decomposition method

8/13/2019 Decomposition Method

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Applied Business Forecasting

and Planning

Time Series Decomposition


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Introduction One approach to the analysis of time series

data is based on smoothing past data in

order to separate the underlying pattern inthe data series from randomness.

The underlying pattern then can be

projected into the future and used as theforecast.


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Introduction The underlying pattern can also be broken down

into sub patterns to identify the component factorsthat influence each of the values in a series.

This procedure is called decomposition.

Decomposition methods usually try to identify twoseparate components of the basic underlying

pattern that tend to characterize economics andbusiness series.

Trend Cycle

Seasonal Factors


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Introduction The trend Cycle represents long term changes in

the level of series.

The Seasonal factor is the periodic fluctuations ofconstant length that is usually caused by knownfactors such as rainfall, month of the year,temperature, timing of the Holidays, etc.

The decomposition model assumes that the datahas the following form:

Data = Pattern + Error

=f(trend-cycle, Seasonality , error)


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Decomposition Model Mathematical representation of the decomposition

approach is:

Yt is the time series value (actual data) at period t.

St is the seasonal component ( index) at period t.

Tt is the trend cycle component at period t.

Et is the irregular (remainder) component at period t.

),,( tttt ETSfY


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Decomposition Model The exact functional form depends on the

decomposition model actually used. Two

common approaches are: Additive Model

Multiplicative Model

tttt ETSY

tttt ETSY


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Decomposition Model An additive model is

appropriate if the

magnitude of the seasonal

fluctuation does not vary

with the level of the series.

Time plot of U.S. retail

Sales of general

merchandise stores foreach month from Jan.

1992 to May 2002.


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Decomposition Model Multiplicative model is

more prevalent witheconomic series since

most seasonal economicseries have seasonalvariation which increaseswith the level of the series.

Time plot of number ofDVD players sold for eachmonth from April 1997 toJune 2002.


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Decomposition Model Transformations can be used to model additively,

when the original data are not additive.

We can fit a multiplicative relationship by fittingan additive relationship to the logarithm of thedata, since if

Then

tttt ETSY

tttt ELogTLogSLogYLog


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Seasonal Adjustment A useful by-product of decomposition is

that it provides an easy way to calculate

seasonally adjusted data.

For additive decomposition, the seasonally

adjusted data are computed by subtracting

the seasonal component.tttt ETSY


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Seasonal Adjustment For Multiplicative decomposition, the

seasonally adjusted data are computed by

dividing the original observation by theseasonal component.

Most published economic series areseasonally adjusted because Seasonalvariation is usually not of primary interest

tt

t

t ETS

Y


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Deseasonalizing the data The process of deseasonalizing the data has

useful results:

The seasonalized data allow us to see better theunderlying pattern in the data.

It provides us with measures of the extent ofseasonality in the form of seasonal indexes.

It provides us with a tool in projecting what onequarters (or months) observation may portendfor the entire year.


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Deseasonalizing the data Fore example, assume you are working for a

manufacturer of major household appliances andheard that housing starts for the first quarter were258.4. Since your sales depend heavily on newconstruction, you want to project this forward forthe year. We know that housing starts show strongseasonal components. To make a more accurate

projection you need to take this into consideration.Suppose that the seasonal index for the firstquarter of the housing start is .797.


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Deseasonalizing the data and Finding

Seasonal Indexes Once the Seasonal indexes are known you can

deseasonalize data by dividing by the appropriate

index that is:Deseasonalized data = Raw data/Seasonal Index

Therefore

Multiplying this deseasonalized value by 4 would give a

projection for the year of 1,296.864.

216.324797.0

4.258dataizedDeseasonal


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Seasonal Indexes In general:

Seasonal adjustment allows reliable comparison of

values at different points in time. It is easier to understand the relationship among

economic or business variables once the complicating

factor of seasonality has been removed from the data.

Seasonal adjustment may be a useful element in theproduction of short term forecasts of future values of a

time series.


