Time Series Analysis
3. ADL Model
Autoregressive Distributed Lag Model
Autoregressive: p lags of dependent variable Yt
distributed lag: q lags of additional regressor Xt
⇒ ADL(p, q)
Yt = β0 + β1Yt−1 + ... + βpYt−p + δ1Xt−1 + ... + δq Xt−q + ut
• β(L)Yt = β0 + δ(L)Xt−1 with lag-polynomials defined by
β(L) = 1 − β1L − ... − βpLp
δ(L) = δ1 + δ2L + ... + δq Lq−1
• k additional predictors: ADL(p, q1, ..., qk )
β(L)Yt = β0 + δ1(L)X1,t−1 + δ2(L)X2,t−1 + ... + δk (L)Xk,t−1
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Time Series Analysis
3. ADL Model
Autoregressive Distributed Lag Model
• Model Assumptions
(1)
E[ut|Yt−1, Yt−2, ..., X1t−1, X1t−1, ..., Xkt−1, Xkt−2, ...] = 0
(2)(a) (Yt, X1t, ..., Xkt) are (strictly) stationary
(b) (Yt, X1t, ..., Xkt) are ergodic, i.e.
(3)
(Yt, X1t, ..., Xkt) and (Yt−j , X1t−j , ..., Xkt−j ) become independent for
j→∞
Yt and X1t, ..., Xkt have nonzero, finite fourth moments
(4)
no perfect multicollinearity
⇒ OLS regression theory applies
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Time Series Analysis
3. ADL Model
Autoregressive Distributed Lag Model
(1) implies
o Cov(ut, Yt−j ) = 0, Cov(ut, Xit−j ) = 0 ∀j > 0 and i = 1, ..., k
o Cov(ut, ut−j ) = 0 ∀j > 0 :
ut’s are not serially correlated
⇒ no further lags of Yt, Xit’s needed
(3) assures that variance estimators are consistent
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Time Series Analysis
3. ADL Model
Granger Causality (“predictability”)
• Asks whether forecast of Yt can be improved by considering lags of Xt
• Ωt: information set in period t
MSFE (Yt+h|Ωt): MSFE of forecasting Yt+h given Ωt
Xt Granger causes Yt:
Gr
Xt → Yt if
MSFE (Yt+h|Ωt) < MSFE (Yt+h|Ωt/{Xs|s ≤ t}))
for at least one forecast horizon h > 0
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Time Series Analysis
3. ADL Model
Granger Causality (“predictability”)
Gr
• if Xt 9 Yt, then Xt has no predictive content for Yt
Gr
Gr
• Test: H0 : Xt 9 Yt vs. H1 : Xt → Yt
=
b F -Test on significance of lags of Xt in ADL(p, q) model
Gr
Example: H0 : Unemploymentt 9 ∆Inft
ADL(4, 4) model: F -Test = 10.45
⇒ p-value < 0.0001
⇒ Reject H0
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Time Series Analysis
3. ADL Model
Forecast Uncertainty
• Forecast uncertainty: unknown uT +1 + estimation error
• RMSFE usual measure → standard error of forecast
• Example ADL(1, 1):
Yt = β0 + β1Yt−1 + δXt−1 + ut
ŶT +1|T = β̂0 + β̂1YT + δ̂1XT
ûT +1|T = YT +1 − ŶT +1|T = uT +1 + [(β̂0 − β0) + (β̂1 − β1)YT + (δ̂1 − δ1)XT ]
MSFE = E[(YT +1 − ŶT +1|T )2]
= σu2 + Var [(β̂0 − β0) + (β̂1 − β1)YT + (δ̂1 − δ1)XT ]
|
{z
}
γ̂
because uT +1 is uncorrelated with second term
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Time Series Analysis
3. ADL Model
Forecast Uncertainty
• How to estimate Var(γ̂)? Problem: dependence of β̂0, β̂1 and δ̂1 on YT , XT
o Pseudo out-of-sample forecasts
o Estimate conditional variance Var(γ̂|YT , XT ) (as for models with
nonstochastic regressors)
o Estimate asymptotic variance lim T · Var(γ̂)
T →∞
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Time Series Analysis
3. ADL Model
Standard Error of Forecast: ADL(1,1)
Var(γ̂|YT , XT ) = σu2 zT (Z 0Z)−1zT0 with zT = (1 YT XT ) and


1 Y1 X1
..
.. 
Z =  ..
1 YT XT
MSFE = σu2 (1 + zT (Z 0Z)−1zT0 )
|
{z
}
Inflation factor
q
RMSFE = σu (1 + zT (Z 0Z)−1zT0 ) is standard error of forecast
q
\ = σ̂u (1 + zT (Z 0Z)−1z 0 ) is estimated standard error of forecast
RMSFE
T
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Time Series Analysis
3. ADL Model
Forecast Interval
• ”Confidence interval for forecast”
• If ut is normally distributed, then (asymptotical) 95% forecast interval for
ŶT +1|T is given by
\
ŶT +1|T ± 1.96 · RMSFE
• If infinitely many FI’s are computed, then 95% of the FI estimates contain
the true value YT +1
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