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}

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2
Q

How to apply/implement ARMAX models in practice?

A
  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
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3
Q

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

A
  • AR(p): OLS

- MA(q): NLS or ML estimation

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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})

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5
Q

Akaike Information Criterion (AIC) formula

A

AIC(k) = T log(σ^^2) + 2k

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6
Q

Schwarz Information Criterion (SIC)

A

SIC(k) = T log(σ^^2) + k log(T)

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7
Q

Misspecification tests

A
  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
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