Post on 26-Dec-2015
Time-Series Analysis and Forecasting – Part IV
To read at home
Method of moving average
Method of moving average consists in replacement of initial levels of
series by the average values, which are calculated for the successively
changing periods of time
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-4
Moving Averages
Example: Five-year moving average
First average:
Second average:
etc.
5
xxxxxx 54321*
5
5
xxxxxx 65432*
6
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-5
(2m+1)-Point Moving Average
A series of arithmetic means over time Result depends upon choice of m (the
number of data values in each average) Examples:
For a 5 year moving average, m = 2 For a 7 year moving average, m = 3 Etc.
Replace each xt with
m
mjjt
*t m)n,2,m1,m(tX
12m
1X
sum sum average
2001 4,5 - - -
2002 4,3 - - -
2003 5,2 25 19,9 4,982004 5,3 26,5 21,35 5,342005 5,7 28,2 22,6 5,652006 6 28,9 23,3 5,832007 6 29,3 23,6 5,92008 5,9 - - -
2009 5,7 - - -
5,78
Five term moving averageYear yt
5,35,64
5,86--
Centered moving average
average
Four term moving average,
sum
--5 19,3
20,522,223
23,623,6
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-7
Year Sales
1
2
3
4
5
6
7
8
9
10
11
etc…
23
40
25
27
32
48
33
37
37
50
40
etc…
Annual Sales
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10 11
Year
Sa
les
…
…
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-8
Calculating Moving Averages
Each moving average is for a consecutive block of (2m+1) years
Year Sales
1 23
2 40
3 25
4 27
5 32
6 48
7 33
8 37
9 37
10 50
11 40
Average Year
5-Year Moving Average
3 29.4
4 34.4
5 33.0
6 35.4
7 37.4
8 41.0
9 39.4
… …
5
322725402329.4
etc…
Let m = 2
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-9
Annual vs. 5-Year Moving Average
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10 11
Year
Sal
es
Annual 5-Year Moving Average
Annual vs. Moving Average
The 5-year moving average smoothes the data and shows the underlying trend
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-10
Centered Moving Averages
Let the time series have period s, where s is even number i.e., s = 4 for quarterly data and s = 12 for monthly data
To obtain a centered s-point moving average series Xt
*:
Form the s-point moving averages
Form the centered s-point moving averages
(continued)
s/2
1(s/2)jjt
*.5t )
2
sn,2,
2
s1,
2
s,
2
s(txx
)2
sn,2,
2
s1,
2
s(t
2
xxx
*.5t
*.5t*
t
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-11
Centered Moving Averages Used when an even number of values is used in the moving
average Average periods of 2.5 or 3.5 don’t match the original
periods, so we average two consecutive moving averages to get centered moving averages
Average Period
4-Quarter Moving
Average
2.5 28.75
3.5 31.00
4.5 33.00
5.5 35.00
6.5 37.50
7.5 38.75
8.5 39.25
9.5 41.00
Centered Period
Centered Moving
Average
3 29.88
4 32.00
5 34.00
6 36.25
7 38.13
8 39.00
9 40.13
etc…
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-12
Calculating the Ratio-to-Moving Average
Now estimate the seasonal impact Divide the actual sales value by the centered
moving average for that period
*t
t
x
x100
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-13
Calculating a Seasonal Index
Quarter Sales
Centered Moving Average
Ratio-to-Moving Average
1
2
3
4
5
6
7
8
9
10
11
…
23
40
25
27
32
48
33
37
37
50
40
…
29.88
32.00
34.00
36.25
38.13
39.00
40.13
etc…
…
…
83.7
84.4
94.1
132.4
86.5
94.9
92.2
etc…
…
…
83.729.88
25(100)
x
x100
*3
3
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-14
Calculating Seasonal Indexes
Quarter Sales
Centered Moving Average
Ratio-to-Moving Average
1
2
3
4
5
6
7
8
9
10
11
…
23
40
25
27
32
48
33
37
37
50
40
…
29.88
32.00
34.00
36.25
38.13
39.00
40.13
etc…
…
…
83.7
84.4
94.1
132.4
86.5
94.9
92.2
etc…
…
…
1. Find the median of all of the same-season values
2. Adjust so that the average over all seasons is 100
Fall
Fall
Fall
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-15
Interpreting Seasonal Indexes
Suppose we get these seasonal indexes:
SeasonSeasonal
Index
Spring 0.825
Summer 1.310
Fall 0.920
Winter 0.945
= 4.000 -- four seasons, so must sum to 4
Spring sales average 82.5% of the annual average sales
Summer sales are 31.0% higher than the annual average sales
etc…
Interpretation:
Analytical smoothing of time series
Levels of time series are considered as the function of time
tˆ f(t)y
The procedure of smoothing has 3 steps choice of the form of function;
determination of parameters of the function; receiving the smoothed values of the levels of series on the basis of the function
Let’s consider this method on the example of linear trend equation
where a & b – parameters; t – time
ty a b t,
The best way to use linear trend in cases, when the preliminary
investigation shows, that levels of series change with approximately the same speed, i.e. when chain absolute
increases are approximately equal
Parameters a & b are determined by the least square method LSM
The usage of LSM gives the following system of equations for
determining the parameters:
tytbna
t)(ytbta t2
The given system of equations can be significantly simplified, if we
enumerate the time in the way, that
0t
If the series contains odd number of levels, then the central level of series is enumerated as 0. Levels to the side of
decrease of time are enumerated by -1;-2;-3..., to the side of increase of time –
by 1;2;3...
If the series contains even number of levels, then the closest levels to the
center are enumerated by -1 and 1, then numeration is the same as with odd
number of levels but only with the step 2: ...-5,-3,-1,+1,+3,+5...
tyna
n
ya t
2
t
t
t)(ybt)(ytb t
2
Year
2005 800 - -2 4 -1600 798,2 0,24
2006 857 57 -1 1 -857 857,7 0,49
2007 915 58 0 0 0 917,2 4,84
2008 976 61 1 1 976 976,7 0,49
2009 1038 62 2 4 2076 1036,2 3,24
Total: 4586 - 0 10 595 4586,0 12,3
ty цtΔ t 2t ty t t
Λ
y2
Λ
tt )y(y
917,25
4586a
59,510
595b
ty 917,2 59,5 t
Year
2005 800 - -2 4 -1600 798,2 0,24
2006 857 57 -1 1 -857 857,7 0,49
2007 915 58 0 0 0 917,2 4,84
2008 976 61 1 1 976 976,7 0,49
2008 1038 62 2 4 2076 1036,2 3,24
Total: 4586 - 0 10 595 4586,0 12,3
ty цtΔ t 2t ty t t
Λ
y 2Λ
tt )y(y
End of Part IV
To be continued…
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-30