One major problem with time series analysis and forecasting is that the past is not necessarily a good predictor of the future… A more serious flaw is that sometimes things happen which have little or no CAUSE relationship but seem predictive. Despite this is it often possible to construct a convincing scenario for things such as skirt lengths or sunspots and the stock market. These are called spurious correlations.
Go to the
spurious correlation site
and produce one chart and short explanation of why (despite everything) one might convince another person it is true.
Feel free to have a little fun with this.
MAKE AS AN ORIGINAL!!!!!
asapfollow directions
© 2019 Cengage. All Rights Reserved.
Time Series Analysis and
Forecasting
Chapter 5
© 2019 Cengage. All Rights Reserved.
Introduction (Slide 1 of 2)
• Forecasting methods can be classified as qualitative or quantitative.
• Qualitative methods generally involve the use of expert judgment to
develop forecasts.
• Quantitative forecasting methods can be used when:
• Past information about the variable being forecast is available.
• The information can be quantified.
• It is reasonable to assume that past is prologue.
© 2019 Cengage. All Rights Reserved.
Introduction (Slide 2 of 2)
• The objective of time series analysis is to uncover a pattern in the
time series and then extrapolate the pattern into the future.
• The forecast is based solely on past values of the variable and/or on
past forecast errors.
• Modern data-collection technologies have enabled individuals,
businesses, and government agencies to collect vast amounts of
data that may be used for causal forecasting.
© 2019 Cengage. All Rights Reserved.
Time Series Patterns
Horizontal Patter
n
Trend Pattern
Seasonal Pattern
Trend and Seasonal Pattern
Cyclical Pattern
Identifying Time Series Patterns
© 2019 Cengage. All Rights Reserved.
Time Series Patterns (Slide 1 of 20)
• Time series: A sequence of observations on a variable measured at
successive points in time or over successive periods of time.
• The measurements may be taken every hour, day, week, month,
year, or any other regular interval. The pattern of the data is
important in understanding the series’ past behavior.
• If the behavior of the times series data of the past is expected to
continue in the future, it can be used as a guide in selecting an
appropriate forecasting method.
© 2019 Cengage. All Rights Reserved.
Time Series Patterns (Slide 2 of 20)
Horizontal Pattern:
• Exists when the data fluctuate randomly around a constant mean over
time.
• Stationary time series: It denotes a time series whose statistical
properties are independent of time:
• The process generating the data has a constant mean.
• The variability of the time series is constant over time.
• A time series plot for a stationary time series will always exhibit a
horizontal pattern with random fluctuations.
© 2019 Cengage. All Rights Reserved.
Time Series Patterns (Slide 3 of 20)
Table 8.1: Gasoline Sales Time Series
Week
Sales (1,000s of
gallons)
1 17
2 2
1
3 19
4 23
5 18
6 16
7 20
8 18
9 22
10 20
11 15
12 22
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Time Series Patterns (Slide 4 of 20)
Figure 8.1: Gasoline Sales
Time Series Plot
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Time Series Patterns (Slide 5 of 20)
Table 8.2: Gasoline
Sales
Time Series
after Obtaining the
Contract with the
Vermont State
Polic
e
Week
Sales (1,000s of
gallons) Week
Sales (1,000s of
gallons)
1 17 12 22
2 21 13 31
3 19 14 3
4
4 23 15 31
5 18 16 33
6 16 17 28
7 20 18 32
8 18 19 30
9 22 20 29
10 20 21 34
11 15 22 33
© 2019 Cengage. All Rights Reserved.
Time Series Patterns (Slide 6 of 20)
Figure 8.2: Gasoline
Sales Time Series Plot
after Obtaining the
Contract with the
Vermont State Police
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Time Series Patterns (Slide 7 of 20)
Trend Pattern:
• A trend pattern shows gradual shifts or movements to relatively higher
or lower values over a longer period of time.
• A trend is usually the result of long-term factors such as:
• Population increases or decreases.
• Shifting demographic characteristics of the population.
• Improving technology.
• Changes in the competitive landscape.
• Changes in consumer preferences.
