Exercise 7.5 (p. 343)
Consider the hotel occupancy data in Table 6.4 of Chapter 6 (p. 297)
t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
yt
501
488
504
578
545
632
728
725
585
542
480
530
518
489
528
599
572
659
739
758
602
587
497
558
t
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
yt
555
523
532
623
598
683
774
780
609
604
531
592
578
543
565
648
615
697
785
830
645
643
551
606
t
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
yt
585
553
576
665
656
720
826
838
652
661
584
644
623
553
599
657
680
759
878
881
705
684
577
656
t
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
yt
645
593
617
686
679
773
906
934
713
710
600
676
645
602
601
709
706
817
930
983
745
735
620
698
t
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
yt
665
626
649
740
729
824
937
994
781
759
643
728
691
649
656
735
748
837
995
1040
809
793
692
763
t
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
yt
723
655
658
761
768
885
1067
1038
812
790
692
782
758
709
715
788
794
893
1046
1075
812
822
714
802
t
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
yt
748
731
748
827
788
937
1076
1125
840
864
717
813
811
732
745
844
833
935
1110
1124
868
860
762
877
a. Analyze this data using the multiplicative decomposition method in an Excel
spreadsheet.
The book contains a CD-ROM with the data sets in different data formats.
Note! Data are given consecutively in column 1 (not tabled as in Table 6.4
of the book)
Add a column with the time variable (month)
In column C, add the formulas for calculating CMAt
Monthly data  Formula can be entered in cells C8:C163 (Row 1 is reserved for
column labels)
yt  6  2   yt  5  y t  4    y t    y t  5   yt  6
CMAt 
2 12
 Enter the following formula in cell C8 and copy it to cells C9:C163
“=(B2+2*(B3+B4+B5+B6+B7+B8+B9+B10+B11+B12+B13)+B14)/24”
Next step: Divide yt with CMAt for t = 7,…,162 (Spreadsheet formula is entered
into cells D8:D163)
This gives the rough seasonal components (y / CMA)
Now we need to estimate the seasonal
component, i.e. the 12 seasonal indices
Average the rough seasonal components for
each calendar month
Cell E8 corresponds with calendar month 7, i.e. July
Enter the following formula in E8
“=(D8+D20+D32+D44+D56+D68+D80+D92+D104+D116+D128+D140+
D152)/13”
Copy the formula to cells E9:E17 (August-April). For the cells E18:E19 (May
and June), remove the last term in the sum of the formula and divide by 12.
The contents of cells E8:E19 are now
close to the seasonal components.
Fine adjust them by dividing with their
average.
In cell F8 enter the following formula
=E8/(AVERAGE($E$8:$E$19))
and copy this to the cells F9:F19
Now, the seasonal components shall
be “copied” to the rest of the
relevant cells in column F.
To make it more dynamic (allowing
original observations to be changed)
use formulas depending on the
values calculated in F8:F19
In Cell F2, enter the formula =F14
(as month 1 is the same calendar month as month 13)
In cell F3, enter formula =F15 and so on until cell F25
For the cells F26:F37 enter successively formulas
=$F$2, =$F$3, =$F$4, =$F$5, =$F$6, =$F$7, =$F$8, =$F$9, =$F$10,
=$F$11, =$F$12, =$F$13
The copy and paste these 12 cells further in the column
Now, deseasonalize the original
observations by dividing them
with the corresponding
seasonal component
In cell G2 enter =B2/F2 ,
copy and paste into all relevant
cells of column G.
From deseasonalized data, the trend function can be estimated.
If a linear trend is to be found, we simply use columns A and G, if a quadratic trend
is to be found a new column with squares of the values in A must be created.
Stick to the linear case here.
In Excel, open the menu Tools (open it while a cell in the spreadsheet is active, not
e.g. a chart)
Select the alternative Data
Analysis…
If that alternative does not exist,
select the alternative Add-Ins…
and check the box for the
Analysis ToolPak add-in, then
Data Analysis will appear on the
Tools menu.
Scroll down to the alternative
Regression and select that one.
Enter the cells with the
deseasonalized data here, i.e. cells
G1:G169 (including column label)
Enter the cells with the time
variable(s) here, in our case cells
A1:A169 (including column label)
Check this box as labels are
present (otherwise you may enter
the ranges G2:G169 and A2:A169
in the Input Y and X Ranges
above)
Enter a cell here that does not
interfere with your previously
entered columns or your
prospectively entered columns
(trend column, cyclical
component etc.)
