Time series Decomposition Farideh Dehkordi-Vakil

Time series Decomposition

Farideh Dehkordi-Vakil

Introduction One approach to the analysis of time series

data is based on smoothing past data in order to separate the underlying pattern in the data series from randomness.

The underlying pattern then can be projected into the future and used as the forecast.

Introduction The underlying pattern can also be broken down

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

This procedure is called decomposition. Decomposition methods usually try to identify

two separate components of the basic underlying pattern that tend to characterize economics and business series. Trend Cycle Seasonal Factors

Introduction The trend Cycle represents long term changes in

the level of series. The Seasonal factor is the periodic fluctuations of

constant length that is usually caused by known factors such as rainfall, month of the year, temperature, timing of the Holidays, etc.

The decomposition model assumes that the data has the following form:

Data = Pattern + Error = f (trend-cycle, Seasonality , error)

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

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

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 for each month from Jan. 1992 to May 2002.

Decomposition Model Multiplicative model is

more prevalent with economic series since most seasonal economic series have seasonal variation which increases with the level of the series.

Time plot of number of DVD players sold for each month from April 1997 to June 2002.

Decomposition Model Transformations can be used to model additively,

when the original data are not additive. We can fit a multiplicative relationship by fitting

an additive relationship to the logarithm of the data, since if

Then

tttt ETSY

tttt ELogTLogSLogYLog

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

Seasonal Adjustment For Multiplicative decomposition, the

seasonally adjusted data are computed by dividing the original observation by the seasonal component.

Most published economic series are seasonally adjusted because Seasonal variation is usually not of primary interest

ttt

t ETS

Y

Deseasonalizing the data

The process of deseasonalizing the data has useful results: The seasonalized data allow us to see better the

underlying pattern in the data. It provides us with measures of the extent of

seasonality in the form of seasonal indexes. It provides us with a tool in projecting what one

quarter’s (or month’s) observation may portend for the entire year.

Deseasonalizing the data

Fore example, assume you are working for a manufacturer of major household appliances and heard that housing starts for the first quarter were 258.4. Since your sales depend heavily new construction, you want to project this forward for the year. We know that housing starts show strong seasonal components.To make a more accurate projection you need to take this into consideration. Suppose that the seasonal index for the first quarter of the housing start is .797.

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

4.258data izedDeseasonal


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 the production of short term forecasts of future values of a time series.

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

smoothing the series to reduce the random variation. There is a range of smoother available. We will look at Moving Average

Simple moving average Centered moving average Double Moving average

Local Regression Smoothing

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.

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 we

use averages of three points. Moving average of order 5 or MA(5) is when we

use averages of five points. The term moving average is used because each

average is computed by dropping the oldest observation and including the next observation.

Simple Moving Average Simple 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 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

mjjtt Y

kT

1

)(3

1

)(3

1

3212

11

YYYT

YYYT tttt

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

smoother?

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.

Example: Weekly Department Store Sales

The weekly sales figures (in millions of dollars) presented in the following table are used by a major department store to determine the need for temporary sales personnel.

Period (t) Sales (y)1 5.32 4.43 5.44 5.85 5.66 4.87 5.68 5.69 5.410 6.511 5.112 5.813 514 6.215 5.616 6.717 5.218 5.519 5.820 5.121 5.822 6.723 5.224 625 5.8


Weekly Sales

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30

Weeks

Sale

s

Sales (y)


Calculation of MA(3) and MA(5) smoother for the weekly department store sales.

In applying a k-term moving average, m=(k-1)/2 neighboring points are needed on either side of the observation.

Therefore it is not possible to estimate the trend-cycle close to the beginning and end of series.

To overcome this problem a shorter length moving average can be used.

Period (t) Sales (y) MA(3) MA(5)1 5.32 4.4 5.033 5.4 5.20 5.34 5.8 5.60 5.25 5.6 5.40 5.446 4.8 5.33 5.487 5.6 5.33 5.48 5.6 5.53 5.589 5.4 5.83 5.6410 6.5 5.67 5.6811 5.1 5.80 5.5612 5.8 5.30 5.7213 5 5.67 5.5414 6.2 5.60 5.8615 5.6 6.17 5.7416 6.7 5.83 5.8417 5.2 5.80 5.7618 5.5 5.50 5.6619 5.8 5.47 5.4820 5.1 5.57 5.7821 5.8 5.87 5.7222 6.7 5.90 5.7623 5.2 5.97 5.924 6 5.6725 5.8

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

We

ek

Sales (y)

MA(3)

MA(5)

Centered Moving Average The simple moving average required an odd

number of observations to be included in each average. This was to ensure that the average was centered at the middle of the data values being averaged.

What about moving average with an even number of observations?

For example MA(4)

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 period early) 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

225.54

8.54.54.43.5

or

Centered Moving Average A centered moving average can be

expressed as a single but weighted moving average, where the weights for each period are unequal.

8

222

)44

(2

1

2

4

4

54321

543243215.35.23

54325.3

43215.2

YYYYY

YYYYYYYYTTT

YYYYT

YYYYT

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

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

Therefore a 2MA(4) smoother is equivalent to a weighted moving average of order 5.