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Time Series Patterns (Slide 8 of 20)
Table 8.3: Bicycle Sales Time Series Year Sales
(1,000s)
1 21.6
2 22.9
3 25.5
4 21.9
5 23.9
6 27.5
7 31.5
8 29.7
9 28.6
10 31.4
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Time Series Patterns (Slide 9 of 20)
Figure 8.3: Bicycle
Sales Time Series Plot
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Time Series Patterns (Slide 10 of 20)
Table 8.4: Cholesterol
Drug Revenue Times
Year Revenue ($ millions)
1 23.1
2 21.3
3 27.4
4 34.6
5 33.8
6 43.2
7 59.5
8 64.4
9 74.2
10 99.3
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Time Series Patterns (Slide 11 of 20)
Figure 8.4: Cholesterol Drug Revenue Times Series Plot ($ millions)
© 2019 Cengage. All Rights Reserved.
Time Series Patterns (Slide 12 of 20)
Seasonal Pattern:
• Seasonal patterns are recurring patterns over successive periods of time.
• Example: A retailer that sells bathing suits expects low sales activity in the
fall and winter months, with peak sales in the spring and summer months to
occur every year.
• The time series plot not only exhibits a seasonal pattern over a one-year
period but also for less than one year in duration.
• Example: daily traffic volume shows within-the-day “seasonal” behavior,
with peak levels occurring during rush hour, moderate flow during the rest
of the day, and light flow from midnight to early morning.
© 2019 Cengage. All Rights Reserved.
Time Series Patterns (Slide 13 of 20)
Table 8.5: Umbrella
Sales Time Series
Year Quarter Sales
1 1 125
2 153
3 106
4 88
2 1 118
2 161
3 133
4 102
3 1 138
2 144
3 113
4 80
© 2019 Cengage. All Rights Reserved.
Time Series Patterns (Slide 14 of 20)
Table 8.5: Umbrella
Sales Time Series
(cont.)
Year Quarter Sales
4 1 109
2 137
3 125
4 109
5 1 130
2 165
3 128
4 96
© 2019 Cengage. All Rights Reserved.
Time Series Patterns (Slide 15 of 20)
Figure 8.5:
Umbrella Sales
Time Series Plot
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Time Series Patterns (Slide 16 of 20)
Trend and Seasonal Pattern:
• Some time series include both a trend and a seasonal pattern.
Table 8.6: Quarterly
Smartphone Sales Time
Series
Year Quarter Sales
($1,000s)
1 1 4.8
2 4.1
3 6.0
4 6.5
2 1 5.8
2 5.2
3 6.8
4 7.4
© 2019 Cengage. All Rights Reserved.
Time Series Patterns (Slide 17 of 20)
Table 8.6: Quarterly
Smartphone Sales Time
Series (cont.)
Year Quarter Sales ($1,000s)
3 1 6.0
2 5.6
3 7.5
4 7.8
4 1 6.3
2 5.9
3 8.0
4 8.4
© 2019 Cengage. All Rights Reserved.
Time Series Patterns (Slide 18 of 20)
Figure 8.6: Quarterly
Smartphone Sales
Time Series Plot
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Time Series Patterns (Slide 19 of 20)
Cyclical Pattern:
• A cyclical pattern exists if the time series plot shows an alternating
sequence of points below and above the trendline that lasts for more
than one year.
• Example: Periods of moderate inflation followed by periods of rapid
inflation can lead to a time series that alternates below and above a
generally increasing trendline.
• Cyclical effects are often combined with long-term trend effects and
referred to as trend-cycle effects.
© 2019 Cengage. All Rights Reserved.
Time Series Patterns (Slide 20 of 20)
Identifying Time Series Patterns:
• The underlying pattern in the time series is an important factor in
selecting a forecasting method.
• A time series plot should be one of the first analytic tools.
• We need to use a forecasting method that is capable of handling the
pattern exhibited by the time series effectively.
© 2019 Cengage. All Rights Reserved.
Forecast Accuracy
© 2019 Cengage. All Rights Reserved.
Forecast Accuracy (Slide 1 of 10)
Table 8.7: Computing Forecasts and Measures of Forecast Accuracy Using the
Most Recent Value as the Forecast for the Next Period
Week
Time Series
Value Forecas
t
Forecast
Error
Absolute
Value of
Forecast
Error
Squared
Forecast
Error
Percentage
Error
Absolute
Value of
Percentage
Error
1 17
2 21 17 4 4 16 19.05 19.05
3 19 21 −2 2 4 −10.53 10.53
4 23 19 4 4 16 17.39 17.39
5 18 23 −5 5 25 −27.78 27.78
6 16 18 −2 2 4 −12.50 12.50
© 2019 Cengage. All Rights Reserved.
