Chapter 16
Time-Series Forecasting and Index Numbers
LEARNING OBJECTIVES
This chapter discusses the general use of forecasting in business, several tools that are available for
making business forecasts, and the nature of time series data, thereby enabling you to:
1.
2.
3.
4.
5.
6.
7.
Gain a general understanding time series forecasting techniques.
Understand the four possible components of time-series data.
Understand stationary forecasting techniques.
Understand how to use regression models for trend analysis.
Learn how to decompose time-series data into their various elements.
Understand the nature of autocorrelation and how to test for it.
Understand autoregression in forecasting.
CHAPTER OUTLINE
16.1
Introduction to Forecasting
Time-Series Components
The Measurement of Forecasting Error
Error
Mean Absolute Deviation (MAD)
Mean Square Error (MSE)
16.2
Smoothing Techniques
Naïve Forecasting Models
Averaging Models
Simple Averages
Moving Averages
Weighted Moving Averages
Exponential Smoothing
16.3
Trend Analysis
Linear Regression Trend Analysis
Regression Trend Analysis Using Quadratic Models
Holt’s Two-Parameter Exponential Smoothing Method
16.4
Seasonal Effects
Decomposition
Finding Seasonal Effects with the Computer
Winters’ Three-Parameter Exponential Smoothing Method
16.5
Autocorrelation and Autoregression
Autocorrelation
Ways to Overcome the Autocorrelation Problem
Addition of Independent Variables
Transforming Variables
Autoregression
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16.6
Index Numbers
Simple Index Numbers and Unweighted Aggregate Price Indexes
Unweighted Aggregate Price Index Numbers
Weighted Aggregate Price Index Numbers
Laspeyres Price Index
Paasche Price Index
KEY WORDS
autocorrelation
autoregression
averaging models
cycles
cyclical effects
decomposition
deseasonalized data
Durbin-Watson test
error of an individual forecast
exponential smoothing
first-differences approach
forecasting
forecasting error
index number
irregular fluctuations
Laspeyres price index
mean absolute deviation (MAD)
mean squared error (MSE)
moving average
naïve forecasting methods
Paasche price index
seasonal effects
serial correlation
simple average
simple average model
simple index number
smoothing techniques
stationary
time series data
trend
unweighted aggregate price index number
weighted aggregate price index number
weighted moving average
STUDY QUESTIONS
1. Shown below are the forecast values and actual values for six months of data:
Month
June
July
Aug.
Sept.
Oct.
Nov.
Actual Values
29
51
60
57
48
53
Forecast Values
40
37
49
55
56
52
The mean absolute deviation of forecasts for these data is __________. The mean square error is
__________________.
2. Data gathered on a given characteristic over a period of time at regular intervals are referred to as
____________________________.
3. Time series data are thought to contain four elements: _______________, _______________,
_______________, and _______________.
4. Patterns of data behavior that occur in periods of time of less than 1 year are called
_____________________ effects.
5. Long-term time series effects are usually referred to as _______________.
Chapter 16: Time Series Forecasting and Index Numbers
281
6. Patterns of data behavior that occur in periods of time of more than 1 year are called
_______________________ effects.
7. Consider the time series data below. The equation of the trend line to fit these data is
__________________________________.
Year
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
Sales
28
31
39
50
55
58
66
72
78
90
97
104
112
8. Time series data are deseasonalized by dividing the each data value by its associated value of
____________.
9. Perhaps the simplest of the time series forecasting techniques are ____________________________
models in which it is assumed that more recent time periods of data represent the best predictions.
10. Consider the time-series data shown below:
Month
Jan.
Feb.
Mar.
Apr.
May
Volume
1230
1211
1204
1189
1195
The forecast volumes for April, May, and June are _______, _______, and _______ using a threemonth moving average on the data shown above and starting in January. Suppose a three-month
weighted moving average is used to predict volume figures for April, May, and June. The weights on
the moving average are 3 for the most current month, 2 for the month before, and 1 for the other
month. The forecasts for April, May, and June are _______, _______, and _______._ using a threemonth moving average starting in January.
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11. Consider the data below:
Month
Jan.
Feb.
Mar.
Apr.
May
Volume
1230
1211
1204
1189
1195
If exponential smoothing is used to forecast the Volume for May using  = .2 and using the January
actual figure as the forecast for February, the forecast is ____________________. If  = .5 is used,
the forecast is ___________________. If  = .7 is used, the forecast is _____________________.
The alpha value of ________ produced the smallest error of forecast.
12. ____________________________ occurs when the error terms of a regression forecasting
model are correlated. Another name for this is _____________________________.
