# Chapter 14

```Week 3
September 15-19
Five Mini-Lectures
QMM 510
Fall 2014
ML 3.1
Chapter Contents
14.1 Time-Series Components
14.2 Trend Forecasting
14.3 Assessing Fit
14.4 Moving Averages
14.5 Exponential Smoothing
14.6 Seasonality
14.7 Index Numbers
14.8 Forecasting: Final Thoughts
14-2
So many
topics, so little
time …
Chapter 14
Time-Series Analysis
Chapter 14
Time-Series Analysis
•
A time-series variable (Y) consists of data observed over n
periods of time.
•
Businesses use time-series data
- to monitor a process to determine if it is stable
- to predict the future (forecasting)
•
Time-series data can also be used to understand economic,
population, health, crime, sports, and social problems.
14-3
Chapter 14
Time-Series Analysis
Time- Series Data
•
Time-series data are usually plotted
as a line graph.
•
Time is on the horizontal (X) axis.
•
Trends and fluctuations are easier to
see on a line graph.
•
The following notation is used:
yt is the value of the time series in
period t (t is an index denoting the
time period t = 1, 2, …, n); n is the
number of time periods; y1, y2, …, yn
is the data set for analysis.
14-4
Chapter 14
Time-Series Components
Time-Series Data
•
To distinguish time-series data from cross-sectional data, use yt
instead of xi for an individual observation.
Measuring Time Series
•
Time-series data may be measured at a point in time.
•
For example, prime rate of interest is measured at a particular
point in time.
•
Time-series data may also be measured over an interval of time.
•
For example, gross domestic product (GDP) is a flow of goods
and services measured over an interval of time.
14-5
Chapter 14
Time-Series Components
Periodicity
•
•
The periodicity is the time interval over which data are collected.
Data can be collected once a year (e.g., 1 observation per year), quarter (e.g., 4
observations per year), month (e.g., 12 observations per year), etc.
Additive versus Multiplicative Models
•
•
Time-series decomposition seeks to separate a time series Y into four
components:
- Trend (T)
- Cycle (C)
- Seasonal (S)
- Irregular (I)
These components are assumed to follow either an additive or a
multiplicative model.
14-6
Additive versus Multiplicative Models
•
14-7
The multiplicative model becomes additive if logarithms are taken (for
nonnegative data):
Chapter 14
Time-Series Components
Trend
•
•
•
Trend (T) is the general
movement over all years
(t = 1, 2, ..., n).
Trends may be steady and
predictable, increasing,
decreasing, or staying the
same.
A mathematical trend can be
fitted to any data but may or
may not be useful for
predictions.
14-8
Chapter 14
Time-Series Components
Chapter 14
Time-Series Components
Cycle
•
•
Cycle (C) is a repetitive up-anddown movement about a trend
that covers several years.
Over a small number of time
periods, cycles are undetectable
or may resemble a trend.
Note: Forecasters generally ignore the cycle so the
multiplicative model is just Y = T x S x I.
14-9
Chapter 14
Time-Series Components
Seasonal
•
•
•
Seasonal (S) is a repetitive cyclical
pattern within a year (or within a
week, day, or other time period).
Over a small number of time
periods, cycles are undetectable
or may resemble a trend.
By definition, annual data have no
seasonality.
14-10
Chapter 14
Time-Series Components
Irregular
•
•
Irregular (I) is a random
disturbance that follows no
pattern.
It is also called the error
component or random noise
reflecting all factors other than
trend, cycle and seasonality.
14-11
Chapter 14
Trend Forecasting
ML 3.2
The main categories of forecasting models are:
We will only look at this one category
of models (more time to surf)
14-12
Chapter 14
Trend Forecasting
Steps in Forecasting:
1. Make a nice Excel graph
a. Highlight the data column (excluding heading)
b. Insert > Line Chart (e.g., with markers)
c. Add a descriptive chart title, etc.
