Forecasting & Time Series
Minggu 6
Learning Objectives
• Understand the three categories of
forecasting techniques available.
• Become aware of the four components that
make up a time series.
• Understand how to identify which
components are present in a specific time
series.
Learning Objectives,
continued
• Recognize the forecasting methods
available for time series with specific
components.
• Learn several ways of identifying the
forecasting methods with the least
forecasting error.
• Forecast for time series with specific
components using stationary methods,
trend methods, and seasonal methods.
Introduction to Forecasting
Forecasting is the art or science of
predicting the future.
Forecasting techniques
(1) Qualitative techniques: Subjective
estimates from informed sources that are
used when historical data are scarce or
non-existent
- Examples: Delphi techniques, scenario writing,
and visionary forecast.
Introduction to Forecasting,
continued
(2) Time Series Techniques:
Quantitative techniques that use
historical data for only the forecast
variable to find patterns.
- Based on the premise that the factors that
influenced patterns of activity in the past will
continue to do so in the future.
- Examples: moving averages, exponential
smoothing, and trend projections
Introduction to Forecasting,
continued
(3) Causal Techniques: Quantitative
techniques based on historical data for the
variable being forecast, and one or more
explanatory variables.
- Based on the supposition that a relationship
exists between the variable to be forecast and
other explanatory time series data.
- Examples: regression models, econometric
models, and leading indicators
Time Series Components
• Trend: Long-term upward or downward
change in a time series
• Seasonal: Periodic increases or
decreases that occur within one year
• Cyclical: Periodic increases or decreases
that occur over more than a single year
• Irregular: Changes not attributable to the
other three components; non-systematic
and unpredictable
Components of Time Series
Data
Trend
Seasonal
Cyclical
Irregular
Components of Time Series
Data
Irregular
fluctuations
Cyclical
Trend
1
2
3
4
5
6
7
Year
8
Seasonal
9
10
11
12
13
Composite Time Series Data
1
2
3
4
5
6
7
Year
8
9
10
11
12
13
Time Series Forecasting
Procedure
Step 1: Identifying Time
Series Form
• Trend component
– time series plot
– trend line
• Seasonal component
– folded annual time series plot
– autocorrelation
Step 2: Select Potential
Methods
• Stationary forecasting methods are effective
for a stationary time series, that is one that
contains only an irregular component. These
methods attempt to eliminate the irregular
through averaging.
• Trend forecasting methods are effective for
time series that contain a trend component.
These methods asses the trend component and
use it to make projections.
• Seasonal forecasting methods are used for a
time series that contains a trend, a seasonal and
an irregular component.
Step 3: Evaluate Potential
Methods
• Once the appropriate method has been chosen,
it is used to forecast the historical data for the
time series. The an evaluation is done of how
close the estimates approach the actual
historical data.
• Forecasting Error: A single measure of the
overall error of a forecast for an entire set of
data.
• Error of an Individual Forecast: The difference
between the actual value and the forecast of that
value.
et = Yt - Ft
Reasons for Forecast Failure
•
•
•
•
•
•
•
•
Failure to examine assumptions
Limited expertise
Lack of imagination
Neglect of constraints
Excessive optimism
Reliance on mechanical extrapolation
Premature closure
Over specification
Measurement of Forecasting
Error
Mean Error (ME): The average of all the errors
of forecast for a group of data.
Mean Absolute Deviation (MAD): The mean, or
average of the absolute values of the errors.
Mean Square Error (MSE): The average of the
squared errors.
Mean Percentage Error (MPE): The average of
the percentage errors of a forecast.
Mean Absolute Percentage Error (MAPE): The
average of the absolute values of the percentage
errors of a forecast.
Example:
Nonfarm
Partnership
Tax
Returns:
Actual and
Forecast
with  = .7
Year
1
2
3
4
5
6
7
8
9
10
11
Actual Forecast Error
1402
1458 1402.0
56.0
1553 1441.2 111.8
1613 1519.5
93.5
1676 1584.9
91.1
1755 1648.7 106.3
1807 1723.1
83.9
1824 1781.8
42.2
1826 1811.3
14.7
1780 1821.6 -41.6
1759 1792.5 -33.5
Mean Error
for the Nonfarm Partnership Forecasted Data
Year
1
2
3
4
5
6
7
8
9
10
11
Actual Forecast Error
1402.0
1458.0 1402.0
56.0
1553.0 1441.2 111.8
1613.0 1519.5
93.5
1676.0 1584.9
91.1
1755.0 1648.7 106.3
1807.0 1723.1
83.9
1824.0 1781.8
42.2
1826.0 1811.3
14.7
1780.0 1821.6 -41.6
1759.0 1792.5 -33.5
524.3
ME 
e
i
number of forecasts
524.3