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Trend-Cycle Estimation The trend-cycle can be estimated by

smoothing the series to reduce the random

variation. There is a range of smootheravailable. We will look at

Moving Average

Simple moving average

Centered moving average

Double Moving average

Local Regression Smoothing


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Simple Moving Average The idea behind the moving averages is that

observations which are nearby in time are also

likely to be close in value. The average of the points near an observation will

provide a reasonable estimate of the trend-cycle at

that observation.

The average eliminate some of the randomness in

the data, and leaves a smooth trend-cycle

component.


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Simple Moving Average The first question is; how many data points to

include in each average.

Moving average of order 3 or MA(3) is when weuse averages of three points.

Moving average of order 5 or MA(5) is when weuse averages of five points.

The term moving average is used because eachaverage is computed by dropping the oldestobservation and including the next observation.


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Simple Moving Average Simple centered moving averages can be defined

for any odd order. A moving average of order k,or MA(k) where k is an odd integer is defined as

the average consisting of an observation and the m= (k-1)/2 points on either side.

For example for MA(3)

m

mj

jtt Yk

T 1

)(

3

1

)(3

1

3212

11

YYYT

YYYTtttt


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Simple Moving Average What is the formula for the MA(5)

smoother?


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Simple Moving Average The number of points included in a moving

average affects the smoothness of the resulting

estimate. As a rule, the larger the value of k the smoother

will be the resulting trend-cycle estimate.

Determining the appropriate length of a moving

average is an important task in decomposition

methods.


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Example: Weekly Department Store Sales

The weekly sales

figures (in millions of

dollars) presented inthe following table are

used by a major

department store to

determine the need fortemporary sales

personnel.

Period (t) Sales (y)

1 5.3

2 4.4

3 5.4

4 5.8

5 5.6

6 4.8

7 5.68 5.6

9 5.4

10 6.5

11 5.1

12 5.8

13 5

14 6.2

15 5.6

16 6.7

17 5.218 5.5

19 5.8

20 5.1

21 5.8

22 6.7

23 5.2

24 6

25 5.8


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Example: Weekly Department Store Sales

Weekly Sales

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30

Weeks

Sales

Sales (y)


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Example: Weekly Department Store Sales Calculation of MA(3) and

MA(5) smoother for the weeklydepartment store sales.

In applying a k-term movingaverage, m=(k-1)/2 neighboring

points are needed on either sideof the observation.

Therefore it is not possible toestimate the trend-cycle close to

the beginning and end of series. To overcome this problem a

shorter length moving averagecan be used.

Period (t) Sales (y) MA(3) MA(5)

1 5.3

2 4.4 5.03

3 5.4 5.20 5.3

4 5.8 5.60 5.2

5 5.6 5.40 5.44

6 4.8 5.33 5.48

7 5.6 5.33 5.4

8 5.6 5.53 5.58

9 5.4 5.83 5.64

10 6.5 5.67 5.68

11 5.1 5.80 5.56

12 5.8 5.30 5.72

13 5 5.67 5.54

14 6.2 5.60 5.86

15 5.6 6.17 5.74

16 6.7 5.83 5.84

17 5.2 5.80 5.76

18 5.5 5.50 5.66

19 5.8 5.47 5.48

20 5.1 5.57 5.78

21 5.8 5.87 5.72

22 6.7 5.90 5.76

23 5.2 5.97 5.9

24 6 5.67

25 5.8


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Example: Weekly Department Store SalesWeekly Department Store Sales

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30

Sales

Week

Sales (y)

MA(3)

MA(5)


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Centered Moving Average The simple moving average required an odd

number of observations to be included in

each average. This was to ensure that theaverage was centered at the middle of thedata values being averaged.

What about moving average with an evennumber of observations?

For example MA(4)


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Centered Moving Average To calculate a MA(4) for the weekly sales data, the trend

cycle at time 3 can be calculated as

The center of the first moving average is at 2.5 (half periodearly) and the center of the second moving average is at

3.5 (half period late). How ever the center of the two moving averages is

centered at 3.

