session 10a

Session 10a

Decision Models -- Prof. Juran

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OverviewForecasting Methods • Exponential Smoothing

– Simple– Trend (Holt’s Method)– Seasonality (Winters’ Method)

• Regression– Trend– Seasonality– Lagged Variables


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Forecasting1. Analysis of Historical Data

• Time Series (Extrapolation)• Regression (Causal)

2. Projecting Historical Patterns into the Future

3. Measurement of Forecast Quality


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Measuring Forecasting Errors• Mean Absolute Error• Mean Absolute Percent Error• Root Mean Squared Error• R-square


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Mean Absolute Error

MAE n

n

ii1


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Mean Absolute Percent Error

MAPE *%100nY

n

i i

i1

Or, alternatively *%100nY

n

i i

i1 ˆ


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Root Mean Squared Error

RMSE

n

n

ii

1

2

nSSE


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R-Square

2R TSSSSR

TSSSSE1


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Trend Analysis• Part of the variation in Y is believed

to be “explained” by the passage of time

• Several convenient models available in an Excel chart


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Example: Revenues at GMGM Revenue

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Right-click on the data seriesSuperimpose a trend line on the graph:


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GM Revenue - Linear Trend

y = 340.23x + 31862R2 = 0.6618

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GM Revenue - Logarithmic Trend

y = 5162.3Ln(x) + 24937R2 = 0.6601

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GM Revenue - Polynomial Trend

y = -5.6121x2 + 604x + 29752R2 = 0.6872

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GM Revenue - Power Trend

y = 26532x0.1372

R2 = 0.6783

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GM Revenue - Exponential Trend

y = 32044e0.0088x

R2 = 0.6505

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You can also show moving-average trend lines, although showing the equation and R-square are no longer options: GM Revenue - 4-Period Moving Average

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GM Revenue - 3-Period Moving Average

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GM Revenue - 2-Period Moving Average

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Simple Exponential Smoothing

B asi ca l l y , th i s m eth o d u ses a f o r eca st f o r m u l a o f th e f o r m : ktF tL

F o r eca s t “ k ” p er i o d s i n th e fu tu r e = C u r r en t “ L ev el ”

= W ei g h ted C u r r en t O b ser v ed V a lu e + W ei g h ted P r ev i o u s L ev el

11 tt LY N o te th a t th e w ei g h ts m u st a d d u p to 1 .0.


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Why is it called “exponential”?

tL 11 tt LY

...111 33

22

1 tttt YYYY


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Example: GM RevenueGM Revenue

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In this spreadsheet model, the forecasts appear in column G.

Note that our model assumes that there is no trend. We use a default alpha of 0.10.

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A B C D E F G H I J K L MAlpha 0.100 GM_Rev SmLevel Forecast Error abs(error) abs(%error) error^2

1-91 29200 29200.0MAE 4014.376 2-91 31300 29410.0 29200.0 2100.0 2100.0 7.2% 4410000.0RMSE 4690.9738 3-91 28900 29359.0 29410.0 -510.0 510.0 1.7% 260100.0MAPE 11.148% 4-91 33600 29783.1 29359.0 4241.0 4241.0 14.4% 17986081.0

1-92 32000 30004.8 29783.1 2216.9 2216.9 7.4% 4914645.62-92 35200 30524.3 30004.8 5195.2 5195.2 17.3% 26990206.83-92 29400 30411.9 30524.3 -1124.3 1124.3 3.7% 1264075.34-92 35800 30950.7 30411.9 5388.1 5388.1 17.7% 29031838.11-93 35000 31355.6 30950.7 4049.3 4049.3 13.1% 16396895.82-93 36658 31885.9 31355.6 5302.4 5302.4 16.9% 28115204.63-93 30138 31711.1 31885.9 -1747.9 1747.9 5.5% 3055016.24-93 37268 32266.8 31711.1 5556.9 5556.9 17.5% 30879421.81-94 37495 32789.6 32266.8 5228.2 5228.2 16.2% 27334420.42-94 40392 33549.8 32789.6 7602.4 7602.4 23.2% 57796633.23-94 34510 33645.8 33549.8 960.2 960.2 2.9% 921924.04-94 42553 34536.6 33645.8 8907.2 8907.2 26.5% 79337354.0