Forecast Accuracy (Slide 2 of 10)
Table 8.7: Computing Forecasts and Measures of Forecast Accuracy Using the
Most Recent Value as the Forecast for the Next Period (cont.)
Week
Time Series
Value Forecast
Forecast
Error
Absolute
Value of
Forecast
Error
Squared
Forecast
Error
Percentage
Error
Absolute
Value of
Percentage
Error
7 20 16 4 4 16 20.00 20.00
8 18 20 −2 2 4 −11.11 11.11
9 22 18 4 4 16 18.18 18.18
10 20 22 −2 2 4 −10.00 10.00
11 15 20 −5 5 25 −33.33 33.33
12 22 15 7 7 49 31.82 31.82
Totals 5 41 179 1.19 211.69
© 2019 Cengage. All Rights Reserved.
Forecast Accuracy (Slide 3 of 10)
• Naïve forecasting method: Using the most recent data to predict future
data.
• The key concept associated with measuring forecast accuracy is forecast
error.
• Measures to determine how well a particular forecasting method is able
to reproduce the time series data that are already available.
• Forecast error.
• Mean forecast error (MFE).
• Mean absolute error (MAE).
• Mean squared error (MSE).
• Mean absolute percentage error (MAPE).
© 2019 Cengage. All Rights Reserved.
Forecast Accuracy (Slide 4 of 10)
Forecast Error: Difference between the actual and the forecasted values for period t.
Mean Forecast Error: Mean or average of the forecast errors.
© 2019 Cengage. All Rights Reserved.
Forecast Accuracy (Slide 5 of 10)
Mean Absolute Error (MAE): Measure of forecast accuracy that avoids the problem of
positive and negative forecast errors offsetting one another.
Mean Squared Error (MSE): Measure that avoids the problem of positive and negative errors
offsetting each other is obtained by computing the average of the squared forecast errors.
© 2019 Cengage. All Rights Reserved.
Forecast Accuracy (Slide 6 of 10)
Mean Absolute Percentage Error (MAPE): Average of the absolute value of
percentage forecast errors.
© 2019 Cengage. All Rights Reserved.
Forecast Accuracy (Slide 7 of 10)
Table 8.8: Computing Forecasts and Measures of Forecast Accuracy
Using the Average of All the Historical Data as the Forecast for the Next
Period
Week
Time Series
Value Forecast
Forecast
Error
Absolute
Value of
Forecast
Error
Squared
Forecast
Error
Percentage
Error
Absolute
Value of
Percentage
Error
1 17
2 21 17.00 4.00 4.00 16.00 19.05 19.05
3 19 19.00 0.00 0.00 0.00 0.00 0.00
4 23 19.00 4.00 4.00 16.00 17.39 17.39
5 18 20.00 −2.00 2.00 4.00 −11.11 11.11
© 2019 Cengage. All Rights Reserved.
Forecast Accuracy (Slide 8 of 10)
Table 8.8: Computing Forecasts and Measures of Forecast Accuracy
Using the Average of All the Historical Data as the Forecast for the Next
Period (cont.)
Week
Time Series
Value Forecast
Forecast
Error
Absolute
Value of
Forecast
Error
Squared
Forecast
Error
Percentage
Error
Absolute
Value of
Percentage
Error
6 16 19.60 −3.60 3.60 12.96 −22.50 22.50
7 20 19.00 1.00 1.00 1.00 5.00 5.00
8 18 19.14 −1.14 1.14 1.31 −6.35 6.35
9 22 19.00 3.00 3.00 9.00 13.64 13.64
© 2019 Cengage. All Rights Reserved.
Forecast Accuracy (Slide 9 of 10)
Table 8.8: Computing Forecasts and Measures of Forecast Accuracy
Using the Average of All the Historical Data as the Forecast for the Next
Period (cont.)
Week
Time Series
Value Forecast
Forecast
Error
Absolute
Value of
Forecast
Error
Squared
Forecast
Error
Percentage
Error
Absolute
Value of
Percentage
Error
10 20 19.33 0.67 0.67 0.44 3.33 3.33
11 15 19.40 −4.40 4.40 19.36 −29.33 29.33
12 22 19.00 3.00 3.00 9.00 13.64 13.64
Totals 4.52 26.81 89.07 2.75 141.34
© 2019 Cengage. All Rights Reserved.