10
 52.43
Mean Absolute Deviation
for the Nonfarm Partnership Forecasted Data
Year
1
2
3
4
5
6
7
8
9
10
11
Actual Forecast Error
1402.0
1458.0
1402.0
56.0
1553.0
1441.2 111.8
1613.0
1519.5
93.5
1676.0
1584.9
91.1
1755.0
1648.7 106.3
1807.0
1723.1
83.9
1824.0
1781.8
42.2
1826.0
1811.3
14.7
1780.0
1821.6 -41.6
1759.0
1792.5 -33.5
|Error|
56.0
111.8
93.5
91.1
106.3
83.9
42.2
14.7
41.6
33.5
674.5
MAD 
e
i
number of forecasts
674.5

10
 67.45
Mean Square Error
for the Nonfarm Partnership Forecasted Data
Year
1
2
3
4
5
6
7
8
9
10
11
Actual Forecast Error Error2
1402
1458 1402.0
56.0
3136.0
1553 1441.2 111.8 12499.2
1613 1519.5
93.5
8749.7
1676 1584.9
91.1
8292.3
1755 1648.7 106.3 11303.6
1807 1723.1
83.9
7038.5
1824 1781.8
42.2
1778.2
1826 1811.3
14.7
214.6
1780 1821.6 -41.6
1731.0
1759 1792.5 -33.5
1121.0
55864.2
e
2
MSE 
i
number of forecasts
55864.2

10
 5586.42
Mean Percentage Error
for the Nonfarm Partnership Forecasted Data
Year
1
2
3
4
5
6
7
8
9
10
11
Actual Forecast Error Error %
1402
1458
1402.0 56.0
3.8%
1553
1441.2 111.8
7.2%
1613
1519.5 93.5
5.8%
1676
1584.9 91.1
5.4%
1755
1648.7 106.3
6.1%
1807
1723.1 83.9
4.6%
1824
1781.8 42.2
2.3%
1826
1811.3 14.7
0.8%
1780
1821.6 -41.6
-2.3%
1759
1792.5 -33.5
-1.9%
31.8%
 ei

  X  100
i
MPE 
number of forecasts
318
.

10
 318%
.
Mean Absolute Percentage Error
for the Nonfarm Partnership Forecasted Data
Year
1
2
3
4
5
6
7
8
9
10
11
Actual Forecast Error |Error %|
1402
1458
1402.0
56.0
3.8%
1553
1441.2 111.8
7.2%
1613
1519.5
93.5
5.8%
1676
1584.9
91.1
5.4%
1755
1648.7 106.3
6.1%
1807
1723.1
83.9
4.6%
1824
1781.8
42.2
2.3%
1826
1811.3
14.7
0.8%
1780
1821.6 -41.6
2.3%
1759
1792.5 -33.5
1.9%
40.3%
 e

i
   100
 Xi

MAPE 
number of forecasts
40.3

10
 4.03%
Use of Error Measures
To identify the best forecasting method
• Use error measure to identify the best
value for the parameters of a specific
method.
• Use error measure to identify the best
method.
• Use MSE and MAD for both of these
situations. Note that MSE tends to
emphasize large errors.
Use of Error Measures,
continued
Forecast bias is the tendency of a
forecasting method to over or under
predict.
The mean error, ME, measures the
forecast bias.
Step 4: Make Required
Forecasts
• The best forecasting method is that with
the smallest overall error measurement.
• Using a stationary method will make a
forecast for one time into the future, Ft+1.
This is also the forecast for all future time
periods.
• Forecasts made using a non-stationary
method will not be the same for all time
periods in the future.
Stationary Forecasting
Methods
• Naive Forecasting Method
• Moving Average Forecasting Method
• Weighted Moving Average Forecasting
Method
• Exponential Smoothing Forecasting
Method
Naive Forecasting
Simplest of the
naive forecasting
models
We sold 532 pairs of shoes last
week, I predict we’ll
sell 532 pairs this week.
t