=AVERAGE(I3:I47)=SQRT(AVERAGE(K3:K47))=AVERAGE(J3:J47)

=$B$1*E7+(1-$B$1)*F6

=F8

=E11-G11

=ABS(H13)

=ABS(H15/G15)

=H17^2


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GM Revenue - Simple Smoothing (alpha 0.10)

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We use Solver to minimize RMSE by manipulating alpha.

After optimizing, we see that alpha is 0.350 (instead of 0.10). This makes an improvement in RMSE, from 4691 to 3653.

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A B C D E F G H I J KAlpha 0.350 GM_Rev SmLevel Forecast Error abs(error)abs(%error) error^2

1-91 29200 29200.0MAE 3275.989 2-91 31300 29935.9 29200.0 2100.0 2100.0 7.2% 4410000.0RMSE 3653.2722 3-91 28900 29572.9 29935.9 -1035.9 1035.9 3.5% 1073072.4MAPE 8.584% 4-91 33600 30984.1 29572.9 4027.1 4027.1 13.6% 16217616.5

1-92 32000 31340.1 30984.1 1015.9 1015.9 3.3% 1032075.02-92 35200 32692.7 31340.1 3859.9 3859.9 12.3% 14898909.33-92 29400 31538.9 32692.7 -3292.7 3292.7 10.1% 10841859.24-92 35800 33032.1 31538.9 4261.1 4261.1 13.5% 18157357.91-93 35000 33721.7 33032.1 1967.9 1967.9 6.0% 3872765.02-93 36658 34750.6 33721.7 2936.3 2936.3 8.7% 8621982.53-93 30138 33134.3 34750.6 -4612.6 4612.6 13.3% 21276434.34-93 37268 34582.8 33134.3 4133.7 4133.7 12.5% 17087842.81-94 37495 35603.3 34582.8 2912.2 2912.2 8.4% 8480779.82-94 40392 37281.4 35603.3 4788.7 4788.7 13.5% 22931441.83-94 34510 36310.2 37281.4 -2771.4 2771.4 7.4% 7680620.34-94 42553 38497.9 36310.2 6242.8 6242.8 17.2% 38972198.5


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GM Revenue - Simple Smoothing (alpha 0.35)

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Exponential Smoothing with Trend:

Holt’s Method

Weighted Current TrendWeighted Current Observation

Weighted Current Level

ktF tt kTL

1111 11 tttttt TLLkTLY


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A B C D E F G H I J K L M NSmoothing constant(s) GM_Rev SmLevel SmTrend Forecast Error abs(error)abs(%error) error^2Level (alpha) 0.266 1-91 29200 29200.000 0.000Trend (beta) 0.048 2-91 31300 29757.957 26.659 29200.000 2100.0 2100.0 7.2% 4410000.0

3-91 28900 29549.579 15.429 29784.616 -884.6 884.6 3.0% 782545.7MAE 3094.683 4-91 33600 30637.081 66.652 29565.008 4035.0 4035.0 13.6% 16281159.9RMSE 3568.391 1-92 32000 31048.143 83.108 30703.733 1296.3 1296.3 4.2% 1680308.2MAPE 8.01% 2-92 35200 32212.293 134.760 31131.251 4068.7 4068.7 13.1% 16554716.2

