Chapter 3.2

Question 1

What is an ARMAX(p,q) model?

Answer 1

An ARMA model with exogenous regressors β_{1}x_{1,t} + … + β_{k}x_{k,t}

Question 2

How to apply/implement ARMAX models in practice?

Answer 2

1. Select regressors x_{1,t}, …, x_{k,t} and AR and MA orders p and q
2. Estimate parameters ϕ = (α, ϕ_{1}, …, ϕ_{p}), β = (β_{1}, …, β_{k}) and θ = (θ_{1}, …, θ_{q})
3. Evaluate the model by performing misspecification tests an other diagnostic measures
4. Modify the model if necessary (go back to step 1)
5. If the model cannot be further improved and is satisfactory, use it for description or forecasting

Question 3

How to estimate parameters of AR(p) and MA(q) models

Answer 3

• AR(p): OLS

- MA(q): NLS or ML estimation

Question 4

Finite sample estimate of ϕ_{1} in AR(1) model

Answer 4

sqrt(T) (ϕ^{1} - ϕ{1}) -> N(0, σ^2 γ_{0}^{-1})

Question 5

Akaike Information Criterion (AIC) formula

Answer 5

AIC(k) = T log(σ^^2) + 2k

Question 6

Schwarz Information Criterion (SIC)

Answer 6

SIC(k) = T log(σ^^2) + k log(T)

Question 7

Misspecification tests

Answer 7

1. Test of no residual autocorrelation
   
   • Ljung-Box (LB) test
   • Lagrange Multiplier (LM) test
2. Test of homoskedasticity (constant variance, often based on autocorrelations of squared residuals)
   
   • If rejected, standard errors of parameters should be adjusted or heteroskedasticity should be modelled explicitly
3. Test of normality (skewness = 0, kurtosis = 3)
   
   • Jarque-Bera (JB) test