t
 the forecast for time period t
t 1
 the value for time period t - 1
F
where: F
X
X
t 1
Simple Average Forecasting Method
The monthly average
last 12 months was
56.45, so I predict
56.45 for September.
Ft 
X
t 1

X
t 2

X
n
Month
Year
January
February
March
April
May
June
July
August
September
October
November
December
1994
t 3

X
t n
Cents
per
Gallon
61.3
63.3
62.1
59.8
58.4
57.6
55.7
55.1
55.7
56.7
57.2
58.0
Month
Year
January
February
March
April
May
June
July
August
September
October
November
December
1995
Cents
per
Gallon
58.2
58.3
57.7
56.7
56.8
55.5
53.8
52.8
Moving Average Forecasting Method
• Updated (recomputed) for every new
time period
• May be difficult to choose optimal
number of periods
• May not adjust for trend, cyclical, or
Update me each period.
seasonal effects
F
t

X
t 1

X
t 2

X
n
t 3
 .... 
X
t n
Weighted Moving Average
Forecasting Method
F
W
X

t 1
t
t 1
 W t  2 X t 2  W t 3 X t 3  ...  W t  n X t  n
t n
W
i t 1
i
Exponential Smoothing
Forecasting Method
where:
t 1
   X t  1     F t
t 1
 the forecast for the next time period (t+1)
t
 the forecast for the present time period (t)
t
 the actual value for the present time period
F
F
F
X
 = a value between 0 and 1
 is the exponential
smoothing constant
Trend Forecasting Methods
• Linear Trend Projection Forecasting
Method: Forecasting by fitting a linear
equation to a time series
• Non-linear Trend Projection
Forecasting Method: Forecasting by
fitting a non-linear equation to a time
series
Average Hours Worked per Week
by Canadian Manufacturing
Workers
Period Hours
1 37.2
2 37.0
3 37.4
4 37.5
5 37.7
6 37.7
7 37.4
8 37.2
9 37.3
10 37.2
Period Hours Period Hours
11 36.9
21 35.6
12 36.7
22 35.2
13 36.7
23 34.8
14 36.5
24 35.3
15 36.3
25 35.6
16 35.9
26 35.6
17 35.8
27 35.6
18 35.9
28 35.9
19 36.0
29 36.0
20 35.7
30 35.7
Period Hours
31 35.7
32 35.5
33 35.6
34 36.3
35 36.5
Excel Regression Output
using Linear Trend
Regression Statistics
Multiple R
0.782
R Square
0.611
Adjusted R Square
0.5600
Standard Error
0.509
Observations
35
Y     X 
Y  data value for period i
X  time period
Y  37.416  0.0614 X
i
where:
0
1
ti
i
i
i
t
ANOVA
df
Regression
Residual
Total
Intercept
Period
SS
MS
F
1 13.4467 13.4467 51.91
33 8.5487 0.2591
34 21.9954
Significance F
.00000003
Coefficients Standard Error t Stat
37.4161
0.17582
212.81
-0.0614
0.00852
-7.20
P-value
.0000000
.00000003
Work Week
Excel Graph of Hours Worked
Data with a Linear Trend Line
38.0
37.5
37.0
36.5
36.0
35.5
35.0
34.5
0
5
10
15
20
25
Time Period
30
35
Excel Regression Output
using Quadratic Trend
Regression Statistics
Multiple R
0.8723
R Square
0.761
Adjusted R Square
0.747
Standard Error
0.405
Observations
Y i   0   1 X ti   2
where:
35
Regression
Residual
Total
Intercept
Period
Period2
2
32
34
Coefficients
38.16442
-0.18272
0.00337
SS
16.7483
5.2472
21.9954
ti
i
data value for period i
i
time period
2
ANOVA
df
Y
X
X 
X 
2
ti
i
the square of the ith period
2
Y  38164
.
 0183
.

