Question

An analyst must decide between two different forecasting techniques for weekly sales of roller blades: a linear trend equation and the naive approach. The linear trendequation is Ft = 125 + 2.2t, and it was developed using data from periods 1 through 10. Based on data for periods 11 through 20 as shown in the table, which of thesetwo methods has the greater accuracy if MAD and MSE are used? (Round your answers to 2 decimal places.)

t Units Sold
11 144
12 148
13 151
14 147
15 154
16 149
17 154
18 156
19 158
20 163


MAD (Naive) ?
MAD (Linear) ?
MSE (Naive) ?
MSE (Linear) ?

Answer 1

1. Mean absolute deviation(MAD) is the mean or average of the sum of the errors for a given data set taken without regard to sign. Theformula for calculating MAD is:

2. Mean squared error(MSE) is a measure of accuracy computed by squaring the individual errors for each item and then finding theaverage value of the sum of those squares. MSE gives greater weight to large errors than to small errors, since the errors are squared before beingsummed. The formula for calculating MSE is:

now usinga linear trend equation :Given Ft = 124 +2t
For t =11 then Ft = 124 + 2(11)=146 ---->|e|=|144-146|=2
For t =12 then Ft = 124 +2(12)=148 ---->|e|= |148-148|=0
For t =13 then Ft = 124 + 2(13)=150---->|e|=|151-150|=1
For t =14 then Ft = 124 +2(14)=152 ---->|e|=|147-152|=5
For t =15 then Ft = 124 + 2(15)=154 ---->|e|=|154-154|=0
For t =16 then Ft = 124 +2(16)=156---->|e|=|149-156|=7
For t =17 then Ft = 124 + 2(17)=158 ---->|e|=|154-158|=4
For t =18 then Ft = 124 +2(18)=160 ---->|e|=|156-160|=4
For t =19 then Ft = 124 + 2(19)=162 ---->|e|=|158-162|=4
For t =20 then Ft = 124 +2(20)=164 ---->|e|=|163-164|=1hence Σ |e| = 2+0+1+5+0+7+4+4+4+1=28Σ e²= 4+0+1+25+0+49+16+16+16+1=128henceMAD (Linear) = 28/10=2.8MSE (Linear)= 128/9=14.2-----> MAD(linear) gives more accuracy than MSE(linear) now,The Naive Forecasting Model ( benchmarkmodel) : In this modelForecasting for next period =Observed value this period i.e,F_t+1= X_thence

F₁₁=F₁₀=125+2.2*10=147 ---->|e|=|144-147|=3
F₁₂=F₁₁=144 ---->|e|= |148-144|=4
F₁₃= F₁₂=148---->|e|=|151-148|=3

F₁₄= F₁₃=151---->|e|=|147-151|=4

F₁₅=F₁₄= 147---->|e|=|154-147|=7
F₁₆=F₁₅ = 154 ---->|e|=|149-154|=5
F₁₇=F₁₆= 149---->|e|=|154-159|=5
F₁₈=F₁₇= 154---->|e|=|156-154|=2
F₁₉= F₁₈=156---->|e|=|158-156|=2
F₂₀= F₁₉= 158---->|e|=|163-158|=5

hence Σ|e|=3+4+3+4+7+5+5+2+2+5=40

Σ e² =9+16+9+16+49+25+25+4+4+25=157

MAD (Naive) =40/10=4

MSE (Naive) =157/9=17.44

∴MAD (Naive) =4
MAD (Linear) =2.8
MSE (Naive) =17.44
MSE (Linear =14.2

Hence MAD (Linear) is the more accurate method since it gives least error as show above

Answer 2

Naive rules are simple but potentially effective time-seriesforecasting techniques. They are rules in the sense that they areprespecified so that noparameter values need be estimated. Thenaivete is implicit in the fact that the basis for any naiveforecast of a time series is the time series itself. Theseries isused to predict itself, that is, historical values of the seriesare used to compose or construct future values of the same series.The technique ofnaive forecasting is therefore extrapolation.
Naive rules, in their simplicity, are relativelylow-cost approaches to forecasting, but if a method is effectiveone should not hesitate to employ itbecause of its simplicity ornaivete. Naive rules are more effective at short-term than atlong-term forecasting. As the forecasting span or gapislengthened, the less accurate is the naive forecast likely to be,and the greater the attendant risk in basing decisions upon suchforecasts In this methodwe get the continous values and thecorrect values
For linear trend method we get these values
given Ft = 124 +2t
For t =11 then Ft = 124 + 2(11)=146
For t =12 then Ft = 124 +2(12)=148
For t =13 then Ft = 124 + 2(13)=150
For t =14 then Ft = 124 +2(14)=152
For t =15 then Ft = 124 + 2(15)=154
For t =16 then Ft = 124 +2(16)=156
For t =17 then Ft = 124 + 2(17)=158
For t =18 then Ft = 124 +2(18)=160
For t =19 then Ft = 124 + 2(19)=162
For t =20 then Ft = 124 +2(20)=164

Therefore we use Naive approach for this
Hope this will help you