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 )(