Time Series and Their Components
Chapter 5
Time Series and Their Components
2
¾
Time series are often recorded at fixed time intervals.
¾
For example, Y might represent sales, and the associated time series
could be a sequence of annual sales figures.
¾
Other examples of time series include quarterly earnings, monthly
inventory levels, and weekly exchange rates.
¾
In general, time series do not behave like a random sample and require
special methods for their analysis.
¾
Observations of a time series are typically related to one another
(autocorrelated).
¾
This dependence produces patterns of variability that can be used to
forecast future values and assist in the management of business
operations.
¾
Consider these situations.
¾ It is important that managers understand the past and use historical data and
sound judgment to make intelligent plans to meet the demands of the future.
Time Series and Their Components
2
¾
Time series are often recorded at fixed time intervals.
¾
For example, Y might represent sales, and the associated time series
could be a sequence of annual sales figures.
¾
Other examples of time series include quarterly earnings, monthly
inventory levels, and weekly exchange rates.
¾
In general, time series do not behave like a random sample and require
special methods for their analysis.
¾
Observations of a time series are typically related to one another
(autocorrelated).
¾
This dependence produces patterns of variability that can be used to
forecast future values and assist in the management of business
operations.
¾
Consider these situations.
¾ It is important that managers understand the past and use historical data and
sound judgment to make intelligent plans to meet the demands of the future.
Time Series and Their Components
3
¾ Properly constructed time series forecasts help eliminate some of the
uncertainty associated with the future and can assist management in
determining alternative strategies.
¾ Forecasting is done by a set of procedures followed by judgments.
Time Series and Their Components
4
Decomposition
¾ It is an approach to the analysis of time series data involves an attempt
to identify the component factors that influence each of the values in a
series.
¾ The components of time series are:
Time-Series
Trend
Component
Seasonal
Component
Cyclical
Component
Random
Component
Time Series and Their Components
5
1. Trend Component
¾ It represents the growth and the decline in a time series, denoted by T.
¾ Long-run increase or decrease over time (overall upward or downward
movement) and they could linear or nonlinear
¾ Data taken over a long period of time
Time Series and Their Components
6
2. Cyclical Component
¾ It represents a long-term wavelike fluctuations or cycles of more than one
yearʹs duration in a time series, denoted by C.
¾ Practically it is difficult to identify and frequently regarded as part of trend.
¾ Regularly occur but may vary in length
¾ Often measured peak to peak
Time Series and Their Components
7
3. Seasonal Component
¾ It represents the seasonal variation in a time series which refers to a more or
less stable pattern of change that appears short-term regular wave-like patterns
and repeats itself season after season, denoted by S.
¾ Observed within 1 year.
¾ Often monthly or quarterly.
Time Series and Their Components
8
4. Irregular Component
¾ It represents the unpredictable or random fluctuations in a time series, denoted
by I.
¾ Unpredictable, random, “residual” fluctuations
¾ Due to random variations of
¾ Nature
¾ Accidents or unusual events
¾
“Noise” in the time series
¾ To study the components of a time series, the analyst must consider how
the components relate to the original series.
Time Series and Their Components
9
Time Series Components Models
Additive Components Model
¾ It is suggested to use when the variability are the same throughout the
length of the series.
Multiplicative Components Model
¾ It is suggested to use when the variability are increasing throughout the
length of the series.
¾ Note that it is possible to convert the multiplicative model to the
additive model using logarithms. i.e.
Time Series and Their Components
10
Time series with constant variability
Time Series and Their Components
11
Time series with increasing variability
Time Series and Their Components
12
Estimation of Time Series Components
Estimation of Trend Component
¾
Trends are long term movements in a time series that can be sometimes
be described by a straight line or a smooth curve.
Remark
¾
Fitting a trend curve helps us in providing some indication of the general
direction of the observed series, and in getting a clear picture of the
seasonality after removing the trend from the original series.
The Linear Trend
The Quadratic Trend
Time Series and Their Components
13
The Exponential Trend
Where Tt is the predicted value of the trend at time t , b0 , b1 and are called
the model parameters.
We can forecast the trend using the above models as and so on.
Note that the Error Sum of Squares (SSE) is measured by
Time Series and Their Components
14
Example 5.1
Data on annual registrations of new passenger cars in the United States
from 1960 to 1992 are shown in the following table and plotted in the later
figure.
Time Series and Their Components
15
We definitely
have a trend
here!
Time Series and Their Components
16
The values from 1960 to 1992 are used to develop the trend equation.
Registrations is the dependent variable, and the independent variable is
time t coded as 1960 = 1, 1961 = 2, and so on. The fitted trend line has the
equation
The slope of the trend equation indicates that registrations are estimated to
increase an average of 68,700 each year.