Preferably: Enter a cell under
your data columns, i.e. G171
Click OK, look at cell G171
and below!
Standard regression output. What we need here are the contents of cells H187
(intercept) and H188 (slope parameter).
Column H should now contain the estimated trend component.
In cell H2 enter the formula
=$H$187+$H$188*A2
(tr1 = 554.0933 + 2.003414  t )
Copy and paste into all relevant cells in column H
Next step is to filter out the combined cyclical and irregular component.
This should be stored in column I.
(cl  ir)t = yt* / trt  Enter the formula =G2/H2 in cell I2. Copy and paste into all
relevant cells in column I.
To separate between the cyclical and the irregular component, moving averages need
to be calculated. The moving averages constitute cl and the ratio between (cl  ir)
and cl constitutes ir
Excel has this function among the Data Analysis methods, but it is not consistent
with the description in this course. Only 3-point moving averages are centred, the
rest are skewed.
To make 3-point centred moving averages, enter the following formula into cell J3:
=average(I2:I4)
Copy and paste into cells J4:J168 (Cell J169 like cell J2 will not have any value)
Then enter the formula =I3/J3 into cell K3 , copy and paste into cells K4:K168 to
produce ir.
To make 5-point centred moving averages, enter the following formula into cell J4:
=average(I2:I6)
Copy and paste into cells L5:L167, calculate ir (=I4/L4) in cells M4:M167
Etc.
In this case, calculate 3-, 5- 7- and 9-point moving averages.
To judge upon which of the moving averages that works best, calculate the
standard deviation and the serial correlation coefficient for all estimated irregular
components. Use e.g. rows 171 and 172 for these outputs.
In cell K171 enter the formula =stdev(K2:K169)
It doesn’t matter for this calculation that some cells are empty. Excel only
calculates with active cells. This formula can therefore be copied to cells M171,
O171 and Q171
In cell K172 enter the formula =correl(K2:K168,K3:K169)
This calculates the correlation coefficient between the series of ir terms and the onestep lagged series of ir terms. Note that be benefit of that cell K169 is empty. Empty
cells are treated as missing values in the Excel calculations.
Copy and paste the formula into the cells M172, O172 and Q172. There are more
empty cells in these columns, but that does not affect the result.
Now we can see that the 3-point moving average gives the lowest standard
deviation (0.042967) of the ir terms and the 5-point moving average gives the
lowest serial correlation (in absolute value).
To help in the decision process, it could be wise to plot (cl  ir) together with cl for
the two cases (use the “line” chart type in the Chart Wizard)).
3-point
1.3
1.2
1.1
1
cl x ir
cl(3)
0.9
0.8
0.7
0.6
1
5-point
13 25 37 49 61 73 85 97 109 121 133 145 157
1.3
1.2
1.1
Gives a smoother impression
1
cl x ir
cl(5)
0.9
0.8
0.7
0.6
1 13 25 37 49 61 73 85 97 109 121 133 145 157
b. Use statistical software, such as SAS or MINITAB, to produce point forecasts and
95% prediction intervals for the deseasonalized hotel room averages in each month
the 15th year.
Copy the column of deseasonalized values (G) together with the time column (A) to
the software (here to MINITAB).
Perform a regression of deseasonalized values against time and ask for prediction in
the 15 added time points.
c. Using the values from part (b), compute point and 95% prediction interval
forecasts of the hotel room averages in each month of the 15th year.
See page 339 in the textbook. The same error bound can be used in the prediction
interval for y as those previously used for y* .
In the Minitab worksheet, calculate the bounds as the differences between column
C6 (PLIM2) and C4 (PFIT1) and the differences between C5 (PLIM1) and C4.
Commands that can be used in Session
Window if not Calculator is used
Now, the forecasts for y (hotel room averages) can be formed by multiplying the
forecasted deseasonalized values (in column C4) with the estimated seasonal
components from the Excel spreadsheet.
Thus copy the first 12 seasonal components to MINITAB and then add the bounds to
these forecasts.
Forecasts
Lower 95% Pred.limits
Upper 95% Pred. Limits