Random Series

1
Time Series and Forecasting

• Random Series

2
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?

3
Time Series Plot of Stereo Sales
4
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.

5
Runs Test

• The runs test is the second check for a random
  series.
• A run is a consecutive sequence of 0s and 1s.
• The runs test checks whether this is about the
  right number of runs for a random series.

6
Calculations

• To do a runs test in Excel we use StatPros 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.

7
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.

8
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 two-sided 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.

9
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.

10

• Random Series

11
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?

12
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.

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.

14
Solution

• We use StatPros 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.

15
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
Lags and Autocorrelations for Stereo Sales
17
Correlogram for Stereo Sales
18

• Random Series

19
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.

20
Time Series Plot of Demand for Parts
21
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.

22
Autocorrelations and Runs Test for Demand Data
23
Correlogram for Demand Data
24
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.

25
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.

26

• The Random Walk Model

27
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.

28
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.

29
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.

30
Differences for Dow Jones Data
31
Time Series Plot of Dow Differences
32
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.

33
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.

34

• Autoregressive Models

35
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.

36
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?

37
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.

38
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

39
Autoregression Output with Three Lagged Variables
40
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.

41
Autoregression Output with a Single Lagged
Variable
42
Forecasts from Aggression

• This graph shows the original Sales variable and
  its forecasts

43
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.

44
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

45
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.

46
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.

47

• Regression-Based Trend Models

48
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?

49
Time Series Plot of Reebok Sales
50
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.

51
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.

52
Solution -- continued

• The Quarter variable simply labels the quarters
  (Q1-86 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.

53
Regression Output for Linear Trend
54
Time Series Plot with Linear Trend Superimposed

• The linear trendline, superimposed on the sales
  data, appears to be a decent fit.

55
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 quarters sales
• Furthermore, there is some regularity to the
  forecast errors shown in the following plot.

56
Time Series Plot of Forecasted Errors
57
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.

58

• Regression-Based Trend Models

59
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.

60
Time Series Plot of Sales with Exponential Trend
Superimposed
61
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 Excels LN
  function.
• The following is a plot of the log data.

62
Time Series Plot of Log Sales with Linear Trend
Superimposed
63
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.

64
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.

65
Data Setup for Regression of Exponential Trend
66
Regression Output for Exponential Trend
67
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 Intels 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 cant
  keep up this fantastic rate forever.

68
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 Excels EXP
  function. The specific formula is
  EXP(J18J19A4).

69
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
  )2.
• 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 Excels 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.

70
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.

71

• Moving Averages

72
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?

73
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.

74
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.

75
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.
• Well go through the entire procedure step by
  step.

76
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.

77
Timing Dialog Box

• In the next dialog box, we specify which
  forecasting method to use and any parameters of
  that method.

78
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 .

79
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.

80
Moving Averages with Output Span 3
81
Moving Averages with Output Span 12
82
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(E2E4) in cell F5,
  which is then copied down.

83
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.

84
Moving Averages Forecasts with Span 3
85
Moving Averages with Forecasts Span 12
86
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.

87

• Exponential Smoothing

88
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?

89
Time Series Plot of Exxon Sales
90
StatPros 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

91
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.

92
StatPros 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. AlphaE6(1-Alpha)F5
  F5 E6-G6

93
StatPros 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.

94
Simple Exponential Smoothing Output
95
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 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.
• A plot of the forecast errors shows some quite
  large errors, yet the errors do appear to be
  fairly random.

96
Plot of Forecasts from Simple Exponential
Smoothing
97
Plot of Forecast Errors from Simple Exponential
Smoothing
98
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.

99
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.

100
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 quarters 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.

101

• Exponential Smoothing

102
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 Holts exponential smoothing
  method can make an improvement.

103
Solution

• This first graph shows how a simple exponential
  smoothing model handles this trend, using alpha
  0.2.
• The graphs 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 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

104
Plot of Forecasts from Simple Exponential
Smoothing
105
Plot of Forecasts from Holts Model
106
Holts 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.
• Holts model rectifies this by dealing with trend
  explicitly.
• 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.

107
Using Holts Method

• 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.
• 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.

108
Portion of Output from Holts Method
109
Smoothing Constants

• It was mentioned that the smoothing constants
  used above are not optimal.
• 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.
• In this case we find 1.0 and 0.0 for the
  smoothing constants.
• Therefore, the best forecast for next months
  value is the months value plus a constant trend.

110

• Exponential Smoothing

111
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 Winters method track this upward
  tend and seasonal pattern?

112
Time Series Plot of Coca Cola Sales
113
Seasonality

• Seasonality if defined as the consistent
  month-to-month (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.

114
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 months value.
• A multiplicative model also finds seasonal
  indexes, but we multiply the monthly average by
  these indexes to get a particular months value.
• Either model can be used but multiplicative
  models are somewhat easier to interpret.

115
Winters Model of Seasonality

• 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.
• 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.

116
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.

117
Portion of Output from Winters Method
118
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.

119
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.

120
In Conclusion

• Some analysts would suggest using more typical
  values for the constants such as alphabeta0.2
  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 B6B8.
• The summary measures get worse, yet the plot
  still indicates a very good fit.

121

• Deseasonalizing The Ratio-to-Moving-Averages
  Method

122
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?

123
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.

124
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 Holts exponential smoothing method
  optimized.

125
Solution -- continued

• 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.
• This is exactly what occurred.
• 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.

126
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.

127
Ratio-to-Moving Averages Output
128
Forecast Plot of Deseasonalized Series

• Here we see only the smooth upward trend with no
  seasonality, which Holts method is able to track
  very well.

129
The Results of Reseasonalizing
130
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.

131

• Estimating Seasonality with Regression

132
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?

133
Solution

• We illustrate a multiplicative approach, although
  an additive approach is also possible.
• The data setup is as follows

134
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

135
Regression Output
136
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.

137
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
  StatPros forecasting procedure to obtain the
  summary measures MAE, RMSE, and MAPE.

138
Forecast Errors and Summary Measures
139
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.