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732G29 Time series analysis
Fall semester 2009
• 7.5 ECTS-credits
• Course tutor and examiner: Anders Nordgaard
• Course web: www.ida.liu.se/~732G29
• Course literature:
• Bowerman, O’Connell, Koehler: Forecasting, Time Series and Regression. 4th ed. Thomson, Brooks/Cole 2005. ISBN 0-534-40977-6.
Organization of this course:
• (Almost) weekly “meetings”: Mixture between lectures and tutorials
• A great portion of self-studying
• Weekly assignments
• Individual project at the end of the course
• Individual oral exam
Access to a computer is necessary. Computer rooms PC1-PC5 in Building E, ground floor can be used when they are not booked for another course
For those of you that have your own PC, software Minitab can be borrowed for installation.
Examination
The course is examined by
1.Homework exercises (assignments) and project work
2.Oral exam
Homework exercises and project work will be marked Passed or Failed. If Failed, corrections must be done for the mark Pass.
Oral exam marks are given according to ECTS grades. To pass the oral exam, all homework exercises and the project work must have been marked Pass.
The final grade will be the same grade as for the oral exam.
Communication
Contact with course tutor is best through e-mail: [email protected].
Office in Building B, Entrance 27, 2nd floor, corridor E (the small one close to Building E), room 3E:485. Telephone: 013-281974
Working hours:
Odd-numbered weeks: Wed-Fri 8.00-16.30
Even-numbered weeks: Thu-Fri 8.00-16.30
E-mail response all weekdays
All necessary information will be communicated through the course web. Always use the English version. The first page contains the most recent information (messages)
Assignments will successively be put on the course web as well as information about the project.
Solutions to assignments can be e-mailed or posted outside office door.
Time series
Sales figures jan 98 - dec 01
05
1015202530354045
jun-
97
jan-
98
jul-9
8
feb-
99
aug-
99
mar
-00
okt-0
0
apr-0
1
nov-
01
maj
-02
Tot-P ug/l, Råån, Helsingborg 1980-2001
0
100
200
300
400
500
600
700
800
900
1000
19
80
-01
-15
19
81
-01
-15
19
82
-01
-15
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-01
-15
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-01
-15
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-15
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-01
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-15
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-01
-15
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-15
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-15
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-15
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-01
-15
Characteristics
• Non-independent observations (correlations structure)
• Systematic variation within a year (seasonal effects)
• Long-term increasing or decreasing level (trend)
• Irregular variation of small magnitude (noise)
Where can time series be found?
• Economic indicators: Sales figures, employment statistics, stock market indices, …
• Meteorological data: precipitation, temperature,…
• Environmental monitoring: concentrations of nutrients and pollutants in air masses, rivers, marine basins,…
Time series analysis
• Purpose: Estimate different parts of a time series in order to– understand the historical pattern– judge upon the current status– make forecasts of the future development
• Methodologies:Method This course?
Time series regression Yes
Classical decomposition Yes
Exponential smoothing Yes
ARIMA modelling (Box-Jenkins) Yes
Non-parametric and semi-parametric analysis No
Transfer function and intervention models No
State space modelling No
Modern econometric methods: ARCH, GARCH, Cointegration
No
Spectral domain analysis No
Time series regression?Let
yt=(Observed) value of times series at time point t
and assume a year is divided into L seasons
Regession model (with linear trend):
yt=0+ 1t+j sj xj,t+t
where xj,t=1 if yt belongs to season j and 0 otherwise, j=1,…,L-1
and {t } are assumed to have zero mean and constant variance (2 )
The parameters 0, 1, s1,…, s,L-1 are estimated by the Ordinary Least Squares method:
(b0, b1, bs1, … ,bs,L-1)=argmin {(yt – (0+ 1t+j sj xj,t)2}
Advantages:
• Simple and robust method
• Easily interpreted components
• Normal inference (conf..intervals, hypothesis testing) directly applicable
• Forecasting with prediction limits directly applicable
•Drawbacks:
•Fixed components in model (mathematical trend function and constant seasonal components)
•No consideration to correlation between observations
Example: Sales figures
jan-98 20.33 jan-99 23.58 jan-00 26.09 jan-01 28.43feb-98 20.96 feb-99 24.61 feb-00 26.66 feb-01 29.92mar-98 23.06 mar-99 27.28 mar-00 29.61 mar-01 33.44apr-98 24.48 apr-99 27.69 apr-00 32.12 apr-01 34.56maj-98 25.47 maj-99 29.99 maj-00 34.01 maj-01 34.22jun-98 28.81 jun-99 30.87 jun-00 32.98 jun-01 38.91jul-98 30.32 jul-99 32.09 jul-00 36.38 jul-01 41.31aug-98 29.56 aug-99 34.53 aug-00 35.90 aug-01 38.89sep-98 30.01 sep-99 30.85 sep-00 36.42 sep-01 40.90okt-98 26.78 okt-99 30.24 okt-00 34.04 okt-01 38.27nov-98 23.75 nov-99 27.86 nov-00 31.29 nov-01 32.02dec-98 24.06 dec-99 24.67 dec-00 28.50 dec-01 29.78
Sales figures January 1998 - December 2001
0
5
10
15
20
25
30
35
40
45
Jun-
97
Jan-
98
Jul-9
8
Feb-9
9
Aug-9
9
Mar
-00
Oct-0
0
Apr-0
1
Nov-0
1
May
-02
month
Construct seasonal indicators: x1, x2, … , x12
January (1998-2001): x1 = 1, x2 = 0, x3 = 0, …, x12 = 0
February (1998-2001): x1 = 0, x2 = 1, x3 = 0, …, x12 = 0
etc.
