Assignment 3: ForecastingQuestion 5 33A major source of revenue
in Texas is a state sales tax on certain types of goods and
services. Data are compiled and the state comptroller uses them to
project future revenues for the state budget. One particular
category of goods is classified as Retail Trade. Four years of
quarterly data for one particular area of southeast Texas
follows:QuarterYear 1Year 2Year 3Year 4
1218225234250
2247254265283
3243255264289
4292299327356
a) Compute seasonal indices for each quarter based on a
CMA.QuarterDataMACMAPercentageSeasonal ratio
1218
2247
3243250250.8896.860.97
4292251.75252.63115.591.16
1225253.5255.0088.240.88
2254256.5257.3898.690.99
3255258.25259.3898.310.98
4299260.5261.88114.181.14
1234263.25264.3888.510.89
2265265.5269.0098.510.99
3264272.5274.5096.170.96
4327276.5278.75117.311.17
1250281284.1387.990.88
2283287.25290.8897.290.97
3289294.5
4356
At the first, we must compute a series of moving averages (MA)
and then average the MA in order to build the seasonal indices
based on a CMA. In addition, the percentage column is simply the
data column, divided by the CMA, and multiplied by 100. Using QM
for Windows, we specify Centered Moving Average and we get:Index
for quarter 1, I1 = (0.88+0.88+0.88)/3 = 0.88Index for quarter 2,
I2 = (0.99+0.98+0.97)/3 = 0.98Index for quarter 3, I3 =
(0.96+0.98+0.96)/3 = 0.97Index for quarter 4, I4 =
(1.16+1.14+1.17)/3 = 1.16
b) Deseasonalize the data and develop a trend line on the
deseasonalized data.
With using Excel, in order to get deseasonalized data, we simply
data/(seasonal ratio). We get:
QuarterDataSeasonal ratioDeseasonalize
12180.88247.73
22470.98252.04
32430.97250.88
42921.16252.63
12250.88255.00
22540.99257.38
32550.98259.38
42991.14261.88
12340.89264.38
22650.99269.00
32640.96274.50
43271.17278.75
12500.88284.13
22830.97290.88
32890.97297.94
43561.16306.90
To compute the trend line, we must we must run a least squares
regression. The 'explanatory' variable here will be simply a time
index. Therefore, calling Y the explained variable (the actual
data) and X the explanatory variable, you would have to run a
regression on the following data (also adding a constant).
YX
247.731
252.042
250.883
252.634
255.005
257.386
259.387
261.888
264.389
269.0010
274.5011
278.7512
284.1313
290.8814
297.9415
306.9016
So, we have to find the coefficients 'a' and 'b' in the
following regression:Y = a + bX
Using excel, we get the intercept and slope. We get that these
values are:a = 237.8226b = 3.663168
So, the trend line is Y = 237.82 + 3.66X
c) Use the trend line to forecast the sales for each quarter of
year 5.
This forecast can be obtained by simply using as "explanatory
variables" the values 17, 18, 19 and 20, which would correspond to
each quarter of the fifth yeard (recall that the 4th quarter of the
4th year would be the 16th value).
17Quarter 1: Y = 237.82 + 3.66(17) = 300.0418Quarter 2: Y =
237.82 + 3.66(18) = 303.719Quarter 3: Y = 237.82 + 3.66(19) =
307.3620Quarter 4: Y = 237.82 + 3.66(20) = 311.02
d) Use the seasonal indices to adjust the forecasts found in
part (c) to obtain the final forecasts.
Since the trend forecasts were done using deseasonalized data,
we must now adjust each forecast to see the actual value for each
quarter. This is simply a matter of undoing what we did in question
a. We must take each value and multiply it by (seasonal index)/100.
