CHAPTER13:FORECASTING

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.

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.

To forecast the sales for the next three time points: projecting the sales trend line.

• projecting the sales trend line is not so simple

• Intuitively any forecast made from this data would be less reliable.

• 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.

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

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

• 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

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.

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

Quarter 1 2 3 4 5 6 7 8 9 10 11

Ex.Rate 2.95 2.97 2.94 2.96 3.01 3.02 2.98 2.96 2.94 2.97 2.88

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

– 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

13.2.1 A Notation & the simple EWMA relationship

• A common abbreviated notation for this time-series:– Xt, t=1,2,…,n.

• The estimate of the level made on the basis of the previous t observations is labelled as Mt , then a weighted average ca

n be calculated as follows:Mt = atXt + at-1Xt-1 + at-2Xt-2 +…+ a1X1

at+ at-1 + at-2 +…+ a1=1

• Simple average– at= at-1 = at-2 =…= a1=1/t

• weighted Average– Heavier weighting is given to more current

data points

at>at-1> at-2 >…>a1

at-j= (1-)j j=1,2,3… 0≤≤1

Mt = Xt + (1-) Xt-1 + (1-)2Xt-2 + (1-)3X

t-3 +…

– 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 informati

on +one minus that proportion of the estimate at time t-1.

13.2.2 Forecasting with the

Simple model • For Xt, t=1,2,…,n,

– Mt = Xt + (1-)Mt-1

– Calculate M2 using t=2

– Calculate M3 using t=3

– Calculate M4 using t=4

– Calculate Mn using t=n

– The forecast of the value of Xn+1= Mn

• Two problems for the process – How to start the calculation – How to choose a value for , the smoothing constant.

13.2.3 How to start the

calculations, A Starting Rule • Starting rule

– let M1 = X1

– Let =0.25

• Calculating

Mt

2.95

2.96

2.95

2.95

2.97

2.98

2.98

2.98

2.97

2.97

2.95

t   Calculation

1 2.95 M1=X1

2 2.97 M2 = 0.25X2 + 0.75M1

3 2.94 M3 = 0.25X3 + 0.75M2

4 2.96 M4 = 0.25X4 + 0.75M3

5 3.01 M5 = 0.25X5 + 0.75M4

6 3.02 M6 = 0.25X6 + 0.75M5

7 2.98 M7 = 0.25X7 + 0.75M6

8 2.96 M8 = 0.25X8 + 0.75M7

9 2.94 M9 = 0.25X9 + 0.75M8

10 2.97 M10= 0.25X10+ 0.75 M9

11 2.88 M11=0.25X11+0.75M10

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.

X12=M11=2.95

• The graph of the data and the Mt series is given in below:

• Define Ft(1) to mean the forecast made on the b

asis of the time-series data values, X1, X2, X3,... X

t of the next value of the time-series, Xt+1

– For the simple exponential smoothing model the forecast function is Ft(1) = Mt

– F11(1) = M11=2.95

• Define Ft(2) to mean the forecast made on the ba

sis of the time-series data values, X1, X2, X3,... Xt

of the value of Xt+2.

– Ft(2) is called the two step forecast.

– Ft(2) = Mt

– F11(2) = M11=2.95 (forecast value of Xt+2)

• Ft(h) means the h step forecast made on the basis of the previous t time-series points. – Ft(h) = Mt

– F11(3) = M11=2.95 (forecast value of X11+3)

– F11(4) = M11=2.95(forecast value of X11+4)

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

• 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%

13.2.5 How to choose a value for ,

the smoothing constant: • Mt = Xt + (1-)Mt-1

– IF =0, Mt = Mt-1 =Mt-1=…=X1

• 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.

• 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.

• Guidance for smoothing parameter– a)The value of should be in the range 0.0

5 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.

13.2.6 Building a Spreadsheet Model of

the EWMA • Model The Conceptual Paper Worksheet:

• 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

– 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

13.3 EXPONENTIAL SMOOTHING

MODEL WITH TREND • A product inventory level at the end of

week over the last 25 weeks: 1 2 3 4 5 6 7 8 9 10 11 12 13

140 159 136 157 173 131 177 188 154 179 180 160 182

14 15 16 17 18 19 20 21 22 23 24 25  

192 224 188 198 206 203 238 228 231 221 259 273  

This time-series exhibits a definite upward trend.

13.3.1 The Exponential Smoothing

Model with Trend • If assuming the trend is locally linear, at ti

me t the level and rate of change of level, (the slope) is known,

• Xt+1 = 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,

– 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.

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.

– 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– M1=146– R1=30

• Choice of smoothing parameter– a)Choose the values of and according t

o 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.

• M1=146, R1=3, = 0.5 and = 0.5, to give the following spreadsheet calculations

13.3.3 Forecasting with the trend

model • The one step forecast, the forecast for tim

e point 26 is:263.37 + 12.92 • The two step forecast, the forecast for time

point 27 is:263.37 + 2*12.92 • the forecast function for the time point h st

eps ahead is: Ft(h) = Mt + hRt

• at all time points – F1(1) = M1 + R1

– F2(1) = M2 + 2R2

• The forecast for the time periods 26, 27 & 28:– F25(1) = M25+ R25 =263.37 + 1*12.92 =276.29

– F25(2) = M25 + 2R25 = 263.37 + 2*12.92 = 289.22

– F25(3) = M25 + 3R25 = 263.37 + 3*12.92 = 302.14

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.

–

– 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

– MSE=367.01

– Summary of the forecasts using differing smoothing parameters:

(,

)

F25(1) F25(2) F25(3) M.S.E.

(0.5, 0.5) 276.29 289.22 302.14 455.59

(0.1,0.01) 238.72 241.89 245.06 367.01

(0.07, 0.99)

259.24 268.93 276.93 270.00

Conceptual Paper Worksheet

13.4 Summary-  The simple exponential smoothin

g 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 is:Ft(h) = Mt

-  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

- All exponential smoothing models require:-a)   A starting Ruleb)   A choice of smoothing parameter

a) For the simple model the simplest starting rule is: M1=X1

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.

- Forecasting- For h=1,2,3,….

Ft(h) = Mt

- Forecasting- For h=1,2,3,….

Ft(h) = Mt + hRt

- 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

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.