3-92 29400 31564.039 97.348 32347.053 -2947.1 2947.1 9.1% 8685120.14-92 35800 32760.991 149.887 31661.387 4138.6 4138.6 13.1% 17128119.61-93 35000 33465.944 176.407 32910.877 2089.1 2089.1 6.3% 4364433.22-93 36658 34443.591 214.690 33642.352 3015.6 3015.6 9.0% 9094133.33-93 30138 33457.270 157.306 34658.282 -4520.3 4520.3 13.0% 20432945.14-93 37268 34585.269 203.686 33614.577 3653.4 3653.4 10.9% 13347499.71-94 37495 35507.934 238.038 34788.955 2706.0 2706.0 7.8% 7322680.12-94 40392 36980.394 297.019 35745.973 4646.0 4646.0 13.0% 21585567.43-94 34510 36542.128 261.887 37277.413 -2767.4 2767.4 7.4% 7658572.14-94 42553 38331.485 334.869 36804.015 5749.0 5749.0 15.6% 33050827.6

=$B$2*E4+(1-$B$2)*(F3+G3)

=$B$3*(F6-F5)+(1-$B$3)*G5

=F7+G7

=E10-H10

=ABS(I12)

=ABS(I14/H14)

=I16^2


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Holt’s model with optimized smoothing constants. This model is slightly better than the simple model (RMSE drops from 3653 to 3568).

GM Revenue - Holts Method (Smoothing with Trend)

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Exponential Smoothing with Seasonality:

Winters’ MethodT h i s m e t h o d i n c l u d e s a n e x p l i c i t t e r m f o r s e a s o n a l i t y , w h e r e M i s t h e n u m b e r o f p e r i o d s i n a s e a s o n . W e w i l l u s e M = 4 b e c a u s e w e h a v e q u a r t e r l y d a t a .

L e v e l : tL 111

ttMt

t TLSY

T r e n d : tT 11 1 ttt TLL

S e a s o n a l i t y : tS Mtt

t SLY

1

N o w , f o r a n y t i m e k p e r i o d s i n t h e f u t u r e , t h e f o r e c a s t i s g i v e n b y : ktF Mkttt SkTL

N o t e t h a t t h e t r e n d t e r m i s a d d i t i v e , a n d t h e s e a s o n a l i t y t e r m i s m u l t i p l i c a t i v e .


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tS Mtt

t SLY

1

Weighted Current Seasonal Factor

Weighted Seasonal Factor from Last Year


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Winters’ model with optimized smoothing constants. This model is better than the simple model and the Holt’s model (as measured by RMSE).

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A B C D E F G H I J K L MSmoothing constant(s) GM_Rev SmLevel SmTrend SmSeason Forecast Error abs(error) abs(%error) error^2Level (alpha) 0.312 1-91 29200 29200.000 0.000 1.000Trend (beta) 0.037 2-91 31300 29855.671 24.178 1.000Seasonality (gamma) 0.202 3-91 28900 29573.917 12.897 1.000

4-91 33600 30839.828 59.102 1.000MAE 2670.440 1-92 32000 31242.711 71.779 1.005 30898.931 1101.1 1101.1 3.6% 1212353.7RMSE 3233.995 2-92 35200 32527.642 116.515 1.017 31314.491 3885.5 3885.5 12.4% 15097181.1MAPE 6.82% 3-92 29400 31631.252 79.164 0.986 32644.157 -3244.2 3244.2 9.9% 10524553.4

4-92 35800 32987.284 126.249 1.017 31710.415 4089.6 4089.6 12.9% 16724701.71-93 35000 33649.376 146.008 1.012 33275.398 1724.6 1724.6 5.2% 2974252.42-93 36658 34502.621 172.088 1.026 34355.316 2302.7 2302.7 6.7% 5302351.43-93 30138 33394.029 124.862 0.969 34181.443 -4043.4 4043.4 11.8% 16349433.84-93 37268 34492.750 160.774 1.030 34095.266 3172.7 3172.7 9.3% 10066242.91-94 37495 35401.921 188.371 1.022 35069.261 2425.7 2425.7 6.9% 5884208.42-94 40392 36772.011 231.948 1.040 36509.418 3882.6 3882.6 10.6% 15074445.5

=$B$2*(E8/H4)+(1-$B$2)*(F7+G7)

=$B$3*(F10-F9)+(1-$B$3)*G9

=$B$4*(E12/F12)+(1-$B$4)*(H8)

