Presentation is loading. Please wait.

Presentation is loading. Please wait.

(1) Time Series Stationary: Toeplitz covariance matrix.

There are copies: 1
(1) Time Series Stationary: Toeplitz covariance matrix.

Similar presentations


Presentation on theme: "(1) Time Series Stationary: Toeplitz covariance matrix."— Presentation transcript:

1 (1) Time Series Stationary: Toeplitz covariance matrix

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

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

4 (4) Backshift if

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

6 (6)

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

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

9 Note: EWMA = winner in CHANCE paper.

10 (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 !

11 (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.

12 (10) Transfer function: Observe Intervention:

13 (11) American Airlines stock volume – 25 years:

14 (12) Near 9/11/2001

15 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

16 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

17 Forecasts from this model shown below:

18 Data before 9/11with transfer function forecast:

19 Residuals before 9/11/01:

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

21 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

22 ** 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


Download ppt "(1) Time Series Stationary: Toeplitz covariance matrix."

Similar presentations


Ads by Google