December (1998-2001): x1 = 0, x2 = 0, x3 = 0, …, x12 = 1
Use 11 indicators, e.g. x1 - x11 in the regression model
sales time x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12
20.33 1 1 0 0 0 0 0 0 0 0 0 0 0
20.96 2 0 1 0 0 0 0 0 0 0 0 0 0
23.06 3 0 0 1 0 0 0 0 0 0 0 0 0
24.48 4 0 0 0 1 0 0 0 0 0 0 0 0
I I I I I I I I I I I I I I
32.02 47 0 0 0 0 0 0 0 0 0 0 1 0
29.78 48 0 0 0 0 0 0 0 0 0 0 0 1
Analysis with software Minitab®
Regression Analysis: sales versus time, x1, ...
The regression equation is
sales = 18.9 + 0.263 time + 0.750 x1 + 1.42 x2 + 3.96 x3 + 5.07 x4 + 6.01 x5
+ 7.72 x6 + 9.59 x7 + 9.02 x8 + 8.58 x9 + 6.11 x10 + 2.24 x11
Predictor Coef SE Coef T P
Constant 18.8583 0.6467 29.16 0.000
time 0.26314 0.01169 22.51 0.000
x1 0.7495 0.7791 0.96 0.343
x2 1.4164 0.7772 1.82 0.077
x3 3.9632 0.7756 5.11 0.000
x4 5.0651 0.7741 6.54 0.000
x5 6.0120 0.7728 7.78 0.000
x6 7.7188 0.7716 10.00 0.000
x7 9.5882 0.7706 12.44 0.000
x8 9.0201 0.7698 11.72 0.000
x9 8.5819 0.7692 11.16 0.000
x10 6.1063 0.7688 7.94 0.000
x11 2.2406 0.7685 2.92 0.006
S = 1.087 R-Sq = 96.6% R-Sq(adj) = 95.5%
Analysis of Variance
Source DF SS MS F P
Regression 12 1179.818 98.318 83.26 0.000
Residual Error 35 41.331 1.181
Total 47 1221.150
Source DF Seq SS
time 1 683.542
x1 1 79.515
x2 1 72.040
x3 1 16.541
x4 1 4.873
x5 1 0.204
x6 1 10.320
x7 1 63.284
x8 1 72.664
x9 1 100.570
x10 1 66.226
x11 1 10.039
Unusual Observations
Obs time sales Fit SE Fit Residual St Resid
12 12.0 24.060 22.016 0.583 2.044 2.23R
21 21.0 30.850 32.966 0.548 -2.116 -2.25R
R denotes an observation with a large standardized residual
Predicted Values for New Observations
New Obs Fit SE Fit 95.0% CI 95.0% PI
1 32.502 0.647 ( 31.189, 33.815) ( 29.934, 35.069)
Values of Predictors for New Observations
New Obs time x1 x2 x3 x4 x5 x6
1 49.0 1.00 0.000000 0.000000 0.000000 0.000000 0.000000
New Obs x7 x8 x9 x10 x11
1 0.000000 0.000000 0.000000 0.000000 0.000000
Sales figures with predicted value
0
5
10
15
20
25
30
35
40
45
Jun-
97
Jan-
98
Jul-9
8
Feb-9
9
Aug-9
9
Mar
-00
Oct-0
0
Apr-0
1
Nov-0
1
May
-02
month
What about serial correlation in data?