The figure shows a straight-line
trend fitted to the actual data. It
also shows forecasts of new car
registrations for the years 1993
and 1994 (t = 34 and t = 35)
obtained by extrapolating the
trend line.
Time Series and Their Components
17
The estimated trend values for passenger car registrations from 1960 to
1992 are shown in the table. For example, the trend equation estimates
registrations in 1992 (t =33) to be
¾
or 10,255,000 registrations.
¾
Registrations of new passenger cars were actually 8,054,000 in 1992.
For 1992, the trend equation overestimates registrations by approximately
2.2 million.
¾
¾
This error and the remaining estimation errors were listed in the table.
The estimation errors were used to compute the measures of fit, MAD,
MSD, and MAPE also were shown in the figure.
¾
Time Series and Their Components
18
Forecasting a Trend
¾ Which trend model is appropriate?
¾ Linear, quadratic or exponential
¾
Linear models assume that a variable is increasing (or decreasing) by a
constant amount each time period.
¾
A quadratic curve is needed to model the trend.
¾ Based on the accuracy measures, a quadratic trend appears to be a better
representation of the general direction of the data.
Time Series and Their Components
19
¾
When a time series starts slowly and then appears to be increasing at an
increasing rate such that the percentage difference from observation to
observation is constant, an exponential trend can he fitted.
¾
The coefficient b1 is related to the growth rate.
¾
If the exponential trend is fit to annual data, the annual growth rate is
estimated to be 100(b1 — 1)%.
¾
The figure next contains the
number of mutual fund salespeople
for several consecutive years.
¾
The increase in the number of
salespeople is not constant.
¾
It appears as if increasingly larger
numbers of people are being added
in the later years.
Time Series and Their Components
20
¾
A linear trend fit to the salespeople data would indicate a constant
average crease of about nine salespeople per year.
¾
This trend overestimates the actual increase in the earlier years and
underestimates the increase in the last year.
¾
It does not model the apparent trend in the data as well as the
exponential curve.
¾
It is clear that extrapolating an exponential trend with a 31 % growth rate
will quickly result in some very big numbers.
¾
This is a potential problem with an exponential trend model.
¾
What happens when the economy cools off and stock prices begin to
retreat?
¾
The demand for mutual fund salespeople will decrease and the number
of salespeople could even decline.
¾
The trend forecast by the exponential curve will be much too high.
Time Series and Their Components
21
¾
Growth curves of the Gompertz and logistic types reflect a situation in
which sales begin low, then increase as the product catches on, and
finally ease off as saturation is reached.
¾
Judgment and common sense are very important in selecting the right
approach.
¾
As we will discuss later, the line or curve that best fits a set of data points
might not make sense when projected as the trend of the future.
Time Series and Their Components
22
¾
Suppose we are presently at time t = n (end of series) and we want to use
a trend model to forecast the value of Y, p steps ahead.
¾
The time period at which we make the forecast, n in this case, is called
the forecast origin.
¾
The value p is called the lead time.
¾
For the linear trend model, we can produce a forecast by evaluating
¾
Using the trend line fitted to the car registration data in Example 5.1 , a
forecast of the trend for 1993 (t = 34) made in 1992 (t = n = 33) would be
the p = 1 step ahead forecast
¾
Similarly, the p = 2 step ahead forecast (1994) is given by
Time Series and Their Components
23
¾
Using the quadratic trend curve for the car registration data, we can
calculate forecasts of the trend for 1993 and 1994 by setting t = 33 + 1 = 34
and t = 33 + 2 = 35.
¾
The forecasts are = 8.690 and = 8.470 (respectively)
¾
Recalling that car registrations are measured in millions, the two
forecasts of trend produced from the quadratic curve are quite different
from the forecasts produced by the linear trend equation.
¾
Moreover, they are headed in the opposite direction.
¾
If we were to extrapolate the linear and quadratic trends for additional
time periods, their differences would be magnified.
¾
This example illustrates why great care must be exercised in using fitted
trend curves for the purpose of forecasting future trends.
¾
The differences can be substantial for large lead times (long-run
forecasting).
Time Series and Their Components
24
¾
Trend curve models are based on the following assumptions:
¾
The correct trend curve has been selected.
¾
The curve that fits the past is indicative of the future.
¾
We must be able to argue that the correct trend has been selected the
future will be like the past.
¾
There are objective criteria for selecting a trend curve. We will discuss
two of these criteria, the Akaike Information Criterion (AIC) and the
Bayesian Information Criterion (BIC), in later chapters.