Forecast Accuracy (Slide 10 of 10)
• Compare the accuracy of the two forecasting methods by comparing
the values of MAE, MSE, and MAPE for each method.
Naïve Method
Average of Past
Values
MAE 3.73 2.44
MSE 16.27 8.10
MAPE 19.24% 12.85%
• The average of past values provides more accurate forecasts for the
next period than using the most recent observation.
© 2019 Cengage. All Rights Reserved.
Moving Averages and Exponential
Smoothing
Moving Averages
Exponential Smoothing
© 2019 Cengage. All Rights Reserved.
Moving Averages and Exponential Smoothing
(Slide 1 of 16)
Moving Averages:
• Moving averages
method: Uses the
average of the most
recent k data values in
the time series as the
forecast for the next
period.
© 2019 Cengage. All Rights Reserved.
Moving Averages and Exponential Smoothing
(Slide 2 of 16)
Table 8.9: Summary of Three-Week Moving Average Calculations
Week
Time Series
Value Forecast
Forecast
Error
Absolute
Value of
Forecast
Error
Squared
Forecast
Error
Percentage
Error
Absolute
Value of
Percentage
Error
1 17
2 21
3 19
4 23 19 4 4 16 17.39 17.39
5 18 21 −3 3 9 −16.67 16.67
6 16 20 −4 4 16 −25.00 25.00
© 2019 Cengage. All Rights Reserved.
Moving Averages and Exponential Smoothing
(Slide 3 of 16)
Table 8.9: Summary of Three-Week Moving Average Calculations (cont.)
Week
Time Series
Value Forecast
Forecast
Error
Absolute
Value of
Forecast
Error
Squared
Forecast
Error
Percentage
Error
Absolute
Value of
Percentage
Error
7 20 19 1 1 1 5.00 5.00
8 18 18 0 0 0 0.00 0.00
9 22 18 4 4 16 18.18 18.18
10 20 20 0 0 0 0.00 0.00
11 15 20 −5 5 25 −33.33 33.33
12 22 19 3 3 9 13.64 13.64
Totals 0 24 92 −20.79 129.21
© 2019 Cengage. All Rights Reserved.
Moving Averages and Exponential Smoothing
(Slide 4 of 16)
Figure 8.7:
Gasoline Sales
Time Series Plot
and Three-Week
Moving Average
Forecasts
© 2019 Cengage. All Rights Reserved.
Moving Averages and Exponential Smoothing
(Slide 5 of 16)
Figure 8.8: Data Analysis Dialog Box
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Moving Averages and Exponential Smoothing
(Slide 6 of 16)
Figure 8.9: Moving Average Dialog Box
© 2019 Cengage. All Rights Reserved.
Moving Averages and Exponential Smoothing
(Slide 7 of 16)
Figure 8.10: Excel Output
for Moving Average
Forecast for Gasoline
Data
© 2019 Cengage. All Rights Reserved.
Moving Averages and Exponential Smoothing
(Slide 8 of 16)
Forecast Accuracy:
The values of the three
measures of forecast
accuracy for the three-week
moving average calculations
in Table 8.9.
12
4
12
2
4
12
4
24
MAE 2.67
3 9
92
MSE 10.22
3 9
100
129.21
MAPE 14.36%
3 9
t
t
t
t
t
t t
e
n
e
n
e
y
n
=
=
=
= = =
−
= = =
−
= = =
−
© 2019 Cengage. All Rights Reserved.
Moving Averages and Exponential Smoothing
(Slide 9 of 16)
Exponential Smoothing:
• Exponential smoothing uses a weighted average of past time series
values as a forecast.
• Smoothing constant ( ) is the weight given to the actual value in
period t; weight given to the forecast in period t is 1 .−
© 2019 Cengage. All Rights Reserved.
Moving Averages and Exponential Smoothing
(Slide 10 of 16)
Illustration of Exponential Smoothing:
Consider a time series involving only three periods of data: 1 2 3, , and .
y
y y
• Let 1ŷ equal the actual value of the time series in period 1; that is, 1 1ˆ .y y=
• Hence, the forecast for period 2 is:
( )
( )
2
1 1
1 1
1
ˆ ˆ1
1
y y y
y y
y
= + −
= + −
=
© 2019 Cengage. All Rights Reserved.
