CHAPTER 10
FORECASTING
SOLUTION TO SOLVED PROBLEMS
10.S1 Forecasting Charitable Donations at the Union Mission
Cash donations (in thousands of dollars) at the Union Mission for 2004-2006 were as shown
below.
Quarter
Q1 2004
Q2 2004
Q3 2004
Q4 2004
Donations
242
282
254
345
Quarter
Q1 2005
Q2 2005
Q3 2005
Q4 2005
Donations
253
290
262
352
Quarter
Q1 2006
Q2 2006
Q3 2006
Q4 2006
Donations
270
286
271
378
a. Ignoring seasonal effects, compare both the MAD and MSE values for the last value method, the
averaging method, the moving average method (based on the most recent 4 quarters), the
exponential smoothing method (with an initial estimate of 275 and a smoothing constant of  =
0.2), and the exponential smoothing method with trend (with initial estimates of 275 for the
average value, 2 for the trend, along with smoothing constants of  = 0.2 and  = 0.2) when they
are applied retrospectively to the years 2004-2006.
Each of these methods are applied below, using the appropriate template from your MS
Courseware.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
B
C
D
E
F
G
H
Template for Last-Value Forecasting Method
Time
Period
1
2
3
4
5
6
7
8
9
10
11
12
13
True
Value
242
282
254
345
253
290
262
352
270
286
271
378
Last-Value
Forecast
Forecasting
Error
242
282
254
345
253
290
262
352
270
286
271
378
40
28
91
92
37
28
90
82
16
15
107
1
Mean Absolute Deviation
MAD =
57
Mean Square Error
MSE =
4,367
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
C
D
E
F
G
H
Template for Averaging Forecasting Method
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
B
Time
Period
1
2
3
4
5
6
7
8
9
10
11
12
13
True
Value
242
282
254
345
253
290
262
352
270
286
271
378
Averaging
Forecast
Forecasting
Error
242
262
259
281
275
278
275
285
283
284
282
290
40
8
86
28
15
16
77
15
3
13
96
B
C
D
E
Mean Absolute Deviation
MAD =
36
Mean Square Error
MSE =
2,329
F
G
H
Template for Moving-Average Forecasting Method
Time
Period
1
2
3
4
5
6
7
8
9
10
11
12
13
True
Value
242
282
254
345
253
290
262
352
270
286
271
378
Moving
Average
Forecast
Forecasting
Error
#N/A
#N/A
#N/A
281
284
286
288
289
294
293
295
301
Number of previous
periods to consider
n=
4
Mean Absolute Deviation
MAD =
32
28
7
24
65
19
8
22
83
2
Mean Square Error
MSE =
1,668
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
C
D
E
F
G
H
Template for Exponential Smoothing Forecasting Method
Time
Period
1
2
3
4
5
6
7
8
9
10
11
12
13
A
1
B
B
Exponential
Smoothing
Forecast
275
268
271
268
283
277
280
276
291
287
287
284
303
True
Value
242
282
254
345
253
290
262
352
270
286
271
378
C
D
Forecasting
Error
33
14
17
77
30
13
18
76
21
1
16
94
E
F
Smoothing Constant

0.2
Initial Estimate
Average =
275
Mean Absolute Deviation
MAD =
34
Mean Square Error
MSE =
G
H
2,024
I
J
Template for Exponential Smoothing Forecasting Method with Trend
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Time
Period
1
2
3
4
5
6
7
8
9
10
11
12
13
True
Value
242
282
254
345
253
290
262
352
270
286
271
378
Latest
Trend
-8.5
3.8
-4.7
23.0
-6.4
4.0
-5.3
21.6
-5.0
-1.2
-6.3
26.0
Estimated
Trend
2.0
-1.2
0.3
-1.2
6.1
2.3
2.8
0.4
6.8
3.2
1.9
-0.6
7.4
Exponential
Smoothing
Forecast
277
265
271
265
295
285
289
281
309
301
298
289
323
3
Forecasting
Error
35
17
17
80
42
5
27
71
39
15
27
89
Smoothing Constants

