Time Series and Forecasting
Random Series
STEREO.XLS

Monthly sales for a chain of stereo retailers are listed
in this file.

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.

This behavior is apparent in the time series chart of
sales shown on the next slide. It is possible that this
series is random.

Does a runs test support this conjecture?
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Time Series Plot of Stereo Sales
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Random Model

The simplest time series is the random model.

In a random model the observations vary around a
constant mean, have a common variance, and are
probabilistically independent of one another.

How can we tell whether a time series is random?

There are several checks that can be done
individually or in tandem.

The first of these is to plot the series on a control
chart. If the series is random it should be “in control”.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Runs Test

The runs test is the second check for a random
series.

A run is a consecutive sequence of 0’s and 1’s.

The runs test checks whether this is about the right
number of runs for a random series.
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Calculations

To do a runs test in Excel we use StatPro’s Runs Test
procedure.

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.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Output

Note that StatPro adds two new variables,
Sales_High and Sales_NewRun, as well as the
elements for the test.

The values in the Sales_High are 1 or 0 depending
on whether the corresponding sales value are above
or below the mean.

The values in the Sales_NewRun column are also 1
or 0, depending on whether a new run starts in that
month.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Output -- continued

The rest of the output is fairly straightforward.

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 twosided p-value.

The output shows that there is some evidence of not
enough runs.

The expected number of runs under randomness is
24.8333 and there are only 20 runs for this series.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Conclusion

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.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Random Series
The Problem

The runs test on the stereo sales data suggests that
the pattern of sales is not completely random.

Large values tend to follow large values, and small
values tend to follow small values.

Do autocorrelations support this conclusion?
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Autocorrelations

Recall that successive observations in a random
series are probabilistically independent of one
another.

Many time series violate this property and are instead
autocorrelated.

The “auto” means that successive observations are
correlated with one other.

To understand autocorrelations it is first necessary to
understand what it means to lag a time series.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Autocorrelations

This concept is easy to understand in spreadsheets.

To lag by 1 month, we simply “push down” the series
by one row.

Lags are simply previous observations, removed by a
certain number of periods from the present time.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Solution

We use StatPro’s Autocorrelation procedure.

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.

How large is a “large” autocorrelation?

If the series is truly random, then only an occasional
autocorrelation should be larger than two standard
errors in magnitude.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Solution -- continued

Therefore, any autocorrelation that is larger than two
standard errors in magnitude is worth our attention.

The only “large” autocorrelation for the sales data is
the first, or lag 1, the autocorrelation is 0.3492.

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.

The autocorrelations are less than two standard
errors in magnitude and can be considered “noise”.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Lags and Autocorrelations for
Stereo Sales
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Correlogram for Stereo Sales
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Random Series
DEMAND.XLS

The dollar demand for a certain class of parts at a
local retail store has been recorded for 82
consecutive days.

This file contains the recorded data.

The store manager wants to forecast future
demands.

In particular, he wants to know whether there is any
significant time pattern to the historical demands or
whether the series is essentially random.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Time Series Plot of Demand for
Parts
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Solution

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).

The variance appears to be constant through time,
and there are no obvious time series patterns.

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.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Autocorrelations and Runs Test
for Demand Data
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Correlogram for Demand Data
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Solution -- continued

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.

These findings are consistent with randomness. For
all practical purposes there is no time series pattern
to these demand data.

The mean is $247.54 and the standard deviation is
$47.78.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Solution -- continued

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.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
The Random Walk Model
DOW.XLS

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.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Random Walk Model

Random series are sometimes building blocks for
other time series models.

The random walk model is an example of this.

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.

This type of behavior is typical of stock price data.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Solution

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.

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.

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.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Differences for Dow Jones Data
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Time Series Plot of Dow
Differences
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Solution -- continued

Similarly, the autocorrelations are all small except for
a random “blip” at lag 11.

Because the values are 11 months apart we would
tend to ignore this autocorrelation.

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.

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.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Additional Forecasting

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.

That is, we just project the trend that far into the
future.

We caution about forecasting too far into the future
for such a volatile series as the Dow.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Autoregressive Models
HAMMERS.XLS

A retailer has recorded its weekly sales of hammers
(units purchased) for the past 42 weeks.

The data are found in the file.

The graph of this time series appears below and
reveals a “meandering” behavior.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
The Plot and Data

The values begin high and stay high awhile, then get
lower and stay lower awhile, then get higher again.