Positive serial correlation:
Values follow a smooth pattern
Negative serial correlation:
Values show a “thorny” pattern
How to obtain it?
Use the residuals.
48,...,1;ˆˆˆˆ11
1,,10
txtyyyej
tjjstttt
Residual plot from the regression analysis:
2
1
0
-1
-2
Month number (from jan 1998)
302010
Smooth or thorny?
Durbin Watson test on residuals:
Thumb rule:
If d < 1 or d > 3, the conclusion is that residuals (and original data) are correlated.
Use shape of figure (smooth or thorny) to decide if positive or negative)
(More thorough rules for comparisons and decisions about positive or negative correlations exist.)
n
tt
n
ttt
e
eed
1
2
2
21)(
Durbin-Watson statistic = 2.05 (Comes in the output )
Value > 1 and < 3 No significant serial correlation in residuals!
What happens when the serial correlation is substantial?
Estimated parameters in a regression model get their special properties regarding variance due to the fundamental conditions for the error terms {t }:
• Mean value zero
• Constant variance
• Uncorrelated
• (Normal distribution)
If any of the first three conditions is violated Estimated variances of estimated parameters are not correct
• Significance tests for parameters are not reliable
• Prediction limits cannot be trusted
How should the problem be handled?
Besides DW-test, carefully perform graphical residual analysis
If the serial correlation is modest (DW-test non-significant, and graphs OK) it is usually OK to proceed
Otherwise, amendments to the model is need, in particular by modelling the serial correlation (will appear later in this course)
• Decompose – Analyse the observed time series in its different components:
– Trend part (TR)
– Seasonal part (SN)
– Cyclical part (CL)
– Irregular part (IR)
Cyclical part: State-of-market in economic time series
In environmental series, usually together with TR
Classical decomposition
• Multiplicative model:
yt=TRt·SNt ·CLt ·IRt
Suitable for economic indicators Level is present in TRt or in TCt=(TR∙CL)t
SNt , IRt (and CLt) works as indices
Seasonal variation increases with level of yt
161412108642
16
14
12
10
8
6
4
2
• Additive model:
yt=TRt+SNt +CLt +IRt
More suitable for environmental data Requires constant seasonal variation SNt , IRt (and CLt) vary around 0
161412108642
10
9
8
7
6
5
4
3
2
1
Example 1: Sales data
Sales figures jan 98 - dec 01
0.005.00
10.0015.0020.0025.0030.0035.0040.0045.00
jun-
97
jan-
98
jul-9
8
feb-
99
aug-
99
mar
-00
okt-0
0
apr-0
1
nov-
01
maj
-02
Observed (blue) and deseasonalised (magenta)
0.00
5.00
10.00
15.00
20.0025.00
30.00
35.00
40.00
45.00
jun-
97
jan-
98
jul-9
8
feb-
99
aug-
99
mar
-00
okt-0
0
apr-
01
nov-
01
maj
-02
Observed (blue) and theoretical trend (magenta)
0.00
5.00
10.00
15.00
20.0025.00
30.00
35.00
40.00
45.00
jun-
97
jan-
98
jul-9
8
feb-
99
aug-
99
mar
-00
okt-0
0
apr-
01
nov-
01
maj
-02
Observed (blue) with estimated trendline (black)
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
40.00
45.00
mar-97 jul-98 dec-99 apr-01 sep-02
Example 2:
Estimation of components, working scheme
1. Seasonally adjustment/Deseasonalisation:• SNt usually has the largest amount of variation among the components.
• The time series is deseasonalised by calculating centred and weighted Moving Averages:
where L=Number of seasons within a year (L=2 for ½-year data, 4 for quaerterly data och 12 för monthly data)
2
2...2...2 )2/()12/()12/()2/()(
L
yyyyyM LtLttLtLtL
t
– Mt becomes a rough estimate of (TR∙CL)t .
– Rough seasonal components are obtained by• yt/Mt in a multiplicative model
• yt – Mt in an additive model
– Mean values of the rough seasonal components are calculated for eacj season separetly. L means.
– The L means are adjusted to• have an exact average of 1 (i.e. their sum equals L ) in a
multiplicative model.
• Have an exact average of 0 (i.e. their sum equals zero) in an additive model.
