Chapter 3.2 Flashcards
1
Q
What is an ARMAX(p,q) model?
A
An ARMA model with exogenous regressors β_{1}x_{1,t} + … + β_{k}x_{k,t}
2
Q
How to apply/implement ARMAX models in practice?
A
- Select regressors x_{1,t}, …, x_{k,t} and AR and MA orders p and q
- Estimate parameters ϕ = (α, ϕ_{1}, …, ϕ_{p}), β = (β_{1}, …, β_{k}) and θ = (θ_{1}, …, θ_{q})
- Evaluate the model by performing misspecification tests an other diagnostic measures
- Modify the model if necessary (go back to step 1)
- If the model cannot be further improved and is satisfactory, use it for description or forecasting
3
Q
How to estimate parameters of AR(p) and MA(q) models
A
- AR(p): OLS
- MA(q): NLS or ML estimation
4
Q
Finite sample estimate of ϕ_{1} in AR(1) model
A
sqrt(T) (ϕ^{1} - ϕ{1}) -> N(0, σ^2 γ_{0}^{-1})
5
Q
Akaike Information Criterion (AIC) formula
A
AIC(k) = T log(σ^^2) + 2k
6
Q
Schwarz Information Criterion (SIC)
A
SIC(k) = T log(σ^^2) + k log(T)
7
Q
Misspecification tests
A
- Test of no residual autocorrelation
- Ljung-Box (LB) test
- Lagrange Multiplier (LM) test
- 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
- Test of normality (skewness = 0, kurtosis = 3)
- Jarque-Bera (JB) test