This behavior could be caused by any number of
things.

How useful is autoregression for modeling these data
and how would it be used for forecasting?
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Autocorrelations

A good place to start is with the autocorrelations of
the series.

These indicate whether the Sales variable is linearly
related to any of its lags.

The first six autocorrelations are shown below.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Autocorrelations -- continued

The first three of them are significantly positive, and
then they decrease.

Based on this information, we create three lags of
Sales and run a regression of Sales versus these
three lags.

Here is the output from this regression
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Autoregression Output with
Three Lagged Variables
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Autocorrelations -- continued

We see that R2 is fairly high, about 57%, and that se
is about 15.7.

However, the p-values for lags 2 and 3 are both quite
large.

It appears that once the first lag is included in the
regression equation, the other two are not really
needed.

Therefore we reran the regression with only the first
lag include.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Autoregression Output with a
Single Lagged Variable
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Forecasts from Aggression

This graph shows the original Sales variable and its
forecasts
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Regression Equation

The estimated regression equation is
Forecasted Salest = 13.763 + 0.793Salest-1

The associated R2 and se values are approximately
65% and 155.4. The R2 is a measure of the
reasonably good fit we see in the previous graph,
whereas the se is a measure of the likely forecast
error for short-term forecasts.

It implies that a short-term forecast could easily be off
by as much as two standard errors, or about 31
hammers.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Regression Equation -- continued

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.

Specifically, the forecast for week 43, the first week
after the data period, is approximately 98.6 using the
equation
ForecastedSales43 = 13.763 + 0.793Sales42

The forecast for week 44 is approximately 92.0 and
requires the forecasted value of sales in week 43 in
the equation:
ForecastedSales44 = 13.763 + 0.793ForecastedSales43
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Forecasts

Perhaps these two forecasts of future sales are on
the mark and perhaps they are not.

The only way to know for certain is to observe future
sales values.

However, it is interesting that in spite of the upward
movement in the series, the forecasts for weeks 43
and 44 are downward movements.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Regression Equation Properties

The downward trend is caused by a combination of
the two properties of the regression equation.

First, the coefficient of Salest-1, 0.793, is positive.
Therefore the equation forecasts that large sales will
be followed by large sales (that is, positive
autocorrelation).

Second, however, this coefficient is less than 1, and
this provides a dampening effect.

The equation forecasts that a large will follow a large,
but not that large.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Regression-Based Trend
Models
REEBOK.XLS

This file includes quarterly sales data for Reebok
from first quarter 1986 through second quarter 1996.

The following screen shows the time series plot of
these data.

Sales increase from $174.52 million in the first
quarter to $817.57 million in the final quarter.

How well does a linear trend fit these data?

Are the residuals from this fit random?
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Time Series Plot of Reebok Sales
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Linear Trend

A linear trend means that the time series variable
changes by a constant amount each time period.

The relevant equation is Yt = a + bt + Et where a is
the intercept, b is the slope and Et is an error term.

If b is positive the trend is upward, if b is negative
then the trend is downward.

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.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Solution

The plot indicates an obvious upward trend with little
or no curvature.

Therefore, a linear trend is certainly plausible.

We use regression to estimate the linear fit, where
Sales is the response variable and Time is the single
explanatory variable.

The Time variable is coded 1-42 and is used as the
explanatory variable in the regression.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Solution -- continued

The Quarter variable simply labels the quarters (Q186 to Q2-96) and is used only to label the horizontal
axis.

The following regression output shows that the
estimated equation is Forecasted Sales = 244.82 +
16.53Time with R2 and se values of 83.8% and
$90.38 million.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Regression Output for Linear
Trend
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Time Series Plot with Linear
Trend Superimposed

The linear trendline, superimposed on the sales data,
appears to be a decent fit.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Solution -- continued

The trendline implies that sales are increasing by
about $16.53 million per quarter during this period.

The fit is far from perfect, however.
– First, the se value $90.38 million is an indication of the typical
forecast error. This is substantial, approximately equal to
11% of the final quarter’s sales
– Furthermore, there is some regularity to the forecast errors
shown in the following plot.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Time Series Plot of Forecasted
Errors
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Plot Interpretation

They zigzag more than a random series.

There is probably some seasonal pattern in the sales
data, which we might be able to pick up with a more
sophisticated forecasting method.