13. The Durbin-Watson statistic is used to test for ______________________________.
14. Examine the data given below.
Year
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
y
126
203
211
223
238
255
269
271
276
286
289
294
305
311
324
338
x
34
51
60
57
64
66
80
93
92
97
101
108
110
107
109
116
The simple regression forecasting model developed from this data is ______________________. The
value of R2 for this model is _________________. The Durbin-Watson D statistic for this model is
__________________. The critical value of dL for this model using  = .05 is _____________ and
the critical value of dU for this model is _____________. This model (does, does not, inconclusive)
_______________ contain significant autocorrelation.
15. One way to overcome the autocorrelation problem is to add __________________________ to the
analysis. Another way to overcome the autocorrelation problem is to transform variables. One such
method is the ___________________________________ approach.
16. A forecasting technique that takes advantage of the relationship of values to previous period values is
______________________________. This technique is a multiple regression technique where the
independent variables are time-lagged versions of the dependent variable.
Chapter 16: Time Series Forecasting and Index Numbers
283
17. Examine the price figures shown below for various years.
Year
1998
1999
2000
2001
2002
Price
23.8
47.3
49.1
55.6
53.0
The simple index number for 2001 using 1998 as a base year is _________________.
The simple index number for 2002 using 1999 as a base year is _________________.
18. Examine the price figures given below for four commodities.
Item
1
2
3
1999
1.89
.41
.76
Year
2000 2001
1.90
1.87
.48
.55
.73
.79
2002
1.84
.69
.82
The unweighted aggregate price index for 2000 using 1999 as a base year is ________________. The
unweighted aggregate price index for 2001 using 1999 as a base year is __________. The unweighted
aggregate price index for 2002 using 1999 as a base year is _______________.
19. Weighted aggregate price indexes that are computed by using the quantities for the year of interest
rather than the base year are called __________________________ price indexes.
20. Weighted aggregate price indexes that are computed by using the quantities for the base year are
called ____________________________ price indexes.
21. Examine the data below.
Item
1
2
3
4
Quantity Quantity Price Price
2001
2002 2001 2002
23
27
1.33 1.45
8
6
5.10 4.89
61
72
.27
.29
17
24
1.88
2.11
Using 2001 as the base year
The Laspeyres price index for 2002 is _____________________.
The Paasche price index for 2002 is ______________________.
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ANSWERS TO STUDY QUESTIONS
1. 1.5, 7.83, 84.5, –0.57, 17.63
13. Autocorrelation
2. Time Series Data
14.
yˆ  93.602  2.023x , .916,
1.004, 1.10, 1.37, Does
3. Seasonal, Cyclical, Trend, Irregular
4. Seasonal
15. Independent Variables,
First-Differences
5. Trend
16. Autoregression
6. Cyclical
17. 233.6, 112.05
7. Yˆ  19.29091  6.84545 X
18. 101.6, 104.9, 109.5
8. S
19. Paasche
9. Naive Forecasting
20. Laspeyres
10. 1215, 1201.3, 1196, 1210.7,
1197.7, 1194.5
21. 105.18, 106.82
11. 1215.21, 1200.63, 1194.64, .7
12. Autocorrelation, Serial Correlation
SOLUTIONS TO ODD-NUMBERED PROBLEMS IN CHAPTER 16
16.1
Period
1
2
3
4
5
6
7
8
9
Total
MAD =
MSE =
2.30
1.60
–1.40
1.10
0.30
–0.90
–1.90
–2.10
0.70
–0.30
2.30
1.60
1.40
1.10
0.30
0.90
1.90
2.10
0.70
12.30
e
no. forecasts
e
e2
e
e

12.30
= 1.367
9

20.43
= 2.27
9
2
no. forecasts
5.29
2.56
1.96
1.21
0.09
0.81
3.61
4.41
0.49
20.43
Chapter 16: Time Series Forecasting and Index Numbers
16.3
Period Value
1
2
3
4
5
6
19.4
23.6
24.0
26.8
29.2
35.5
Total
MAD =
MSE =
16.5
F
e
e
e2
16.6
19.1
22.0
24.8
25.9
28.6
2.8
4.5
2.0
2.0
3.3
6.9
21.5
2.8
4.5
2.0
2.0
3.3
6.9
21.5
7.84
20.25
4.00
4.00
10.89
47.61
94.59
e

21.5
= 5.375
4

94.59
= 23.65
4
no. forecasts
e
2
no. forecasts
a.)
4-mo. mov. avg.
44.75
52.75
61.50
64.75
70.50
81.00
error
14.25
13.25
9.50
21.25
30.50
16.00
b.)