We then get:
17Quarter 1: 300.04(0.88) = 264.035218Quarter 2: 303.7(0.98) =
297.62619Quarter 3: 307.36(0.97) = 298.139220Quarter 4:
311.02(1.16) = 360.7832
Question 5 34Yxxxx
SalesT (time period)Q1Q2Q3
2181100
2472010
2433001
2924000
2255100
2546010
2557001
2998000
2349100
26510010
26411001
32712000
25013100
28314010
28915001
35616000
SUMMARY OUTPUT
Regression Statistics
Multiple R0.984243958
R Square0.968736169
Adjusted R Square0.957367504
Standard Error7.67070875
Observations16
ANOVA
dfSSMSFSignificance F
Regression420055.25013.885.211073.34E-08
Residual11647.237558.83977
Total1520702.44
CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower
95.0%Upper 95.0%
Intercept281.56255.75303248.941593.18E-14268.9002294.2248268.9002294.2248
t3.693750.4288068.6140423.21E-062.7499554.6375452.7499554.637545
Q1-75.668755.574474-13.57423.25E-08-87.9381-63.3994-87.9381-63.3994
Q2-48.86255.491392-8.898022.34E-06-60.949-36.776-60.949-36.776
Q3-52.056255.440934-9.567521.15E-06-64.0317-40.0808-64.0317-40.0808
Using Excel, we get:Y = 281.6 + 3.7t 75.7Q1 48.9Q2 52.1Q3The
forecast for the next 4 quarters are:Y = 281.6 + 3.7(17) 75.7(1)
48.9(0) 52.1(0) = 268.7Y = 281.6 + 3.7(18) 75.7(0) 48.9(1) 52.1(0)
= 299.2Y = 281.6 + 3.7(19) 75.7(0) 48.9(0) 52.1(1) = 299.7Y = 281.6
+ 3.7(20) 75.7(0) 48.9(0) 52.1(0) = 355.4
Question 5 - 35xy
QuarterDataTrend Line
1274197.26
2172196.93
3130196.59
4162196.26
5282195.92
6178195.59
7136195.25
8168194.92
9282194.58
10182194.25
11134193.91
12170193.58
13296193.24
14210192.91
15158192.57
16182192.24
Intercept197.6
Slope-0.34
SUMMARY OUTPUT
Regression Statistics
Multiple R0.028
R Square0.001
Adjusted R Square-0.071
Standard Error58.65
Observations16
ANOVA
DfSSMSFSignificance F
Regression138.2235338.223530.0111110.917546
Residual1448160.783440.055
Total1548199
CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower
95.0%Upper 95.0%
Intercept197.630.757366.4244781.59E-05131.632263.568131.632263.568
X Variable
1-0.3352943.180851-0.105410.917546-7.157546.486952-7.157546.486952
a) Using Excel, we get Y = 197.6 0.34X, where X = time
periodBesides that, the slope is -0.34 specify a small negative
trend. In addition, the result that we get are not statically
significant and r2 = 0.001b) QuarterDataMACMAPercentagesSeasonal
RatioDeseasonalized
112741.47186.6021
221720.96178.8708
33130184.50185.5070.080.70185.5
44162186.50187.2586.520.87187.25
15282188.00188.75149.401.49188.75
26178189.50190.2593.560.94190.25
37136191.00191.0071.200.71191
48168191.00191.5087.730.88191.5
19282192.00191.75147.071.47191.75
210182191.50191.7594.920.95191.75
311134192.00193.7569.160.69193.75
412170195.50199.0085.430.85199
113296202.50205.50144.041.44205.5
214210208.50210.00100.001.00210
315158211.500.70225.2356
416182183.330.87210.2661
Intercept176.90
Slope2.18
Using Excel, the seasonal indices are:Quarter 1: 1.47Quarter 2:
0.96Quarter 3: 0.70Quarter 4: 0.87The trend equation found with the
deseasonalized data is Y = 176.90 + 2.18X. The slope indicates a
positive trend of 2.18 per time period. However, the results are
statistically significant.
c) The negative slope that we get in part (a) was found when the
seasonality was ignored. The quarter 1 has a high seasonal ratio,
so the first observation was very large relative to the last
observation. According raw data, which was used for the trend line
in a part (a), it appeared that there was a negative trend line but
in reality this was due to the seasonal variation and not due to
trend. In addition, the decomposition method is better to use when
there is a sesonal pattern present.