¾
However, although these and other criteria help to determine an
appropriate model, they do not replace good judgment.
Time Series and Their Components
25
Estimation of Seasonal Component
¾
A seasonal pattern is one that repeats itself year after year.
¾
For annual data, seasonality is not an issue because there is no chance to
model a within year pattern with data recorded once per year.
¾
Time series consisting of weekly, monthly, or quarterly observations
often exhibit seasonality.
¾
The analysis of the seasonal component of a time series has direct shortterm implications and is of greatest importance to mid- and lower-level
management.
¾
Marketing plans have to take into consideration expected seasonal
patterns in consumer purchases.
¾
Several methods for measuring seasonal variation have been developed.
The basic idea in all of these methods is to first estimate and remove the
trend from the original series and then smooth out the irregular
component.
¾
Time Series and Their Components
26
¾
The seasonal values are collected and summarized to produce a number
(generally an index number) for each observed interval of the year (week,
month, quarter, and so on).
¾
The identification of the seasonal component in a time series differs
from trend analysis in at least two ways:
1 . The trend is determined directly from the original data, but the seasonal
component is determined indirectly after eliminating the other components
from the data so that only the seasonality remains.
2. The trend is represented by one best-fitting curve, or equation, but a separate
seasonal value has to be computed for each observed interval (week, month,
quarter) of the year and is often in the form of an index number.
¾
Always we estimate the seasonality in form of index numbers,
percentages that show changes over time, are called seasonal index.
¾
If an additive decomposition is used, estimates of the trend, seasonal,
and irregular components are added together to produce the original
series.
Time Series and Their Components
27
¾
If a multiplicative decomposition is used, the individual components
must be multiplied together to reconstruct the original series, and in this
formulation, the seasonal component is represented by a collection of
index numbers.
¾
These numbers show which periods within the year are relatively low
and which periods are relatively high.
¾
The seasonal indices trace out the seasonal pattern.
¾
Index numbers are percentages that show changes over time.
Remark
¾ In this chapter we study the multiplicative model and leave the additive
model to chapter 8 if we have time.
¾ In multiplicative decomposition model, the ratio to moving average is a
popular method for measuring seasonal variation.
Time Series and Their Components
28
Finding Seasonal Indexes
Ratio-to-moving average method:
¾
Begin by removing the seasonal and irregular components (St and It),
leaving the trend and cyclical components (Tt and Ct)
Example: Four-quarter moving average
¾
First average:
Moving average 1 =
¾
Second average:
Moving average
¾
Q1+Q2+Q3+Q4
4
etc…
2
=
Q2+Q3+Q4+Q5
4
Time Series and Their Components
29
Quarter
Sales
1
23
2
40
60
3
25
50
4
27
40
5
32
30
6
48
20
7
33
10
8
37
0
9
10
37
50
11
40
etc…
etc…
Quarterly Sales
1
2
3
4
5
6
Quarter
7
8
9
10
11
Time Series and Their Components
30
Centered Seasonal Index
Average
Period
4-Quarter
Moving
Average
Quarter
Sales
1
23
2.5
28.75
2
40
3.5
31.00
3
25
4.5
33.00
4
27
5.5
35.00
5
32
6.5
37.50
6
48
7.5
38.75
7
33
8.5
39.25
8
37
9.5
41.00
9
37
10
50
11
40
etc…
2.5=
28.75=
1+2+3+4
4
23+40+25+27
4
Each moving average is for a consecutive block of 4
quarters
Time Series and Their Components
31
¾
¾
Average periods of 2.5 or 3.5 don’t match the original quarters, so we
average two consecutive moving averages to get centered moving averages
Average
Period
4-Quarter
Moving
Average
Centered
Period
Centered
Moving
Average
2.5
28.75
3
29.88
3.5
31.00
4
32.00
4.5
33.00
5
34.00
5.5
6
36.25
6.5
35.00 etc…
37.50
7
38.13
7.5
38.75
8
39.00
8.5
39.25
9
40.13
9.5
41.00
Now estimate the St x It value by dividing the actual sales value by the
centered moving average for that quarter.