4-mo. wt. mov. avg.
53.25
56.375
62.875
67.25
76.375
89.125
error
5.75
9.625
8.125
18.75
24.625
7.875
c.)
difference in errors
14.25 – 5.75 = 8.5
3.626
1.375
2.5
5.875
8.125
285
In each time period, the four-month moving average produces greater errors of forecast than the
four-month weighted moving average.
286
16.7
Solutions Manual and Study Guide
Period
1
2
3
4
5
6
7
8
9
Value
9.4
8.2
7.9
9.0
9.8
11.0
10.3
9.5
9.1
 =.3
9.4
9.0
8.7
8.8
9.1
9.7
9.9
9.8
Error  =.7
–1.2
–1.1
0.3
1.0
1.9
0.6
–0.4
–0.7
9.4
8.6
8.1
8.7
9.5
10.6
10.4
9.8
Error
3-mo.avg. Error
–1.2
–0.7
0.9
1.1
1.5
–0.3
–0.9
–0.7
8.5
8.4
8.9
9.9
10.4
9.6
0.5
1.4
1.1
0.4
–0.9
–0.5
An examination of the forecast errors reveals that for periods 4 through 9,
the 3-month moving average has the smallest error for two periods,  = .3 has the smallest error
for three periods, and  = .7 has the smallest error for one period. The results are mixed.
16.9
Year
1
2
3
4
5
6
7
8
9
10
11
12
13
No.Issues
332
694
518
222
209
172
366
512
667
571
575
865
609
F( = .2)
e
F( = .9)
e
–
332.0
404.4
427.1
386.1
350.7
315.0
325.2
362.6
423.5
453.0
477.4
554.9
362.0
113.6
205.1
177.1
178.7
51.0
186.8
304.4
147.5
122.0
387.6
54.1
332.0
657.8
532.0
253.0
213.4
176.1
347.0
495.5
649.9
578.9
575.4
836.0
362.0
139.8
310.0
44.0
41.4
189.9
165.0
171.5
78.9
3.9
289.6
227.0
 e = 2289.9
For  = .2, MAD =
2289.9
= 190.8
12
For  = .9, MAD =
2023.0
= 168.6
12
 = .9 produces a smaller mean average error.
 e =2023.0
Chapter 16: Time Series Forecasting and Index Numbers
Trend line: Members = 17,206 – 62.7 Year
R2 = 80.9%
se = 158.8
F = 63.54, reject the null hypothesis.
Regression Plot
Members = 17206.2 - 62.6814 Year
S = 158.837
R-Sq = 80.9 %
R-Sq(adj) = 79.6 %
17400
17200
17000
Members
16.11
16800
16600
16400
16200
16000
0
5
10
Year
15
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16.13
Month
Jan.(yr. 1)
Feb.
Mar.
Apr.
May
June
Broccoli
132.5
164.8
141.2
133.8
138.4
150.9
12-Mo. Mov.Tot.
2-Yr.Tot.
TC
1655.2
July
146.6
3282.8136.78
93.30
3189.7132.90
90.47
3085.0128.54
92.67
3034.4126.43
98.77
2996.7124.86
111.09
2927.9122.00
100.83
2857.8119.08
113.52
2802.3116.76
117.58
2750.6114.61
112.36
2704.8112.70
92.08
2682.1111.75
99.69
2672.7111.36
102.73
1627.6
Aug.
146.9
1562.1
Sept.
138.7
1522.9
Oct.
128.0
1511.5
Nov.
112.4
1485.2
Dec.
121.0
1442.7
Jan.(yr. 2)
104.9
1415.1
Feb.
99.3
1387.2
Mar.
102.0
1363.4
Apr.
122.4
May
112.1
1341.4
1340.7
June
108.4
1332.0
July
119.0
Aug.
119.0
Sept.
114.9
Oct.
106.0
Nov.
111.7
Dec.