2. Click on the line in the graph to select the variable.
3. Right-click and choose Add Trendline.
4. Select a trend (e.g., linear). Try several.
5. Make forecasts (if desired).
6. If quarterly or monthly data, calculate seasonal
factors (using MegaStat or Minitab).
7. Multiply each numerical forecast by its seasonal
factor to get seasonally adjusted forecasts.
Detailed examples follow…
14-13
Chapter 14
Trend Forecasting
Three Trend Models
•
Three trend models are especially useful in business applications:
•
All three models can be fitted by Excel, MegaStat, or MINITAB.
Linear Trend Model
•
The linear trend model has the form yt = a + bt
•
It is the simplest and may suffice for short-run forecasting or as a
baseline model.
14-14
Chapter 14
Trend Forecasting
Linear Trend Model
Linear Trend Calculations
•
Linear trend is fitted by using ordinary least squares formulas.
•
Note: Instead of using the actual time values (e.g., years), use an index
xt = 1, 2, 3, …. as the independent variable.
14-15
Chapter 14
Trend Forecasting
Linear Trend Calculations
these
calculations
are done by
Excel (whew!)
Forecasting a Linear Trend
•
Once the slope and intercept have been calculated, a forecast can be
made for any future time period by inserting t = n+1, n+2, n+3, etc into
the fitted trend equation.
14-16
Chapter 14
Trend Forecasting
Linear Trend: Calculating R2
•
R2 can be calculated as
•
An R2 close to 1.0 would indicate a good fit to the past data.
•
However, a high R2 does not guarantee a good forecast. Projecting a
trend assumes that nothing changes.
14-17
Chapter 14
Trend Forecasting
Exponential Trend Model
•
The exponential trend model has the form yt = aebt.
•
Useful for a time series that grows or declines at the same rate (b) in
each time period.
14-18
Chapter 14
Trend Forecasting
When to Use the Exponential Model
•
The exponential model (yt = aebt) is often used for data that may be assumed to
grow at a steady percent growth rate (e.g., financial data).
•
You can compare growth rates in two time-series variables with dissimilar data
units by comparing their b estimates (i.e., the fitted growth rate b is unit-free)
•
There may not be much difference between a linear and exponential model
when the data set covers only a few time periods.
•
The linear model yt = a + bt and the
exponential model yt = aebt are
equally simple because they are
two-parameter models, and a logtransformed exponential model is
actually linear.
14-19
Chapter 14
Trend Forecasting
Exponential Trend Calculations
Calculations of the exponential trend are done by using a
transformed variable zt = ln(yt) to produce a linear equation so that
the least squares formulas can be used.
Excel does all this. Once the least
squares calculations are
completed, Excel transforms the
intercept back to the original
units by exponentiation to get
the correct intercept.
Caution: You can’t fit an exponential model if
any data values are zero or negative.
14-20
Chapter 14
Trend Forecasting
•
A quadratic trend model has the form yt = a + bt + ct2
•
If c = 0, then the quadratic model becomes a linear model (i.e., the
linear model is a special case of the quadratic model).
yt = a + bt + ct2
•
Fitting a quadratic model is a way of checking for nonlinearity. If c does
not differ significantly from zero, then the linear model would suffice.
Note: A quadratic equation is unfamiliar to many, and has
no simple interpretation. Use it only when your data has a
peak or trough and no other model suffices .
14-21
Chapter 14
Trend Forecasting
Depending on the
values of b and c,
model can assume
any of four shapes:
Note: Use the quadratic only for short term
forecasts when no other model suffices .
14-22
Which Trend Model?
… or maybe none of the
above will give reasonable
forecasts
14-23
Chapter 14
Trend Forecasting
Chapter 14
Trend Forecasting
Five Measures of Fit
We usually refer to R2
because it is familiar.
“Fit” refers to how well the estimated trend model matches the observed
historical past data. We usually look at R2 because it is familiar.
14-24
Chapter 14
Trend Forecasting
Example: Revenue of Amazon.com Inc (AMZN)
•
Eyeball the data – see anything unusual?
