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Consumer Price Index - Clothing and Footwear. Consumer Price Index - Clothing and Footwear. Seasonally differenced Consumer Price Index - Clothing and Footwear. Seasonally differenced Consumer Price Index - Clothing and Footwear. CPI Clothing and Footwear SARIMA (1, 0, 0, 0, 1, 0). - PowerPoint PPT Presentation
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Time series analysis - lecture 4
Consumer Price Index
- Clothing and Footwear
YearMonth
20082004200019961992198819841980JanJanJanJanJanJanJanJan
180
160
140
120
100
CPI_
Clo
thin
g
Time Series Plot of CPI_Clothing
Time series analysis - lecture 4
Consumer Price Index
- Clothing and Footwear
YearMonth
19941992199019881986198419821980JanJanJanJanJanJanJanJan
180
170
160
150
140
CPI_
Clo
thin
g
Time Series Plot of CPI_Clothing
Time series analysis - lecture 4
Seasonally differenced Consumer Price Index
- Clothing and Footwear
YearMonth
19941992199019881986198419821980JanJanJanJanJanJanJanJan
12.5
10.0
7.5
5.0
2.5
0.0
-2.5
-5.0
Seas_
dif
f
Time Series Plot of Seas_diff
Time series analysis - lecture 4
Seasonally differenced Consumer Price Index
- Clothing and Footwear
4035302520151051
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
Lag
Auto
corr
ela
tion
Autocorrelation Function for Seas_diff(with 5% significance limits for the autocorrelations)
4035302520151051
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
Lag
Part
ial Auto
corr
ela
tion
Partial Autocorrelation Function for Seas_diff(with 5% significance limits for the partial autocorrelations)
Time series analysis - lecture 4
CPI Clothing and Footwear
SARIMA (1, 0, 0, 0, 1, 0)
Final Estimates of Parameters
Type Coef SE Coef T PAR 1 0.7457 0.0522 14.29 0.000Constant 0.2222 0.1783 1.25 0.214
Differencing: 0 regular, 1 seasonal of order 12Number of observations: Original series 178, after
differencing 166Residuals: SS = 865.115 (backforecasts excluded) MS = 5.275 DF = 164
192168144120967248241
190
180
170
160
150
140
Time
CPI_
Clo
thin
g
Time Series Plot for CPI_Clothing(with forecasts and their 95% confidence limits)
Time series analysis - lecture 4
CPI Clothing and Footwear
SARIMA (1, 0, 0, 2, 1, 0)
Final Estimates of Parameters
Type Coef SE Coef T PAR 1 0.8145 0.0460 17.72 0.000SAR 12 -0.6092 0.0830 -7.34 0.000SAR 24 -0.2429 0.0843 -2.88 0.005Constant 0.3275 0.1557 2.10 0.037
Differencing: 0 regular, 1 seasonal of order 12Number of observations: Original series 178, after
differencing 166Residuals: SS = 651.883 (backforecasts excluded) MS = 4.024 DF = 162
192168144120967248241
190
180
170
160
150
140
Time
CPI_
Clo
thin
g
Time Series Plot for CPI_Clothing(with forecasts and their 95% confidence limits)
Time series analysis - lecture 4
CPI Clothing and Footwear
SARIMA (1, 0, 0, 2, 1, 0) residuals
160140120100806040201
5.0
2.5
0.0
-2.5
-5.0
Observation Order
Resi
dual
Versus Order(response is CPI_Clothing)
39363330272421181512963
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
Lag
Auto
corr
ela
tion
ACF of Residuals for CPI_Clothing(with 5% significance limits for the autocorrelations)
Time series analysis - lecture 4
Models for multiple time series of data
Dynamic regression models General input-output models Models for intervention analysis
Response surface methodologies Smoothing of multiple time series Change-point detection
Time series analysis - lecture 4
Percentage of carbon dioxide in the output from a gas furnace
0
10
20
30
40
50
60
70
1 21 41 61 81 101 121 141 161 181 201 221 241 261 281
-3
-2
-1
0
1
2
3
4
CO2 Gasrate
Time series analysis - lecture 4
The dynamic regression model
where Yt = the forecast variable (output series);
Xt = the explanatory variable (input series);
Nt = the combined effect of all other factors influencing Yt (the noise);
(B) = ( 0 + 1B + 2B2 + … + kBk), where k is the order of the transfer function
ttt NXBaY )(
Time series analysis - lecture 4
Using the SAS procedure AUTOREG- regression in which the noise is modelled as an autoregressive sequence
Consider a dataset with one input variable (gasrate) and one output variable (CO2)
data newdata;set mining.gasfurnace;gasrate1= lag1(gasrate);gasrate2= lag2(gasrate);gasrate3=lag3(gasrate);gasrate4= lag4(gasrate);
run;
proc autoreg data=newdata;model CO2 = gasrate/nlag=1;model CO2 = gasrate gasrate1/nlag=1;model CO2 = gasrate gasrate1 gasrate2/nlag=1;model CO2 = gasrate gasrate1 gasrate 2 gasrate3/nlag=1;model CO2 = gasrate gasrate1 gasrate2 gasrate3 gasrate4/nlag=1;output out=model4 residual=res;
run;
Time series analysis - lecture 4
Predicted and observed levels of carbon dioxide in the output
from a gas furnace- dynamic regression model with inputs time-lagged up to 4 steps
45474951535557596163
1 21 41 61 81 101 121 141 161 181 201 221 241 261 281
pred CO2
Time series analysis - lecture 4
No. air passengers by week in Sweden
-original series and seasonally differenced data
12.2
12.4
12.6
12.8
13.0
13.2
13.4
13.6
1992 1996 2000 2004No
.pas
sen
ger
s at
Sw
edis
h a
irp
ort
s (t
ho
usa
nd
s)
No. passengers
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
1992 1996 2000 2004D
iffe
ren
ce la
g 5
2 (t
ho
usa
nd
s)
Difference lag 52
Time series analysis - lecture 4
Intervention analysis
where Yt = the forecast variable (output series);
Xt = the explanatory variable (step or pulse function);
Nt = the combined effect of all other factors influencing Yt (the noise);
(B) = ( 0 + 1B + 2B2 + … + kBk), where k is the order of the transfer function
ttt NXBaY )(