112.3
SI
Chapter 16: Time Series Forecasting and Index Numbers
16.15 Regression Analysis
The regression equation is:
Predictor
Coef
Constant
0.6283
Shelter
0.6905
s = 2.018
Food
14.3
8.5
3.0
6.3
9.9
11.0
8.6
7.8
4.1
2.1
3.8
2.3
3.2
4.1
4.1
5.8
5.8
2.9
1.2
2.2
2.4
2.8
3.3
2.6
2.2
2.1
R-sq = 64.1%
Yˆ
Shelter
9.6
9.9
5.5
6.6
10.2
13.9
17.6
11.7
7.1
2.3
4.9
5.6
5.5
4.7
4.8
4.5
5.4
4.5
3.3
3.0
3.1
3.2
3.2
3.1
3.3
2.9
(e  e
t
Food = 0.628 + 0.690 Shelter
Stdev
t-ratio
p
0.7583
0.83
0.416
0.1055
6.54
0.000
t 1
7.2570
7.4642
4.4260
5.1855
7.6713
10.2262
12.7810
8.7071
5.5308
2.2164
4.0117
4.4950
4.4260
3.8736
3.9426
3.7355
4.3569
3.7355
2.9069
2.6997
2.7688
2.8378
2.8378
2.7688
2.9069
2.6307
R-sq(adj) = 62.6%
e
7.04296
1.03581
–1.42599
1.11446
2.22866
0.77382
–4.18103
–0.90709
–1.43079
–0.11640
–0.21169
–2.19504
–1.22599
0.22641
0.15736
2.06451
1.44306
–0.83549
–1.70690
–0.49975
–0.36880
–0.03785
0.46215
–0.16880
–0.70690
–0.53070
e2
49.6033
1.0729
2.0335
1.2420
4.9669
0.5988
17.4810
0.8228
2.0472
0.0135
0.0448
4.8182
1.5031
0.0513
0.0248
4.2622
2.0824
0.6981
2.9135
0.2497
0.1360
0.0014
0.2136
0.0285
0.4997
0.2816
)2 = 36.09 + 6.06 + 6.45 + 1.24 + 2.12 + 24.55 + 10.72 +
0.27 + 1.73 + 0.01 + 3.93 + 0.94 + 2.11 + 0.00 + 3.64 + 0.39 + 5.19 +
0.76 + 1.46 + 0. 17 + 0.11 + 0.25 + 0.40 + 0.29 + 0.31 = 109.19
D =
 (e  e
e
t
2
t 1
)2

109.19
= 1.12
97.69
Since D = 1.12 is less than dL, the decision is to reject the null hypothesis. There is significant
autocorrelation.
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Solutions Manual and Study Guide
16.17
The regression equation is:
Failed Bank Assets = 1,379 + 136.68 Number of Failures
ŷ = 21,881 (million $)
for x= 150:
R2 = 37.9%
adjusted R2 = 34.1%
se = 13,833
F = 9.78, p = .006
The Durbin Watson statistic for this model is:
D = 2.49
The critical table values for k = 1 and n = 18 are dL = 1.16 and dU = 1.39. Since the observed
value of D = 2.49 is above dU, the decision is to fail to reject the null hypothesis. There is no
significant autocorrelation.
Failed Bank Assets
8,189
104
1,862
4,137
36,394
3,034
7,609
7,538
56,620
28,507
10,739
43,552
16,915
2,588
825
753
186
27
Number of Failures
11
7
34
45
79
118
144
201
221
206
159
108
100
42
11
6
5
1
ŷ
2,882.8
2,336.1
6,026.5
7,530.1
12,177.3
17,507.9
21,061.7
28,852.6
31,586.3
29,536.0
23,111.9
16,141.1
15,047.6
7,120.0
2,882.8
2,199.4
2,062.7
1,516.0
e
5,306.2
–2,232.1
–4,164.5
–3,393.1
24,216.7
–14,473.9
–13,452.7
–21,314.6
25,033.7
– 1,029.0
–12,372.9
27,410.9
1,867.4
–4,532.0
–2,057.8
–1,446.4
–1,876.7
–1,489.0
e2
28,155,356
4,982,296
17,343,453
11,512,859
586,449,390
209,494,371
180,974,565
454,312,622
626,687,597
1,058,894
153,089,247
751,357,974
3,487,085
20,539,127
4,234,697
2,092,139
3,522,152
2,217,144
Chapter 16: Time Series Forecasting and Index Numbers
16.19
Starts
311
486
527
429
285
275
400
538
545
470
306
240
205
382
436
468
483
420
404
396
329
254
288
302
351
331
361
364
lag1 lag2
*
*
311
*
486
311
527
486
429
527
285
429
275
285
400
275
538
400
545
538
470
545
306
470
240
306
205
240
382
205
436
382
468
436
483
468
420
483
404
420
396
404
329
396
254
329
288
254
302
288
351 302
331 351
361 331
The model with 1 lag:
Housing Starts = 158 + 0.589 lag 1
F = 13.66 p = .001 R2 = 35.3% adjusted R2 = 32.7% se = 77.55
The model with 2 lags:
Housing Starts = 401 – 0.065 lag 2
F = 0.11 p = .744 R2 = 0.5% adjusted R2 = 0.0% Se = 95.73
The model with 1 lag is the best model with a very modest R2 32.7%. The model
with 2 lags has no predictive ability.