Moving Averages and Exponential Smoothing
(Slide 11 of 16)
Table 8.10:
Summary of the
Exponential
Smoothing
Forecasts and
Forecast Errors
for the Gasoline
Sales Time Series
with Smoothing
Constant 0.2 =
Week
Time Series
Value Forecast
Forecast Error
Squared
Forecast Error
1 17
2 21 17.00 4.00 16.00
3 19 17.80 1.20 1.44
4 23 18.04 4.96 24.60
5 18 19.03 −1.03 1.06
6 16 18.83 −2.83 8.01
© 2019 Cengage. All Rights Reserved.
Moving Averages and Exponential Smoothing
(Slide 12 of 16)
Table 8.10:
Summary of the
Exponential
Smoothing
Forecasts and
Forecast Errors
for the Gasoline
Sales Time Series
with Smoothing
Constant 0.2 =
(cont.)
Week
Time Series
Value Forecast Forecast Error
Squared
Forecast Error
7 20 18.26 1.74 3.03
8 18 18.61 −0.61 0.37
9 22 18.49 3.51 12.32
10 20 19.19 0.81 0.66
11 15 19.35 −4.35 18.92
12 22 18.48 3.52 12.39
Totals 10.92 98.80
© 2019 Cengage. All Rights Reserved.
Moving Averages and Exponential Smoothing
(Slide 13 of 16)
Figure 8.11:
Actual and
Forecast Gasoline
Time Series with
Smoothing
Constant 0.2 =
© 2019 Cengage. All Rights Reserved.
Moving Averages and Exponential Smoothing
(Slide 14 of 16)
Figure 8.13:
Exponential
Smoothing
Dialog Box
© 2019 Cengage. All Rights Reserved.
Moving Averages and Exponential Smoothing
(Slide 15 of 16)
Figure 8.14: Excel
Output for
Exponential
Smoothing
Forecast for
Gasoline Data
© 2019 Cengage. All Rights Reserved.
Moving Averages and Exponential Smoothing
(Slide 16 of 16)
Forecast Accuracy:
• Insight into choosing a good value for can be obtained by rewriting
the basic exponential smoothing model as:
( )
( )
1
ˆ ˆ1
ˆ
ˆ
ˆ ˆ
ˆ
t t t
t t t
t t t
t t
y y y
y y y
y y y
y e
+ = + −
= + −
= + −
= +
• If the time series contains substantial random variability, a small value
of the smoothing constant is preferred and vice-versa.
• Choose the value of that minimizes the MSE.
© 2019 Cengage. All Rights Reserved.
Using Regression Analysis for
Forecasting
Linear Trend Projection
Seasonality without Trend
Seasonality with Trend
Using Regression Analysis as a Causal
Forecasting Method
Combining Causal Variables with Trend
and Seasonality Effects
Considerations in Using Regression in
Forecasting
© 2019 Cengage. All Rights Reserved.
Using Regression Analysis for Forecasting
(Slide 1 of 19)
Linear Trend Projection:
• Regression analysis can be used to forecast a time series with a linear trend.
• Simple linear regression analysis yields the linear relationship between the
independent variable and the dependent variable that minimizes the MSE.
• Use this approach to find a best-fitting line to a set of data that exhibits a linear
trend.
• The variable to be forecasted ( ,ty the actual value of the time series period t) is
the dependent variable.
• Trend variable (time period t) is the independent variable.
© 2019 Cengage. All Rights Reserved.
Using Regression Analysis for Forecasting
(Slide 2 of 19)
Linear Trend Projection (cont.):
• Equation for the trendline:
• Trend equation for the bicycle sales time series:
• Substituting t = 11 into the above equation yields next year’s trend projection,
11
ˆ :y
( )11ˆ 20.4 1.1 11 32.5y = + =
• Thus, the linear trend model yields a sales forecast of 32,500 bicycles for the next
year.
© 2019 Cengage. All Rights Reserved.
Using Regression Analysis for Forecasting
(Slide 3 of 19)
Figure 8.15: Excel
Simple Linear
Regression
Output for
Trendline Model
for Bicycle Sales
Data
© 2019 Cengage. All Rights Reserved.