3.54

6.58.54.54.4

5

225.5

4

8.54.54.43.5

4

5432

4321

yyyy

or

yyyy


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Centered Moving Average A centered moving average can be

expressed as a single but weighted moving

average, where the weights for each periodare unequal.

8

222

)44

(2

1

2

4

4

54321

543243215.35.23

54325.3

43215.2

YYYYY

YYYYYYYYTTT

YYYYT

YYYYT


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Centered Moving Average The first and the last term in this average have

weights of 1/8 and all the other terms haveweights of 1/4.

Therefore a double MA(4) smoother is equivalentto a weighted moving average of order 5.

In general a double MA(k) smoother is equivalent

to a weightedmoving average of order k+1 withweights 1/k for all observations except for the firstand the last observation in the average, whichhave weights 1/2k.


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Least squares estimates The general procedure for estimating the pattern

of a relationship is through fitting some functional

form in such a way as to minimize the errorcomponent of equation

data = pattern + Error

The name least squares is based on the fact that

this estimation procedure seeks to minimize the

sum of the squared errors in the above equation.


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Least squares estimates A major consideration in forecasting is to identify

and fit the most appropriate pattern (functionalform) so as to minimize the MSE.

A possible functional form is a straight line.

Recall that a straight line is represented by theequation

Where the two parameters a, and b represent theintercept and the slope respectively.

bXaY


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Least squares estimates The values a and b can be chosen by minimizing

the MSE.

This procedure is known as simple linear

regression and will be examined in detail inchapter 6.

One way to estimate trend-cycle is throughextending the idea of moving averages to moving

lines. That is instead of taking average of the points, we

may fit a straight line to these points and estimatetrend-cycle that way.


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Least squares estimates A straight trend line can be represented by the equation

The values of a and b can be found by minimizing the sum of squarederrors where the errors are the differences between the data values ofthe time series and the corresponding trend line values. That is:

A straight trend line is sometimes appropriate, but there are many timeseries where some curved trend is better.

btaTt

n

t

t btaY

1

2)(


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Least squares estimates Local regression is a way of fitting a much

more flexible trend-cycle curve to the data.

Instead of fitting a straight line to the entiredataset, a series of straight lines will be

fitted to sections of the data.


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Classical Decomposition Multiplicative Decomposition

We assume the time series is multiplicative.

This method is often called the ratio-tomoving averages method.


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Seasonal Indexes First the trend-cycle Ttis computed using a

centered moving average. This removes the short-

term fluctuations from the data so that the longer-term trend-cycle components can be more clearly

identified.

These short-term fluctuations include both

seasonal and irregular variations.

An appropriate moving average (MA) can do the

job.


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Seasonal Indexes The moving average should contain the

same number of periods as there are in the

seasonality that you want to identify. To identify monthly patter use MA(12)

To identify quarterly pattern use MA(4).

The moving average represents a typicallevel of Y for the year that is centered onthat moving average.


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Seasonal Indexes Through the following hypothetical

example we will see how this procedure

works.Year Quarter Time inde Y MA CMA

1 1 1 10

2 2 18

3 3 20

4 4 12

2 1 5 12

2 6 20

3 7 24

4 8 13

3 1 9 14

2 10 22

3 11 28

4 12 16


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Seasonal Indexes The centered moving averages represent the

deseasonalized data.

The degree of seasonality, called seasonalfactor (SF), is the ratio of the actual value to

the deseasonalized value. That is

t

tt

CMAYSF


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Seasonal Indexes A seasonal factor greater than 1 indicates a

period in which Y is greater than the yearly

average, while a seasonal factor less than 1indicates a period in which y is less than the

yearly average.

In our example:

76.075.15

12

31.125.15

20

4

44

3

33

CMA

YSF

CMA

YSF


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Seasonal Indexes The seasonal indexes are calculated as follows:

The seasonal factors for each of the four quarters (or12 months) are summed and divided by the number of

observations to arrive at the average seasonal factorsfor each quarter (or month).

