MS6215: Forecasting Methods for Business
Mid-session test
16 February 2012, 8:00pm – 9:30pm
The time series St in Output 1 contains 28 quarterly observations of sales volume (in millions of U.S.
dollars) between 2005 Quarter 1 and 2011 Quarter 4. A plot of the series is given in Figure 1.
1)
Comment on the behaviour of St as observed from Figure 1.
(3)
In light of the behaviour of St, the investigator decides to analyse the series by a multiplicative
decomposition model. He uses the first 24 observations to estimate the model, and the last 4
observations for forecast evaluation purposes. Output 1 summarises the steps of the analysis.
2)
3)
4)
5)
6)
7)
8)
9)
State the data generating process that underlies the multiplicative decomposition model.
Briefly explain the meaning of each of the four components forming the process.
(4)
Calculate the values of A, B, C, D, E and F in Output 1.
(4)
Suppose that SN-tilde’s shown in Output 1 for Quarters 1, 3 and 4 have means of 0.981926,
0.987807 and 1.017846 respectively. Work out the values of the seasonal indices for the four
quarters.
(5)
Let D_t be the seasonally adjusted observations. Compute D_t for Quarter 3 of 2009.
(1)
The trend is estimated by linear regression. State the independent and dependent variables of
the regression.
(2)
2
2
The regression output is shown in Output 2. Compute the R , adjusted R , F-statistic for testing
the overall significance of the model, and the t-statistic for testing the significance of the slope
coefficient.
(6)
Comment on the model’s goodness of fit using the four measures computed in Question 7. (2)
Forecast St for 2011 Quarter 4. Does this forecast under-predict or over-predict the actual St?
Is this an ex-post or ex-ante forecast? Explain.
(2)
The investigator then analyses the same data series (up to observation 24) using a Fourier series model.
The SAS program used to perform the analysis and the associated output are shown in Output 3.
10)
11)
12)
13)
14)
What is the maximum allowable number of harmonics for this data series?
(2)
Perform F-tests to test the significance of the three most influential harmonics. Conduct your
tests at the 0.01 level of significance. What do you conclude?
(9)
State the frequency and wavelength of each of the harmonics found to be significant in
Question 11).
(2)
Forecast St for 2011 Quarter 4 using the Fourier series model that incorporates the significant
harmonics found in Question 11). Compare your forecast with that obtained in Question 9).
What do you conclude?
(5)
State a drawback that is common to both the mean squared error and mean absolute deviation
when measuring forecast accuracy.
(3)
1
Figure 1: Plot of St
290000
270000
S_t
250000
230000
210000
0
5
10
15
20
25
30
Output 1: Summary of multiplicative decomposition
Year Quarter t
2005
1
1
2
2
3
3
4
4
2006
1
5
2
6
3
7
4
8
2007
1
9
2 10
3 11
4 12
2008
1 13
2 14
3 15
4 16
2009
1 17
2 18
3 19
4 20
2010
1 21
2 22
3 23
4 24
2011
1 25
2 26
3 27
4 28
S_t (millions of
dollars)
moving average
(MA)
214862
223968
221454
231917
229911
240976
237828
245465
236989
245126
241536
251776
248862
258913
255727
268409
262781
276057
272419
283327
270415
281313
276712
280370
277182
284432
280098
288543
223050.25
226812.5
231064.5
235158
238545
A
241352
242279
243856.75
246825
250271.75
253819.5
257977.75
261457.5
265743.5
269916.5
273646
275554.5
276868.5
B
277202.5
centered moving
average (CMA)
224931.375
228938.5
233111.25
236851.5
239429.75
C
241815.5
243067.875
245340.875
248548.375
252045.625
255898.625
259717.625
263600.5
267830
271781.25
274600.25
276211.5
277405.125
D
SN-tilde
0.98454028
1.01301004
0.98627158
1.01741386
0.99331015
E
0.98004057
1.00846729
0.98449148
1.0129859
0.98736885
1.01177957
0.98463475
1.01824162
0.98114849
1.01573232
0.99205664
1.02576106
0.97480175
F
2
Output 2: Regression results
SUMMARY OUTPUT
Regression Statistics
R Square
Adjusted R Square
Standard Error
Observations
??????????
??????????
3424.66222
24
ANOVA
df
1
22
23
SS
8883286748
258022848.5
9141309596
MS
8883286748
11728311.3
F
?????????
Coefficients
217615.517
2779.31595
Standard Error
1442.983089
?????????
t Stat
150.8094714
?????????
P-value
1.15502E-34
1.54396E-18
Regression
Residual
Total
Intercept
X Variable 1
Significance F
1.54396E-18
Output 3: SAS program and results
data test2012;
input sales;
cards;
214862
223968
221454
231917
229911
240976
237828
245465
236989
245126
241536
251776
248862
258913
255727
268409
262781
276057
272419
283327
3
270415
281313
276712
280370
;
proc spectra data=test2012 coeff out=out1;
var sales;
run;
data out1;
set out 1;
sq=p_01;
if period=. then sq=0;
if round (freq, .0001)=3.1416 sq=.5*p_01;
run;
proc print data=out1;
sum sq;
run;
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
FREQ
0.00000
0.26180
0.52360
0.78540
1.04720
1.30900
1.57080
1.83260
2.09440
2.35619
2.61799
2.87979
3.14159
PERIOD
.
24.0000
12.0000
8.0000
6.0000
4.8000
4.0000
3.4286
3.0000
2.6667
2.4000
2.1818
2.0000
COS_01
504759.42
-3214.21
-6294.03
-3399.80
-1677.54
-2770.49
-3488.00
-2980.18
-2287.71
-2359.53
-1848.72
-2275.79
-9843.42
SIN_01
0.00
-22085.33
-8744.25
-6633.35
-4608.77
-3396.89
-2909.25
-1413.31
-804.75
-823.68
-693.25
-963.74
0.00
P_01
3057384824564
5977115227.92
1392921081.52
666719637.35
288658960.08
230573578.09
247558554.75
130547024.27
70574863.08
74950042.15
46780326.98
73296320.73
1162714220.08
sq
0.00
5977115227.92
1392921081.52
666719637.35
288658960.08
230573578.09
247558554.75
130547024.27
70574863.08
74950042.15
46780326.98
73296320.73
581357110.04
=============
9781052726.96
4