Question 5 39YearxDJIATrend SRF ErrorMADMSEMAPE
1994137545769.2142015.212015.21406108953.6818
1995238346166.5812332.582332.58544093560.8394
1996351176563.9481446.951446.95209365928.2773
1997464486961.315513.32513.322634927.9608
1998579087358.682-549.32549.323017506.9464
1999692137756.049-1456.951456.95212270715.8141
20007115028153.416-3348.583348.581121301629.1131
20018107918550.783-2240.222240.22501857420.7601
20029100228948.15-1073.851073.85115315510.7149
20031083429345.5171003.521003.52100704512.0297
200411104539742.883-710.12710.125042666.7934
2005121078410140.25-643.75643.754144145.9695
2006131071810537.62-180.38180.38325381.6830
2007141246010934.98-1525.021525.02232567312.2393
2008151326211332.35-1929.651929.65372354514.5502
200916877211729.722957.722957.72874809633.7177
2010171043112127.081696.081696.08287670416.2600
2011181157712524.45947.45947.458976658.1839
2012191239212921.82529.82529.822807084.2755
2013201310413319.19215.19215.19463051.6421
5.46E-131365.78262626717.5726
Intercept5371.85
Slope397.37
The trend equation is Y = 5371.85 + 397.37XFor 2014, X = 21; Y =
5371.85 + 397.37(21) = 13716.62For 2015, X = 22; Y = 5371.85 +
397.37(22) = 14113.99For 2016, X = 23; Y = 5371.85 + 397.37(23) =
14511.36The MSE from Excel output is 2626267.Question 5
40Exponential Smoothing0.8SE1693.325303
0.2MSE2867351
YearDJIAFTFITErrorMSE
19943754375403754
19953834375403754806400
19965117381813383112861654310
199764484859.76219507813701875936
199879086174.07438661212961680119
199992137648.7616458294919844767
2000115029029.178792982116812824543
20011079111165.87106112227-14362061979
20021002211078.1983111910-18883562751
2003834210399.5152910929-25876691711
2004104538859.367115897514782185068
20051078410157.363521050927575456
20061071810729.0639611125-407165616
20071246010799.393311113013301768431
20081326212194.0454412738524275004
2009877213157.1262713785-501325125958
2010104319774.516-1759600831690619
20111157710264.79-421022313541832754
20121239211306.2417511481911829442
20131310412209.8532112531573328797
Using Excel, the MSE is 2,867,351. As we can see, this MSE is
higher than the MSE that we found using a trend line. So, the trend
line provides better forecasts than exponential smoothing. But,
other values for the two smoothing constants might result in better
forecasts and a lower MSE.Question 5 41 (a)Exponential
Smoothing0.4SE1942.656717
MSE3773915
YearDJIAFTFITErrorMSE
19943754375403754
19953834375403754806400
1996511737860378613311771561
199764484318.40431821304535196
199879085170.240517027387495330
199992136265.3440626529488688676
2000115027444.40607444405816464066
2001107919067.4440906717242970646
2002100229756.8660975726570296
200383429862.9209863-15212313197
2004104539254.5520925511981436278
2005107849733.9310973410501102645
20061071810153.96010154564318143
20071246010379.5801038020804328167
20081326211211.7501121220504203545
2009877212031.85012032-326010626603
20101043110727.91010728-29788155
20111157710609.14010609968936743
20121239210996.2901099613961948015
20131310411554.5701155515492400727
20141310412174.34012174
Using Excel, with a smoothing constant of 0.4, the MSE =
3,773,915.
(b)Exponential Smoothing0.9904588SE1623.168907
MSE2634677
YearDJIAFTFITErrorMSE
19943754375403754
19953834375403754806400
199651173833.2370383312841648048
199764485104.7510510513431804317
199879086435.1840643514732169187
199992137893.9480789413191739899
2000115029200.4150920023025297295
20011079111480.04011480-689474776
20021002210797.57010798-776601515
2003834210029.4010029-16872847318
2004104538358.10835820954388607
20051078410433.01010433351123192
20061071810780.65010781-633925
20071246010718.601071917413032482
20081326212443.38012443819670131
2009877213254.19013254-448220090022
2010104318814.7650881516162612215
20111157710415.5801041611611348898
20121239211565.92011566826682410
20131310412384.12012384720518230
20141310413097.13013097
Using Excel, the best smoothing constant is 0.99. According this
results the lowest MSE of 2,632,477
