EBK STATISTICS FOR BUSINESS & ECONOMICS
EBK STATISTICS FOR BUSINESS & ECONOMICS
12th Edition
ISBN: 9780100460461
Author: Anderson
Publisher: YUZU
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Chapter 17.4, Problem 18E
To determine

Develop a forecast for t=6 by using the Holt’s linear exponential smoothing method with α=0.3 and β=0.5.

Expert Solution & Answer
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Answer to Problem 18E

The forecast for t=6 by using the Holt’s linear exponential smoothing method with α=0.3 and β=0.5 is 21.16.

Explanation of Solution

Calculation:

The given data represents time series data.

The equations for Holt’s linear exponential smoothing method is,

Lt=αYt+(1α)(Lt1+bt1)

bt=β(LtLt1)+(1β)bt1

Ft+1=Lt+btk

Where Lt represents the estimate of the level of the time series in period t, bt represents estimate of the slope of the time series in period t, α represents the smoothing constant for the level of the time series, Yt is the value of time series in period t, β is the smoothing constant for the slope the time series, Ft+1 is the forecast for k periods ahead and k represents the number of periods ahead to be forecast.

For week 1:

Consider

L1=Y1

From given data, it can be observed that Y1=6.

Therefore,

L1=6

Now,

b1=Y2Y1=116=5

Consider k=1 the Ft+1 is

Ft+1=Lt+bt

The equation Ft+1=Lt+bt at t=1 is

F1+1=L1+b1F2=6+5=11

For week 2:

The equation Lt=αYt+(1α)(Lt1+bt1) at t=2 is,

L2=αY2+(1α)(L21+b21)=αY2+(1α)(L1+b1)

Substitute α=0.3, Y2=11, L1=6 and b1=5

L2=(0.3)(11)+(10.3)(6+5)=3.3+7.7=11

The equation bt=β(LtLt1)+(1β)bt1 at t=2 is,

b2=β(L2L21)+(1β)b21=β(L2L1)+(1β)b1

Substitute L1=6, L2=11, β=0.5 and b1=5

b2=0.5(116)+(10.5)5=0.5(5)+0.5(5)=2.5+2.5=5

The equation Ft+1=Lt+bt at t=2 is,

F2+1=L2+b2=11+5=16

Similarly the remaining estimated levels, estimated trend and forecasts are obtained as follows:

tValueEstimated LevelEstimated trendForecast
1665 
21111511
3913.903.9516
41416.703.3717.85
51518.552.6120.07

The forecast for week 6 is,

Ft+1=Lt+btF5+1=L5+b5F6=L5+b5

Substitute L5=18.55 and b5=2.61

F6=L5+b5=18.55+2.61=21.16

Thus, the forecast for t=6 by using the Holt’s linear exponential smoothing method with α=0.3 and β=0.5 is 21.16.

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Chapter 17 Solutions

EBK STATISTICS FOR BUSINESS & ECONOMICS

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