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Solutions Manual and Study Guide
16.21 Year
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
Price a.) Index1950
22.45
100.0
31.40
139.9
32.33
144.0
36.50
162.6
44.90
200.0
61.24
272.8
69.75
310.7
73.44
327.1
80.05
356.6
84.61
376.9
87.28
388.8
16.23
Totals
b.) Index1980
32.2
45.0
46.4
52.3
64.4
87.8
100.0
105.3
114.8
121.3
125.1
1985
1.31
1.99
2.14
2.89
Year
1992
1.53
2.21
1.92
3.38
1997
1.40
2.15
2.68
3.10
8.33
9.04
9.33
Index1987 =
8.33
(100) = 100.0
8.33
Index1992 =
9.04
(100) = 108.5
8.33
Index1997 =
9.33
(100) = 112.0
8.33
16.25
Item
1
2
3
4
Quantity
1995
21
6
17
43
P1995Q1995
10.50
7.38
14.28
6.45
Totals
Price
1995
0.50
1.23
0.84
0.15
P2000Q1995
14.07
11.10
12.75
9.03
38.61
Index1997 =
46.95
P
P
Q1995
2000
Q1995
1995
Price
2000
0.67
1.85
0.75
0.21
Price
2001
0.68
1.90
0.75
0.25
Price
2002
0.71
1.91
0.80
0.25
P2001Q1995
14.28
11.40
12.75
10.75
P2002Q1995
14.91
11.46
13.60
10.75
49.18
50.72
(100) =
46.95
(100) = 121.6
38.61
Chapter 16: Time Series Forecasting and Index Numbers
P
P
Q1995
2001
Index1998 =
Q1995
(100) =
49.18
(100) = 127.4
38.61
(100) =
50.72
(100) = 131.4
38.61
1995
P
P
Q1995
2002
Index1999 =
Q1995
1995
16.27
a) The linear model:
Yield = 9.96 – 0.14 Month
F = 219.24 p = .000
The quadratic model:
F = 176.21
R2 = 90.9 s = .3212
Yield = 10.4 – 0.252 Month + .00445 Month2
p = .000 R2 = 94.4% se = .2582
Both t ratios are significant, for x,
t = –7.93, p = .000 and for x, t = 3.61, p = .002
The linear model is a strong model. The quadratic term adds some
predictability but has a smaller t ratio than does the linear term.
b)
x
10.08
10.05
9.24
9.23
9.69
9.55
9.37
8.55
8.36
8.59
7.99
8.12
7.91
7.73
7.39
7.48
7.52
7.48
7.35
7.04
6.88
6.88
7.17
7.22

F
–
–
–
–
9.65
9.55
9.43
9.46
9.29
8.96
8.72
8.37
8.27
8.15
7.94
7.79
7.63
7.53
7.47
7.46
7.35
7.19
7.04
6.99
e = 6.77
MAD =
e
–
–
–
–
.04
.00
.06
.91
.93
.37
.73
.25
.36
.42
.55
.31
.11
.05
.12
.42
.47
.31
.13
.23
6.77
= .3385
20
293
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Solutions Manual and Study Guide
c)
x
10.08
10.05
9.24
9.23
9.69
9.55
9.37
8.55
8.36
8.59
7.99
8.12
7.91
7.73
7.39
7.48
7.52
7.48
7.35
7.04
6.88
6.88
7.17
7.22
 = .3
F
 = .7
e
–
–
10.08
.03
10.07
.83
9.82
.59
9.64
.05
9.66
.11
9.63
.26
9.55 1.00
9.25
.89
8.98
.39
8.86
.87
8.60
.48
8.46
.55
8.30
.57
8.13
.74
7.91
.43
7.78
.26
7.70
.22
7.63
.28
7.55
.51
7.40
.52
7.24
.36
7.13
.04
7.14
.08
e = 10.06

MAD=.3 =
F
e
–
–
10.08 .03
10.06 .82
9.49 .26
9.31 .38
9.58 .03
9.56 .19
9.43 .88
8.81 .45
8.50 .09
8.56 .57
8.16 .04
8.13 .22
7.98 .25
7.81 .42
7.52 .04
7.49 .03
7.51 .03
7.49 .14
7.39 .35
7.15 .27
6.96 .08
6.90 .27
7.09 .13
e = 5.97

10.06
= .4374
23
MAD=.7 =
5.97
= .2596
23
 = .7 produces better forecasts based on MAD.
d) MAD for b) .3385, c) .4374 and .2596. Exponential smoothing with  = .7 produces the
lowest error (.2596 from part c).