•
Make a nice graph.
•
Fit several trend models using Excel.
Quarter
t
yt
Quarter
t
yt
03/31/2004
06/30/2004
09/30/2004
12/31/2004
03/31/2005
06/30/2005
09/30/2005
12/31/2005
03/31/2006
06/30/2006
09/30/2006
12/31/2006
03/31/2007
06/30/2007
09/30/2007
12/31/2007
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1530
1387
1462
2541
1902
1753
1858
2977
2279
2139
2307
3986
3015
2886
3262
5672
03/31/2008
06/30/2008
09/30/2008
12/31/2008
03/31/2009
06/30/2009
09/30/2009
12/31/2009
03/31/2010
06/30/2010
09/30/2010
12/31/2010
03/31/2011
06/30/2011
09/30/2011
12/31/2011
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
4135
4063
4264
6704
4889
4651
5449
9520
7131
6566
7560
12947
Revenue is in
\$millions (e.g.,
first data value
is 1.53 billion)
Objective:
Fill in these
4 boxes
14-25
Chapter 14
Trend Forecasting
Excel’s “Polynomial
Order 2” is a
Example: Revenue of Amazon.com Inc (AMZN)
Make nice
graph, then
click on the
data series
Be sure to
click these 2
boxes
14-26
Chapter 14
Example: Amazon Revenue
14000
Net Sales (millions)
12000
y = 284.58x + 117.77
R² = 0.7396
10000
8000
6000
4000
2000
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
Note: Excel will show forecasts on the
graph but no numbers are given.
14-27
Chapter 14
Example: Amazon Revenue
Fitted Trends: Amazon Revenue
If necessary, format the fitted trend label
Moving average
(not really a trend
model)
14-28
Chapter 14
Example: Amazon Revenue
14000
Net Sales (millions)
12000
y = 284.58x + 117.77
R² = 0.7396
10000
8000
6000
4000
2000
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
Interpretation: growing at
\$284.58 million per quarter, 74%
of variation explained by linear
trend model, forecasts seem
low?
Interpretation:
complex nonlinear
equation, 82% of
variation explained
14000
12000
Net Sales (millions)
Net Sales (millions)
12000
14000
y = 12.752x2 - 85.227x + 1966.8
R² = 0.8168
10000
8000
6000
4000
2000
Interpretation: growing
6.86% per quarter, 89% of
variation explained by
exponential trend,
believable forecasts
y = 1316.5e0.0686x
R² = 0.8861
10000
8000
6000
4000
2000
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
14-29
Chapter 14
Example: Amazon Revenue
14000
Net Sales (millions)
12000
y = 284.58x + 117.77
R² = 0.7396
Excel formula for t = 32 forecast:
=284.58*32 + 117.77
10000
8000
Excel formula for t = 32 forecast:
=12.75*32^2 - 85.227*32 + 1966.8
6000
4000
Excel formula for t = 32 forecast:
=1316.5*EXP(.0686*32)
2000
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
14000
12000
Net Sales (millions)
Net Sales (millions)
12000
14000
y = 12.752x2 - 85.227x + 1966.8
R² = 0.8168
10000
8000
6000
4000
2000
y = 1316.5e0.0686x
R² = 0.8861
10000
8000
6000
4000
2000
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
14-30
Chapter 14
Example: Amazon Revenue
t
29
30
31
32
Linear
Expon
8370
8655
8939
9224
10220
10887
11579
12298
9625
10309
11041
11825
Note: To make these forecasts,
the formulas from fitted trends
were entered into cells beside the
time index t = 29, 30, 31, 32 (as
shown below).
t
29
30
31
32
Linear
=117.11+284.58*J11
=117.11+284.58*J12
=117.11+284.58*J13
=117.11+284.58*J14
=1966.8-85.227*J11+12.752*J11^2
=1966.8-85.227*J12+12.752*J12^2
=1966.8-85.227*J13+12.752*J13^2
=1966.8-85.227*J14+12.752*J14^2
Expon
=1316.5*EXP(0.0686*J11)
=1316.5*EXP(0.0686*J12)
=1316.5*EXP(0.0686*J13)
=1316.5*EXP(0.0686*J14)
Note: In this example, the time index t = 29,
30, 31, 32 is in cells J11, J12, J13, J14
14-31
Chapter 14
Example: Amazon Revenue
to make future forecasts, insert formula in each cell for all three
fitted models using time index in column A (or wherever it is)
Note: The year and quarter are just labels – they
are not used in any of the calculations.