However, the basic linear trend is sufficient as a first
approximation to the behavior of sales.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Regression-Based Trend
Models
INTEL.XLS

This file contains quarterly sales data for the chip
manufacturing firm Intel from the beginning of 1986
through the second quarter of 1996.

Each sales value is expressed in millions of dollars.

Check that an exponential trend fits these sales data
fairly well.

Then estimate the relationship and interpret it.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Time Series Plot of Sales with
Exponential Trend Superimposed
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
The Time Series Plot of Sales

The time series plot shows that sales are clearly
increasing at an increasing rate, which a linear trend
would not capture.

The smooth curve of the plot is an exponential
trendline, which appears to be an adequate fit.

Alternatively, we can try to “straighten out” the data
by taking the log of sales with Excel’s LN function.

The following is a plot of the log data.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Time Series Plot of Log Sales
with Linear Trend Superimposed
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
The Time Series Plot of Log
Sales

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.

Either is evidence of an exponential trend in the sales
data.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Estimating the Exponential Trend

To estimate the exponential trend, we run a
regression of the log of sales, LnSales, versus Time.

A portion of the resulting data and output appears
below.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Data Setup for Regression of
Exponential Trend
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Regression Output for
Exponential Trend
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Regression Output

The regression output shows that the estimated log
of sales is given by
Forecasted LnSales = 5.6883 + 0.0657Time

Looking at the coefficient of Time, we can say that
Intel’s sales are increasing by approximately 6.6%
per quarter during this period.

This translates to an annual percentage increase of
about 29%. Perhaps the slight tailing off that we see
at the right indicates that Intel can’t keep up this
fantastic rate forever.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Regression Output -- continued

It is important to view the R2 and se values with
caution. Each is based in log units not original units.

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 Excel’s EXP function. The
specific formula is =EXP($J$18+$J$19*A4).
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Regression Output -- continued

As usual, R2 is the square of the correlation between
actual and fitted sales values, so the formula in cell
J22 is =CORREL(Sales,FittedSales)ˆ2.

Then se is the square root of the sum of squared
residuals divided by n-2. We can calculate this in cell
J23 by using Excel’s SUMSQ(sum of squares)
function: =SQRT(SUMSQ(ResidSAles)/40).

The R2 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.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Regression Output -- continued

However, the se value if 159.698 (in millions of
dollars) indicates the forecasts based on this
exponential fit could still be fairly far off.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Moving Averages
DOW.XLS

We again look at the Dow Jones monthly data from
January 1988 through March 1992 contained in this
file.

How well do moving averages track this series when
the span is 23 months; when the span is 12 months?

What about future forecasts, that is, beyond March
1992?
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Moving Averages

Perhaps the simplest and one of the most frequently
used extrapolation methods is the method of moving
averages.

To implement the moving averages method, we first
choose a span, the number of terms in each moving
average.

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.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Moving Averages -- continued

The resulting series forecasts will be much smoother
than the original series.

For this reason the moving average method is called
a smoothing method.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Moving Averages Method in
Excel

Although the moving averages method is quite easy
to implement with Excel, it can be tedious.

Therefore we can use the Forecasting procedure of
StatPro. This procedure lets us forecast with many
methods.

We’ll go through the entire procedure step by step.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Forecasting Procedure

To use the StatPro Forecasting procedure, the cursor
needs to be in a data set with time series data.

We use the StatPro/Forecasting menu item and
eventually choose Dow as the variable to analyze.

We then see several dialog boxes, the first of which
is where we specify the timing.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Timing Dialog Box

In the next dialog box, we specify which forecasting
method to use and any parameters of that method.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Method Dialog Box

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 .
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
The Output

The output consists of several parts.

First, the forecasts and forecast errors are shown for
the historical period of data.

Actually, with moving averages we lose some
forecasts at the beginning of the period.

If we ask for future forecasts, they are shown in red
at the bottom of the data series.

There are no forecast errors and to the left we see
the summary measures.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Moving Averages with Output
Span 3
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Moving Averages with Output
Span 12
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
The Output -- continued

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.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
The Plots

The plots show the behavior of the forecasts.

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.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Moving Averages Forecasts with
Span 3
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Moving Averages with Forecasts
Span 12
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
In Summary

The summary measures MAE, RMSE, and MAPE
confirm that moving averages with span 3 forecast
the known observations better.