Chapter 16: Time Series Forecasting and Index Numbers
e)
TCSI
10.08
4 period
moving tots
8 period
moving tots
TC
SI
10.05
38.60
9.24
76.81
9.60
96.25
75.92
9.49
97.26
75.55
9.44
102.65
75.00
9.38
101.81
72.99
9.12
102.74
70.70
8.84
96.72
68.36
8.55
97.78
66.55
8.32
103.25
65.67
8.21
97.32
64.36
8.05
100.87
62.90
7.86
100.64
61.66
7.71
100.26
60.63
7.58
97.49
59.99
7.50
99.73
59.70
7.46
100.80
59.22
7.40
101.08
58.14
7.27
101.10
56.90
7.11
99.02
56.12
7.02
98.01
56.12
7.02
98.01
38.21
9.23
37.71
9.69
37.84
9.55
37.16
9.37
35.83
8.55
34.87
8.36
33.49
8.59
33.06
7.99
32.61
8.12
31.75
7.91
31.15
7.73
30.51
7.39
30.12
7.48
29.87
7.52
29.83
7.48
29.39
7.35
28.75
7.04
28.15
6.88
27.97
6.88
28.15
7.17
7.22
295
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Solutions Manual and Study Guide
1st Period
2nd Period
3rd Period
4th Period
102.65 97.78 100.64
101.81 103.25 100.26
96.25 102.74 97.32
97.26 96.72 100.87
100.80
101.08
97.49
99.73
98.01
98.01
101.10
99.02
The highs and lows of each period (underlined) are eliminated and the others are
averaged resulting in:
1st
2nd
3rd
4th
total
Seasonal Indexes:
99.82
101.05
98.64
98.67
398.18
Since the total is not 400, adjust each seasonal index by multiplying by
resulting in the final seasonal indexes of:
1st 100.28
2nd 101.51
3rd 99.09
4th 99.12
16.29
Item
1
2
3
4
5
6
1998
3.21
0.51
0.83
1.30
1.67
0.62
1999
3.37
0.55
0.90
1.32
1.72
0.67
2000
3.80
0.68
0.91
1.33
1.90
0.70
2001
3.73
0.62
1.02
1.32
1.99
0.72
2002
3.65
0.59
1.06
1.30
1.98
0.71
Totals
8.14
8.53
9.32
9.40
9.29
Index1998 =
P
P
(100) 
8.14
(100) = 100.0
8.14
P
P
(100) 
8.53
(100) = 104.8
8.14
P
P
(100) 
9.32
(100) = 114.5
8.14
P
P
(100) 
9.40
(100) = 115.5
8.14
P
P
(100) 
9.29
(100) = 114.1
8.14
1998
1998
Index1999 =
1999
1998
Index2000 =
2000
1998
Index2001 =
2001
1998
Index2002 =
2002
1998
400
= 1.004571
398.18
Chapter 16: Time Series Forecasting and Index Numbers
16.31
a) moving average
Year
Quantity
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
3654
3547
3285
3238
3320
3294
3393
3946
4588
6204
7041
7031
7618
8214
7936
7667
7474
7244
7173
6832
6912
b) = .2
F
3495.33
3356.67
3281.00
3284.00
3335.67
3544.33
3975.67
4912.67
5944.33
6758.67
7230.00
7621.00
7922.67
7939.00
7692.33
7461.67
7297.00
7083.00
e
257.33
36.67
13.00
109.00
610.33
1043.67
2228.33
2128.33
1086.67
859.33
984.00
315.00
255.67
465.00
448.33
288.67
465.00
171.00
F
3654.00
3632.60
3563.08
3498.06
3462.45
3428.76
3421.61
3526.49
3738.79
4231.83
4793.67
5241.14
5716.51
6216.01
6560.01
6781.41
6919.93
6984.74
7022.39
6984.31
 e =11,765.33
MADmoving average =
MAD=.2 =
c)
e
numberforecasts
e
numberforecasts
=
=
e
325.08
178.06
168.45
35.76
524.39
1061.51
2465.21
2809.17
2237.33
2376.86
2497.49
1719.99
1106.99
692.59
324.07
188.26
190.39
72.31
 e =18,973.91
11,765.33
= 653.63
18
18,973.91
= 1054.11
18
The three-year moving average produced a smaller MAD (653.63) than did
exponential smoothing with  = .2 (MAD = 1054.11). Using MAD as the criterion, the threeyear moving average was a better forecasting tool than the exponential smoothing with  = .2.