14-32
Chapter 14
Example: Amazon Revenue
t
29
30
31
32
Linear
Expon
8370
8655
8939
9224
10220
10887
11579
12298
9625
10309
11041
11825
Comment: The linear forecasts
are much more conservative than
the other two trend models.
Quadratic forecasts are the most
aggressive, though only slightly
more than the exponential
forecasts.
Comment: These are quarterly
data, so now we should adjust
the forecasts for seasonality.
14-33
Four Trend-Fitting Criteria
Criteria for selecting a trend forecasting model:
Criterion
• Occam’s Razor
• Overall fit
• Believability
• Fit to recent data.
14-34
Would a simpler model
suffice?
How does the trend fit the
past data?
Does the extrapolated trend
“look right”?
Does the fitted trend match the last few
data points?
Chapter 14
Trend Forecasting
Chapter 14
Forecasting with Seasonality
ML 3.3
When and How to Deseasonalize
•
When the data periodicity is monthly or quarterly, calculate a seasonal
index and use it to deseasonalize the data.
•
For the multiplicative model, a seasonal index is a ratio.
•
The seasonal indexes must sum to 12 for monthly data or to 4 for
quarterly data.
•
In a multiplicative model, seasonal indexes near 1.00 suggest a lack of
seasonality:
Y=TxSxI
 if S = 1.00 then S disappears
14-35
Chapter 14
Seasonality
When and How to Deseasonalize
Step 1: Calculate a centered moving average
(CMA) for each month (quarter).
Step 2: Divide each observed yt value by the
MA to obtain seasonal ratios.
Step 3: Average the seasonal ratios by the
month (quarter) to get raw seasonal indexes.
Step 4: Adjust the raw seasonal indexes so they sum
to 12 (monthly) or 4 (quarterly).
Step 5: Divide each yt by its seasonal index to get
deseasonalized data.
Note: We rely on MegaStat or another computer
package for these complex calculations.
14-36
Chapter 14
Seasonality
to label the data by
year and quarter
14-37
Chapter 14
Seasonality
MegaStat’s Seasonal Analysis
t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Year
2004
2004
2004
2004
2005
2005
2005
2005
2006
2006
2006
2006
2007
2007
2007
2007
2008
2008
2008
2008
2009
2009
2009
2009
2010
2010
2010
2010
Net
Quarter
sales
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
Centered
Moving
(millions) Average
1,530
1,387
1,462 1776.737
2,541 1868.901
1,902 1964.049
1,753 2067.995
1,858 2169.625
2,977 2265.000
2,279 2369.375
2,139 2551.625
2,307 2769.750
3,986 2955.125
3,015 3167.875
2,886 3498.000
3,262 3848.750
5,672 4135.875
4,135 4408.250
4,063 4662.500
4,264 4885.750
6,704 5053.500
4,889 5275.125
4,651 5775.250
5,449 6407.500
9,520 6927.125
7,131 7430.375
6,566 8122.625
7,560
12,947
Ratio to Seasonal Net sales (millions)
CMA Indexes Deseasonalized
0.956
1,601.1
0.837
1,657.8
0.823
0.851
1,717.7
1.360
1.356
1,874.0
0.968
0.956
1,989.9
0.848
0.837
2,094.8
0.856
0.851
2,182.3
1.314
1.356
2,195.6
0.962
0.956
2,384.3
0.838
0.837
2,556.0
0.833
0.851
2,709.6
1.349
1.356
2,939.7
0.952
0.956
3,154.3
0.825
0.837
3,448.7
0.848
0.851
3,831.3
1.371
1.356
4,183.2
0.938
0.956
4,326.1
0.871
0.837
4,855.2
0.873
0.851
5,008.2
1.327
1.356
4,944.3
0.927
0.956
5,114.9
0.805
0.837
