CHAPTER13: FORECASTING

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CHAPTER13FORECASTING
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13.1 INTRODUCTION

• Typical business forecasting situations
• A company wishes to forecast the sales of its
  products
• Forecast the returns resulting to the company
  from the purchase of new equipment.
• A local authority forecasts the number of
  children for the next ten years
• The Treasury has a large economic model that
  allows the investigation of the likely effects on
  the economy if the Chancellor changes the income
  tax rate, or alters the interest rate.

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13.1.1 Approaches to Forecasting

• If the company has available the monthly sales
  figures for its products for the previous twelve
  months then this information can be used to make
  a forecast of sales for the next month.

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To forecast the sales for the next three time
points projecting the sales trend line.
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• projecting the sales trend line is not so simple
• Intuitively any forecast made from this data
  would be less reliable.

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• Time-series method
• Use historical data collected over time and use
  this data to project forward to make a forecast
• Other methods of forecasting
• For local authority example, to predict the
  number of couples within age bands, the birth
  rate for each age band hence the forecast number
  of children as required.
• The Treasury has a large econometric model that
  allows the investigation of the likely effects on
  the economy if the Chancellor changes the income
  tax rate, or alters the interest rate.

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13.1.2 Time-Series

• A time-series may be formally defined as
• A set of observations made on a particular
  variable at equidistant time intervals.
• Some examples of time-series
• The sales data used in the two examples above.
• The number of people recorded as unemployed at
  the end of each month.
• The daily closing price for a company shares
  quoted by London Stock Exchange
• The temperature of a hospital patient recorded on
  an hourly basis.
• Measure of the accuracy of the forecast

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13.1.3 Time-Series Graphs

• Time-series plot
• A visual inspection useful information about
  the nature of the time-series.
• well-defined trend
• seasonal structure.
• EXAMPLE 1
• well-defined trend having little variability
  about the trend.
• give relatively precise forecasts.
• forecasts for time points 13, 14 15
• measure of the forecast accuracy for different
  forecasting methods

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• EXAMPLE 2
• more problematical
• forecasts produced from time-series data less
  reliable.
• forecasts for time points 13, 14 15
• The measure of forecast accuracy in this
  situation would suggest the forecasts were not
  very reliable.
• Forecasting method
• calculating the forecast for each required time
  point
• calculating measure of forecast accuracy

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13.1.4 Exponential Smoothing Methods

• Methodology for exponential smoothing is based on
  intuitive ideas,
• a set of 'custom and practice methods' rather
  than having a well defined underlying theoretical
  structure.
• Exponential smoothing model
• simple exponential smoothing model
• model to deal with time-series that contain a
  trend
• Model to deal with time-series that contain both
  trend and seasonality.

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13.2 THE SIMPLE EXPONENTIAL SMOOTHING MODEL

• Exchange rate between Pound Sterling and German
  Mark
• To forecast the exchange rate for time periods
  12, 13 14

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• No well-defined trend or seasonal variation
• Using simple exponential smoothing model

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• This type of time-series data is described as a
  stationary time-series.
• For a stationary time-series the forecast for
  the next time point is the average value of the
  'time-series variable' over the length of the
  series.
• The estimate of the Exchange Rate at time point
  12
• Simple average
• Weighted average

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13.2.1 A Notation the simple EWMA
relationship

• A common abbreviated notation for this
  time-series
• Xt, t1,2,,n.
• The estimate of the level made on the basis of
  the previous t observations is labelled as Mt ,
  then a weighted average can be calculated as
  follows
• Mt atXt at-1Xt-1 at-2Xt-2 a1X1
• at at-1 at-2 a11

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• Simple average
• at at-1 at-2 a11/t
• weighted Average
• Heavier weighting is given to more current data
  points
• atgtat-1gt at-2 gtgta1
• at-j ?(1-?)j j1,2,3 0?1
• Mt ? Xt ?(1-?) Xt-1 ?(1-?)2Xt-2
  ?(1-?)3Xt-3

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• The series ?, ?(1-?), ?(1- ?)2, ?(1- ?)3, ?(1-
  ?)4 an exponential series (geometric series)
• Mt ? Xt ?(1-?) Xt-1 ?(1-?)2Xt-2
  ?(1-?)3Xt-3
• Mt ? Xt (1-?) ? Xt-1 ?(1-?)Xt-2
  ?(1-?)2Xt-3
• Mt ? Xt (1-?)Mt-1
• This is the basic exponential smoothing equation
• The estimate at time t a proportion of the new
  information one minus that proportion of the
  estimate at time t-1.