297
298
Solutions Manual and Study Guide
16.35
1999
P Q
0.83 21
0.89 5
1.43 70
1.05 12
3.01 27
7.21
Item
Marg.
Short.
Milk
Coffee
Chips
Total
Index1999 =
2000
P Q
0.81 23
0.87 3
1.56 68
1.02 13
3.06 29
7.32
P
P
(100) 
7.21
(100) = 100.0
7.21
P
P
(100) 
7.32
(100) = 101.5
7.21
P
P
(100) 
7.43
(100) = 103.05
7.21
1999
1999
Index2000 =
2000
1999
Index2001 =
2001
1999
P1999Q1999
17.43
4.45
100.10
12.60
81.27
215.85
Totals
2001
P
Q
0.83 22
0.87 4
1.59 65
1.01 11
3.13 28
7.43
P2000Q1999
17.01
4.35
109.20
12.24
82.62
225.42
IndexLaspeyres2000 =
P
P
P
P
Q1999
2000
Q1999
P2001Q1999
17.43
4.35
111.30
12.24
82.62
229.71
(100) =
225.42
(100) = 104.4
215.85
(100) =
229.71
(100) = 106.4
215.85
1999
IndexLaspeyres2001 =
Q1999
2001
Q1999
1999
Total
P1999Q2000
P1999Q2001
P2000Q2000
P2001Q2001
19.09
2.67
97.24
13.65
87.29
219.94
18.26
3.56
92.95
11.55
84.28
210.60
18.63
2.61
106.08
13.26
88.74
229.32
18.26
3.48
103.35
11.11
87.64
223.84
IndexPaasche2000 =
P
P
Q2000
2000
Q2000
(100) =
229.32
(100) = 104.3
219.94
(100) =
223.84
(100) = 106.3
210.60
1999
IndexPaasche2001 =
P
P
Q2001
2001
Q2001
1999
Chapter 16: Time Series Forecasting and Index Numbers
16.37
Year
x
Fma
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
100.2
102.1
105.0
105.9
110.6
115.4
118.6
124.1
128.7
131.9
133.7
133.4
132.0
131.7
132.9
133.0
131.3
129.6
Fwma
103.3
105.9
109.2
112.6
117.2
121.7
125.8
129.6
131.9
132.8
132.7
132.5
132.4
132.2
SEMA
SEWMA
104.3 53.29 39.69
107.2 90.25 67.24
111.0 88.36 57.76
114.8 132.25 86.49
119.3 132.25 88.36
124.0 104.04 62.41
128.1 62.41 31.36
131.2 14.44 4.84
132.7
0.01 0.49
132.8
1.21 1.21
132.3
0.04 0.36
132.4
0.25 0.36
132.6
1.21 1.69
132.2
6.76 6.76
SE = 678.80 440.57
MSEma =
MSEwma =
 SE
numberforecasts

 SE
numberforecasts
686.77
= 49.06
14

449.02
= 32.07
14
The weighted moving average does a better job of forecasting the data using MSE as the
criterion.
299
300
Solutions Manual and Study Guide
16.39-16.41:
Qtr TSCI 4qrtot
Year1 1
8qrtot
TC
SI
TCI
T
54.019
2
56.495
3
50.169
213.574
425.044 53.131 94.43 51.699 53.722
211.470
4
52.891
Year2 1
51.915
421.546 52.693 100.38 52.341 55.945
210.076
423.402 52.925 98.09 52.937 58.274
213.326
2
55.101
430.997 53.875 102.28 53.063 60.709
217.671
3
53.419
4
57.236
440.490 55.061 97.02 55.048 63.249
222.819
453.025 56.628 101.07 56.641 65.895
230.206
Year3 1 57.063
467.366 58.421 97.68 58.186 68.646
237.160
2 62.488
480.418 60.052 104.06 60.177 71.503
243.258
3 60.373
492.176 61.522 98.13 62.215 74.466
248.918
4 63.334
503.728 62.966 100.58 62.676 77.534
254.810
Year4 1 62.723
512.503 64.063 97.91 63.957 80.708
257.693
2 68.380
518.498 64.812 105.51 65.851 83.988
260.805
3 63.256
524.332 65.542 96.51 65.185 87.373
263.527
4 66.446
526.685 65.836 100.93 65.756 90.864
263.158
Year5 1 65.445
526.305 65.788 99.48 66.733 94.461
263.147
2 68.011
526.720 65.840 103.30 65.496 98.163
263.573
3 63.245
521.415 65.177 97.04 65.174 101.971
257.842
4 66.872
511.263 63.908 104.64 66.177 105.885
253.421
Year6 1 59.714
501.685 62.711 95.22 60.889 109.904
248.264