Using Regression Analysis for Forecasting
(Slide 4 of 19)
Linear Trend Projection (cont.):
• We can also use more complex regression models to fit nonlinear trends:
2 3
0 1 2 3t̂y b b t b t b t= + + +
• Autoregressive models: Regression models in which the independent
variables are previous values of the time series.
0 1 1 2 2 3 3t̂ t t ty b b y b y b y− − −= + + +
© 2019 Cengage. All Rights Reserved.
Using Regression Analysis for Forecasting
(Slide 5 of 19)
Seasonality without Trend:
• We can model a time series with a seasonal pattern by treating the
season as a dummy variable.
• Illustration:
• Consider the data on the number of umbrellas sold in Table 8.5.
• The time series plot corresponding to this data in Figure 8.5 does not
suggest any long-term trend in sales.
© 2019 Cengage. All Rights Reserved.
Using Regression Analysis for Forecasting
(Slide 6 of 19)
• Illustration (cont.):
• Closer inspection of the time series plot suggests that a quarterly seasonal
pattern is present.
• 1k − dummy variables are required to model a categorical variable that has
k levels.
• Thus, to model the seasonal effects in the umbrella time series we need
4 1 3− = dummy variables.
© 2019 Cengage. All Rights Reserved.
Using Regression Analysis for Forecasting
(Slide 7 of 19)
• Seasonality without Trend Illustration (cont.):
• The three dummy variables can be coded as follows:
Qtr1 1 if period is quarter 1; 0 otherwise.
Qtr2 1 if period is quarter 2; 0 otherwise.
Qtr3 1 if period is quarter 3; 0 otherwise.
t
t
t
t
t
t
=
=
=
• General form of the equation relating the number of umbrellas sold to the
quarter the sales take place:
© 2019 Cengage. All Rights Reserved.
Using Regression Analysis for Forecasting
(Slide 8 of 19)
Table 8.11: Umbrella Sales Time Series with Dummy Variables
Period Year Quarter Qtr1 Qtr2 Qtr3 Sales
1 1 1 1 0 0 125
2 2 0 1 0 153
3 3 0 0 1 106
4 4 0 0 0 88
5 2 1 1 0 0 118
6 2 0 1 0 161
7 3 0 0 1 133
8 4 0 0 0 102
© 2019 Cengage. All Rights Reserved.
Using Regression Analysis for Forecasting
(Slide 9 of 19)
Table 8.11: Umbrella Sales Time Series with Dummy Variables (cont.)
Period Year Quarter Qtr1 Qtr2 Qtr3 Sales
9 3 1 1 0 0 138
10 2 0 1 0 144
11 3 0 0 1 113
12 4 0 0 0 80
13 4 1 1 0 0 109
14 2 0 1 0 137
15 3 0 0 1 125
16 4 0 0 0 109
© 2019 Cengage. All Rights Reserved.
Using Regression Analysis for Forecasting
(Slide 10 of 19)
Table 8.11: Umbrella Sales Time Series with Dummy Variables (cont.)
Period Year Quarter Qtr1 Qtr2 Qtr3 Sales
17 5 1 1 0 0 130
18 2 0 1 0 165
19 3 0 0 1 128
20 4 0 0 0 96
© 2019 Cengage. All Rights Reserved.
Using Regression Analysis for Forecasting
(Slide 11 of 19)
Seasonality with Trend:
• The time series contains both seasonal effects and a linear trend.
• Consider the data for the smartphone time series in Table 8.6.
• The time series plot corresponding to this data (Figure 8.6) indicates that
there is both linear trend and seasonal pattern.
• The general form of the regression equation takes the form.
© 2019 Cengage. All Rights Reserved.
Using Regression Analysis for Forecasting
(Slide 12 of 19)
Table 8.12: Smartphone Sales Time Series with Dummy Variables and
Time Period
Period Year Quarter Qtr1 Qtr2 Qtr3 Sales (1,000s)
1 1 1 1 0 0 4.8
2 2 0 1 0 4.1
3 3 0 0 1 6.0
4 4 0 0 0 6.5
5 2 1 1 0 0 5.8
6 2 0 1 0 5.2
7 3 0 0 1 6.8
8 4 0 0 0 7.4
© 2019 Cengage. All Rights Reserved.