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13.2.2 Forecasting with the Simple model

• For Xt, t1,2,,n,
• Mt ? Xt (1-?)Mt-1
• Calculate M2 using t2
• Calculate M3 using t3
• Calculate M4 using t4
• Calculate Mn using tn
• The forecast of the value of Xn1 Mn
• Two problems for the process
• How to start the calculation
• How to choose a value for ?, the smoothing
  constant.

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13.2.3 How to start the calculations, A
Starting Rule

• Starting rule
• let M1 X1
• Let ? 0.25
• Calculating

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Forecasting is the last value of M, (M11 ) is
2.95, this is a weighted moving average based on
all the previous 11 time-series data
points. X12M112.95
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• The graph of the data and the Mt series is given
  in below

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• Define Ft(1) to mean the forecast made on the
  basis of the time-series data values, X1, X2,
  X3,... Xt of the next value of the time-series,
  Xt1
• For the simple exponential smoothing model the
  forecast function is Ft(1) Mt
• F11(1) M112.95
• Define Ft(2) to mean the forecast made on the
  basis of the time-series data values, X1, X2,
  X3,... Xt of the value of Xt2.
• Ft(2) is called the two step forecast.
• Ft(2) Mt
• F11(2) M112.95 (forecast value of Xt2)

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• Ft(h) means the h step forecast made on the
  basis of the previous t time-series points.
• Ft(h) Mt
• F11(3) M112.95 (forecast value of X113)
• F11(4) M112.95(forecast value of X114)

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13.2.4 Measuring Forecast Precision

• Ft(1) Mt
• F10(1) M10
• F9(1) M9
• F8(1) M8
• Common Measures of forecast precision
• Mean Absolute Deviation
• Mean Square error
• Mean Percentage Error

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• Mean Absolute Deviation
• MAD ? Et/n
• Exponentially weighted MAD
• MADt ? Et (1- ? ) MAD t-1
• Mean Square Error
• MSE? (Et)2/n
• Mean Percentage Error
• MPE? (Et /Xt)100/n
• Example
• MAD 0.030
• MSE 0.002
• MPE 1.02

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13.2.5 How to choose a value for ?, the
smoothing constant

• Mt ? Xt (1-?)Mt-1
• IF ?0, Mt Mt-1 Mt-1X1
• For a very small(near 0) value of ? we get very
  heavy smoothing, very little weight is given to
  the new data, and a heavy weighting given to the
  history of the series.
• IF ?1, Mt Xt
• For large values of ? (near 1) a high weighting
  is given to the current data and very little to
  the past history of the series.

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• Smoothing constant ? determines the level of
  smoothing.
• A small value of ? gives heavy smoothing
• A large value of ? gives less smoothing
• In practice the value of ? used to make a
  forecast represents a trade-off between these two
  extremes.

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• Guidance for smoothing parameter
• a)The value of ? should be in the range 0.05 to
  0.3 (suggestion by C D Lewis)
• Choose ? small if a plot of the series suggests
  a stable series.
• Choose ? large if a plot of the series suggests
  a more dynamic series.
• b)Choose the value of ? to minimise one of the
  measures of forecast precision.

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13.2.6 Building a Spreadsheet Model of the
EWMA

• Model The Conceptual Paper Worksheet

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• The choice of ?
• Using 'Table' command on Excel
• i. Estimate the value of ? that gives the
  smallest MSE.
• ii. Enter this estimated value of ? into cell Dl.
• iii.The forecasts for time period 12 13 14 are
  read from cells D15, D16 D17.
• Using Solver command on Excel
• ?0.667825
• Final spreadsheet model(?0.7) as following table

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• Mean Square Error by definition is the average
  squared error, as such is measured in squared
  units, this does not make for sensible
  interpretation. The Root Mean Square Error, RMSE,
  which is the square root of the MSE is in the
  correct units. For this data RMSE ?0.00149
  0.0386.
• 1 2.91 Marks, with a RMSE 0.0386, the
  implication being that the forecast error is
  likely to /- 0.04 Marks

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13.3 EXPONENTIAL SMOOTHING MODEL WITH TREND

• A product inventory level at the end of week over
  the last 25 weeks

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This time-series exhibits a definite upward
trend.
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13.3.1 The Exponential Smoothing Model with
Trend

• If assuming the trend is locally linear, at time
  t the level and rate of change of level, (the
  slope) is known,
• Xt1 Level(t) Slope(t) error
• Level(t) Mt
• Slope or Gradient at time t Rt Mt Mt-1
• The estimate at time t proportion of the new
  information one minus that proportion of the
  estimate at time t-1,

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• Estimate of the level at time t ? new
  information (1- ?)Estimate of the level based
  on time t-1 information
• Mt ? Xt (1- ?)(Mt-1 Rt-1)
• Rt ?(Mt-Mt-1) (1- ?)Rt-1
• one step forecast Mt Rt
• two-step forecast Mt 2Rt
• h-step forecast Ft(h) Mt hRt
• Using this model to forecast presents the
  following problems
• a) A starting rule is required, initial values
  for M1 and R1 need to be estimated.
• b)Values for the two smoothing parameters ? and ?
  need to be specified.