2 63.590
3 58.088
4 61.443
491.099 61.387 103.59 61.238 114.029
Chapter 16: Time Series Forecasting and Index Numbers
Quarter Year1
1
2
3
94.43
4
100.38
Year2 Year3 Year4 Year5 Year6 Index
98.09
97.68
97.91
99.48 95.22
97.89
102.28 104.06 105.51 103.30 103.59 103.65
97.02
98.13
96.51
97.04
96.86
101.07 100.58 100.93 104.64
100.86
Total
399.26
400
Adjust the seasonal indexes by:
= 1.00185343
399.26
Adjusted Seasonal Indexes:
16.43
Quarter
Index
1
2
3
4
98.07
103.84
97.04
101.05
Total
400.00
The regression equation is:
Equity Funds = –591 + 3.01 Taxable Money Markets
R2 = 97.1%
Equity TaxMkts
44.4
41.2
53.7
77.0
83.1
116.9
161.5
180.7
194.8
249.0
245.8
411.6
522.8
749.0
866.4
1,269.0
1,750.9
2,399.3
2,978.2
4,041.9
3,962.3
74.5
181.9
206.6
162.5
209.7
207.5
228.3
254.7
272.3
358.7
414.7
452.6
451.4
461.9
500.4
629.7
761.8
898.1
1,163.2
1,408.7
1,607.2
se = 225.9
ŷ
et
–366.69
411.091
– 43.64
84.837
30.66
23.040
–101.99
178.991
39.98
43.116
33.37
83.533
95.93
65.568
175.34
5.358
228.28
–33.482
488.17 –239.170
656.62 –410.815
770.62 –359.017
767.01 –244.207
798.59 – 49.591
914.40 – 47.997
1,303.33 –34.325
1,700.68
50.224
2,110.66 288.639
2,908.07
70.131
3,646.52 395.378
4,243.60 –281.301
e
t
2
= 969,697
et – et–1
e t2
168,996
7,197
531
32,038
1859
6,978
4,299
29
1,121
57,202
168,769
128,893
59,637
2,459
2,304
1,178
2,522
83,313
4,918
156,323
79,131
(et – et–1)2
–326.254
– 61.797
155.951
–135.875
40.417
–17.965
–60.210
–38.840
–205.688
–171.645
51.798
114.810
194.616
1.594
13.672
84.549
238.415
–218.508
325.247
–676.679
 (e  e
t
t 1
106,441.673
3,818.869
24,320.714
18,462.016
1,633.534
322.741
3,625.244
1,508.546
42,307.553
29,462.006
2,683.033
13,181.336
37,875.387
2.541
186.924
7,148.533
56,841.712
47,745.746
105,785.611
457,894.469
)2 = 961,248.188
301
302
Solutions Manual and Study Guide
D =
 (e  e
e
t 1
t
)2
2

t
961,248.188
= 0.99
969,697
For n = 21 and  = .01, dL = 0.97 and dU = 1.16.
Since dL = 0.97 < D = 0.99 < dU = 1.16, the Durbin-Watson test is inconclusive.
16.45
The model is: Bankruptcies = 75,532.436 – 0.016 Year
Since R2 = .28 and the adjusted R2 = .23, this is a weak model.
et – et–1
et
–1,338.58
–8,588.28
–7,050.61
1,115.01
12,772.28
14,712.75
–3,029.45
–2,599.05
622.39
9,747.30
9,288.84
–434.76
–10,875.36
–9,808.01
–4,277.69
–256.80
(et – et–1)2
–7,249.7
1,537.7
8,165.6
11,657.3
1,940.5
–17,742.2
430.4
3,221.4
9,124.9
–458.5
–9,723.6
–10,440.6
1,067.4
5,530.3
4,020.9
52,558,150
2,364,521
66,677,023
135,892,643
3,765,540
314,785,661
185,244
10,377,418
83,263,800
210,222
94,548,397
109,006,128
1,139,343
30,584,218
16,167,637
 (e  e
t
D =
 (e  e
e
t 1
2
t
t
)2

et2
1,791,796
73,758,553
49,711,101
1,243,247
163,131,136
216,465,013
9,177,567
6,755,061
387,369
95,009,857
86,282,549
189,016
118,273,455
96,197.060
18,298,632
65,946
t 1
)2 =921,525,945
e
t
2
=936,737,358
921,525,945
= 0.98
936,737,358
For n = 16,  = .05, dL = 1.10 and dU = 1.37
Since D = 0.98 < dL = 1.10, the decision is to reject the null hypothesis and conclude that
there is significant autocorrelation.