1 decomposition method. 2 types of data time series data: a sequence of observations measured over...
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Decomposition Method
2
Types of Data
Time series data: a sequence of observations measured over time (usually at equally spaced intervals, e.g., weekly, monthly and annually). Examples of time series data include:Gross Domestic Product each quarter;annual rainfall;daily stock market index
Cross sectional data: data on one or more variables collected at the same point in time
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Time Series vs Causal Modeling
Causal (regression) models: the investigator specifies some behavioural relationship and estimates the parameters using regression techniques;
Time series models: the investigator uses the past data of the target variable to forecast the present and future values of the variable
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Time Series vs Causal Modeling
On the other hand, there are many cases when one cannot, or one prefers not to, build causal models:
1. insufficient information is known about the behavioural relationship;
2. lack of, or conflicting, theories;3. insufficient data on explanatory variables;4. expertise may be unavailable;5. time series models may be more accurate
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Time Series vs Causal Modeling
Direct benefits of using time series models:1. Little storage capacity is needed;
2. some time series models are automatic in that user intervention is not required to update the forecasts each period;
3. some time series models are evolutionary in that the models adapt as new information is received;
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Classical Decomposition of Time Series
Trend – does not necessarily imply a monotonically increasing or decreasing series but simply a lack of constant mean, though in practice, we often use a linear or quadratic function to predict the trend;
Cycle – refers to patterns or waves in the data that are repeated after approximately equal intervals with approximately equal intensity. For example, some economists believe that “business cycles” repeat themselves every 4 or 5 years;
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Classical Decomposition of Time Series
Seasonal – refers to a cycle of one year duration;
Random (irregular) – refers to the (unpredictable) variation not covered by the above
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Decomposition Method
Multiplicative Models
ttttt IRCLSNTRY
ttttt IRCLSNTRY
Additive Models
Find the estimates of these four components.
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Examples:
(1) US Retail and Food Services Sales from 1996 Q1 to 2008 Q1
Multiplicative Decomposition
(2) Quarterly Number of Visitor Arrivals in Hong Kong from 2002 Q1 to 2008 Q1
Figure 2.1
Figure 2.2
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Figure 2.1 US Retail Sales
Back
US Retail & Food Services Sales
0
50,000
100,000
150,000
200,000
250,000
300,000
350,000
400,000
450,000
500,000
Q1-
96
Q3-
96
Q1-
97
Q3-
97
Q1-
98
Q3-
98
Q1-
99
Q3-
99
Q1-
00
Q3-
00
Q1-
01
Q3-
01
Q1-
02
Q3-
02
Q1-
03
Q3-
03
Q1-
04
Q3-
04
Q1-
05
Q3-
05
Q1-
06
Q3-
06
Q1-
07
Q3-
07
Q1-
08
Time
Sal
es Y
(t)
(in
MN
US
$)
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Figure 2.2 Visitor Arrivals
Number of Visitor Arrivals in Hong Kong
0
500000
1000000
1500000
2000000
2500000
3000000
Q1-
02
Q3-
02
Q1-
03
Q3-
03
Q1-
04
Q3-
04
Q1-
05
Q3-
05
Q1-
06
Q3-
06
Q1-
07
Q3-
07
Q1-
08
Time
Nu
mb
er o
f V
isit
ors
Y(t
)
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Cycles are often difficult to identify with a short time series.
Classical decomposition typically combines cycles and trend as one entity:
tttt IRSNTCY
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Illustration : Consider the following 4-year quarterly time series on sales volume:
Period (t) Year Quarter Sales
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
72
110
117
172
76
112
130
194
78
119
128
201
81
134
141
216
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Figure 2.3
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Step 1 : Estimation of seasonal component (SNt)
Yt = TCt SNt IRt
Moving Average
for periods 1 – 4
Moving Average
for periods 2 – 5
tt
tt IRTC
YNS
ˆ
75.1174
17211711072
75.1184
76172117110
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Period (t) Year Quarter Sales MA (t)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
72
110
117
172
76
112
130
194
78
119
128
201
81
134
141
216
117.75
118.75
119.25
122.5
128
128.5
130.25
129.75
131.5
132.25
136
139.25
143
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Assuming the average of the observations is also the median of the observations, the MA for periods 1 – 4, 2 – 5, 3 – 6 are centered at positions 2.5, 3.5 and 4.5 respectively.
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To get an average centered at periods 3, 4, 5 etc. the means of two consecutive moving averages are calculated:
Centered Moving
Average for period 3
Centered Moving
Average for period 4
25.1182
75.11875.117
1192
25.11975.118
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Period (t) Year Quarter Sales MA (t) CMA(t)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
72
110
117
172
76
112
130
194
78
119
128
201
81
134
141
216
117.75
118.75
119.25
122.5
128
128.5
130.25
129.75
131.5
132.25
136
139.25
143
118.25
119
120.875
125.25
128.25
129.375
130
130.625
131.875
134.125
137.625
141.125
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Because the CMAt contains no seasonality and irregularity, the seasonal component may be estimated by
t
tt CMA
YNS ~
445.1119
172~
989.025.118
117~ example,For
4
3
NS
NS
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Period (t)
Year
Quarter
Sales MA (t) CMA(t)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
72
110
117
172
76
112
130
194
78
119
128
201
81
134
141
216
117.75
118.75
119.25
122.5
128
128.5
130.25
129.75
131.5
132.25
136
139.25
143
118.25
119
120.875
125.25
128.25
129.375
130
130.625
131.875
134.125
137.625
141.125
0.989429175
1.445378151
0.628748707
0.894211577
1.013645224
1.499516908
0.6
0.911004785
0.970616114
1.49860205
0.588555858
0.949512843
)(~
tSN
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After all have been computed, they are further averaged to eliminate irregularities in the series. We also adjust the seasonal indices so that they sum to the number of seasons in a year (i.e., 4 for quarterly data, 12 for monthly data). Why?)
