ARMA Models

Question 1

AR model

Answer 1

Expresses the current value of a time series as a linear combination of its past values and a random error term.
For an AR model of order p (AR(p)):
* X_t = ϕ₁X_t-1 + ϕ₂X_t-2 + … + ϕ_pX_t-p + ϵ_t

Question 2

Explanation of MA model

Answer 2

Relates the current value of a time series to past forecast errors (residuals)
For an MA model of order q (MA(q)):
* X_t = μ + θ₁ϵ_t-1 + θ₂ϵ_t-2 + θ_qϵ_t-q + ϵ_t

Question 3

ARMA(p,q)

Answer 3

Combines AR(p) and MA(q) components
* X_t = ϕ₁X_t-1 + ϕ₂X_t-2 + … + ϕ_pX_t-p + θ₁ϵ_t-1 + θ₂ϵ_t-2 + … + θ_qϵ_t-q + ϵ_t

Question 4

Behaviour based on ϕ₁

Answer 4

• ϕ₁ = 0: no relationship; purely random process
• |ϕ₁| < 1: stationary process; shocks decay over time
• ϕ₁ = 1: non-stationary process (random walk)
• |ϕ₁| > 1: Explosive process; values grow without bound

Question 5

Behaviour based on θ₁

Answer 5

• θ₁ = 0: no memory of past errors
• θ₁ != 0: memory of past errors influences the current value

Question 6

AR Stationarity Conditions

Answer 6

the roots of the characteristic equation lie outside the unit circle
* |ϕ₁| < 1

Question 7

Patterns for model identification

Answer 7

• AR(p) Process: PACF cuts off after lag p, ACF decays gradually
• MA(q) Process: ACF cuts off after lag q, PACF decays gradually
• ARMA(p,q) Process: ACF and PACF both decay gradually

Question 8

MA Stationarity Conditions

Answer 8

Always stationarity because it depends on a finite number of past errors

Question 9

Box-Ljung Test

Answer 9

Tests if residual autocorrelations are statistically significant
* Null Hypothesis H₀: Residuals are uncorrelated (white noise)

Question 10

Box-Jenkins Approach

Answer 10

• Model identification
• Parameter estimation
• Model diagnostics
• Forecasting

Question 11

Akaike Information Criterion (AIC)

Answer 11

Measures the relative quality of a statistical model for a given dataset by penalizing poor model fit ( -2ln(L) ) and number of parameters ( 2k )
* AIC = -2ln(L) + 2k
* Lower AIC is better
* Useful when comparing models with the same data but different numbers of parameters

Question 12

Bayesian Information Criterion (BIC)

Answer 12

Similar to AIC but includes a stronger penalty for model complexity (parameters)
* BIC = -2ln(L) + kln(n)