5,557.8
0.850
0.851
6,400.0
1.374
1.356
7,021.1
0.960
0.956
7,460.5
0.808
0.837
7,846.2
0.851
8,879.4
1.356
9,548.5
14-38
Chapter 14
Seasonality
MegaStat’s Seasonal Indexes
2004
2005
2006
2007
2008
2009
2010
mean:
Calculation of Seasonal Indexes
1
2
3
0.823
0.968
0.848
0.856
0.962
0.838
0.833
0.952
0.825
0.848
0.938
0.871
0.873
0.927
0.805
0.850
0.960
0.808
0.951
0.833
0.847
0.956
0.837
0.851
4
1.360
1.314
1.349
1.371
1.327
1.374
1.349
1.356
3.980
4.000 <--- adjusted so they add to 4.000
14-39
Chapter 14
Seasonality
MegaStat’s Graph
Note: MegaStat’s graph
does not show any
forecasts (only the
deseasonalized time
series
So … we have to make
our own numerical
forecasts
14-40
Chapter 14
Seasonality
t
29
30
31
32
Now, multiply
each trend
forecast by its
quarterly
seasonal factor
Linear
Expon
8370
8655
8939
9224
10220
10887
11579
12298
9625
10309
11041
11825
t
Linear
29
30
31
32
8018
7215
7607
12520
Expon
9790
9076
9854
16693
9221
8594
9395
16051
MegaStat Seasonals
0.956
Qtr 1
0.837
Qtr 2
0.851
Qtr 3
1.356
Qtr 4
14-41
Chapter 14
Seasonality
Compare Forecasts and Choose One
14-42
ML 3.4
Chapter Contents
14.1
14.2
14.3
14.4
14.5
14.6
14.7
14.8
14-43
Time-Series Components
Trend Forecasting
Assessing Fit
Moving Averages
Exponential Smoothing
Seasonality
Index Numbers
Forecasting: Final Thoughts
… when no
trend model
works
Chapter 14
More Time Series Methods
Chapter 14
Moving Averages
Trendless or Erratic Data
•
In cases where the time series y1, y2, …, yn is erratic or has no
consistent trend, there may be little point in fitting a trend line.
•
A simple approach is to calculate either a trailing or centered moving
average.
Trailing Moving Average (TMA)
•
The TMA simply averages the data over the last m periods.
•
The TMA smoothes the past fluctuations in
the time series in order to see the pattern
more clearly.
•
Choosing a larger m yields a “smoother” TMA
but requires more data.
Note: Excel
uses the
TMA
method.
14-44
Chapter 14
Moving Averages
Centered Moving Average (CMA)
•
The CMA smoothing method calculates the mean of the current
observation and observations on either side of the current data. For
example, for m = 3:
•
•
When m is odd (m = 3, 5, etc.), the CMA is easy to calculate.
When m is even, the mean would lie between two data points and
would not be correctly centered, so we would take a double
moving average.
Caution: Excel does not offer the
CMA method (only TMA).
14-45
Chapter 14
Exponential Smoothing
Forecast Updating
•
The exponential smoothing model is a special kind of moving average.
•
This one-period-ahead forecasting technique is utilized for data that
have up-and-down movements but no consistent trend.
•
The updating formula is
where
14-46
Chapter 14
Exponential Smoothing
Smoothing Constant ()
•
•
•
•
•
The forecast Ft+1 is a weighted average of yt (the current data) and Ft (the
previous forecast).
The value of  (the smoothing constant) is the weight given to the latest
data.
A small value of  would give low weight to the most recent observation.
A large value of  would give heavy weight to the previous forecast.
