Page 135 number 12
General American Investors Co., a closed-end regulated investment management
company, invest primarily in medium-and high-quality stocks. Jim Campbell is studying the
asset value per share for this company and would like to forecast this variable for the
remaining quarters of 1996. The data are presented in Table P-12.
Evaluate the ability to forecast the asset value per share variable using the following
forecasting methods: naïve, moving averages and exponential smoothing. When you
compare this tecsniques, take into consideration that the actual asset value per share for
the second quarter of 1996 was 26.47. write a report for Jim indicating which method he
should use and why.
Answer:
Plot
Time Series Plot of yt
32.5
30.0
27.5
yt
25.0
22.5
20.0
17.5
15.0
4
8
12
16
20
24
Index
28
32
36
40
44
1. With NAÏVE Method
t
naive1
yt
(stationary
)
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
naive2
naive3
(trend)
(seasonal)
e1
e2
e3
16.98
-
-
-
-
-
-
18.47
16.98
-
-
1.49
-
-
17.63
18.47
19.96
20.09075
-0.84
-2.33
-2.46075
20.65
17.63
16.79
16.8282
3.02
3.86
3.821798
21.95
20.65
23.67
24.18732
1.3
-1.72
-2.23732
23.85
21.95
23.25
23.33184
1.9
0.6
0.51816
20.44
23.85
25.75
25.91446
-3.41
-5.31
-5.47446
19.29
20.44
17.03
17.51755
-1.15
2.26
1.772449
22.75
19.29
18.14
18.2047
3.46
4.61
4.545298
23.94
22.75
26.21
26.83061
1.19
-2.27
-2.89061
24.84
23.94
25.13
25.19225
0.9
-0.29
-0.35225
16.7
24.84
25.74
25.77383
-8.14
-9.04
-9.07383
18.04
16.7
8.56
11.22746
1.34
9.48
6.812544
19.19
18.04
19.38
19.48752
1.15
-0.19
-0.29752
18.97
19.19
20.34
20.41331
-0.22
-1.37
-1.44331
17.03
18.97
18.75
18.75252
-1.94
-1.72
-1.72252
18.23
17.03
15.09
15.2884
1.2
3.14
2.941603
19.8
18.23
19.43
19.51456
1.57
0.37
0.285443
22.89
19.8
21.37
21.50521
3.09
1.52
1.384789
21.41
22.89
25.98
26.46223
-1.48
-4.57
-5.05223
21.5
21.41
19.93
20.02569
0.09
1.57
1.474308
25.05
21.5
21.59
21.59038
3.55
3.46
3.459622
20.33
25.05
28.6
29.18616
-4.72
-8.27
-8.85616
20.6
20.33
15.61
16.49936
0.27
4.99
4.100643
25.33
20.6
20.87
20.87359
4.73
4.46
4.456414
26.06
25.33
30.06
31.14606
0.73
-4
-5.08606
28.89
26.06
26.79
26.81104
2.83
2.1
2.078962
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
forecas
t
30.6
28.89
31.72
32.02733
1.71
-1.12
-1.42733
27.44
30.6
32.31
32.41121
-3.16
-4.87
-4.97121
26.69
27.44
24.28
24.60633
-0.75
2.41
2.083673
28.71
26.69
25.94
25.9605
2.02
2.77
2.749501
28.56
28.71
30.73
30.88288
-0.15
-2.17
-2.32288
25.87
28.56
28.41
28.41078
-2.69
-2.54
-2.54078
24.96
25.87
23.18
23.43336
-0.91
1.78
1.526635
27.61
24.96
24.05
24.08201
2.65
3.56
3.52799
24.75
27.61
30.26
30.54135
-2.86
-5.51
-5.79135
23.32
24.75
21.89
22.18625
-1.43
1.43
1.133745
22.61
23.32
21.89
21.97262
-0.71
0.72
0.637378
24.08
22.61
21.9
21.92162
1.47
2.18
2.158383
22.31
24.08
25.55
25.64557
-1.77
-3.24
-3.33557
22.67
22.31
20.54
20.6701
0.36
2.13
1.999896
23.52
22.67
23.03
23.03581
0.85
0.49
0.484191
25.41
23.52
24.37
24.40187
1.89
1.04
1.00813
23.94
25.41
27.3
27.45188
-1.47
-3.36
-3.51188
25.68
23.94
22.47
22.55504
1.74
3.21
3.124959
26.47
25.68
27.42
27.54647
0.79
Best Naive
-0.95
-1.07647
Autocorrelation Function for naive1
(with 5% significance limits for the autocorrelations)
1.0
0.8
Autocorrelation
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
3
4
5
6
Lag
7
8
9
10
11
9
10
11
Autocorrelation Function for n2
(with 5% significance limits for the autocorrelations)
1.0
0.8
Autocorrelation
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
3
4
5
6
Lag
7
8
Autocorrelation Function for n3
(with 5% significance limits for the autocorrelations)
1.0
0.8
Autocorrelation
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
3
4
5
6
Lag
7
8
9
10
11
 But from the ACF plot we see that the seasonal naïve give the good forecast, all of the first
six tome lags into the confidence interval.