For example, the forecasts are off by about 3.6% with
span 3, versus 7.7% with span 12.

Nevertheless, there is no guarantee that a span of 3
is better for forecasting future observations.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Exponential Smoothing
EXXON.XLS

This file contains data on quarterly sales (in millions
of dollars) for the period from 1986 through the
second quarter of 1996.

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.

Does a simple exponential smoothing model track
these data well? How do the forecasts depend on the
smoothing constant, alpha?
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Time Series Plot of Exxon Sales
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
StatPro’s Exponential Smoothing
Model

We start by selecting the StatPro/Forecasting menu
item.

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.

We then fill out the next dialog box like this:
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Method Dialog Box

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.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
StatPro’s Exponential Smoothing
Model -- continued

On the next dialog sheet we ask for time series
charts of the series with the forecasts superimposed
and the series of forecast errors.

The results appear in the following three figures.

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
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
StatPro’s Exponential Smoothing
Model -- continued

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.

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.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Simple Exponential Smoothing
Output
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Forecast Series & Error Charts

The next figure shows the forecast series
superimposed on the original series.

We see the obvious smoothing effect of a relatively
small alpha level.

The forecasts don’t track the series well; but if the zig
zags are just random noise, then we don’t want the
forecasts to track these random ups and downs too
closely.

A plot of the forecast errors shows some quite large
errors, yet the errors do appear to be fairly random.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Plot of Forecasts from Simple
Exponential Smoothing
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Plot of Forecast Errors from
Simple Exponential Smoothing
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Summary Measures

We see several summary measures of the forecast
errors.

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.

This is a fairly sizable error. One way to try to reduce
it is to use a different smoothing constant.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Summary Measures -- continued

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.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Summary Measures -- continued

The forecast series now appears to tack the original
series very well - or does it?

A closer look shows that we are essentially
forecasting each quarter’s sales value by the
previous sales value.

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.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Exponential Smoothing
DOW.XLS

We return to the Dow Jones data found in this file.

Again, these are average monthly closing prices from
January 1988 through March 1992.

Recall that there is a definite upward trend in this
series.

In this example, we investigate whether simple
exponential smoothing can capture the upward trend.

The we see whether Holt’s exponential smoothing
method can make an improvement.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Solution

This first graph shows how a simple exponential
smoothing model handles this trend, using alpha =
0.2.

The graph’s summary error messages are not bad
(MAPE is 5.38%), but the forecasted series is
obviously lagging behind the original series.

Also, the forecasts for the next 12 months are
constant, because no trend is built into the model.

In contrast, the following graph shows forecasts from
Holt’s 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
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Plot of Forecasts from Simple
Exponential Smoothing
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Plot of Forecasts from Holt’s
Model
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Holt’s Method

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.

Holt’s model rectifies this by dealing with trend
explicitly.

Holt’s 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.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Using Holt’s Method

To produce the output from Holt’s 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.

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.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Portion of Output from Holt’s
Method
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Smoothing Constants

It was mentioned that the smoothing constants used
above are not optimal.

If we use an StatPro’s optimize option to find the best
alpha for simple exponential smoothing or the best
alpha and beta for the Holt’s method.

In this case we find 1.0 and 0.0 for the smoothing
constants.

Therefore, the best forecast for next month’s value is
the month’s value plus a constant trend.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Exponential Smoothing
COCACOLA.XLS

The data in this spreadsheet represents quarterly
sales for Coca Cola from the first quarter of 1986
through the second quarter of 1996.

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.

Sales in warmer quarters, 2 and 3, are consistently
higher than in the colder quarters, 1 and 4.

How well can Winter’s method track this upward tend
and seasonal pattern?
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Time Series Plot of Coca Cola
Sales
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Seasonality

Seasonality if defined as the consistent month-tomonth (or quarter-to-quarter) differences that occur
each year.

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.

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.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Seasonality -- continued

Winters’ model is of the first type. It attacks
seasonality directly.

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
month’s value.
– A multiplicative model also finds seasonal indexes, but we
multiply the monthly average by these indexes to get a
particular month’s value.

Either model can be used but multiplicative models
are somewhat easier to interpret.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Winter’s Model of Seasonality

Winters’ model is very similar to Holt’s 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.

The new smoothing constant controls how quickly the
method reacts to perceived changes in the pattern of
seasonality.

