Chapter 3.1

Question 1

Formula for k-th order covariance

Answer 1

γ_{k} = E[(y_{t} - E(y_{t}) (y_{t-k} - E(y_{t-k})]

Question 2

Formula for k-th order autocorrelation

Answer 2

ρ^{k} = γ^{k} / γ^_{0}

ρ_{0} = 1, ρ_{-k} = ρ_{k}

Question 3

What is the empirical autocorrelation function?

Answer 3

The set of all autocorrelations ρ^_{k} for k = 1,2,…

Question 4

What do non-zero autocorrelations imply?

Answer 4

That y_{t-k} may be a useful variable to include in the regression

Question 5

What are the 3 properties of a white noise series?

Answer 5

1. E(ε_{t}) = 0
2. E(ε_{t}^2) = σ^2
3. E(ε_{s}ε_{t}) = 0

Residuals being a while noise process is a minimum requirement for a time series model to qualify as “adequate”

Question 6

How to rewrite AR(1) model in terms of shocks?

Answer 6

y_{t} = ϕ_{1}^t y_{0} + Σ{i=0, t-1} ϕ_{1}^i ε_{t-i}

The effect of the shock is determined by ϕ_{1}^i

Question 7

What does the shock have a transitory effect? What is a transitory effect?

Answer 7

When |ϕ_{1}| < 1, ϕ_{1}^i -> 0 as i increases

Effect of shock decays and dissapears

Question 8

What does the shock have an explosive effect? What is an explosive effect?

Answer 8

When |ϕ_{1}| > 1, the effect of shocks ε_{t-i} on y_{t} increases with i

Effect of shock grows with time - typically exclude explosive case in TSA

Question 9

What is the effect of the shock when ϕ_{1} = 1?

Answer 9

AR model simplifies to y_{t} = y_{0} + Σ{i=0, t-1} ε_{t-i}
Shock ε_{t-i} has same impact on all observations y_{t-i+h}
Shock is said to have permanent effects

Question 10

Conditional mean formula

Answer 10

E(y_{t} | Υ_{t-1})

Question 11

What is a necessary and sufficient condition (<=>) for stationarity in the AR(1) model?

Answer 11

|ϕ_{1}| < 1

Question 12

What are the correlations between y and ε? (Important in derivations)

Answer 12

• E(y_{t}ε_{t}) = σ^2
• E(y_{t}ε_{t+j}) = 0 (current observation not correlated with future shocks)
• E(y_{t}ε_{t-k}) ≠ 0 (current observation is correlated with past shocks)

Question 13

What is a property of the unconditional variance in AR(1) models?

Answer 13

The unconditional variance is constant:

E[(y_{t} - E(y_{t})^2] = E[(y_{t-1} - E(y_{t-1})^2]

Question 14

What is the conditional variance of an AR(1) model?

Answer 14

E(y_{t} | Y_{t-1}) = σ^2

Question 15

What is the trend in the autocorrelations of an AR(1) model with |ϕ_{1}| < 1?

Answer 15

They decline exponentially towards zero

Question 16

What is the autocovariance in the case of an AR(1) model with a unit root (ϕ=1)?

Answer 16

γ_{0,t} = E[y_{t}^2] = tσ^2
γ_{1,t} = E[y_{t} y{t-1}] = (t-1)σ^2

ρ_{1,t} = (t-1)/t, k>0

When t becomes large, all (theoretical) autocorrelations ρ_{k,t} = 1

Question 17

Limitations of the AR(1) model

Answer 17

• If ϕ > 0, all correlations are positive and decline monotonically towards zero
• If ϕ < 0, all even autocorrelations are positive, all odd autocorrelations are negative, and decline monotonically towards zero

Hence, the AR(1) model cannot capture EACF patterns

Question 18

Lag operator definition

Answer 18

L^k y_{t} = y_{t-k}

Question 19

How to write AR(p) model using lag operator?

Answer 19

ϕ_{p}(L) y_{t} = ε_{t},

where ϕ_{p}(L) = 1 - ϕ_{1} L - … - ϕ_{p} L^p

Question 20

What is the characteristic polynomial?

Answer 20

The lag polynomial ϕ_{p}(L) but considered as a function of z:
ϕ_{p}(z) = 1 - ϕ_{1}z - … - ϕ_{p}z^p

Its roots are the solutions to ϕ_{p}(z) = 0, and they determine whether the effects of shocks are transitory or permanent

Question 21

Interpretation of roots

Answer 21

Note: Root is z = …

• When ϕ_{1} = 1, the AR(1) polynomial is said to have a unit root (and shocks have permanent effects)
• When |ϕ_{1}| < 1, the root exceeds 1 (and shocks have transitory effects)

The solution is said to be outside the unit circle when |ϕ_{1}| < 1

Question 22

How to rewrite an AR(1) model as MA model (simple, no lag operator)?

Answer 22

y_{t} = ε_{t} + π_{1} ε_{t-1} + π_{2} ε_{t-2} + …

Question 23

How to rewrite an AR(p) model as MA model (with lag operator)?

Answer 23

y_{t} = ϕ_{p}(L)^{-1} ε_{t}

Question 24

What are the autocovariances of an MA(q) model?

Answer 24

γ_{k} = Σ{i=0, q-k} θ_{i}θ_{i+k} σ^2, for k = 0, 1, …, q
= 0, for k > q

with θ_{0} = 1

Question 25

Lag versions of AR(p) and MA(q) models

Answer 25

AR(p): ϕ_{p}(L) = 1 - ϕ_{1}L - … - ϕ_{p}L^{p}

MA(q): θ_{q}(L) = 1 + θ_{1}L + … + θ_{q}L^{q}

Question 26

Combine AR(p) and MA(q) (using lag operators)

Answer 26

ϕ_{p}(L) y_{t} = θ_{q}(L) ε_{t}

Question 27

How to write ARMA(p,q) model as “pure” AR or “pure” MA?

Answer 27

“Pure” AR: [ϕ_{p}(L) / θ_{q}(L)] y_{t} = ε_{t}

“Pure” MA: y_{t} = [θ_{q}(L) / ϕ_{p}(L)] ε_{t}

Question 28

True or false: A non-zero kth order autocorrelation implies that we need to include y_{t-k} in a time series model for y{t}

Answer 28

False

Question 29

What is k-th order partial autocorrelation?

Answer 29

It measures the dependence or correlation between y_{t} and y_{t-k}, after their common dependence on the intermediate observations y_{t-1}, …, y_{t-k+1} has been removed

It can be defined (and obtained) as the coefficient ψ_{k} in the regression:
y_{t} = η_{1} y_{t-1} + η_{2} y_{t-2} + … + η_{k-1} y_{t-k+1} + ψ_{k} y_{t-k} + u_{t}

Question 30

Values for PAC (ψ_{k}) in AR and MA models

Answer 30

• AR(p) model: ψ_{k} = 0 for all k > p
• MA(q) model: ψ_{k} declines exponentially towards zero
  (follows from writing MA(q) as θ_{q}(L)^{-1} y_{t} = ε_{t}