2. With Moving averages method we use 4 quarter in length
Moving Average Plot for yt
32.5
Variable
Actual
Fits
Forecasts
95.0% PI
30.0
27.5
Mov ing A v erage
Length 4
25.0
yt
A ccuracy
MAPE
MAD
MSD
22.5
20.0
17.5
15.0
1
5
Moving Average for yt
Data
Length
NMissing
yt
45
0
Moving Average
Length
4
Accuracy Measures
MAPE
MAD
MSD
9.31334
2.12848
6.92939
Moving Average Plot for yt
MTB > MA 'yt' 4;
SUBC>
Forecasts 1.
Moving Average for yt
Data
Length
NMissing
yt
45
0
Moving Average
10
15
20
25
Index
30
35
40
45
Measures
9.31334
2.12848
6.92939
Length
4
Accuracy Measures
MAPE
MAD
MSD
9.31334
2.12848
6.92939
Forecasts
Period
46
Forecast
24.6375
Lower
19.4781
Upper
29.7969
3. With single exponential smoothing
Smoothing Plot for yt
Single Exponential Method
32.5
Variable
Actual
Fits
Forecasts
95.0% PI
30.0
27.5
Smoothing Constant
Alpha
0.696098
25.0
yt
Accuracy
MAPE
MAD
MSD
22.5
20.0
17.5
15.0
1
5
10
15
Single Exponential Smoothing for yt
Data
Length
yt
45
Smoothing Constant
Alpha
0.696098
Accuracy Measures
MAPE
MAD
MSD
8.42548
1.89435
5.46249
20
25
Index
30
35
40
45
Measures
8.42548
1.89435
5.46249
Forecasts
Period
46
Forecast
25.2269
Lower
20.5858
Upper
29.8679
4. With Double Exponential Smoothing
Smoothing Plot for yt
Double Exponential Method
32.5
Variable
Actual
Fits
Forecasts
95.0% PI
30.0
27.5
Smoothing Constants
Alpha (lev el)
0.951666
Gamma (trend)
0.037978
yt
25.0
Accuracy
MAPE
MAD
MSD
22.5
20.0
17.5
15.0
1
5
10
15
20
25
Index
Double Exponential Smoothing for yt
Data
Length
yt
45
Smoothing Constants
Alpha (level)
Gamma (trend)
0.951666
0.037978
Accuracy Measures
MAPE
MAD
MSD
8.50550
1.87142
5.94963
Forecasts
Period
46
Forecast
25.8330
Lower
21.2481
Upper
30.4178
30
35
40
45
Measures
8.50550
1.87142
5.94963
5. With Winter’s Method (multiplikatif)
Winters' Method Plot for yt
Multiplicative Method
32.5
Variable
Actual
Fits
Forecasts
95.0% PI
30.0
27.5
Smoothing C onstants
A lpha (lev el)
0.2
Gamma (trend)
0.2
Delta (seasonal)
0.2
yt
25.0
22.5
Accuracy Measures
MA PE
11.6253
MA D
2.5842
MSD
10.2970
20.0
17.5
15.0
1
5
10
15
20
Winters' Method for yt
Multiplicative Method
Data
Length
yt
45
Smoothing Constants
Alpha (level)
Gamma (trend)
Delta (seasonal)
0.2
0.2
0.2
Accuracy Measures
MAPE
MAD
MSD
11.6253
2.5842
10.2970
Forecasts
Period
46
Forecast
23.5163
Lower
17.1851
Upper
29.8474
25
Index
30
35
40
45
Moving average
MSD
6.92939
MAPE
9.31334
MAD
2.12848
Forecast
24.6375
Single Exponential
5.46249
8.42548
1.89435
25.2269
Hol’t Method
5.94963
8.50550
1.87142
25.8330
Winter’s Method
10.2970
11.6253
2.5842
23.5163
Conclusion : according to the result of error and value of forecast we can
see that the best method is Double exponential smoothing with forecast
25.83.
Error = 26.47 – 25.83 = 0.64