=(F13+G13)*H10


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GM Revenue - Winters Method (Smoothing with Trend and Seasonality)

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Forecasting with Regression

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A B C D E F G HGM_Rev GM_EPS Trend 1Q 2Q 3Q

1-91 29200 -1.28 1 1 0 02-91 31300 -1.44 2 0 1 03-91 28900 -1.88 3 0 0 14-91 33600 -4.25 4 0 0 01-92 32000 -0.53 5 1 0 02-92 35200 -1.18 6 0 1 03-92 29400 -1.86 7 0 0 14-92 35800 -1.25 8 0 0 01-93 35000 0.42 9 1 0 02-93 36658 0.92 10 0 1 03-93 30138 -0.49 11 0 0 14-93 37268 1.28 12 0 0 01-94 37495 1.86 13 1 0 02-94 40392 2.23 14 0 1 03-94 34510 0.4 15 0 0 14-94 42553 1.74 16 0 0 01-95 43285 2.51 17 1 0 02-95 42204 2.39 18 0 1 0


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65666768697071727374757677787980818283

B C D E F GRegression Statistics

Multiple R 0.8852R Square 0.7835Adjusted R Square 0.7624Standard Error 2736.1392Observations 46

ANOVAdf SS MS F Significance F

Regression 4 1111067275.3109 277766818.8277 37.1026 0.0000Residual 41 306944777.4065 7486457.9855Total 45 1418012052.7174

Coefficients Standard Error t Stat P-valueIntercept 33286.7628 1101.4629 30.2205 0.0000Trend 335.8508 30.4091 11.0444 0.00001Q -1289.9144 1142.5337 -1.1290 0.26552Q 423.4015 1142.1290 0.3707 0.71283Q -4582.6038 1167.0899 -3.9265 0.0003


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The most reasonable statistic for comparison is probably RMSE for smoothing models vs. standard error for regression models, as is reported here:

The regression models are superior most of the time (6 out of 10 revenue models and 7 out of 10 EPS models).

Revenue Mattel McD Lilly GM MSFT ATT Nike GE Coke Ford Regression $99.37 $112.23 $109.22 $2,736.14 $154.25 $12,836.41 $279.45 $1,164.02 $164.20 $969.14 Winters' $76.44 $84.92 $135.33 $3234.00 $103.91 $14,622.26 $191.94 $1,184.06 $258.02 $1,648.61

EPS

Regression $0.0874 $0.0205 $0.3727 $1.3882 $0.0687 $0.6879 $0.2080 $0.6591 $0.0164 $0.4988 Winters' $0.1327 $0.0295 $0.5285 $1.2603 $0.0635 $0.5719 $0.2687 $0.7587 $0.0228 $1.3475

Which Method is Better?


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For GM , a regression model seems best for forecasting revenue, but a Winters model seems best for earnings:

GM Revenue - W inters M ethod (Sm oothing with Trend and Seasonality)

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GM Revenue - Regression

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GM EPS - W inters M ethod (Sm oothing with Trend and Seasonality)

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-1

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GM EPS - Regression

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For N ike, the W inters model is better for revenue, and the regression model is best for earnings.

Nike Revenue (Winters)

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Nike Revenue (Regression)

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Nike EPS (W inters)

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Nike EPS (Regression)

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Time series characterized by relatively consistent trends and seasonality favor the regression model.If the trend and seasonality are not stable over time, then Winters’ method does a better job of responding to their changing patterns.


