Time Series Analysis 3. ADL Model Autoregressive...
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Time Series Analysis 3. ADL Model Autoregressive Distributed Lag Model Autoregressive: p lags of dependent variable Y t distributed lag: q lags of additional regressor X t ⇒ ADL(p, q ) Y t = β 0 + β 1 Y t-1 + ... + β p Y t-p + δ 1 X t-1 + ... + δ q X t-q + u t • β (L)Y t = β 0 + δ (L)X t-1 with lag-polynomials defined by β (L)=1 - β 1 L - ... - β p L p δ (L)= δ 1 + δ 2 L + ... + δ q L q-1 • k additional predictors: ADL(p, q 1 , ..., q k ) β (L)Y t = β 0 + δ 1 (L)X 1,t-1 + δ 2 (L)X 2,t-1 + ... + δ k (L)X k,t-1 1
Transcript of Time Series Analysis 3. ADL Model Autoregressive...
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 + ... + δqXt−q + ut
• β(L)Yt = β0 + δ(L)Xt−1 with lag-polynomials defined by
β(L) = 1− β1L− ...− βpLp
δ(L) = δ1 + δ2L + ... + δqLq−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.
(Yt, X1t, ..., Xkt) and (Yt−j, X1t−j, ..., Xkt−j) become independent forj →∞
(3) 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: XtGr→ 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”)
• if XtGr9 Yt, then Xt has no predictive content for Yt
• Test: H0 : XtGr9 Yt vs. H1 : Xt
Gr→ Yt
= F -Test on significance of lags of Xt in ADL(p, q) model
Example: H0 : UnemploymenttGr9 ∆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
YT+1|T = β0 + β1YT + δ1XT
uT+1|T = YT+1− YT+1|T = uT+1 + [(β0− β0) + (β1− β1)YT + (δ1− δ1)XT ]
MSFE = E[(YT+1 − YT+1|T )2]
= σ2u + Var [(β0 − β0) + (β1 − β1)YT + (δ1 − δ1)XT ]︸ ︷︷ ︸
γ
because uT+1 is uncorrelated with second term
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Time Series Analysis 3. ADL Model
Forecast Uncertainty
• How to estimate Var(γ)? Problem: dependence of β0, β1 and δ1 on YT , XT
o Pseudo out-of-sample forecasts
o Estimate conditional variance Var(γ|YT , XT ) (as for models withnonstochastic regressors)
o Estimate asymptotic variance limT→∞
T · Var(γ)
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Time Series Analysis 3. ADL Model
Standard Error of Forecast: ADL(1,1)
Var(γ|YT , XT ) = σ2uzT (Z ′Z)−1z′T with zT = (1 YT XT ) and
Z =
1 Y1 X1... ... ...1 YT XT
MSFE = σ2u (1 + zT (Z ′Z)−1z′T )︸ ︷︷ ︸
Inflation factor
RMSFE = σu
√(1 + zT (Z ′Z)−1z′T ) is standard error of forecast
RMSFE = σu
√(1 + zT (Z ′Z)−1z′T ) is estimated standard error of forecast
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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 forYT+1|T is given by
YT+1|T ± 1.96 · RMSFE
• If infinitely many FI’s are computed, then 95% of the FI estimates containthe true value YT+1
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