chapter 1: decomposition methodspersonal.cb.cityu.edu.hk/msawan/teaching/ms6215/ms6215ch1.pdf ·...
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
CHAPTER 1: Decomposition Methods
Prof. Alan Wan
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Table of contents
1. Data Types and Causal vs.Time Series Models
2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model
4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Types of Data
I Time series data: a sequence of observations measured overtime, usually at equally spaced intervals, e.g., weekly, monthlyand annually).Examples of time series data include: Quarterly GrossDomestic Product (GDP), Annual rainfall volume, daily stockmarket index, etc.
I Cross sectional data: data on one or more variables collectedat the same point in time.
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Types of Data
I Time series data: a sequence of observations measured overtime, usually at equally spaced intervals, e.g., weekly, monthlyand annually).Examples of time series data include: Quarterly GrossDomestic Product (GDP), Annual rainfall volume, daily stockmarket index, etc.
I Cross sectional data: data on one or more variables collectedat the same point in time.
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Causal vs. Time Series Models
I Causal (regression) models: the investigator specifies somebehaviourial relationship and estimates the unknownparameters using regression techniques.
I Time series models: the investigator uses past data of thetarget variable to forecast the present and future values of thevariable.
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Causal vs. Time Series Models
I Causal (regression) models: the investigator specifies somebehaviourial relationship and estimates the unknownparameters using regression techniques.
I Time series models: the investigator uses past data of thetarget variable to forecast the present and future values of thevariable.
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Causal vs. Time Series Models
I Causal models provide information on the causal relationshipbetween the target variable and its determinants (theregressors).
I On the other hand, there are many instances when onecannot, or prefers not to, construct causal models due toreasons such as
1. insufficient information on the behaviourial relationship2. lack of, or conflicting, theories3. insufficient data on the explanatory variables4. superior forecasts produced by time series models
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Causal vs. Time Series Models
I Causal models provide information on the causal relationshipbetween the target variable and its determinants (theregressors).
I On the other hand, there are many instances when onecannot, or prefers not to, construct causal models due toreasons such as
1. insufficient information on the behaviourial relationship2. lack of, or conflicting, theories3. insufficient data on the explanatory variables4. superior forecasts produced by time series models
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Causal vs. Time Series Models
I Here are some of the direct benefits of using time seriesmodels:
1. little storage capacity is needed2. some time series models are automatic in that user
intervention is not required to update the forecasts each period3. some time series models are evolutionary in that the models
adapt as new information is received
I This course is mainly concerned with forecasting using timeseries models
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Causal vs. Time Series Models
I Here are some of the direct benefits of using time seriesmodels:
1. little storage capacity is needed2. some time series models are automatic in that user
intervention is not required to update the forecasts each period3. some time series models are evolutionary in that the models
adapt as new information is received
I This course is mainly concerned with forecasting using timeseries models
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Classical Decomposition of Time Series
I Trend (TC) - does not necessarily imply a monotonicallyincreasing or decreasing series but simply a lack of constantmean, although in practice, we often use a linear or quadraticfunction to predict the trend.
I Cycle (CL) - refers to patterns or waves in the data that arerepeated after approximately equal intervals withapproximately equal intensity. For example, some economistsbelieve that business cycles repeat themselves every 4 or 5years.
I Seasonal (SN) - refers to a cycle of one year’s duration.
I Random (Irregular) (IR) - refers to the (unpredictable)variations not covered by the three components above.
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Classical Decomposition of Time Series
I Trend (TC) - does not necessarily imply a monotonicallyincreasing or decreasing series but simply a lack of constantmean, although in practice, we often use a linear or quadraticfunction to predict the trend.
I Cycle (CL) - refers to patterns or waves in the data that arerepeated after approximately equal intervals withapproximately equal intensity. For example, some economistsbelieve that business cycles repeat themselves every 4 or 5years.
I Seasonal (SN) - refers to a cycle of one year’s duration.
I Random (Irregular) (IR) - refers to the (unpredictable)variations not covered by the three components above.
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Classical Decomposition of Time Series
I Trend (TC) - does not necessarily imply a monotonicallyincreasing or decreasing series but simply a lack of constantmean, although in practice, we often use a linear or quadraticfunction to predict the trend.
I Cycle (CL) - refers to patterns or waves in the data that arerepeated after approximately equal intervals withapproximately equal intensity. For example, some economistsbelieve that business cycles repeat themselves every 4 or 5years.
I Seasonal (SN) - refers to a cycle of one year’s duration.
I Random (Irregular) (IR) - refers to the (unpredictable)variations not covered by the three components above.
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Classical Decomposition of Time Series
I Trend (TC) - does not necessarily imply a monotonicallyincreasing or decreasing series but simply a lack of constantmean, although in practice, we often use a linear or quadraticfunction to predict the trend.