0
.
003
Xt
Xt
MS
8.3741
0.1640
F
51.07
Standard Error t Stat
0.21766
175.34
0.02788
-6.55
0.00075
4.49
Significance F
1.10021E-10
P-value
2.61E-49
2.21E-07
8.76E-05
Work Week
Excel Graph of Hourly Data
with Quadratic Trend Line
38.0
37.5
37.0
36.5
36.0
35.5
35.0
34.5
0
5
10
15
20
Period
25
30
35
Exponential Smoothing
with Trend Effects: Holt’s
Model
Holt’s Model adds consideration of a trend
component to the basic exponential smoothing
relation.
E   X  (1   )( E  T
Trend Term Update: T   ( E  E )  (1   ) T
Forecast for Next Period: F  E  T
for k periods in the future: F  E  k T
Smoothed Values:
t
t
t
t
t 1
t
t k
t
t 1
t 1
t
t
)
t 1
t 1
Trend Autoregression Method
A multiple regression technique in which the
independent variables are time-lagged versions of
the dependent variable.
Autoregression Model with two lagged variables
Y  b0  b1Y t 1  b2 Y t 2
Autoregression Model with three lagged variables
Y  b0  b1Y t 1  b2 Y t 2  b3Y t 3
Durbin-Watson Test
for Autocorrelation
H 0:   0
Ha:   0
  et  et 1
n
D
2
t 2
n
e
2
t 1
t
where: n = the number of observations
If D >
If D <
If
d
L
d
d
U
, do not reject H0 (there is no significant autocorrelation).
L
, reject H0 (there is significant autocorrelation).
 D  d U , the test is inconclusive.
Overcoming the
Autocorrelation Problem
• Addition of Independent Variables
• Transforming Variables
– First-differences approach
– Percentage change from period to
period
– Use autoregression
Seasonal Forecasting
Methods
• Seasonal Multiple Regression Forecasting
Method
• Seasonal Autoregression Forecasting
Method
• Winter’s Exponential Smoothing
Forecasting Model
• Time Series Decomposition Forecasting
Method
Smoothed Values:
Exponential
Smoothing
with Trend
and
Seasonality:
Winter’s
Model
E
  ( X t / S t  L  (1   )( E t 1  T t 1)
t
Trend Term Update:
T
t
  ( E t  E t 1)  (1   ) T t 1
SeasonalityUpdate:
S
t
  ( X t / E t )  (1   ) S t  L
Forecast for Next Period:
F
t 1
 ( E t  T t ) S t  L 1
for k periods in the future:
F
t k
 ( E t  k T t ) S t  L k
Time Series Decomposition
Forecasting Method
Basis for analysis is the multiplicative
model
Y=T·C·S·I
where:
T = trend component
C = cyclical component
S = seasonal component
I = irregular component
Time Series Decomposition
• Determine the seasonality of the time series by
computing a seasonal index for each season
(each quarter, each month, and so on.
• Divide each time series data value by the
appropriate seasonal index to deseasonalize it.
• Identify a trend model appropriate for the
deseasonalized trend model.
• Forecast deseasonalized values with the trend
model
• Multiply the deseasonalized forecasts times the
appropriate seasonal index to compute the final
seasonalized forecasts.
Demonstration Problem 14.6:
Household Appliance Shipment
Data
Year Quarter Shipments
1
1
4009
2
4321
3
4224
4
3944
2
1
4123
2
4522
3
4657
4
4030
3
1
4493
2
4806
3
4551
4
4485
Year
4
5
Quarter Shipments
1
4595
2
4799
3
4417
4
4258
1
4245
2
4900
3
4585
4
4533
Shipments in $1,000,000.
Demonstration Problem 14.6: Graph of
Household Appliance Shipment Data
Shipments
4950
4800
4650
4500
4350
4200
4050
3900
0
4
8
12
Quarter
16
20