QMIS 320, CH 5 by M. Zainal
Time Series and Their Components
32
¾
Ratio-to-Moving Average formula: S×I
t
Centered
Moving
Average
t
=
Yt
Tt×Ct
Ratio-toMoving
Average
Quarter
Sales
1
2
3
23
40
25
29.88
0.837
4
27
32.00
5
32
34.00
6
48
36.25
7
33
38.13
8
37
39.00
9
37
40.13
10
50
etc…
11
40
…
…
…
…
0.844
0.941
1.324
0.865
0.949
0.922
etc…
…
…
Example
0.837 =
25
29.88
Time Series and Their Components
33
Fall
Fall
Fall
Centered
Moving
Average
Ratio-toMoving
Average
29.88
0.837
32.00
0.844
34.00
0.941
36.25
1.324
38.13
0.865
Quarter
Sales
1
2
3
4
5
6
7
23
40
25
27
32
48
33
8
9
10
37
37
50
39.00
0.949
40.13
0.922
11
40
…
…
…
…
…
…
etc…
etc…
Average all of the Fall
values to get Fall’s seasonal
index
Do the same for the other
three seasons to get the
other seasonal indexes
Time Series and Their Components
34
¾
Suppose we get these seasonal indices:
Season
Seasonal
Index
Spring
0.825
Spring sales average 82.5% of the annual
average sales
Summer
1.310
Summer sales are 31.0% higher than the
annual average sales
Fall
0.920
Winter
0.945
Interpretation:
etc…
Σ = 4.000 -- four seasons, so must sum to 4
QMIS 320, CH 5 by M. Zainal
Time Series and Their Components
35
¾
The data is deseasonalized by dividing the observed value by its
seasonal index
Tt×Ct×It=
¾
Yt
St
This smoothes the data by removing seasonal variation
Quarter
Sales
Seasonal Index
Deseasonalized Sales
1
2
3
23
40
25
0.825
1.310
0.920
27.88
4
5
6
7
27
32
48
33
0.945
0.825
1.310
0.920
8
9
10
37
37
50
0.945
0.825
1.310
11
…
40
0.920
…
30.53
27.17
28.57
38.79
36.64
35.87
39.15
44.85
38.17
43.48
…
27.88 =
23
0.825
Time Series and Their Components
36
Time Series and Their Components
37
Example 5.3
In Example 3.5 the analyst for the Outboard Marine Corporation, used
autocorrelation analysis to determine that sales were seasonal on a quarterly
basis. Now, he uses decomposition to understand the quarterly sales
variable. Minitab was used to produce the following table and figure. To
keep the seasonal pattern current, only the last seven years (1990 to 1996) of
sales data (Y), were analyzed.
The trend is computed using the linear model:
Time Series and Their Components
38
I =CI /C
=1.187 /1.146
=1.036
Tˆ1 = 253.742+1.284(1) = 255.026 SCI YT=
TCI Y S=
.7796= =232.7
7=0.780
255.026==232.7S=(.912+.788+...+.812)
.912
298.486 CI =Y TS = 232.7 (255.026×.7796) C =(1.170+1.187+1.080) 3=1.146
Time Series and Their Components
39
¾
The cyclical indices can be used to answer the following questions:
¾
The series cycle?
¾
How extreme is the cycle?
¾
The series follow the general state of the economy (business cycle)?
¾
One way to investigate cyclical patterns is through the study of business
indicators.
¾
A business indicator is a business-related time series that is used to help
assess the general state of the economy.
¾
The most important list of statistical indicators originated during the
sharp business setback of 1937 to 1938.
¾
Leading indicators.
¾
Coincident indicators.
¾
Lagging indicators.
Time Series and Their Components
40
Forecasting A Seasonal Time Series
¾
In forecasting a seasonal time series, the decomposition process is
reversed.
¾
Instead of separating the series into individual components for
examination, the components are recombined to develop the forecasts for
future periods.
Example 5.4
Forecasts of Outboard Marine Corporation sales for the four quarters of 1997
can he developed using the previous table.
1. Trend. The quarterly trend equation is: T = 253.742 + 1.284t.
9
9
9
9
The forecast origin is the fourth quarter of 1996, or time period t = n = 28.
Sales for the first quarter of 1997 occurred in time period t = 28 + 1 = 29.
This notation shows we are forecasting p = 1 period ahead from the end of the time series.
Setting t = 29, the trend projection is then T29 = 253.742 + 1.284(29) = 290.978
Time Series and Their Components
41
2. Seasonal. The seasonal index for the first quarter is .7796.
3. Cyclical. The cyclical projection must be determined from the estimated cyclical
pattern (if any) and any other information generated by indicators of the general
economy for 1997.
9
Projecting the cyclical pattern for future time periods is fraught with uncertainty, and as
we indicated earlier, is generally assumed for forecasting purposes to be included in the
trend.
9
To demonstrate the completion of this example, we set the cyclical index to 1.0.
4. Irregular. Irregular fluctuations represent random variation that can’t be
explained by the other components.
9
For forecasting, the irregular component is set to the average value 1.0
¾
The forecast for the first quarter of 1997 is
¾
The forecasts for the rest of 1997 are