16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 STEREO.XLS n
Monthly sales for a chain of stereo retailers are listed in this
file. n They cover the period form the beginning of 1995 to the end
of 1998, during which there was no upward or downward trend in
sales and no clear seasonal peaks or valleys. n This behavior is
apparent in the time series chart of sales shown on the next slide.
It is possible that this series is random. n Does a runs test
support this conjecture?
Slide 3
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Time Series
Plot of Stereo Sales
Slide 4
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Random Model
n The simplest time series is the random model. n In a random model
the observations vary around a constant mean, have a common
variance, and are probabilistically independent of one another. n
How can we tell whether a time series is random? n There are
several checks that can be done individually or in tandem. n The
first of these is to plot the series on a control chart. If the
series is random it should be in control.
Slide 5
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Runs Test n
The runs test is the second check for a random series. n A run is a
consecutive sequence of 0s and 1s. n The runs test checks whether
this is about the right number of runs for a random series.
Slide 6
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Calculations
n To do a runs test in Excel we use StatPros Runs Test procedure. n
We must specify the time series variable (Sales) and the cutoff
value for the test, which can be the mean, median or a user
specified value. In this case we select the mean to obtain this
sample of output.
Slide 7
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Output n Note
that StatPro adds two new variables, Sales_High and Sales_NewRun,
as well as the elements for the test. n The values in the
Sales_High are 1 or 0 depending on whether the corresponding sales
value are above or below the mean. n The values in the Sales_NewRun
column are also 1 or 0, depending on whether a new run starts in
that month.
Slide 8
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Output --
continued n The rest of the output is fairly straightforward. n We
find the number of observations above the mean, number of runs,
mean for the observed number of runs, the standard deviation for
the observed number of runs and the Z-value. We then can find the
two- sided p-value. n The output shows that there is some evidence
of not enough runs. n The expected number of runs under randomness
is 24.8333 and there are only 20 runs for this series.
Slide 9
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Conclusion n
The conclusion is that sales do not tend to zigzag as much as a
random series - highs tend to follow highs and lows tend to follow
lows - but the evidence in favor of nonrandomness is not
overwhelming.
Slide 10
Random Series
Slide 11
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 The Problem n
The runs test on the stereo sales data suggests that the pattern of
sales is not completely random. n Large values tend to follow large
values, and small values tend to follow small values. n Do
autocorrelations support this conclusion?
Slide 12
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13
Autocorrelations n Recall that successive observations in a random
series are probabilistically independent of one another. n Many
time series violate this property and are instead autocorrelated. n
The auto means that successive observations are correlated with one
other. n To understand autocorrelations it is first necessary to
understand what it means to lag a time series.
Slide 13
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13
Autocorrelations n This concept is easy to understand in
spreadsheets. n To lag by 1 month, we simply push down the series
by one row. n Lags are simply previous observations, removed by a
certain number of periods from the present time.
Slide 14
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Solution n We
use StatPros Autocorrelation procedure. n This procedure requires
us to specify a time series variable (Sales), the number of lags we
want (we chose 6), and whether we want a chart of the
autocorrelations. This chart is called a correlogram. n How large
is a large autocorrelation? n If the series is truly random, then
only an occasional autocorrelation should be larger than two
standard errors in magnitude.
Slide 15
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Solution --
continued n Therefore, any autocorrelation that is larger than two
standard errors in magnitude is worth our attention. n The only
large autocorrelation for the sales data is the first, or lag 1,
the autocorrelation is 0.3492. n The fact that it is positive
indicates once again that there is some tendency for large sales
values to follow large sales values and for small sales values to
follow small sales values. n The autocorrelations are less than two
standard errors in magnitude and can be considered noise.
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 DEMAND.XLS n
The dollar demand for a certain class of parts at a local retail
store has been recorded for 82 consecutive days. n This file
contains the recorded data. n The store manager wants to forecast
future demands. n In particular, he wants to know whether there is
any significant time pattern to the historical demands or whether
the series is essentially random.
Slide 20
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Time Series
Plot of Demand for Parts
Slide 21
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Solution n A
visual inspection of the time series graph shows that demands vary
randomly around the sample mean of $247.54 (shown as the horizontal
centerline). n The variance appears to be constant through time,
and there are no obvious time series patterns. n To check formally
whether this apparent randomness holds, we perform the runs test
and calculate the first 10 autocorrelations. The numerical output
and associated correlogram are shown on the next slides.
Slide 22
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13
Autocorrelations and Runs Test for Demand Data
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Solution --
continued n The p-value for the run test is relatively large, 0.118
- although these are somewhat more runs than expected - and none of
the autocorrelations is significantly large. n These findings are
consistent with randomness. For all practical purposes there is no
time series pattern to these demand data. n The mean is $247.54 and
the standard deviation is $47.78.
Slide 25
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Solution --
continued n The manager might as well forecast that demand for any
day in the future will be $247.54. If he does so about 95% of his
forecast should be within two standard deviations (about $95) of
the actual demands.
Slide 26
The Random Walk Model
Slide 27
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 DOW.XLS n
Given the monthly Dow Jones data in this file, check that it
satisfies the assumptions of a random walk, and use the random walk
model to forecast the value for April 1992.
Slide 28
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Random Walk
Model n Random series are sometimes building blocks for other time
series models. n The random walk model is an example of this. n In
the random walk model the series itself is not random. However, its
differences - that is the changes from one period to the next - are
random. n This type of behavior is typical of stock price
data.
Slide 29
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Solution n
The Dow Jones series itself is not random, due to upward trend, so
we form the differences in Column C with the formula =B7-B6 which
is copied down column C. The difference can be seen on the next
slide. n A graph of the differences (see graph following data) show
the series to be a much more random series, varying around the mean
difference 26.00. n The runs test appears in column H and shows
that there is absolutely no evidence of nonrandom differences; the
observed number of runs is almost identical to the expected
number.
Slide 30
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Differences
for Dow Jones Data
Slide 31
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Time Series
Plot of Dow Differences
Slide 32
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Solution --
continued n Similarly, the autocorrelations are all small except
for a random blip at lag 11. n Because the values are 11 months
apart we would tend to ignore this autocorrelation. n Assuming the
random walk model is adequate, the forecast of April 1992 made in
March 1992 is the observed March value, 3247.42, plus the mean
difference, 26.00 or 3273.42. n A measure of the forecast accuracy
is the standard deviation of 84.65. We can be 95% certain that our
forecast will be within the standard deviations.
Slide 33
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Additional
Forecasting n If we wanted to forecast further into the future, say
3 months, based on the data through March 1992, we would add the
most recent value, 3247.42, to three times the mean difference,
26.00. n That is, we just project the trend that far into the
future. n We caution about forecasting too far into the future for
such a volatile series as the Dow.
Slide 34
Autoregressive Models
Slide 35
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 HAMMERS.XLS n
A retailer has recorded its weekly sales of hammers (units
purchased) for the past 42 weeks. n The data are found in the file.
n The graph of this time series appears below and reveals a
meandering behavior.
Slide 36
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 The Plot and
Data n The values begin high and stay high awhile, then get lower
and stay lower awhile, then get higher again. n This behavior could
be caused by any number of things. n How useful is autoregression
for modeling these data and how would it be used for
forecasting?
Slide 37
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13
Autocorrelations n A good place to start is with the
autocorrelations of the series. n These indicate whether the Sales
variable is linearly related to any of its lags. n The first six
autocorrelations are shown below.
Slide 38
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13
Autocorrelations -- continued n The first three of them are
significantly positive, and then they decrease. n Based on this
information, we create three lags of Sales and run a regression of
Sales versus these three lags. n Here is the output from this
regression
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13
Autocorrelations -- continued n We see that R 2 is fairly high,
about 57%, and that s e is about 15.7. n However, the p-values for
lags 2 and 3 are both quite large. n It appears that once the first
lag is included in the regression equation, the other two are not
really needed. n Therefore we reran the regression with only the
first lag include.
Slide 41
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13
Autoregression Output with a Single Lagged Variable
Slide 42
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Forecasts
from Aggression n This graph shows the original Sales variable and
its forecasts
Slide 43
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Regression
Equation n The estimated regression equation is Forecasted Sales t
= 13.763 + 0.793Sales t-1 n The associated R 2 and s e values are
approximately 65% and 155.4. The R 2 is a measure of the reasonably
good fit we see in the previous graph, whereas the s e is a measure
of the likely forecast error for short-term forecasts. n It implies
that a short-term forecast could easily be off by as much as two
standard errors, or about 31 hammers.
Slide 44
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Regression
Equation -- continued n To use the regression equation for
forecasting future sales values, we substitute known or forecasted
sales values in the right hand side of the equation. n
Specifically, the forecast for week 43, the first week after the
data period, is approximately 98.6 using the equation
ForecastedSales 43 = 13.763 + 0.793Sales 42 n The forecast for week
44 is approximately 92.0 and requires the forecasted value of sales
in week 43 in the equation: ForecastedSales 44 = 13.763 +
0.793ForecastedSales 43
Slide 45
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Forecasts n
Perhaps these two forecasts of future sales are on the mark and
perhaps they are not. n The only way to know for certain is to
observe future sales values. n However, it is interesting that in
spite of the upward movement in the series, the forecasts for weeks
43 and 44 are downward movements.
Slide 46
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Regression
Equation Properties n The downward trend is caused by a combination
of the two properties of the regression equation. n First, the
coefficient of Sales t-1, 0.793, is positive. Therefore the
equation forecasts that large sales will be followed by large sales
(that is, positive autocorrelation). n Second, however, this
coefficient is less than 1, and this provides a dampening effect. n
The equation forecasts that a large will follow a large, but not
that large.
Slide 47
Regression-Based Trend Models
Slide 48
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 REEBOK.XLS n
This file includes quarterly sales data for Reebok from first
quarter 1986 through second quarter 1996. n The following screen
shows the time series plot of these data. n Sales increase from
$174.52 million in the first quarter to $817.57 million in the
final quarter. n How well does a linear trend fit these data? n Are
the residuals from this fit random?
Slide 49
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Time Series
Plot of Reebok Sales
Slide 50
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Linear Trend
n A linear trend means that the time series variable changes by a
constant amount each time period. n The relevant equation is Y t =
a + bt + E t where a is the intercept, b is the slope and E t is an
error term. n If b is positive the trend is upward, if b is
negative then the trend is downward. n The graph of the time series
is a good place to start. It indicates whether a linear trend model
is likely to provide a good fit.
Slide 51
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Solution n
The plot indicates an obvious upward trend with little or no
curvature. n Therefore, a linear trend is certainly plausible. n We
use regression to estimate the linear fit, where Sales is the
response variable and Time is the single explanatory variable. n
The Time variable is coded 1-42 and is used as the explanatory
variable in the regression.
Slide 52
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Solution --
continued n The Quarter variable simply labels the quarters (Q1- 86
to Q2-96) and is used only to label the horizontal axis. n The
following regression output shows that the estimated equation is
Forecasted Sales = 244.82 + 16.53Time with R 2 and s e values of
83.8% and $90.38 million.
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Time Series
Plot with Linear Trend Superimposed n The linear trendline,
superimposed on the sales data, appears to be a decent fit.
Slide 55
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Solution --
continued n The trendline implies that sales are increasing by
about $16.53 million per quarter during this period. n The fit is
far from perfect, however. First, the s e value $90.38 million is
an indication of the typical forecast error. This is substantial,
approximately equal to 11% of the final quarters sales Furthermore,
there is some regularity to the forecast errors shown in the
following plot.
Slide 56
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Time Series
Plot of Forecasted Errors
Slide 57
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Plot
Interpretation n They zigzag more than a random series. n There is
probably some seasonal pattern in the sales data, which we might be
able to pick up with a more sophisticated forecasting method. n
However, the basic linear trend is sufficient as a first
approximation to the behavior of sales.
Slide 58
Regression-Based Trend Models
Slide 59
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 INTEL.XLS n
This file contains quarterly sales data for the chip manufacturing
firm Intel from the beginning of 1986 through the second quarter of
1996. n Each sales value is expressed in millions of dollars. n
Check that an exponential trend fits these sales data fairly well.
n Then estimate the relationship and interpret it.
Slide 60
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Time Series
Plot of Sales with Exponential Trend Superimposed
Slide 61
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 The Time
Series Plot of Sales n The time series plot shows that sales are
clearly increasing at an increasing rate, which a linear trend
would not capture. n The smooth curve of the plot is an exponential
trendline, which appears to be an adequate fit. n Alternatively, we
can try to straighten out the data by taking the log of sales with
Excels LN function. n The following is a plot of the log data.
Slide 62
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Time Series
Plot of Log Sales with Linear Trend Superimposed
Slide 63
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 The Time
Series Plot of Log Sales n This plot goes together logically with
the time series plot of Sales in the sense that if an exponential
trendline fits the original data well, then a linear trendline will
fit the transformed data well, and vice versa. n Either is evidence
of an exponential trend in the sales data.
Slide 64
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Estimating
the Exponential Trend n To estimate the exponential trend, we run a
regression of the log of sales, LnSales, versus Time. n A portion
of the resulting data and output appears below.
Slide 65
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Data Setup
for Regression of Exponential Trend
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Regression
Output n The regression output shows that the estimated log of
sales is given by Forecasted LnSales = 5.6883 + 0.0657Time n
Looking at the coefficient of Time, we can say that Intels sales
are increasing by approximately 6.6% per quarter during this
period. n This translates to an annual percentage increase of about
29%. Perhaps the slight tailing off that we see at the right
indicates that Intel cant keep up this fantastic rate forever.
Slide 68
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Regression
Output -- continued n It is important to view the R 2 and s e
values with caution. Each is based in log units not original units.
n To produce similar measures in original units, we need to
forecast sales in Column E. This is a two step process. First, we
forecast the log sales. Then we take the antilog with Excels EXP
function. The specific formula is =EXP($J$18+$J$19*A4).
Slide 69
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Regression
Output -- continued n As usual, R 2 is the square of the
correlation between actual and fitted sales values, so the formula
in cell J22 is =CORREL(Sales,FittedSales)2. n Then s e is the
square root of the sum of squared residuals divided by n-2. We can
calculate this in cell J23 by using Excels SUMSQ(sum of squares)
function: =SQRT(SUMSQ(ResidSAles)/40). n The R 2 value of 0.988
indicates that there is a very high correlation between the actual
and fitted sales values. In other words, the exponential fit is a
very good one.
Slide 70
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Regression
Output -- continued n However, the s e value if 159.698 (in
millions of dollars) indicates the forecasts based on this
exponential fit could still be fairly far off.
Slide 71
Moving Averages
Slide 72
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 DOW.XLS n We
again look at the Dow Jones monthly data from January 1988 through
March 1992 contained in this file. n How well do moving averages
track this series when the span is 23 months; when the span is 12
months? n What about future forecasts, that is, beyond March
1992?
Slide 73
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Moving
Averages n Perhaps the simplest and one of the most frequently used
extrapolation methods is the method of moving averages. n To
implement the moving averages method, we first choose a span, the
number of terms in each moving average. n The role of span is very
important. If the span is large - say 12 months - then many
observations go into each average, and extreme values have
relatively little effect on the forecasts.
Slide 74
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Moving
Averages -- continued n The resulting series forecasts will be much
smoother than the original series. n For this reason the moving
average method is called a smoothing method.
Slide 75
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Moving
Averages Method in Excel n Although the moving averages method is
quite easy to implement with Excel, it can be tedious. n Therefore
we can use the Forecasting procedure of StatPro. This procedure
lets us forecast with many methods. n Well go through the entire
procedure step by step.
Slide 76
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Forecasting
Procedure n To use the StatPro Forecasting procedure, the cursor
needs to be in a data set with time series data. n We use the
StatPro/Forecasting menu item and eventually choose Dow as the
variable to analyze. n We then see several dialog boxes, the first
of which is where we specify the timing.
Slide 77
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Timing Dialog
Box n In the next dialog box, we specify which forecasting method
to use and any parameters of that method.
Slide 78
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Method Dialog
Box n We next see a dialog box that allows us to request various
time series plots, and finally we get the usual choice of where to
report the output.
Slide 79
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 The Output n
The output consists of several parts. n First, the forecasts and
forecast errors are shown for the historical period of data. n
Actually, with moving averages we lose some forecasts at the
beginning of the period. n If we ask for future forecasts, they are
shown in red at the bottom of the data series. n There are no
forecast errors and to the left we see the summary measures.
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 The Output --
continued n The essence of the forecasting method is very simple
and is captured in column F of the output. It used the formula
=AVERAGE($E2:$E4) in cell F5, which is then copied down.
Slide 83
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 The Plots n
The plots show the behavior of the forecasts. n The forecasts with
span 3 appear to track the data better, whereas the forecasts with
span 12 is considerably smoother - it reacts less to ups and downs
of the series.
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 In Summary n
The summary measures MAE, RMSE, and MAPE confirm that moving
averages with span 3 forecast the known observations better. n For
example, the forecasts are off by about 3.6% with span 3, versus
7.7% with span 12. n Nevertheless, there is no guarantee that a
span of 3 is better for forecasting future observations.
Slide 87
Exponential Smoothing
Slide 88
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 EXXON.XLS n
This file contains data on quarterly sales (in millions of dollars)
for the period from 1986 through the second quarter of 1996. n The
following chart is the time series chart of these sales and shows
that there is some evidence of an upward trend in the early years,
but that there is no obvious trend during the 1990s. n Does a
simple exponential smoothing model track these data well? How do
the forecasts depend on the smoothing constant, alpha?
Slide 89
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Time Series
Plot of Exxon Sales
Slide 90
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 StatPros
Exponential Smoothing Model n We start by selecting the
StatPro/Forecasting menu item. n We first specify that the data are
quarterly, beginning in quarter 1 of 1986, we do not hold out any
of the data for validation, and we ask for 8 quarters of future
forecasts. n We then fill out the next dialog box like this:
Slide 91
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Method Dialog
Box n That is, we select the exponential smoothing option, elect
the Simple option choose smoothing constant (0.2 was chosen here)
and elect not to optimize, and specify that the data are not
seasonal.
Slide 92
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 StatPros
Exponential Smoothing Model -- continued n On the next dialog sheet
we ask for time series charts of the series with the forecasts
superimposed and the series of forecast errors. n The results
appear in the following three figures. n The heart of the method
takes place in the columns F, G, and H of the first figure. The
following formulas are used in row 6 of these columns.
=Alpha*E6+(1-Alpha)*F5 =F5 =E6-G6
Slide 93
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 StatPros
Exponential Smoothing Model -- continued n The one exception to
this scheme is in row 2. Every exponential smoothing method
requires initial values, in this case the initial smoothed level in
cell F2. There is no way to calculate this value because the
previous value is unknown. n Note that 8 future forecasts are all
equal to the last calculated smoothed level in cell F43. The fact
that these remain constant is a consequence of the assumption
behind simple exponential smoothing, namely, that the series is not
really going anywhere. Therefore, the last smoothed level is the
best indication of future values of the series we have.
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Forecast
Series & Error Charts n The next figure shows the forecast
series superimposed on the original series. n We see the obvious
smoothing effect of a relatively small alpha level. n The forecasts
dont track the series well; but if the zig zags are just random
noise, then we dont want the forecasts to track these random ups
and downs too closely. n A plot of the forecast errors shows some
quite large errors, yet the errors do appear to be fairly
random.
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Summary
Measures n We see several summary measures of the forecast errors.
n The RMSE and MAE indicate that the forecasts from this model are
typically off by a magnitude of about 2300, and the MAPE indicates
that this magnitude is about 7.4% of sales. n This is a fairly
sizable error. One way to try to reduce it is to use a different
smoothing constant.
Slide 99
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Summary
Measures -- continued n The optimal alpha level for this example is
somewhere between 0.8 and 0.9. This figure shows the forecast
series with alpha = 0.85.
Slide 100
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Summary
Measures -- continued n The forecast series now appears to tack the
original series very well - or does it? n A closer look shows that
we are essentially forecasting each quarters sales value by the
previous sales value. n There is not doubt that this gives lower
summary measures for the forecast errors, but it is possibly
reacting too quickly to random noise and might not really be
showing us the basic underlying patter of sales that we see with
alpha = 0.2.
Slide 101
Exponential Smoothing
Slide 102
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 DOW.XLS n We
return to the Dow Jones data found in this file. n Again, these are
average monthly closing prices from January 1988 through March
1992. n Recall that there is a definite upward trend in this
series. n In this example, we investigate whether simple
exponential smoothing can capture the upward trend. n The we see
whether Holts exponential smoothing method can make an
improvement.
Slide 103
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Solution n
This first graph shows how a simple exponential smoothing model
handles this trend, using alpha = 0.2. n The graphs summary error
messages are not bad (MAPE is 5.38%), but the forecasted series is
obviously lagging behind the original series. n Also, the forecasts
for the next 12 months are constant, because no trend is built into
the model. n In contrast, the following graph shows forecasts from
Holts model with alpha = beta = 0.2. The forecasts are still far
from perfect (MAPE is now 4.01%), but at least the upward trend has
been captured
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Plot of
Forecasts from Holts Model
Slide 106
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Holts Method
n The exponential smoothing method generally works well if there is
no obvious trend in the series. But if there is a trend, then this
method lags behind. n Holts model rectifies this by dealing with
trend explicitly. n Holts model includes a trend term and a
corresponding smoothing constant. This new smoothing constant
(beta) controls how quickly the method reacts to perceived changes
in the trend.
Slide 107
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Using Holts
Method n To produce the output from Holts method with StatPro we
proceed exactly as with the simple exponential procedure. The only
difference is that we now get to choose two smoothing parameters. n
The output is also very similar to simple exponential smoothing
output, except that there is now an extra column (column G) for the
estimated trend.
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Smoothing
Constants n It was mentioned that the smoothing constants used
above are not optimal. n If we use an StatPros optimize option to
find the best alpha for simple exponential smoothing or the best
alpha and beta for the Holts method. n In this case we find 1.0 and
0.0 for the smoothing constants. n Therefore, the best forecast for
next months value is the months value plus a constant trend.
Slide 110
Exponential Smoothing
Slide 111
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 COCACOLA.XLS
n The data in this spreadsheet represents quarterly sales for Coca
Cola from the first quarter of 1986 through the second quarter of
1996. n As we might expect there has been an upward trend in sales
during this period and there is also a fairly regular seasonal
pattern as shown in the time series plot of sales. n Sales in
warmer quarters, 2 and 3, are consistently higher than in the
colder quarters, 1 and 4. n How well can Winters method track this
upward tend and seasonal pattern?
Slide 112
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Time Series
Plot of Coca Cola Sales
Slide 113
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Seasonality n
Seasonality if defined as the consistent month-to- month (or
quarter-to-quarter) differences that occur each year. n The easiest
way to check if there is seasonality in a time series is to look at
a plot of the times series to see if it has a regular pattern of up
and/or downs in particular months or quarters. n There are
basically two extrapolation methods for dealing with seasonality:
We can use a model that takes seasonality into account or; We can
deseasonalize the data, forecast the data, and then adjust the
forecasts for seasonality.
Slide 114
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Seasonality
-- continued n Winters model is of the first type. It attacks
seasonality directly. n Seasonality models are usually classified
as additive or multiplicative. An additive model finds seasonal
indexes, one for each month, that we add to the monthly average to
get a particular months value. A multiplicative model also finds
seasonal indexes, but we multiply the monthly average by these
indexes to get a particular months value. n Either model can be
used but multiplicative models are somewhat easier to
interpret.
Slide 115
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Winters Model
of Seasonality n Winters model is very similar to Holts model - it
has level and trend terms and corresponding smoothing constants
alpha and beta - but it also has seasonal indexes and a
corresponding smoothing constant. n The new smoothing constant
controls how quickly the method reacts to perceived changes in the
pattern of seasonality. n If the constant is small, the method
reacts slowly; if the constant is large, it reacts more
quickly.
Slide 116
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Using Winters
Method n To produce the output from Winters method with StatPro we
proceed exactly as with the other exponential methods. n In
particular, we fill out the second main dialog box as shown
below.
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 The Output n
The optimal smoothing constants (those that minimize RMSE) are 1.0,
0.0 and 0.244. Intuitively, these mean react right away to changes
in level, never react to changes in trend, and react fairly slowly
to changes in the seasonal pattern. n If we ignore seasonality, the
series is trending upward at a rate of 67.107 per quarter. n The
seasonal pattern stays constant throughout this 10-year period. n
The forecast series tracks the actual series quite well.
Slide 119
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Plot of the
Forecasts from Winters Method n The plot indicates that Winters
method clearly picks up the seasonal pattern and the upward trend
and projects both of these into the future.