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Lagged Variables• Only applicable in a causal model• Effects of independent variables

might not be felt immediately• Used for advertising’s effect on

sales


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Example: Motel Chain123456789

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B C D E F G H ISales Quarter Adv Adv-Lag1 Qtr_1 Qtr_2 Qtr_31200 1 30 * 1 0 0880 2 20 30 0 1 0

1800 3 15 20 0 0 11050 4 40 15 0 0 01700 5 10 40 1 0 0350 6 50 10 0 1 0

2500 7 5 50 0 0 1760 8 40 5 0 0 0

2300 9 20 40 1 0 01000 10 10 20 0 1 01570 11 60 10 0 0 12430 12 5 60 0 0 01320 13 35 5 1 0 01400 14 15 35 0 1 01890 15 70 15 0 0 13200 16 25 70 0 0 02200 17 30 25 1 0 01440 18 60 30 0 1 04000 19 80 60 0 0 14100 20 50 80 0 0 0


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A B C D E F GSummary measuresMultiple R 0.9856R-Square 0.9714Adj R-Square 0.9571StErr of Est 213.2

ANOVA Tabledf SS MS F p-value

Regression 6 18515047.32 3085841.22 67.88 0.0000Residual 12 545531.63 45460.97

Regression coefficientsCoefficient Std Err t-value p-value Lower limit Upper limit

Constant 98.36 174.96 0.5622 0.5843 -282.9 479.6Quarter 41.58 13.56 3.0672 0.0098 12.0 71.1Advertising 4.53 3.25 1.3959 0.1880 -2.5 11.6Advertising_Lag1 34.03 3.13 10.8759 0.0000 27.2 40.9Qtr_1 280.62 157.66 1.7799 0.1004 -62.9 624.1Qtr_2 -491.59 145.37 -3.3817 0.0055 -808.3 -174.9Qtr_3 532.60 143.04 3.7235 0.0029 221.0 844.2


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A B C D E F G H I J K L M N O P Q R SQtr Sales Quarter Adv Adv-Lag1 Qtr_1 Qtr_2 Qtr_3 Forecast Constant Quarter Adv Adv-Lag1 Qtr_1 Qtr_2 Qtr_3

1 1200 1 30 * 1 0 0 Coefficient 98.36 41.58 4.53 34.03 280.62 -491.59 532.602 880 2 20 30 0 1 0 8023 1800 3 15 20 0 0 1 15044 1050 4 40 15 0 0 0 9571 1700 5 10 40 1 0 0 19942 350 6 50 10 0 1 0 4233 2500 7 5 50 0 0 1 26464 760 8 40 5 0 0 0 7831 2300 9 20 40 1 0 0 22052 1000 10 10 20 0 1 0 7493 1570 11 60 10 0 0 1 17014 2430 12 5 60 0 0 0 26621 1320 13 35 5 1 0 0 12482 1400 14 15 35 0 1 0 14483 1890 15 70 15 0 0 1 20834 3200 16 25 70 0 0 0 32591 2200 17 30 25 1 0 0 20732 1440 18 60 30 0 1 0 16483 4000 19 80 60 0 0 1 38264 4100 20 50 80 0 0 0 38791 21 50 50 1 0 0 31812 22 50 50 0 1 0 24503 23 50 50 0 0 1 35164 24 50 50 0 0 0 3025

=$M$2+SUMPRODUCT($N$2:$S$2,D5:I5)


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Simple Exponential Smoothing Forecast

-

750

1,500

2,250

3,000

3,750

4,500

5,250

6,000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Quarter

ObservationForecast


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Holt's Forecast

-

750

1,500

2,250

3,000

3,750

4,500

5,250

6,000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Quarter

ObservationForecast


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Winters' Forecast

-

750

1,500

2,250

3,000

3,750

4,500

5,250

6,000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Quarter

ObservationForecast


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Multiple Regression Forecast (with Lagged Advertising)

-

750

1,500

2,250

3,000

3,750

4,500

5,250

6,000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Quarter

ObservationForecast


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Simple Holt's Winters' MAE 769.6 766.8 708.0

RMSE 939.9 866.6 845.6 MAPE 50.5% 36.7% 47.3%

Here are measures of model fit for the non-regression models:

The regression model has a standard error of only 213, which is much better than any of the other models.


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SummaryForecasting Methods • Exponential Smoothing

– Simple– Trend (Holt’s Method)– Seasonality (Winters’ Method)

• Regression– Trend– Seasonality– Lagged Variables

session 10a

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