If the constant is small, the method reacts slowly; if
the constant is large, it reacts more quickly.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Using Winters’ Method

To produce the output from Winters’ method with
StatPro we proceed exactly as with the other
exponential methods.

In particular, we fill out the second main dialog box as
shown below.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Portion of Output from Winters’
Method
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
The Output

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.

If we ignore seasonality, the series is trending upward
at a rate of 67.107 per quarter.

The seasonal pattern stays constant throughout this
10-year period.

The forecast series tracks the actual series quite well.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Plot of the Forecasts from
Winters’ Method

The plot indicates that Winters’ method clearly picks
up the seasonal pattern and the upward trend and
projects both of these into the future.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
In Conclusion

Some analysts would suggest using more “typical”
values for the constants such as alpha=beta=0.2 and
0.5 for the seasonality constant.

To see how these smoothing constants would affect
the results, we can simply substitute their values into
the range B6:B8.

The summary measures get worse, yet the plot still
indicates a very good fit.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Deseasonalizing:
The Ratio-to-Moving-Averages
Method
COCACOLA.XLS

We return to this data file that contains the sales
history from 1986 to quarter 2 of 1996.

Is it possible to obtain the same forecast accuracy
with the ratio-to-moving-averages method as we
obtained with the Winters’ method?
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Ratio-to-Moving-Averages
Method

There are many varieties of sophisticated methods
for deseasonalizing time series data but they are all
variations of the ratio-to-moving-averages method.

This method is applicable when we believe that
seasonality is multiplicative.

The goal is to find the seasonal indexes, which can
then be used to deseasonalize the data.

The method is not meant for hand calculations and is
straightforward to implement with StatPro.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Solution

The answer to the question posed earlier depends on
which forecasting method we use to forecast the
deseasonalized data.

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.

For this example, we will compare two methods: the
moving averages method with a span of 4 quarters,
and Holt’s exponential smoothing method optimized.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Solution -- continued

Because the deseasonalized data still has a a clear
upward trend, we would expect Holt’s method to do
well and we would expect the moving averages
forecasts to lag behind the trend.

This is exactly what occurred.

To implement the latter method in StatPro, we
proceed exactly as before, but this time select Holt’s
method and be sure to check “Use this
deseasonalizing method”. We get a large selection of
optional charts.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Ration-to-Moving-Averages
Output

Here are the summary measures for forecast errors.

This output shows the seasonal indexes from the
ratio-to-moving-averages method. They are virtually
identical to the indexes found using Winters’ method.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Ratio-to-Moving Averages
Output
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Forecast Plot of Deseasonalized
Series

Here we see only the smooth upward trend with no
seasonality, which Holt’s method is able to track very
well.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
The Results of Reseasonalizing
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Summary Measures

The summary measures of forecast errors below are
quite comparable to those from Winters’ method.

The reason is that both arrive at virtually the same
seasonal pattern.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Estimating Seasonality with
Regression
COCACOLA.XLS

We return to this data file which contains the sales
history of Coca Cola from 1986 to quarter 2 of 1996.

Does a regression approach provide forecasts that
are as accurate as those provided by the other
seasonal methods in this chapter?
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Solution

We illustrate a multiplicative approach, although an
additive approach is also possible.

The data setup is as follows:
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Solution

Besides the Sales and Time variables, we need
dummy variables for three of the four quarters and a
Log_Sales variable.

We then can use multiple regression, with the
Log_sales as the response variable and Time, Q1,
Q2, and Q3 as the explanatory variables.

The regression output appears as follows:
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Regression Output
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Interpreting the Output

Of particular interest are the coefficients of the
explanatory variables.

Recall that for a log response variable, these
coefficients can be interpreted as percent changes in
the original sales variable.

Specifically, the coefficient of Time means that
deseasonalized sales increase by 2.4% per quarter.

This pattern is quite comparable to the pattern of
seasonal indexes we saw in the last two examples.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Forecast Accuracy

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 StatPro’s
forecasting procedure to obtain the summary measures
MAE, RMSE, and MAPE.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Forecast Errors and Summary
Measures
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13
Forecast Accuracy -- continued

From the summary measures it appears that the
forecast are not quite as accurate.

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.
16.2 | 16.3 | 16.4 | 16.5 | 16.6 | 16.7 | 16.8 | 16.9 | 16.10 | 16.11 | 16.12 | 16.13