The larger the value of , the more quickly the forecasts adapt to recent
data.
Choosing the Value of 
•
If  = 1, there is no smoothing at all and the forecast for the next
period is the same as the latest data point.
14-47
Chapter 14
Exponential Smoothing
Initializing the Process
•
•
•
•
•
Where do we get the initial forecast F1 (i.e., how do we initialize the
process)?
Method A
Use the first data value. Set F1 = y1
Although simple, if y1 is unusual, it could take a few iterations for the
forecasts to stabilize.
Method B
Average the first six data values. Set
F1 = (y1 + y2 + y3 + y4 + y5 + y6)/6
This method consumes more data and is still somewhat vulnerable to
unusual y-values.
14-48
Chapter 14
Exponential Smoothing
Effect of Past Data
•
•
Note that Ft  1 depends on Ft, which in turn depends on Ft 
1, and so on all the way back to F1.
The effect of past data diminishes as time increases.
•
To see this, replace Ft with Ft  1 and repeat this type of
substitution indefinitely to obtain
•
The coefficients diminish so older yt values have less effect on
the current forecast.
14-49
Chapter 14
Exponential Smoothing
Example from LearningStats:
14-50
Chapter 14
•
The Box-Jenkins method uses several different types of timeseries modeling techniques that fall into a class called ARIMA
(Autoregressive Integrated Moving Average) models
•
AR (autoregressive) models take advantage of the
dependency that may exist between values in the time
series.
•
MA (moving average) models take advantage of the
dependency that may exist between errors in the forecasts.
•
Although they are more powerful and general than trend
models, these methods require sophisticated software and
additional training. Excel and MegaStat do not offer these
methods.
14-51
Assignments
ML 3.5
• Project P-1 (data, tasks, questions)
•
•
•
•
Review instructions
Look at the data
Length is up to you
• Project P-2 (data, tasks, questions)
•
•
•
•
0-52
Review instructions
Look at the data
Length is up to you
General Instructions
For each team project, submit a short (5-10 page) report (using Microsoft Word
or equivalent) that answers the questions posed. Strive for effective writing (see
textbook Appendix I). Creativity and initiative will be rewarded. Avoid careless
spelling and grammar. Paste graphs and computer tables or output into your
written report (it may be easier to format tables in Excel and then use Paste
Special > Picture to avoid weird formatting and permit sizing within Word).
Allocate tasks among team members as you see fit, but all should review and
proofread the report (submit only one report).
0-53
Chapter 0
Projects: General Instructions
Project P-2
You will be assigned a company and team members (see Moodle). Delegate
tasks and collaborate as seems appropriate, based on your various skills. Submit
one report. Data: Download your company’s quarterly revenue data 2006-2012
from Moodle or from Doane’s teaching web page. Analysis: (a) Briefly describe
the company’s history, products, services, competition, and market conditions
(e.g., Yahoo or Mergent or Google). (b) Fit several trends (e.g., linear, quadratic,
exponential) using Excel. (c) Interpret each fitted trend equation. Discuss and
compare the R2 statistics. (d) Forecast the next 4 quarters (t = 29, 30, 31, 32)
based on trend alone using each fitted trend model (i.e., plug in the time index
for periods n+1, n+2, n+3, n+4). (e) Use MegaStat (or Minitab) to calculate
quarterly seasonal factors. Is there noticeable seasonality? (f) Ambitious
students: Multiply each quarterly trend forecast by its seasonal factor. Discuss
the effect of the seasonal adjustment. (g) Using the four criteria for assessing
forecasts (see p. 614), which trend model (if any) would yield credible
forecasts? If none, then what?
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Project P-2
A Few Examples
These are from Doane’s teaching web page (only the “cleaned” files were posted to
Moodle). You can look up your company’s stock ticker symbol on the internet (e.g., EMC)
Note: Use the “cleaned”: spreadsheets for your forecasting
project. The others are the “raw data” from Mergent Online.
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Project P-2
Note: Use only the “cleaned”: