1
My interest is in the future because I am going to spend the rest of my life there.—
Charles F. Kettering

2

A time series is numerical sequence of values generated over regular time intervals.

The classical time-series model involves four components:

Secular trend ( T t
).

Cyclical movement ( C t
).

Seasonal fluctuation ( S t
).

Irregular variation ( I t
).

The multiplicative model determine the level of the forecast variable Y t
:
Y t
= T t
× C t
× S t
× I t

3

4

Finding the components is difficult.

A direct approach averages past Y t values by exponential smoothing .

The forecast value is computed from
F
t+1
= a
Y t
+ (1
a
)F t

The above involves a single parameter, the smoothing constant
( a ) alpha.

All previous time periods are reflected in the F s, and greater weight is given to the more recent.

5


The preceding slide shows single-parameter forecasts of Blitz Beer sales. These were generated by computer.

The level for a was .20. A greater a will assign more weight to the present.

Quality of forecasts may be measured. Most common is the mean squared error :
6
(Y t
– F t
) 2
MSE = n which averages errors over all forecasts made.

Other measures are the mean absolute deviation
( MAD ) and mean absolute percent error ( MAPE ).


The smoothing constant can be tuned to the past, possibly providing better forecasts.

But single-parameter forecasts may still lead or lag actuals , as seen for Blitz Beer, because the impact of trends is delayed.

Trend T t can be incorporated with a second trend smoothing constant g
(gamma):
T t b t
=
= a
Y g
(T t t
+ (1 – a
)(T
t –1
– T
t –1
) + (1 –
+ b
t –1
) g
)b
t –1
7
F
t+1
= T t
+ b t
 That greatly reduces Blitz Beer’s
MSE .

8

9

Many time series have regular seasonal patterns to be incorporated into forecasts.

The three-parameter model incorporates a seasonal smoothing constant b
(beta):
T t
= a(
Y t
/S
t –p
) + (1 – a
)(T
t –1
+ b
t –1
) b t
= g
(T t
– T
t –1
) + (1 – g
)b
t –1
S t
= b(
Y t
/T t
) + (1 – b
)S
t –p
F
t+1
= (T t
+ b t
) S
t –p+1

10

11

The above works for p = 4 quarters or p =
12 months .

The preceding slide needs 6 quarters to generate the first (very bad) forecast.

The process settles quickly, providing good forecasts p periods into the future.


To forecast years in advance, regression analysis provides a trend line .
Ŷ(X) = a + bX

The independent variable X is years beyond the base period, the forecast or dependent variable is Y.

The regression coefficients a (intercept) and b (slope) are found (with the computer) by applying the least squares method .
12

The following applies to toothpaste sales Y .

13

14

Regression analysis can make forecasts with with a non-time independent variable.

A simple regression employs a straight line.
Ŷ(X) = a + bX

The dependent variable is not time periods, such as:
 store size
 order amount
 weight

For 10 rail shipments, the transportation time Y was forecast for specific distance X .

15

16

Regression fits data employing a multiple regression equation with several predictors:
Ŷ = a + b
1
X
1
+ b
2
X
2

Floorspace X
1 and advertising expense X
2 make forecasts of hardware outlet sales Y :
Ŷ
=
-
22,979 + 11.42
X
1
+ 23.41
X
2

The above was obtained in a computer run using 10 data points.

Forecast with X
1
=2,500 sq.ft. and X
2
=$750:
Ŷ =
-
22,979+11.42(2,500) + 23.41(750)=$23,129


The classical time-series model provides a rationale for isolating seasonal components:
17
Y t
Moving average
=
T t
× C t
× S t
× I t
T t
× C t
= S t
× I t

The procedure is the ratio-to-movingaverage method :

(1) Compute moving averages from Y s.

(2) Center above.

(3) Express Y s as % of moving average.

(4) Determine medians by season and adjust.

Above (done on computer) leaves only S t
.

18

Single-parameter exponential smoothing

Two-parameter exponential smoothing

Three-parameter exponential smoothing

Regression

Classical time series analysis

19
Single-parameter exponential smoothing is easy with Excel’s ToolPak. Click on Tools on the menu bar, select the Data Analysis option, and then in the Data Analysis dialog box, click on Exponential Smoothing.

(Figure 7-4 )
1. Enter the smoothing constant in D2.
3. If more than
20 periods of data are available, expand columns
B, C, and D to include all the data by inserting rows. Then copy the formula in E26 down to all the cells in the expanded table.
1
2
3
4
5
26
27
28
29
30
31
22
23
24
25
18
19
20
21
11
12
13
14
15
16
17
6
7
8
9
10
A B C D E F
Forecast of Blitz Beer Sales by Single-Parameter Smoothing a
= 0.2
Month
Period Actual Forecast t Sales, Y t
Sales, F t
January 2000
February
March
April
May
June
July
August
September
October
November
December
January 2001 13
February 14
March
April
15
16
May
June
July
August
September
17
18
19
20
21
8
9
6
7
10
11
12
3
4
5
1
2
4,890
4,910
4,970
5,010
5,060
5,100
5,050
5,170
5,180
5,240
5,220
5,280
5,330
5,380
5,440
5,460
5,520
5,490
5,550
5,600
5153.5
5188.8
5227.0
5269.6
5307.7
5350.2
5378.1
5412.5
5450.0
4890.0
4894.0
4909.2
4929.4
4955.5
4984.4
4997.5
5032.0
5061.6
5097.3
5121.8
MSE = 24,254
2. Enter problem information in
B6:D25. Notice
D26 does not have a value because it is to be forecast.
4. Click on Tool,
Data Analysis, and the
Exponential
Smoothing to get the
Exponential
Smoothing dialog box shown next.
20
28
D
=SUMXMY2(D7:D25,E7:E25)/COUNT(E7:E25)

1. In the Input
Range line enter the range of the data. The result shown is
$D$6:$D$25
2. Enter the
Damping factor.
It is 1 a.
3. In the Output
Range enter the location of the results.
(Figure 7-5)
4. Click the OK button to get the results shown previously in Figure 7-4.
21

1. Enter the smoothing constants in C2 and E2.
(Figure 7-6)
2. Enter problem information in
A6:C25. Notice
C26:E25 do not have values because they correspond to the period for which the forecast is being made.
22
A B
Period t
13
14
15
16
9
10
11
12
7
8
5
6
3
4
1
2
17
18
19
20
21
C
Actual
Sales, Y t
4,890
4,910
4,970
5,010
5,060
5,100
5,050
5,170
5,180
5,240
5,220
5,280
5,330
5,380
5,440
5,460
5,520
5,490
5,550
5,600
D E
Forecast of Blitz Beer Sales by Two-Parameter Smoothing a
= 0.20
g
= 0.30
F
1
2
3
4
5
18
19
20
21
14
15
16
17
10
11
12
13
8
9
6
7
26
27
28
29
30
22
23
24
25
31
32
33
Month
January 2000
February
March
April
May
June
July
August
September
October
November
December
January 2001
February
March
April
May
June
July
August
September
7
8
9
=C6
D
MSE =
=$C$2*C8+(1-$C$2)*(D7+E7)
=$C$2*C9+(1-$C$2)*(D8+E8)
Trend
4,890
4,922
4,958
5,001
5,046
5,076
5,122
5,164
5,210
5,245
5,283
5,324
5,367
5,414
5,457
5,504
5,536
5,571
5,608
1,734.64
=C7-C6
T t
E
Trend
Slope, b
20.0
23.6
27.5
31.9
35.9
34.0
37.6
38.9
41.1
39.3
39.0
39.5
40.5
42.5
42.7
43.9
40.5
38.9
38.3
t
=$E$2*(D8-D7)+(1-$E$2)*E7
=$E$2*(D9-D8)+(1-$E$2)*E8
Forecast
Sales, F
F
=D7+E7
=D8+E8 t
4,910.0
4,945.6
4,985.9
5,032.7
5,082.1
5,109.7
5,159.4
5,202.4
5,251.0
5,284.1
5,322.3
5,363.3
5,407.2
5,456.2
5,499.6
5,547.6
5,576.5
5,610.1
5,646.3
3. If more than
20 periods of data are available, expand columns A, B, and C to include all the data by inserting rows.
Copy the formulas in
D25:E25 down to the next to last row in the expanded table. Finally, copy F26 down to all the rows in the expanded table.

2. To minimize the MSE,
Min is selected in the Equal
To line.
3. The
Changing
Cells line has the location of the two smoothing parameters.
23
(Figure 7-7)
NOTE: Normally all these entries appear in the Solver Parameter dialog box so you only need to click on the Solve button. However, you should always check to make sure the entries are correct for the problem you are solving. 1. The location of the MSE is in the Set Target Cell line.
6. Click the
Solve button to get Figure
7-6, shown previously.
4. In the
Subject to the
Constraints box the two smoothing parameters are restricted to being equal to or less than one.
5. Click the
Options button to verify that nonnegative smoothing parameters are specified.

Normally, all these entries already appear.
You will need to use this dialog box only if you need to add a constraint.
If you need to change a constraint, the
Change
Constraint dialog box functions just like this one.
24
(Figure 7-8)
The Add Constraint dialog box is used to specify the constraints on the variable in the
Changing Cells line. Enter a smoothing parameter such as C2 in the Cell Reference line. Select a sign by clicking on the down arrow in the middle, and then enter the right-hand side in the Constraint line (1 here). Click OK when finished.

1. Enter the smoothing constants in C2,
E2, and G2.
2. Enter problem information in
A6:C33.
3. If less or more than 7 years of data are available, shorten or expand columns
A, B, and C to include all the data by deleting or inserting rows. Copy the formulas in
D33:G33 down through the last quarter of the next to last year in the expanded
25 table.
(Figure 7-11)
A B
Seasonal Exponential Smoothing Results for Stationer's Supply a
= 0.4
C D g
= 0.5
E F b
= 0.7
G
1
2
3
4
5
Actual
Quarter t Sales, Y t
Trend
T t
Slope b t
Seasonal Forecast
S t
F t
16
17
30
31
32
33
34
9
10
11
12
13
14
15
6
7
8
39
40
41
42
43
35
36
37
38
44
45
46
47
1994 W 1 90,640
1988 S
1988 S
2 115,540
3 99,190
1988 F 4 128,800
1995 W 5 102,350
1988 S
1988 S
1988 F
6 127,440
7 112,530
8 145,080
1996 W 9 110,490
1988 S 10 143,000
1988 S 11 119,700
1988 F 12 157,760
90,640.00
109,000.00
129,898.00
131,637.20
125,873.45
126,709.16
135,663.60
141,482.77
140,679.34
24,900.00
21,630.00
21,264.00
11,501.60
2,868.93
1,852.32
5,403.38
5,611.27
2,403.92
1.27
0.91
0.99
0.78
139,367.04
144,273.63
545.81
2,726.20
2000 W 25 157,500
1988 S 26 192,240
1988 S 27 168,330
1988 F 28 218,960
2001 W 29
1988
1988
1988
S
S
F
30
31
32
1 1
1 2
1 3
189,364.54
193,811.82
197,257.56
200,406.97
3,818.46
4,132.87
3,789.30
3,469.36
0.83
0.99
0.85
1.09
F G
=$G$ 2*(C 11/ D11)+(1-$ G$2)* F7 =(D1 0+E 10)* F7
=$G$ 2*(C 12/ D12)+(1-$ G$2)* F8 =(D1 1+E 11)* F8
=$G$ 2*(C 13/ D13)+(1-$ G$2)* F9 =(D1 2+E 12)* F9
1.09
0.89
1.05
0.78
1.04
0.87
1.08
11
12
13
7
8
9
10
MSE = 182,250,569
=C 6
D
=$ C$ 2 *C8+(1 -$ C$ 2)*(D 7 +E7 )
=C 7-C 6
E
=$ E$ 2 *(D 8 -D 7 )+(1 -$ E$ 2)*E7
F
=C 7 /D 7
=C 8 /D 8
=$ C$ 2 *C9+(1 -$ C$ 2)*(D 8 +E8 ) =$ E$ 2 *(D 9 -D 8 )+(1 -$ E$ 2)*E8 =C 9 /D 9
=$ C $2 *C 10+(1 -$ C$ 2)*(D 9 +E9 ) = $E$ 2 *(D1 0 -D 9 )+(1 -$ E$ 2)*E9 =C1 0/D 10
D
=$C $2*(C 11 /F 7)+(1 -$C $2)*(D 10+E 10 )
=$C $2*(C 12 /F 8)+(1 -$C $2)*(D 11+E 11 )
=$C $2*(C 13 /F 9)+(1 -$C $2)*(D 12+E 12 )
182,460.91
117,155.57
127,474.78
109,681.80
160,498.09
128,011.99
146,355.98
153,529.94
190,688.25
169,803.61
220,716.23
168,800.51
205,365.30
180,182.79
234,460.18
39
E
= SUM XMY2(C11:C33,G11:G 33)
/COUNT (G11:G 33)
4. Finally, copy the four forecasting formulas in
G34:G37 to the last four rows of the expanded table (the last year).
Make sure that these formulas refer back to last row of data. In this spreadsheet the forecasting formulas in cells G34:G37 refer back to row 33 which contains the data for the fall of
2000, the last period.
Thus, if the four forecasting formulas in the expanded table are in
G44:G47 then they should refer back to row
43, the last row of data.
11
12
13
F G
=$G$ 2*(C1 1/D11) +(1-$G$2 )*F7 = (D10 +E10 )*F7
=$G$ 2*(C1 2/D12) +(1-$G$2 )*F8 = (D11 +E11 )*F8
=$G$ 2*(C1 3/D13) +(1-$G$2 )*F9 = (D12 +E12 )*F9
11
12
13
7
8
9
10
D
= C6
=$C$2*C8+ (1-$C $2)*(D 7+E 7)
=C7-C6
E
=$E$2*(D8-D7) +(1- $E$2 )*E7
F
=C7/D7
=C8/D8
=$C$2*C9+ (1-$C $2)*(D 8+E 8) =$E$2*(D9-D8) +(1- $E$2 )*E8 =C9/D9
=$C$2*C10+ (1-$C $2)*(D 9+E 9) =$E$2*(D10-D9) +(1- $E$2 )*E9 = C10/D10
D
=$C$2*(C 11/F7)+(1-$C$2)*(D10+E10)
=$C$2*(C 12/F8)+(1-$C$2)*(D11+E11)
=$C$2*(C 13/F9)+(1-$C$2)*(D12+E12)
39
E
= SUMXMY2(C11:C33,G 11:G33)
/CO UNT(G 11:G33)

26
Regression is easy with Excel’s Regression
Tool. Click on Tools on the menu bar, select the Data Analysis option, and then in the Data
Analysis dialog box select Regression. This yields the Regression dialog box shown next.

1. In the Input Y
Range line enter the range of the Y data.
The result shown here is $C$7:$C$16
Regression Dialog Box
(Figure 7-18)
3. Click on the
OK button to get the Regression
Summary
Output shown next.
2. In the Input X
Range line enter the range of the
X data. The result here shown is
$B$7:$B$16
27

(Figure 7-16)
The slope and intercept are read from E15:E16 and yield the regression equation below. The multiple R,
R squared, adjusted R, standard error, and F and t
28
7
8
9
10
11
12
13
14
15
16
1
2
3
4
5
6 statistics are shown also.
A B C D E F
Fitting Trend Line to BriDent Toothpaste Sales Using Regression
Year
Year in
Transformed
Units
X
Unit
Sales
(thousands)
Y
SUMMARY OUTPUT
Regressi on S tat ist ics
Multiple R 0.98
R Square
Adjusted R Square
Standard Er ror
Observat ions
0.96
0.96
0.90
10
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2
3
0
1
4
5
8
9
6
7
72.9
74.4
75.9
77.9
78.6
79.1
81.7
84.4
Y

85.9
84.8

ANO VA
Regr ession
Resi dual
Tot al
I nter cept
X
72 ..
96
 df
1
8
9
1 .
47 X
SS
177.47
6.46
183.92
M S F
177.47
219.86
0.81
C oeff icients St andard Er ror t Stat
72.96
0.53
138.17
1.47
0.10
14.83
G
S ignificance F
0.00

29
(Figure 7-21)
26
27
28
29
30
31
32
20
21
22
23
24
25
33
34
35
16
17
18
19
12
13
14
15
7
8
9
10
11
36
1
2
3
4
5
6
A
Store
8
9
6
7
10
1
4
5
2
3
ANOVA
Regression
Residual
Total
Intercept
X
1
X
2
B C D E
Multiple Regression for the Deuce Hardware Store
Monthly
Sales
Y
20,100
14,900
16,800
9,100
15,500
26,700
34,600
7,200
21,800
23,400
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.89332611
R Square 0.798031538
Standard Error 4168.371133
Observations 10 df
Floorspace
Monthly
Advertising
(square feet) Expenditure
X
1
X
2
Y
3,050
1,300
1,890
1,750
1,010
2,690
4,210
1,950
2,830
2,030

 -
350
980
830
760
930
770
440
570
310
920
22 , 979

11
F
Excel’s
-47917.10
5.99
2.99
G regression tool can be used to do multiple regression. Just list ALL the X variables when designating the
Input X Range;
C7:D16 in this example.
.
42
SS MS F Significance F
2 480581774.7
240290887.3
13.8294383
0.003702478
7 121627225.3
17375317.9
9 602209000
Coefficients Standard Error
-22979 10546.50
11.42
2.29
23.41
8.64
t Stat
-2.18
4.98
2.71
P-value
0.07
0.00
0.03
X
1

Lower 95% Upper 95%
1959.87
16.84
43.84
23 .
41 X
2

1. Enter problem information in A19:C46.
2. If more than 7 years of data are available, expand columns A, B, and C to include all the data by inserting rows.
Copy the formulas in
D44:F44 down through the first two quarters of the last year of the data in the expanded table. The
Seasonal Indices in column G are the same every year so enter their values in the expanded table (for the rows corresponding to data and also the rows for the year for which forecasts are being calculated).
30
(Figure 7-22)
1
4
5
2
3
10
11
12
13
8
9
6
7
14
15
16
17
18
1994 W
1988 S
1988 S
1988 F
1995 W
1988 S
1988 S
1988 F
1996 W
1988 F
2000 W
1988 S
1988 S
1988 F
2001 W
1988 S
1988 S
1988 F
46
47
48
49
42
43
44
45
50
51
52
23
24
25
26
27
19
20
21
22
53
54
55
A B C D E F G H
Quarter
CLASSICAL TIME SERIES ANALYSIS
C
PROBLEM: Stationer's Supply
Ratio Index
6
7
8
9
= MEDIAN(F23,F27,F31,F35,F39,F43)
= MEDIAN(F24,F28,F32,F36,F40,F44)
= MEDIAN(F21,F25,F29,F33,F37,F41)
= MEDIAN(F22,F26,F30,F34,F38,F42)
Winter
Spring
Summer
Fall
88.15% 88.18%
105.92% 105.95%
90.43% 90.46%
115.36% 115.40%
8
9
6
7
D
=C6*4/SUM($C$6:$C$9)
=C7*4/SUM($C$6:$C$9)
=C8*4/SUM($C$6:$C$9)
=C9*4/SUM($C$6:$C$9)
19
20
21
H
=C19/G19
=C20/G20
=C21/G21
PROBLEM: Stationer's Supply
PROBLEM DATA
20
21
22
D
=AVERAGE(C19:C22)
E F
=AVERAGE(C20:C23) =AVERAGE(D20:D21) =C21/E21
=AVERAGE(C21:C24) =AVERAGE(D21:D22) =C22/E22
Quarter t
Four-
Quarter Centered Original as a
Moving Moving Percentage of Seasonal Seasonally
Average Average Moving Average Index Adjusted
28
29
30
31
32
24
25
26
27
7
8
5
6
9
3
4
1
2
Actual
Sales, Y t
90,640
115,540
99,190
128,800
102,350
127,440
112,530
145,080
110,490
202,960
157,500
192,240
168,330
218,960
168,488
205,759
178,505
231,317
108,543
111,470 110,006
114,445 112,958
117,780 116,113
121,850 119,815
123,885 122,868
127,775 125,830
129,568 128,671
177,045 175,010
180,258 178,651
184,258 182,258
90.17%
114.03%
88.15%
106.36%
91.59%
115.30%
85.87%
115.97%
88.16%
105.48%
88.18% 102,786.7
105.95% 109,046.3
90.46% 109,646.5
115.40% 111,612.8
88.18% 116,066.0
105.95% 120,277.5
90.46% 124,392.8
115.40% 125,720.3
88.18% 125,296.9
115.40% 175,876.7
88.18% 178,606.7
105.95% 181,435.6
90.46% 186,075.1
115.40% 189,741.7
88.18% 191,066.8
105.95% 194,194.5
90.46% 197,322.3
115.40% 200,450.1
47
48
49
50
C
=H47*G47
=H48*G48
=H49*G49
=H50*G50
47
48
49
50
H
=FORECAST(B47,$H$19:$H$46,$B$19:$B$46)
=FORECAST(B48,$H$19:$H$46,$B$19:$B$46)
=FORECAST(B49,$H$19:$H$46,$B$19:$B$46)
=FORECAST(B50,$H$19:$H$46,$B$19:$B$46)
3. (cont’d)
Copy the formula in H44 down through the end of the data in the expanded table. Make sure that the forecasting formulas in the expanded table (the ones currently in H47:H50) refer back to the last period of data (like the formulas in H47:H50 refer back to
H46).

31
The three parameter-exponential smoothing and classical time series analysis templates described previously are for quarterly data.
The CD-ROM also contains threeparameter exponential smoothing and classical time series analysis templates for monthly data.