Slide 120
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 In Conclusion
n Some analysts would suggest using more typical values for the
constants such as alpha=beta=0.2 and 0.5 for the seasonality
constant. n To see how these smoothing constants would affect the
results, we can simply substitute their values into the range
B6:B8. n The summary measures get worse, yet the plot still
indicates a very good fit.
Slide 121
Deseasonalizing: The Ratio-to-Moving- Averages Method
Slide 122
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 COCACOLA.XLS
n We return to this data file that contains the sales history from
1986 to quarter 2 of 1996. n Is it possible to obtain the same
forecast accuracy with the ratio-to-moving-averages method as we
obtained with the Winters method?
Slide 123
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13
Ratio-to-Moving-Averages Method n There are many varieties of
sophisticated methods for deseasonalizing time series data but they
are all variations of the ratio-to-moving-averages method. n This
method is applicable when we believe that seasonality is
multiplicative. n The goal is to find the seasonal indexes, which
can then be used to deseasonalize the data. n The method is not
meant for hand calculations and is straightforward to implement
with StatPro.
Slide 124
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Solution n
The answer to the question posed earlier depends on which
forecasting method we use to forecast the deseasonalized data. n
The ratio-to-moving-averages method only provides a means for
deseasonalizing the data and providing seasonal indexes. Beyond
this, any method can be used to forecast the deseasonalized data,
and some methods work better than others. n For this example, we
will compare two methods: the moving averages method with a span of
4 quarters, and Holts exponential smoothing method optimized.
Slide 125
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Solution --
continued n Because the deseasonalized data still has a a clear
upward trend, we would expect Holts method to do well and we would
expect the moving averages forecasts to lag behind the trend. n
This is exactly what occurred. n To implement the latter method in
StatPro, we proceed exactly as before, but this time select Holts
method and be sure to check Use this deseasonalizing method. We get
a large selection of optional charts.
Slide 126
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16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13
Ration-to-Moving-Averages Output n This output shows the seasonal
indexes from the ratio-to-moving-averages method. They are
virtually identical to the indexes found using Winters method. n
Here are the summary measures for forecast errors.
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Forecast Plot
of Deseasonalized Series n Here we see only the smooth upward trend
with no seasonality, which Holts method is able to track very
well.
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Summary
Measures n The summary measures of forecast errors below are quite
comparable to those from Winters method. n The reason is that both
arrive at virtually the same seasonal pattern.
Slide 131
Estimating Seasonality with Regression
Slide 132
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 COCACOLA.XLS
n We return to this data file which contains the sales history of
Coca Cola from 1986 to quarter 2 of 1996. n Does a regression
approach provide forecasts that are as accurate as those provided
by the other seasonal methods in this chapter?
Slide 133
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Solution n We
illustrate a multiplicative approach, although an additive approach
is also possible. n The data setup is as follows:
Slide 134
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Solution n
Besides the Sales and Time variables, we need dummy variables for
three of the four quarters and a Log_Sales variable. n We then can
use multiple regression, with the Log_sales as the response
variable and Time, Q1, Q2, and Q3 as the explanatory variables. n
The regression output appears as follows:
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Interpreting
the Output n Of particular interest are the coefficients of the
explanatory variables. n Recall that for a log response variable,
these coefficients can be interpreted as percent changes in the
original sales variable. n Specifically, the coefficient of Time
means that deseasonalized sales increase by 2.4% per quarter. n
This pattern is quite comparable to the pattern of seasonal indexes
we saw in the last two examples.
Slide 137
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Forecast
Accuracy n To compare the forecast accuracy of this method with
earlier examples, we must go through several steps manually. The
multiple regression procedure in StatPRo provide fitted values and
residuals for the log of sales. We need to take these antilogs and
obtain forecasts of the original sales data, and subtract these
from the sales data to obtain forecast errors in Column K. We can
then use the formulas that were used in StatPros forecasting
procedure to obtain the summary measures MAE, RMSE, and MAPE.
16.216.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 |
16.10 | 16.11 | 16.12 |
16.1316.316.416.516.616.716.816.916.1016.1116.1216.13 Forecast
Accuracy -- continued n From the summary measures it appears that
the forecast are not quite as accurate. n However, looking at the
plot below of the forecasts superimposed on the original data shows
us that the method again tracks the data very well.