I Cycle (CL) - refers to patterns or waves in the data that arerepeated after approximately equal intervals withapproximately equal intensity. For example, some economistsbelieve that business cycles repeat themselves every 4 or 5years.
I Seasonal (SN) - refers to a cycle of one year’s duration.
I Random (Irregular) (IR) - refers to the (unpredictable)variations not covered by the three components above.
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Classical Decomposition of Time Series
We are concerned with two types of Decomposition Models:
I Multiplicative Model:
Yt = TCt × SNt × CLt × IRt
I Additive Model:
Yt = TCt + SNt + CLt + IRt
The goal is to find estimates of the four components.
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Classical Decomposition of Time Series
We are concerned with two types of Decomposition Models:
I Multiplicative Model:
Yt = TCt × SNt × CLt × IRt
I Additive Model:
Yt = TCt + SNt + CLt + IRt
The goal is to find estimates of the four components.
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Classical Decomposition of Time Series
We are concerned with two types of Decomposition Models:
I Multiplicative Model:
Yt = TCt × SNt × CLt × IRt
I Additive Model:
Yt = TCt + SNt + CLt + IRt
The goal is to find estimates of the four components.
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
Example 1: U.S. Retail and Food Services Sales from 1996Q1 ro2008Q1:
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
Sale
s Y
(t)
(in
MN
US
$)
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
Example 2: Quarterly Number of Visitor Arrivals in Hong Kongfrom 2002Q1 to 2008Q1:
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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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
I Cycles are often difficult to identify with a short time series.
I Classical decomposition typically combines cycles and trend asone entity, that is,
Yt = TCt × SNt × IRt
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
Consider the following 4-year quarterly time series on sales volume:
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
A plot of the series reveals the following pattern:
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
I We first estimate the seasonal component (SNt).
I Note that Yt = TCt × SNt × IRt
∴ SNt = YtTCt×IRt
I
Moving Average for periods 1− 4 =72 + 110 + 117 + 172
4= 117.75
Moving Average for periods 2− 5 =110 + 117 + 172 + 76
4= 118.75
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
I We first estimate the seasonal component (SNt).
I Note that Yt = TCt × SNt × IRt
∴ SNt = YtTCt×IRt
I
Moving Average for periods 1− 4 =72 + 110 + 117 + 172
4= 117.75
Moving Average for periods 2− 5 =110 + 117 + 172 + 76
4= 118.75
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
I We first estimate the seasonal component (SNt).
I Note that Yt = TCt × SNt × IRt
∴ SNt = YtTCt×IRt
I
Moving Average for periods 1− 4 =72 + 110 + 117 + 172
4= 117.75
Moving Average for periods 2− 5 =110 + 117 + 172 + 76
4= 118.75
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
Assuming that the average of the observations is also the medianof the observations, the moving average (MA) forperiods 1-4, 2-5, 3-6 are centered at t = 2.5, 3.5 and 4.5 respectively.
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
I To obtain averages that center at periods 3, 4, 5, etc. wecalculate the mean of every two consecutive moving averagesas follows:
I
Centered Moving Average for period 3 =117.75 + 118.75
2= 118.25
Centered Moving Average for period 4 =118.75 + 119.25
2= 119
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
I To obtain averages that center at periods 3, 4, 5, etc. wecalculate the mean of every two consecutive moving averagesas follows:
I
Centered Moving Average for period 3 =117.75 + 118.75
2= 118.25
Centered Moving Average for period 4 =118.75 + 119.25
2= 119
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
I Because the centered moving average (CMA) contains noseasonality and no or little irregularity, the seasonalcomponent may be estimated by
SNt = YtCMAt
I For example,
SN3 = 117118.25 = 0.989
SN4 = 172119 = 1.445
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
I Because the centered moving average (CMA) contains noseasonality and no or little irregularity, the seasonalcomponent may be estimated by
SNt = YtCMAt
I For example,
SN3 = 117118.25 = 0.989
SN4 = 172119 = 1.445
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
I After all the SNt′s have been computed, they are further
averaged to eliminate irregularities in the series.
I We also adjust the seasonal indices so that they sum to thenumber of seasons in a year, i.e., 4 for quarterly data, 12 formonthly data. (Why?)
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition ModelJ~ ... ., ~r r :-.. ·.·-~::.~;if~":~· "';i;:; -~;''.;~1:}~{'.'