stNS ~
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Quarter Average
1 (0.628748707 + 0.6 + 0.588555858)/3=2 (0.894211577 + 0.911004785 + 0.949512843)/3=3 (0.989429175 + 1.013645224 + 0.970616114)/3=4 (1.445378151 + 1.499516908 + 1.49860205)/3=
Sum =
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Step 2 : Estimation of Trend/Cycle
Define deseasonalized (or seasonally adjusted) series as
for example, D1 = 72/0.6063 = 118.7506
ttt NSYD ˆ
25
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TCt may be estimated by regression using a linear trend:
where b0 and b1 are least squares estimates of
0 and 1 respectively.
,ˆˆ
3,2,1
10
10
tbbDCT
t
tD
tt
tt
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EXCEL regression output :
tCT t 854638009.16997914.113ˆ
So,
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For example,
4090674.117
2854638009.16997914.113ˆ
5544294.115
1854638009.16997914.113ˆ
2
1
CT
CT
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Step 3 : Computation of fitted values and out-of-sample forecasts
5516.2124825.13740.143ˆ
0621.706063.05544.115ˆ
:fit sample-In
ˆˆˆ
16
1
Y
Y
NSCTY ttt
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054.88
6063.02286.145
6063.017855.1670.113
ˆˆˆ171717
NSCTY
Out of sample forecast :
1796.135
9191.00833.147
9191.018855.1670.113
ˆˆˆ181818
NSCTY
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Figure 2.4
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Measuring Forecast Accuracy :
1) Mean Squared Error
forecast. of errors thebe ˆLet ttt YYe
MSERMSE
neMSEn
tt
1
2
MADRMAD
neMADn
tt
1
2) Mean Absolute Deviation
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Method A Method B
et = – 2 – 4
1.5 0.7
–1 0.5
2.1 1.4
0.7 0.1
Method A : MSE = 2.43
MAD = 1.46
Method B : MSE = 3.742
MAD = 1.34
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Naive Prediction
if U = 1 Forecasts produced are no better than naive forecast
U = 0 Forecasts produced perfect fit
The smaller the value of U, the better the forecasts.
nYY
nYYU
YY
tt
tt
tt
21
2
1
ˆ
ˆ
Theil’s u Statistics
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MSE = 11.932 MAD = 2.892 Theil’s U = 0.0546
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Out-of-Sample Forecasts
1) Expost forecast Prediction for the period in which actual
observations are available
2) Exante forecast Prediction for the period in which actual
observations are not available.
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T1 T2 T3
estimation period (today)Time
“back” casting in-sample simulation
Ex-post forecast
Ex-ante forecast
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Additive Decomposition
tttt IRSNTCY
Time Time
Trend Trend
YtYt
(Multiplicative Seasonality) (Additive Seasonality)
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Multiplicative decomposition is used when the time series exhibits increasing or decreasing seasonal variation (Yt=TCt SNt IRt)
TCt SNt Yt Yt – Yt-1
Yr 1 Q1
Q2
Q3
Q4
11.5
13
14.5
16
1.5
0.5
0.8
1.2
17.25
6.5
11.6
19.2
–10.75
5.1
7.6
Yr 2 Q1
Q2
Q3
Q4
17.5
19
20.5
22
1.5
0.5
0.8
1.2
26.25
9.5
16.4
26.4
–16.75
6.9
10
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Additive decomposition is used when the time series exhibits constant seasonal variation (Yt=TCt + SNt + IRt)
TCt SNt Yt Yt – Yt-1
Yr 1 Q1
Q2
Q3
Q4
11.5
13
14.5
16
1.8
–1
–1.5
0.7
13.3
12
13
16.7
–1.3
1
3.7
Yr 2 Q1
Q2
Q3
Q4
17.5
19
20.5
22
1.8
–1
–1.5
0.7
19.3
18
19
22.7
–1.3
1
3.7
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Step 1 : Estimation of seasonal component (SNt)
Calculation of MAt and CMAt is the same as per multiplicative decomposition
Initial seasonal component may be estimated by
For example,
ttt CMAYNS ~
53119172~
25.125.118117~
4
3
NS
NS
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Seasonal indices are averaged and adjusted so that they sum to zero (Why?)
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Step 2 : Estimation of Trend/Cycle
Deseasonalized series is defined as
TCt may be estimated by regression as per multiplicative decomposition
ttt NSYD ˆ
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i.e., Dt = o + 1t + t
and
Multiplicative decomposition
per asˆˆ10 tbbDCT tt
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So,
and
For example,
and
tCT t 980637255.12270833.113ˆ
ttt NSCTY ˆˆˆ
2077206.115
1980637255.12270833.113ˆ1
CT
40563725.64
80208333.502077206.11151̂
Y
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MSE = 27.911
MAD = 4.477