Using Regression Analysis for Forecasting
(Slide 13 of 19)
Table 8.12: Smartphone Sales Time Series with Dummy Variables and
Time Period (cont.)
Period Year Quarter Qtr1 Qtr2 Qtr3 Sales (1,000s)
9 3 1 1 0 0 6.0
10 2 0 1 0 5.6
11 3 0 0 1 7.5
12 4 0 0 0 7.8
13 4 1 1 0 0 6.3
14 2 0 1 0 5.9
15 3 0 0 1 8.0
16 4 0 0 0 8.4
© 2019 Cengage. All Rights Reserved.
Using Regression Analysis for Forecasting
(Slide 14 of 19)
Seasonality with Trend (cont.):
• The dummy variables in the equation for Smartphone Sales time series
provide four equations given time period t corresponds to quarters 1, 2, 3,
and 4.
• Quarter 1: Sales = 4.71 + 0.146t.
• Quarter 2: Sales = 4.04 + 0.146t.
• Quarter 3: Sales = 5.77 + 0.146t.
• Quarter 4: Sales = 6.07 + 0.146t.
© 2019 Cengage. All Rights Reserved.
Using Regression Analysis for Forecasting
(Slide 15 of 19)
Using Regression Analysis as a Causal Forecasting Method:
• The relationship of the variable to be forecast with other variables may also be used to
develop a forecasting model.
• Advertising expenditures when sales are to be forecast.
• The mortgage rate when new housing construction is to be forecast.
• Grade point average when starting salaries for recent college graduates are to be forecast.
• The price of a product when the demand for the product is to be forecast.
• The value of the Dow Jones Industrial Average when the value of an individual stock is to
be forecast.
• Daily high temperature when electricity usage is to be forecast.
• Causal models: Models that include only variables that are believed to cause changes
in the variable to be forecast.
© 2019 Cengage. All Rights Reserved.
Using Regression Analysis for Forecasting
(Slide 16 of 19)
Table 8.13: Student
Population and
Quarterly Sales Data
for 10 Armand’s
Pizza Parlors
Restaurant
Student Population
(1,000s)
Quarterly Sales
($1,000s)
1 2 58
2 6 105
3 8 88
4 8 118
5 12 117
6 16 137
7 20 157
8 20 169
9 22 149
10 26 202
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Using Regression Analysis for Forecasting
(Slide 17 of 19)
Figure 8.16: Scatter
Chart of Student
Population and
Quarterly Sales for
Armand’s Pizza Parlors
© 2019 Cengage. All Rights Reserved.
Using Regression Analysis for Forecasting
(Slide 18 of 19)
Figure 8.17: Graph of
the Estimated
Regression Equation for
Armand’s Pizza Parlors:
60 5y x= +
© 2019 Cengage. All Rights Reserved.
Using Regression Analysis for Forecasting
(Slide 19 of 19)
Combining Causal Variables with Trend and Seasonality Effects:
• Regression models are very flexible and can incorporate both causal
variables and time series effects.
Considerations in Using Regression in Forecasting:
• Whether a regression approach provides a good forecast depends largely
on how well we are able to identify and obtain data for independent
variables that are closely related to the time series.
• Part of the regression analysis procedure should focus on the selection of
the set of independent variables that provides the best forecasting
model.
© 2019 Cengage. All Rights Reserved.
Determining the Best Forecasting
Model to Use
© 2019 Cengage. All Rights Reserved.
Determining the Best Forecasting
Model to Use (Slide 1 of 2)
• A visual inspection can indicate whether seasonality appears to be a
factor and whether a linear or nonlinear trend seems to exist.
• For causal modeling, scatter charts can indicate whether strong
linear or nonlinear relationships exist between the independent and
dependent variables.
• If certain relationships appear totally random, this may lead you to
exclude these variables from the model.
© 2019 Cengage. All Rights Reserved.
Determining the Best Forecasting
Model to Use (Slide 2 of 2)
• While working with large data sets, it is recommended to divide
your data into training and validation sets.
• Based on the errors produced by the different models for the
validation set, you can pick the model that minimizes some forecast
error measure, such as MAE, MSE or MAPE.
• There are software packages that will automatically select the best
model to use.
• Ultimately, the user should decide which model to use based on the
software output and his managerial knowledge.