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Published byRyan Manning Modified over 2 years ago

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(1) Time Series Stationary: Toeplitz covariance matrix

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(2) e t : white noiseuncorrelated (0, σ 2 ) [Wold representation]

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(3) Example: if | Autoregressive order 1Stationary? Yes if. Forecast L periods ahead: + L (Y n - )

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(4) Backshift if

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(5) Moving Average Order 1 Clearly stationary One step ahead forecast:

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(6)

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(7) Random Walk No Mean Revision Extends to higher order and mixed models

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(8) ARIMA(p, d, q) so

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Note: EWMA = winner in CHANCE paper.

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(9) Roots Y t - = 1.2 (Y t-1 - ) -.32(Y t-2 - )+e t (1-1.2B+.32B 2 )(Y t - )= e t Division & partial fractions: (Y t - ) = ( 2/(1-.8B) -1/(1-.4B) ) e t convergent !

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(1-1.2B+.32B 2 ) roots: 1/.8, 1/.4 Y t - = 1.2 (Y t-1 - ) -.20(Y t-2 - )+e t (1-1.2B+.2B2)(Y t - ) = e t (1-0.2B)(1-B) (1-0.2B)(Y t -Y t-1 ) = e t Unit root, not stationary, no mean reversion. Studentized unit root tests (nonstandard) extend to higher order.

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(10) Transfer function: Observe Intervention:

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(11) American Airlines stock volume – 25 years:

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(12) Near 9/11/2001

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proc arima data=AMERICAN; i var=volume crosscor=(WTC CRASH) noprint; e input = ( (1)/(1,2) WTC (1)/(1,2) CRASH) plot p=2 q=1; f lead=0 out=out1 id=date; where '01jan01'd < date < '01jan03'd; Standard Approx Parameter Estimate Error t Value Pr>|t| Lag Variable MU < volume MA1, < volume AR1, < volume AR1, < volume NUM < WTC NUM1, WTC DEN1, WTC DEN1, < WTC NUM < CRASH NUM1, CRASH DEN1, CRASH DEN1, CRASH

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Interpretation: X t = WTC indicator … Similarly for X t = second crash indicator Error term is ARMA(2,1) Residual Checks Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq Autocorrelations

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Forecasts from this model shown below:

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Data before 9/11with transfer function forecast:

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Residuals before 9/11/01:

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Example 2: Nenana Ice Classic Start = 1917 pot is now $285,000

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The ARIMA Procedure Conditional Least Squares Estimation Standard Approx Parameter Estimate Error t Value Pr > |t| Lag Variable MU < break AR1, break NUM < ramp Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq Autocorrelations

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** Using NLIN to estimate ramp start point **; proc nlin data=all; parms C=1960 a=126 b=-.2 ; X = (year-C)*(year>c); model break = a + b*X; run; NOTE: Convergence criterion met. Sum of Mean Approx Source DF Squares Square F Value Pr > F Model Error Corrected Total Approx Approximate 95% Confidence Parameter Estimate Std Error Limits C a b

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