The sum of the average seasonal factors should equalthe number of periods (4 for quarters and 12 formonths).

If it does not, the average seasonal factors should benormalized by multiplying each by the ratio of thenumber of periods to the sum of the average seasonalfactors.


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Seasonal Indexes In our example For the third quarter

Seasonal factors are 1.311475, 1.371429.

Therefore the average is:

The average of SF for the rest of the

quarters is:

341.12

371429.1311475.13

ASF

ASF4 0.742063

ASF1 0.73697

ASF2 1.144451


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Seasonal Indexes The seasonal indexes for the four quarters

are:Year Quarter SF ASF SI

1 1

2

3 1.3114754 1.341452 1.353315

4 0.7619048 0.742063 0.748626

2 1 0.7272727 0.73697 0.743487

2 1.1678832 1.144451 1.154572

3 1.3714286

4 0.72222223 1 0.7466667

2 1.1210191

3

4

Total 3.964936 4


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Finding the Long-Term Trend The long term movements or trend in a series can

be described by a straight line or a smooth curve.

The long-term trend is estimated from thedeseasonalized data for the variable to be forecast.

To find the long-term trend, we estimate a simplelinear equation as

Where Time =1 for the first period in the data set andincreased by 1each quarter(or month) thereafter.

)()(

TimebaCMATimefCMA


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Finding the Long-Term Trend The method of least squares can be used to

estimate a and b.

a and b values can be used to determine thetrend equation.

The trend equation can be used to estimate thetrend value of the centered moving average for

the historical and forecast periods. This new series is the centered moving-average

trend (CMAT).


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Finding the Long-Term Trend For our example,The values of a and b are estimated by

using EXCEL regression program.

SUMMARY OUTPUT

Regression Statistics

Multiple R 0.995666021

R Square 0.991350826

Adjusted R Square 0.989909297

Standard Error 0.148571238

Observations 8

ANOVA

df SS MS F Significance F Regression 1 15.18005952 15.18006 687.7079 2.02856E-07

Residual 6 0.132440476 0.022073

Total 7 15.3125

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 13.4047619 0.157999932 84.8403 1.81E-10 13.01814971 13.79137409

X Variable 1 0.601190476 0.02292504 26.22418 2.03E-07 0.545094884 0.657286069


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Finding the Long-Term Trend The centered moving-

average trend equation

for this example is

This line is shown

along with the graph

of Y and the

deseasonalized data.0

5

10

15

20

25

30

0 2 4 6 8 10 12 14

Y

Centered moving average

Trend

)(6.040.13 TIMECMAT


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Measuring the Cyclical Component The cyclical component of a time series is

measured by a cycle factor (CF), which is the ratioof the centered moving average (CMA) to the

Centered moving average trend (CMAT).

A cycle factor greater than 1 indicates that thedeseasonalized value for that period is above thelong-term trend of the data. If CF is less than 1,the reverse is true.

CMAT

CMACF


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Measuring the Cyclical component If the cycle factor analyzed carefully, it can be the

component that has the most to offer in terms ofunderstanding where the industry may be headed.

The length and the amplitude of previous cycles mayenable us to anticipate the next tuning point in the currentcycle.

An individual familiar with an industry can often explaincyclic movements around trend line in terms of variables

or events that can be seen to have had some import.

By looking at those variables or events in the present, onecan sometimes get some hint of the likely future directionof the cycle movement.


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Business Cycles Business cycles are

wavelike fluctuations

in the general level ofeconomic activity.

They are often

described by a

diagram such as this.


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Business Cycles Expansion Phase: The period

between the begging trough (A)

and the Peak (B).

Recession, or Contractionphase: the period from peak (B)

to the ending trough (C).

The vertical distance between A

and B` provides a measure of

the degree of expansion The severity of a recession is

measured by the vertical

distance between B`` and C.