In general a 2 MA(k) smoother is equivalent to a weighted moving average of order k+1 with weights 1/k for all observations except for the first and the last observation in the average, which have weights 1/2k.

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 error component 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.

Least squares estimates A major consideration in forecasting is to identify

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

A possible functional form is a straight line. Recall that a straight line is represented by the

equation

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

bXaY

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 in chapter 5.

One way to estimate trend-cycle is through extending 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 estimate trend-cycle that way.

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 squared errors where the errors are the differences between the data values of the time series and the corresponding trend line values. That is:

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

btaTt

n

tt btaY

1

2)(

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 entire

dataset, a series of straight lines will be fitted to sections of the data.

Classical Decomposition Multiplicative Decomposition

We assume the time series is multiplicative. This method is often called the “ratio-to

moving averages” method.


First the trend-cycle Tt is 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.


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 “typical” level of Y for the year that is centered on that moving average.


Through the following hypothetical example we will see how this procedure works.

Year Quarter Time indexY MA CMA1 1 1 10

2 2 183 3 204 4 12

2 1 5 122 6 203 7 244 8 13

3 1 9 142 10 223 11 284 12 16


The centered moving averages represent the deseasonalized data.

The degree of seasonality, called seasonal factor (SF), is the ratio of the actual value to the deseasonalized value. That is

t

tt CMA

YSF


A seasonal factor greater than 1 indicates a period in which Y is greater than the yearly average, while a seasonal factor less than 1 indicates 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


The seasonal indexes are calculated as follows: The seasonal factors for each of the four quarters (or 12

months) are summed and divided by the number of observations to arrive at the average seasonal factors for each quarter (or month).

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

If it does not, the average seasonal factors should be normalized by multiplying each by the ratio of the number of periods to the sum of the average seasonal factors.


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


The seasonal indexes for the four quarters are:

Year Quarter SF ASF SI1 1

23 1.3114754 1.341452 1.3533154 0.7619048 0.742063 0.748626

2 1 0.7272727 0.73697 0.7434872 1.1678832 1.144451 1.1545723 1.37142864 0.7222222

3 1 0.74666672 1.121019134

Total 3.964936 4

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 the

deseasonalized data for the variable to be forecast. To find the long-term trend, we estimate a simple linear

equation as

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

)(

)(

TimebaCMA

TimefCMA

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 the

trend equation. The trend equation can be used to estimate the

trend value of the centered moving average for the historical and forecast periods.

This new series is the centered moving-average trend (CMAT).

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

using EXCEL regression program.SUMMARY OUTPUT

Regression StatisticsMultiple R 0.995666021R Square 0.991350826Adjusted R Square 0.989909297Standard Error 0.148571238Observations 8

ANOVAdf SS MS F Significance F

Regression 1 15.18005952 15.18006 687.7079 2.02856E-07Residual 6 0.132440476 0.022073Total 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 13.79137409X Variable 1 0.601190476 0.02292504 26.22418 2.03E-07 0.545094884 0.657286069

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

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

a cycle factor (CF), which is the ratio of the centered moving average (CMA) to the Centered moving average trend (CMAT).

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

CMAT

CMACF

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

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

The length and the amplitude of previous cycles may enable us to anticipate the next tuning point in the current cycle.

An individual familiar with an industry can often explain cyclic 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, one can sometimes get some hint of the likely future direction of the cycle movement.

Business Cycles Business cycles are

wavelike fluctuations in the general level of economic activity.

They are often described by a diagram such as this.

Business Cycles Expansion Phase: The period

between the begging trough (A) and the Peak (B).

Recession, or Contraction phase: 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.

Business Cycles If business cycles were

true cycles, then they would have a

constant amplitude (The vertical distance from trough to peak).

they would have a constant periodicity (the length of time between successive peaks or trough).

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

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


Coincident Index Employees on nonagricultural payrolls Personal income less transfer payments Industrial production Manufacturing and trade sales


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

Business Cycle Indicators It is possible that one of these indexes, or

one of the series that make up an index may be useful in predicting the cycle factor in a time 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 used

as independent variable in a regression model.

Finding the Cyclical Factor The cyclical factors

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

1 1 1 10 142 2 18 14.63 3 20 15.3 15.2 1.0034 4 12 15.8 15.8 0.997

2 1 5 12 16.5 16.4 1.0062 6 20 17.1 17 1.0073 7 24 17.5 17.6 0.9944 8 13 18 18.2 0.989

3 1 9 14 18.8 18.8 0.9972 10 22 19.6 19.4 1.0123 11 284 12 16

003.115.2

15.3

CMAT

CMACF

CMAT

CMACF

3

33

t

tt

Classical Decomposition Additive Decomposition

We assume that the time series is additive. A classical decomposition can be carried out using the following steps.

Step 1: The trend cycle is computed using a centered MA of order k.

Step2: The detrended series is computed by subtracting the trend-cycle component from the data

tttt ESTY

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

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Time series Decomposition Farideh Dehkordi-Vakil