~ I\ Period (t) Year Quarter Sales MA(t) CMA(t) SN(t) SN(t)
1 1 1 72 0.606312789 2 2 110 0.919068591 3 3 117 117.75 118.25 0.989429175 0.992121312 4 4 172 118.75 119 1.445378151 1.482497308 5 2 1 76 119.25 120.875 0.628748707 0.606312789 6 2 112 122.5 125.25 0.894211577 0.919068591 7 3 130 128 128.25 1.013645224 0.992121312 8 4 194 128.5 129.375 1.499516908 1.482497308 9 3 1 78 130.25 130 0.6 0.606312789
10 2 119 129.75 130.625 0.911004785 0.919068591 11 3 128 131.5 131.875 0.970616114 0.992121312 12 4 201 132.25 134.125 1.49860205 1.482497308 13 4 1 81 136 137.625 0.588555858 0.606312789 14 2 134 139.25 141.125 0.949512843 0.919068591 15 3 141 143 0.992121312 16 4 216 1.482497308
A. Quarter Average Final SN(t)
1 (0.628748707 + 0.6 + 0.588555858)/3 = 0.605768 0.606312789 2 f0.894211577 + 0.911004185 + 6.94951284~ = 0.918243 0.919968591 ----~--
3 (0.989429175 + 1.013645224 + 0.970616114)/3 = 0.99123 0.992121312 4 (1.445378151+1.499516908 + 1.49860205)/3 = 1.481166 1.482497308
Sum= 3.996407 4
Normalizing Factor: 4/3.996407 = 1.000899
'~~""-- i --~~~ ·~ --,----diiiil!il1ii i' iiiBiiiilli'iiil·'· :l
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
I We next estimate the trend (TCt).
I Define the deseasonalized or seasonally adjusted series as:
Dt = Yt
SNt
I For example,
D1 = 720.6063 = 118.7506
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
I We next estimate the trend (TCt).
I Define the deseasonalized or seasonally adjusted series as:
Dt = Yt
SNt
I For example,
D1 = 720.6063 = 118.7506
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
100
105
110
115
120
125
130
135
140
145
150
0 2 4 6 8 10 12 14 16 18
Plot of D(t) against t
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
I TCt may be estimated by regression based on a linear trend.Write
Dt = β0 + β1t + ε, t = 1, 2, · · · , n.
I Then the estimated trend is
TCt = Dt = b0 + b1t,
where b0 and b1 are the least squares estimators of β0 and β1respectively.
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
I TCt may be estimated by regression based on a linear trend.Write
Dt = β0 + β1t + ε, t = 1, 2, · · · , n.
I Then the estimated trend is
TCt = Dt = b0 + b1t,
where b0 and b1 are the least squares estimators of β0 and β1respectively.
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
I For this data set,
TCt = 113.6997914 + 1.854638009t
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3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
I The predicted values of TC may be computed by substitutingthe relevant values of t into the estimated trend equation.
I For example,
TC1 = 113.6997914 + 1.854638009(1) = 115.5544294
TC2 = 113.6997914 + 1.854638009(2) = 117.4090674
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3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
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3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
I One can then compute the forecasted values of Yt by:
Yt = TCt × SNt
I In-sample fitted values:
Y1 = 115.5544× 0.6063 = 70.0621..Y16 = 143.3740× 1.4825 = 212.5516
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
I One can then compute the forecasted values of Yt by:
Yt = TCt × SNt
I In-sample fitted values:
Y1 = 115.5544× 0.6063 = 70.0621..Y16 = 143.3740× 1.4825 = 212.5516
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3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
Out-of-sample forecasts:
Y17 = TC17 × SN17
= [113.670 + 1.855(17)]× 0.6063
= 145.2286× 0.6063
= 88.054
Y18 = TC18 × SN18
= [113.670 + 1.855(18)]× 0.9191
= 147.0833× 0.9191
= 135.1796
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3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
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3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Multiplicative Decomposition Model
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3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Measuring Forecast Accuracy
Let et = Yt − Yt be the forecast error.
I Mean Squared Error (MSE)
MSE =∑n
t=1 e2t /n
RMSE =√MSE
I Mean Absolute Deviation (MAD)
MAD =∑n
t=1 |et |/n
RMAD =√MAD
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3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Measuring Forecast Accuracy
Let et = Yt − Yt be the forecast error.
I Mean Squared Error (MSE)
MSE =∑n
t=1 e2t /n
RMSE =√MSE
I Mean Absolute Deviation (MAD)
MAD =∑n
t=1 |et |/n
RMAD =√MAD
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3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Measuring Forecast Accuracy
Let et = Yt − Yt be the forecast error.
I Mean Squared Error (MSE)
MSE =∑n
t=1 e2t /n
RMSE =√MSE
I Mean Absolute Deviation (MAD)
MAD =∑n
t=1 |et |/n
RMAD =√MAD
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3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Measuring Forecast Accuracy
Method A Method Bet = -2 -4
1.5 0.7-1 0.52.1 1.40.7 0.1
Method A: MSE = 2.43, MAD = 1.46
Method B: MSE = 3.742, MAD = 1.34
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3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Measuring Forecast Accuracy
I Naive prediction - use the last period actual value to predictthe next period’s unknown, i.e.,use Yt−1 to predict Yt .