Final estimates of the seasonal components are set to these adjusted means and are denoted:
Lsnsn ,,1
– The time series is now deaseasonalised by
• in a multiplicative model
• in an additive model
where is one of
depending on which of the seasons t represents.
ttt snyy /*
ttt snyy *
tsn Lsnsn ,,1
2. Seasonally adjusted values are used to estimate the trend component and occasionally the cyclical component.
If no cyclical component is present:• Apply simple linear regression on the seasonally adjusted values
Estimates trt of linear or quadratic trend component. • The residuals from the regression fit constitutes estimates, irt of
the irregular component
If cyclical component is present:• Estimate trend and cyclical component as a whole (do not split
them) by
i.e. A non-weighted centred Moving Average with length 2m+1 caclulated over the seasonally adjusted values
12
**1
**)1(
*
m
yyyyytc mtttmtmt
t
– Common values for 2m+1: 3, 5, 7, 9, 11, 13– Choice of m is based on properties of the final
estimate of IRt which is calculated as
• in a multiplicative model
• in an additive model
– m is chosen so to minimise the serial correlation and the variance of irt .
– 2m+1 is called (number of) points of the Moving Average.
)/(*ttt tcyir
)(*ttt tcyir
Example, cont: Home sales data
Minitab can be used for decomposition by
StatTime seriesDecomposition Choice of model
Option to choose between two models
Time Series Decomposition
Data Sold
Length 47,0000
NMissing 0
Trend Line Equation
Yt = 5,77613 + 4,30E-02*t
Seasonal Indices
Period Index
1 -4,09028
2 -4,13194
3 0,909722
4 -1,09028
5 3,70139
6 0,618056
7 4,70139
8 4,70139
9 -1,96528
10 0,118056
11 -1,29861
12 -2,17361
Accuracy of Model
MAPE: 16,4122
MAD: 0,9025
MSD: 1,6902
Deseasonalised data have been stored in a column with head DESE1.
Moving Averages on these column can be calculated by
StatTime seriesMoving average
Choice of 2m+1
MSD should be kept as small as possible
TC component with 2m +1 = 3 (blue)
By saving residuals from the moving averages we can calculate MSD and serial correlations for each choice of 2m+1.
2m+1 MSD Corr(et,et-1)
3 1.817 -0.444
5 1.577 -0.473
7 1.564 -0.424
9 1.602 -0.396
11 1.542 -0.431
13 1.612 -0.405
A 7-points or 9-points moving average seems most reasonable.
Serial correlations are simply calculated by
StatTime seriesLag
and further
StatBasic statisticsCorrelation
Or manually in Session window:
MTB > lag ’RESI4’ c50
MTB > corr ’RESI4’ c50
Analysis with multiplicative model:
Time Series Decomposition
Data Sold
Length 47,0000
NMissing 0
Trend Line Equation
Yt = 5,77613 + 4,30E-02*t
Seasonal Indices
Period Index
1 0,425997
2 0,425278
3 1,14238
4 0,856404
5 1,52471
6 1,10138
7 1,65646
8 1,65053
9 0,670985
10 1,02048
11 0,825072
12 0,700325
Accuracy of Model
MAPE: 16,8643
MAD: 0,9057
MSD: 1,6388
additive
additive
additive
Classical decomposition, summary
ttttt IRCLSNTRy
Multiplicative model:
Additive model:
ttttt IRCLSNTRy
Deseasonalisation
• Estimate trend+cyclical component by a centred moving average:
2
2...2...2 )2/()12/()12/()2/(
L
yyyyyCMA LtLttLtLt
t
where L is the number of seasons (e.g. 12, 4, 2)
• Filter out seasonal and error (irregular) components:– Multiplicative model:
t
ttt CMA
yirsn
-- Additive model:
tttt CMAyirsn
Calculate monthly averages
mm n llnm irsnsn )(1
Multiplicative model:
for seasons m=1,…,L
Additive model:
mm n llnm irsnsn )(1
Normalise the monhtly means
L
ll
L
llL
m
msn
L
sn
snsn
111
Multiplicative model:
Additive model:
L
llLmm snsnsn
11
Deseasonalise
t
tt sn
yd
ttt snyd
Multiplicative model:
Additive model:
where snt = snm for current month m
Fit trend function, detrend (deaseasonalised) data
)(tftrt
t
ttt tr
dircl
tttt trdircl
Multiplicative model:
Additive model:
Estimate cyclical component and separate from error component
t
tt
kttktktt
cl
irclir
k
irclirclirclirclcl
)(12
)(...)(...)()( )1(
ttt
kttktktt
clirclirk
irclirclirclirclcl
)(12
)(...)(...)()( )1(
Multiplicative model:
Additive model:
