(1) Time Series

1
(1) Time Series
Stationary

Toeplitz covariance matrix

2
(2) et white noiseuncorrelated (0, s2)

Wold representation
3
(3) Example

if rlt1
Autoregressive order 1Stationary? Yes if
. Forecast L periods ahead m rL(Yn-m)
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 Yt-m 1.2 (Yt-1-m) -
.32(Yt-2-m)et (1-1.2B.32B2)(Yt-m ) et
Division partial fractions
(Yt-m ) ( 2/(1-.8B) -1/(1-.4B) ) et
convergent!

11
(1-1.2B.32B2) roots 1/.8,
1/.4 Yt-m 1.2 (Yt-1-m) -
.20(Yt-2-m)et (1-1.2B.2B2)(Yt-m ) et
(1-0.2B)(1-B)
(1-0.2B)(Yt-Yt-1) et 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 dataAMERICAN i varvolume
crosscor(WTC CRASH) noprint e input (
(1)/(1,2) WTC (1)/(1,2) CRASH) plot p2 q1 f
lead0 outout1 iddate where '01jan01'd lt date
lt '01jan03'd
Standard
Approx Parameter Estimate Error t Value
Prgtt Lag Variable MU 1895175.1
241956.3 7.83 lt.0001 0 volume MA1,1
0.80798 0.06487 12.45 lt.0001 1
volume AR1,1 1.30707 0.08535 15.31
lt.0001 1 volume AR1,2 -0.33545
0.07374 -4.55 lt.0001 2 volume NUM1
15624717 831551.1 18.79 lt.0001 0 WTC
NUM1,1 -9564494.0 3086912.7 -3.10 0.0021
1 WTC DEN1,1 -0.21308 0.18425 -1.16
0.2481 1 WTC DEN1,2 0.38677
0.07282 5.31 lt.0001 2 WTC NUM2
7534484.3 842654.2 8.94 lt.0001 0 CRASH
NUM1,1 4260018.6 2401054.1 1.77 0.0767
1 CRASH DEN1,1 0.79740 0.31164 2.56
0.0108 1 CRASH DEN1,2 0.04974 0.17505
0.28 0.7764 2 CRASH
16
Interpretation Xt WTC indicator 0 0 0 1 0 0 0

Similarly for Xt second crash indicator Error
term is ARMA(2,1) Residual Checks
Autocorrelation Check of Residuals To Chi-
Pr gt Lag Square DF ChiSq
---------Autocorrelations-------------- 6
11.07 3 0.0114 -0.002 -0.008 0.016 0.060
0.059 -0.121 12 13.69 9 0.1336 0.032
-0.019 -0.042 -0.030 -0.034 -0.002 18 20.49
15 0.1541 -0.082 -0.017 0.005 0.013 -0.074
0.023 24 37.27 21 0.0157 0.116 0.088
-0.087 -0.036 0.045 0.011 30 43.28 27
0.0245 -0.046 -0.016 0.059 -0.041 0.061
0.008 36 51.68 33 0.0203 0.069 0.013
-0.042 0.085 0.032 0.029 42 55.44 39
0.0425 -0.045 0.020 0.066 -0.002 -0.011
-0.002 48 56.92 45 0.1096 0.020 -0.011
-0.009 0.002 -0.005 0.046
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 gt t Lag Variable
MU 126.01962 0.59155 213.03
lt.0001 0 break AR1,1 -0.21784
0.10929 -1.99 0.0494 6 break
NUM1 -0.18686 0.04326 -4.32
lt.0001 0 ramp
Autocorrelation Check of Residuals To Chi-
Pr gt Lag Square DF ChiSq
-------------Autocorrelations------------- 6
4.81 5 0.4399 -0.040 0.043 -0.064 0.054
-0.197 -0.034 12 14.37 11 0.2134 0.056
-0.089 -0.090 0.215 -0.023 -0.167 18 18.06
17 0.3850 -0.044 0.079 0.033 -0.060 0.129
-0.063 24 21.22 23 0.5679 -0.054 0.031
0.016 -0.123 -0.036 -0.075
22
Using NLIN to estimate ramp start point
proc nlin dataall parms C1960 a126
b-.2 X (year-C)(yeargtc) model
break a bX run
NOTE Convergence criterion met.
Sum of Mean
Approx Source DF Squares
Square F Value Pr gt F Model 2
403.5 201.7 6.35 0.0027
Error 85 2702.1 31.7894
Corrected Total 87 3105.6
Approx Approximate 95
Confidence Parameter Estimate Std
Error Limits C
1967.6 10.7354 1946.2 1988.9 a
126.0 0.7895 124.4
127.6 b -0.1873 0.0868
-0.3599 -0.0147
23
(No Transcript)
24

• Model log(passengers) 9.08 plus
• WTC shock 0,0,0,1,0,0 coeff -0.1673 (0.0115)
• 
  ( p-value )
• WTC shift 0,0,0,1,1,1 coeff -0.4076
  (0.0001)
• Linear trend coeff0.0003 per day
  (0.0001)
• Trendshift interaction (D slope)-.00016 (0.0245)
• Seasonal dummies
• -.17,-.16,.02,.03,.08,.10,.12,.05,-.10,.07,0,0
• J F M A M J J A S O
  N D
• Add ARMA(1,1) error term GLS (without is OLS)
• (1-.83B)r(t) (1-.44B)e(t)
• Box-Ljung lack of fit p-values
• 6 lags 0.22, 12 lags 0.60, 18 lags 0.58, 24 lags
  0.12
• Excellent fit

25
(No Transcript)
26
(No Transcript)
27
(No Transcript)
28
(No Transcript)