I Theil’s U Statistic:
U =
√ ∑(Yt−Yt)2/n∑
(Yt−Yt−1)2/n
I if U = 1⇒ forecasts produced are no better than naiveforecasts;if U = 0⇒ forecasts produced perfect fit
I U is expected to lie between 0 and 1 - the smaller the value ofU, the better the forecasts
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Measuring Forecast Accuracy
I Naive prediction - use the last period actual value to predictthe next period’s unknown, i.e.,use Yt−1 to predict Yt .
I Theil’s U Statistic:
U =
√ ∑(Yt−Yt)2/n∑
(Yt−Yt−1)2/n
I if U = 1⇒ forecasts produced are no better than naiveforecasts;if U = 0⇒ forecasts produced perfect fit
I U is expected to lie between 0 and 1 - the smaller the value ofU, the better the forecasts
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Measuring Forecast Accuracy
I Naive prediction - use the last period actual value to predictthe next period’s unknown, i.e.,use Yt−1 to predict Yt .
I Theil’s U Statistic:
U =
√ ∑(Yt−Yt)2/n∑
(Yt−Yt−1)2/n
I if U = 1⇒ forecasts produced are no better than naiveforecasts;if U = 0⇒ forecasts produced perfect fit
I U is expected to lie between 0 and 1 - the smaller the value ofU, the better the forecasts
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Measuring Forecast Accuracy
I Naive prediction - use the last period actual value to predictthe next period’s unknown, i.e.,use Yt−1 to predict Yt .
I Theil’s U Statistic:
U =
√ ∑(Yt−Yt)2/n∑
(Yt−Yt−1)2/n
I if U = 1⇒ forecasts produced are no better than naiveforecasts;if U = 0⇒ forecasts produced perfect fit
I U is expected to lie between 0 and 1 - the smaller the value ofU, the better the forecasts
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Measuring Forecast Accuracy
For the model used in our last example, MSE = 11.932, MAD =2.892 and U = 0.0546.
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Types of Forecasts
I Expost forecast - Prediction for the period in which theactual observation is available
I Exante forecast - Prediction for the period in which theactual observation is not available
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Types of Forecasts
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Additive Decomposition Model
The diagrams in the top and bottom panels depict situations ofmultiplicative seasonality and additive seasonality respectively.
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Additive Decomposition Model
Multiplicative decomposition (Yt = TCt × SNt × IRt) is used whenthe time series exhibits seasonal variations that follow the trend(multiplicative seasonality). For example,
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Additive Decomposition Model
Additive decomposition (Yt = TCt + SNt + IRt) is used when thetime series exhibits seasonal variations that are constant and donot follow the trend (additive seasonality). For example,
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Additive Decomposition Model
I To construct the additive model, we first calculate MAt andCMAt as per multiplicative decomposition.
I The initial seasonal component may be estimated by
SNt = Yt − CMAt .
I For example, using our previous data set,
SN3 = 117− 118.25 = −1.25SN4 = 172− 119 = 53
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Additive Decomposition Model
I To construct the additive model, we first calculate MAt andCMAt as per multiplicative decomposition.
I The initial seasonal component may be estimated by
SNt = Yt − CMAt .
I For example, using our previous data set,
SN3 = 117− 118.25 = −1.25SN4 = 172− 119 = 53
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Additive Decomposition Model
The initial seasonal indices are then averaged and adjusted so thatthey sum to zero (Why?)
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Additive Decomposition Model
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Additive Decomposition Model
I The seasonally adjusted series is Dt = Yt − SNt .
I TCt may be estimated by regression as per multiplicativedecomposition, i.e.,
Dt = β0 + β1t + ε, t = 1, 2, · · · , n.
and
TCt = Dt = b0 + b1t
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Additive Decomposition Model
I The seasonally adjusted series is Dt = Yt − SNt .
I TCt may be estimated by regression as per multiplicativedecomposition, i.e.,
Dt = β0 + β1t + ε, t = 1, 2, · · · , n.
and
TCt = Dt = b0 + b1t
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Additive Decomposition Model
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Additive Decomposition Model
I So, TCt = 113.2270833 + 1.980637255t
and
Yt = TCt + SNt
I For example,
TC1 = 113.2270833 + 1.980637255(1) = 115.2077206
and
Y1 = 115.2077206− 50.80208333 = 64.40563725
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1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series
3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification
5. Additive Decomposition Model
Additive Decomposition Model
MSE = 27.911 and MAD = 4.477
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