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Business Cycles

If business cycleswere true cycles, then

they would have a

constant amplitude(The vertical distancefrom trough to peak).

they would have a

constant periodicity(the length of timebetween successivepeaks or trough).


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Business Cycle Indicators

There are a number of possible business

cycle indicators, but the following three are

noteworthy The index of leading economic indicators

The index of coincident economic indicators

The index of lagging economic indicators


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Components of the Composite Indexes

Leading Index

Average weekly hours, manufacturing

Average weekly initial claims for unemployment insurance

Manufacturers' new orders, consumer goods and materials

Vendors performance, slower deliveries diffusion index

Manufacturers new orders, nondefense goods

Building permits, new private housing units

Stock prices, 500 common stocks Money supply, M2

Interest rate spread, 10 year treasury bonds less federal funds

Index of consumer expectation


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Coincident Index

Employees on nonagricultural payrolls

Personal income less transfer payments

Industrial production

Manufacturing and trade sales


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Lagging Index

Average duration of unemployment

Inventories to sales ratio, manufacturing and trade Labor cost per unit of output, manufacturing

Average prime rate

Commercial and industrial loans

Consumer installment credit to personal income ratio.

Consumer price index for services.

Source: www.globalindicators.org


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Business Cycle Indicators

It is possible that one of these indexes, orone of the series that make up an index may

be useful in predicting the cycle factor in atime series decomposition.

These could be done in

Regression analysis with the cycle factor (CF)

as the dependent variable.

These indexes or their components may be usedas independent variable in a regression model.


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Finding the Cyclical Factor

The cyclical factors

for our example are:Year QuarterTime inde Y CMA CMAT CF

1 1 1 10 14

2 2 18 14.6

3 3 20 15.3 15.2 1.003

4 4 12 15.8 15.8 0.997

2 1 5 12 16.5 16.4 1.006

2 6 20 17.1 17 1.007

3 7 24 17.5 17.6 0.994

4 8 13 18 18.2 0.989

3 1 9 14 18.8 18.8 0.997

2 10 22 19.6 19.4 1.012

3 11 28

4 12 16

003.115.2

15.3

CMAT

CMACF

CMATCMACF

3

33

t

tt


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Classical Decomposition

Additive Decomposition Step3: In classical decomposition we assume the

seasonal component is constant from year to year.So we the average of the detrended value for a given

month (for monthly data) and given quarter (for

quarterly data) will be the seasonal index for the

corresponding month or quarter.

Step4: the irregular series Etis computed by simply

subtracting the estimated seasonality, and trend-

cycle from the original data.


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The Time-Series Decomposition Forecast

We know how to isolate and measure thesecomponents.

To prepare a forecast based on the timeseries decomposition model, we mustreassemble the components.

The forecast for Y (FY) is:

(CF)(E)(CMAT)(SI)FY


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Example:Private Housing Start


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The centered movingaverage series, shown bythe solid line, is much

smoother than the originalseries of private housingstarts data (dashed line)

because the seasonalpattern and the irregular or

random fluctuation in thedata are removed by theprocess of calculating thecentered moving average.


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The long term trend in private

housing starts is shown by the

straight dotted line

(PHSCMAT).

The dashed line is the raw data

(PHS), while the wavelike solid

line is the deseasonalized data

(PHSCMA).

The long-term trend is positive.

The equation for the trend line

is:)(313.051.237 TimePHSCMAT


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The cyclical factor isthe ratio of thecentered moving

average to the long-term trend in the data.

As this plot shows, thecycle factor moves

slowly around the baseline (1.0) with littleregularity


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The actual values for

private housing starts

are shown by the

dashed line, and the

forecast values based

on the timeseries

decomposition modelare shown by the solid

line.


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The Time-Series Decomposition Forecast Because the time series decomposition

models do not involve a lot of mathematics

or statistics, they are relatively easy toexplain to the end user.

This is a major advantage because if the end

user has an appreciation of how the forecastwas developed, he or she may have more

confidence in its use for decision making.

decomposition method

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