt-2) equal to?
- E(Zt)=0
- V(Zt)= (1+Theta^2)sigma^2
- Cov(Zt,Zt-1)= Gamma 1 = Theta Sigma^2
- Cov(Zt,Zt-2)= Gamma 2 = 0, see notes for full derivation
What does the ACF look like in an MA(1) model?
See notes
In an MA(2) model what is Zt equal to?
Zt = Theta 1 et-1 + Theta 2 et-2 + et
In an MA(2) model, What is the E(Zt) V(Zt) and the Cov(Zt,Zt-1) Cov(Zt,Zt-2) and Cov(Zt,Zt-3) equal to?
- E(Zt)=0
- V(Zt) = ( 1 + Theta 1^2 + Theta 2^2)Sigma^2
- Cov(Zt,Zt-1)=Gamma 1 = (Theta 1 +Theta1 x Theta 2)Sigma^2
- Cov(Zt,Zt-2)=Gamma 2 = Theta 1 x Sigma^2
- Cov(Zt,Zt-3)=Gamma 3 = 0
What does the ACF look like for the MA(2) model?
See notes