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13.3.2 The Starting Rule and smoothing
parameters

• a) The starting Rule. The simplest starting rule
  is to fit a straight line to the first few data
  points. This can be done by fitting a straight
  line by eye to the time-series graph and
  measuring the intercept and slope.

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• When t 1 the value of inventory is 146 (as
  estimated from the graph)
• When t 11 the value of inventory is 176 (as
  estimated from the graph)
• ?Y/ ? X (176-146)/(11-1)30
• M1146
• R130

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• Choice of smoothing parameter
• a)Choose the values of ? and ? according to
  advice offered by experienced users
• Woodward Goldsmith4, suggest values of ? 0.1
  and ? 0.01.
• b) Choose the values of ? and ? to minimise one
  of the measures of forecast precision.
• M1146, R13, ? 0.5 and ? 0.5, to give the
  following spreadsheet calculations

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13.3.3 Forecasting with the trend model

• The one step forecast, the forecast for time
  point 26 is263.37 12.92
• The two step forecast, the forecast for time
  point 27 is263.37 212.92
• the forecast function for the time point h steps
  ahead is Ft(h) Mt hRt
• at all time points
• F1(1) M1 R1
• F2(1) M2 2R2

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• The forecast for the time periods 26, 27 28
• F25(1) M25 R25 263.37 112.92 276.29
• F25(2) M25 2R25 263.37 212.92 289.22
• F25(3) M25 3R25 263.37 312.92 302.14

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13.3.4 A forecasting process

• a)Choose the values of ? and ? according to
  advice offered by experienced users
• Woodward Goldsmith3, suggest values of ? 0.1
  and ? 0.01.
• F25(I) M25 R25 238.72
• F25(2) M25 2R25 241.89
• F25(3) M25 3R25 245.06
• MSE.367.01.

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• b)Choose the values of ? and ? to minimise one of
  the measures of forecast precision.
• ? 0.07 and ? 0.99 to minimise MSE
• F25(l) M25 R25 259.24
• F25(2) M25 2R25 268.09
• F25(3) M25 3R25 276.93
• MSE367.01

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• Summary of the forecasts using differing
  smoothing parameters

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Conceptual Paper Worksheet
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13.4 Summary

• -  The exponential smoothing model with trend
  applies to time-series where the time-series plot
  shows a markedunderlying trend
• -  The basic recurrence relationships are
• Mt ? Xt (l- ?)(Mt-1 Rt-1)
• Rt ?(Mt-Mt-1) (l- ?)Rt-1
• - The forecast function is Ft(h) Mt hRt
  

• -  The simple exponential smoothing model applies
  to time-series where the time-series plot shows
  no evidenceof trend or seasonal factors.
• -  The basic recurrence relationship is
• Mt ? Xt (1- ?)Mt-1
• The forecast function isFt(h) Mt

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• - All exponential smoothing models require-
• a)   A starting Rule
• b)   A choice of smoothing parameter

a) For the simple model the simplest starting
rule is M1X1
a)For the trend model the simplest starting rule
is to fit a line to the first few data points,
and from this line estimate the value of Mt and
Rt
b) Choice of smoothing parameter i)
By experience ii) By minimising a
measure of precision.
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• Forecasting
• For h1,2,3,.
• Ft(h) Mt hRt

• Forecasting
• For h1,2,3,.
• Ft(h) Mt

• - The measures of forecast precision
• i) Mean Absolute Deviation
• MAD ? Et/n
• ii) Mean Square Error
• MSE? (Et)2/n
• iii) Mean Percentage Error
• MPE? (Et /Xt)100/n

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Group Work

• Collect the daily closing price for anyone
  company shares quoted by Shenzhen Stock Exchange
  or Shanghai Stock Exchange, and Use Exponential
  Smoothing Model(Simple or With Tread) to forecast
  the closing price of Next Day. Compare your
  forecasting closing price and the actual closing
  price of Next Day.
• Remarks
• 1) Work in the group ( Total 10 groups)
• 2)  Preparing PPT document and Excel spreadsheet
  model, and selecting 2-3 representatives of the
  group by yourself, and making the presentation of
  your results in the next class (Monday, 17 March,
  2008).
• 3) The presentation time for each group is 15
  minutes.

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