0.3

0.3
Initial Estimates
Average =
Trend =
275
2
Mean Absolute Deviation
MAD =
38.6
Mean Square Error
MSE =
2,174.2
b. Determine the seasonal factors for the four quarters.
The seasonal factors are determined using the template in your MS Courseware, as shown below.
The seasonal factors are 0.8780, 0.9848, 0.9033, and 1.2339 for quarters 1, 2, 3, and 4,
respectively.
A
B
C
D
E
F
G
Template for Seasonal Factors
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Year
1
1
1
1
2
2
2
2
3
3
3
3
Quarter
1
2
3
4
1
2
3
4
1
2
3
4
True
Value
242
282
254
345
253
290
262
352
270
286
271
378
Type of Seasonality
Quarterly
Estimate for
Seasonal Factor
0.8780
0.9848
0.9033
1.2339
Quarter
1
2
3
4
c. Repeat part a, but now considering the seasonal effects.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
B
C
D
E
F
G
H
I
J
K
Template for Last-Value Forecasting Method with Seasonality
Year
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
Quarter
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
True
Value
242
282
254
345
253
290
262
352
270
286
271
378
Seasonally
Adjusted
Value
276
286
281
280
288
294
290
285
308
290
300
306
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
Seasonally
Adjusted
Forecast
Actual
Forecast
Forecasting
Error
271
259
347
245
284
266
358
250
303
262
370
269
11
5
2
8
6
4
6
20
17
9
8
276
286
281
280
288
294
290
285
308
290
300
306
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
Type of Seasonality
Quarterly
Quarter
1
2
3
4
Seasonal Factor
0.878
0.985
0.903
1.234
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
Mean Absolute Deviation
MAD =
8.5
Mean Square Error
MSE =
4
99
A
B
C
D
E
F
G
H
I
J
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Template for Averaging Forecasting Method with Seasonality
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
Template for Moving-Average Forecasting Method with Seasonality
A
Year
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
Quarter
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
B
C
Year
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
Quarter
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
True
Value
242
282
254
345
253
290
262
352
270
286
271
378
D
True
Value
242
282
254
345
253
290
262
352
270
286
271
378
Seasonally
Adjusted
Value
276
286
281
280
288
294
290
285
308
290
300
306
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
Seasonally
Adjusted
Forecast
E
Seasonally
Adjusted
Value
276
286
281
280
288
294
290
285
308
290
300
306
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
Actual
Forecast
Forecasting
Error
276
281
281
281
282
284
285
285
288
288
289
290
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
271
254
347
246
278
257
352
250
283
260
357
255
11
0
2
7
12
5
0
20
3
11
21
F
G
Seasonally
Adjusted
Forecast
K
Type of Seasonality
Quarterly
Quarter
1
2
3
4
Seasonal Factor
0.878
0.985
0.903
1.234
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
Mean Absolute Deviation
MAD =
8.33
Mean Square Error
MSE =
118.86
Actual
Forecast
#N/A
#N/A
#N/A
281
284
286
288
289
294
293
296
301
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
H
Forecasting
Error
I
J
K
Number of previous
periods to consider
n=
4
Type of Seasonality
Quarterly
246
280
258
355
254
290
265
365
264
7
10
4
3
16
4
6
13
Quarter
1
2
3
4
Seasonal Factor
0.878
0.985
0.903
1.234
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
Mean Absolute Deviation
MAD =
7.9
Mean Square Error
MSE =
5
81.3
A
B
C
D
E
F
G
H
I
J
K
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Template for Exponential Smoothing Forecasting Method with Seasonality
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
Template for Exponential-Smoothing with Trend Forecasting Method with Seasonality
A
Year
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
Quarter
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
B
C
Year
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
Quarter
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
Seasonally
Adjusted
Value
276
286
281
280
288
294
290
285
308
290
300
306
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
True
Value
242
282
254
345
253
290
262
352
270
286
271
378
D
True
Value
242
282
254
345
253
290
262
352
270
286
271
378
E
Seasonally
Adjusted
Value
276
286
281
280
288
294
290
285
308
290
300
306
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
F
Latest
Trend
1.7
3.5
2.0
1.2
2.5
3.4
2.0
0.6
4.7
0.7
2.2
3.1
Seasonally
Adjusted
Forecast
275
275
277
278
278
280
283
285
285
289
289
292
295
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
Actual
Forecast
241
271
251
343
244
276
256
351
250
285
262
360
259
Smoothing Constant

0.2
Initial Estimate
Average =
275
Type of Seasonality
Quarterly
Quarter
1
2
3
4
Seasonal Factor
0.878
0.985
0.903
1.234
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
Mean Absolute Deviation
MAD =
7.9
Mean Square Error
MSE =
G
Estimated
Trend
2.0
1.9
2.3
2.2
2.0
2.1
2.4
2.3
2.0
2.5
2.2
2.2
2.4
Forecasting
Error
1
11
3
2
9
14
6
1
20
1
9
18
H
Seasonally
Adjusted
Forecast
277
279
282
284
285
288
292
294
294
299
300
302
305
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
6
I
Actual
Forecast
243
274
255
351
251
284
264
362
258
295
271
372
268
J
Forecasting
Error
1
8
1
6
2
6
2
10
12
9
0
6
K
105.5
L
M
Smoothing Constant


0.2
0.2
Initial Estimate
Average =
Trend =
275
2
Type of Seasonality
Quarterly
Quarter
1
2
3
4
Seasonal Factor
0.878
0.985
0.903
1.234
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
Mean Absolute Deviation
MAD =
5
Mean Square Error
MSE =
41
d. Using the forecasting method from part a or c with the lowest MAD value, make long-range
forecasts for charitable donations in each of the quarters of 2007.
The forecasting method from part a or c with the lowest MAD value is the exponentialsmoothing with trend forecasting method with seasonality.
From part c, the seasonally-adjusted forecast for Quarter 1 of 2007 is 305, leading to an actual
forecast of (0.878)(305) = $268 thousand.
The last estimated trend is 2.4. Therefore, the seasonally adjusted forecast for Quarter 2 of 2007
would be 305 + 2.4 = 307.4, leading to an actual forecast of (0.985)(307.4) = $303 thousand.
Similarly, the seasonally adjusted forecast for Quarter 3 of 2007 would be 307.4 + 2.4 = 309.8,
leading to an actual forecast of (0.903)(309.8) = $280 thousand.
Similarly, the seasonally adjusted forecast for Quarter 4 of 2007 would be 309.8 + 2.4 = 312.2,
leading to an actual forecast of (1.234)(312.2) = $385 thousand.
7