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8/8/2019 6313f07timeseries

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Time SeriesForecasting

Outline:

1. Measuring forecast error

2. The multiplicative time series model

3. Nave extrapolation4. The mean forecast model

5. Moving average models

6. Weighted moving average models

7. Constructing a seasonal index using a centeredmoving average

8. Exponential smoothing


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Forecast error

Month/Year

(1)Forecasted

Value

(2)ActualValue

(3) = (2) (1)

Error July 2000 $390 $423 $33

Aug 2000 450 429 -21

Sept 2000 289 301 12

Forecasting Convenience Store Ice Sales


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3 measures of forecast error

Mean absolute deviationMean square error

Root mean square error.


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Actual

Predicted

Time Average Absolute Error ( AAE ) is given by:

!

!m

t

t t Y Y

m

AAE 1

1

Where Yt is the actual value of variable that we seek toforecast and is the fitted or forecasted value of thevariable.

t Y


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Actual

Predicted

Time R oot M ean Square Error (root M SE ) is given by:

2

1

)(1

t

m

t

t Y Y

m

root M SE !

!

R oot MSE is a statisticthat is typically is reported by forecasting software

applications


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T he time path of a variable(such as monthly sales of building materials by supply stores) is produced by theinteraction of 4 factors or components.T hesecomponents are:

1. T he trend component(T )2. T he seasonal component(S)3. T he cyclical component(C); and4. T he irregular component(I)


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Th e trend component ( T)

T rend is the gradual, long-run (or secular) evolutionof the variables that we are

seeking to forecast.


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F actors affecting t he trend component of atime series

P opulation changesDemographic changes. For example, spending for

healthcare services is likely to rise due to the aging

of the population.Sales of fast food are up due tothe secular increase in the female labor forceparticipation rate.

T echnological change.Sales of music on DVD have

slumped due toIpods.T ypewriter sales haveplumetted.

C hanges in consumer tastes and preferences.


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-60

-40

-20

0

20

40

10 20 30 40 50 60 70 80 90 100

Linear tre nd s

Trend = 10 25t

Trend = -50 + .8t


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0

1000

2000

3000

4000

10 20 30 40 50 60 70 80 90 100

Non-lin ear, increas ing tre nd

Trend = 10 + .3t + .3t2


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-5000

-4000

-3000

-2000

-1000

0

1000

10 20 30 40 50 60 70 80 90 100

Non-lin ear, decreas ing tre nd

Trend = 10 - .4t - .4t 2


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The seasonal component (S)

M any series display a regular pattern of variability depending on the time of year.For example, sales of toys and scotch

whiskey peak in December each year.

Ice cream sales are higher in summer

months than in winter months.Car sales tend typically to be strong

in May and June and weaker inNovember and December.


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The cyclical component (C)

T he time path of a series can be influenced by businesscycle fluctuations.

For example, we expect housing starts to decline in thecontractionary phase of the business cycle.T he same holds true for federal or state tax receiptsT he time path of spending for consumer durable goods

is also shaped by cyclical forces.Spending for capital goods is likewise cyclical.T he movie industry has the reputation for being

counter-cyclicalfor example, it flourished duringthe Depression.


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Th e irregular component (I )

T he irregular component of the series, sometimescalledw hite noise , is the remaining variability(relativeto trend) that cannot be explained by seasonal orcyclical factors.T he irregular component is anunexpected, non-recurring factor that affects the series.For example, hamburger sales plunge due to panic

about E-C oli bacteria.P roduction of trucks slumps because of a strike at a

GM parts plant in Ohio.Airline slump after 9/11.A cold snap affects July ice cream sales in upstate NY.


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If you have a well-designedforecasting model, then forecastingerrors should be mainly accounted

for by irregular factors


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Th e model

t t t t t I C S T Y vvv!

Where:Y

t is the value of the time series variable in period t(month t, quarter t, etc.)T t trend component of the series in period t S t is the seasonal component of the series in period t

C t is the cylical component of the series at period t; and I t is the irregular component of the series in period t.


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Th e Problem: F orecast Sales of HomeF urnis hing Stores, October -D ecember, 2007

T he data:We have monthly data of sales of home

furniture stores January 1992 to July 2007(187 monthly observations).

T he data are expressed in millions of current dollars, not seasonally adjusted


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t Yr Mo $1 1992 1 14602 1992 2 14533 1992 3 15564 1992 4 16225 1992 5 16756 1992 6 17597 1992 7 1789

8 1992 8 18149 1992 9 1721

10 1992 10 183911 1992 11 192512 1992 12 2246

" " " "" " " "

187 2007 7 4803

The D ata


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1000

2000

3000

4000

5000

6000

7000

92 94 96 98 00 02 04 06

Sales of Home Furnishing Stores, 1992-2007(millions of dollars, NSA)

Year/Month

Source: Economagic.com


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OLS is a method of finding the line, or curve, of best fit.

T he trend function of best fit is the one thatminimizes the squared sum of the vertical distancesof the sample points(the actual monthly values of home furnishing sales) from the trend line(fittedvalues of monthly building materials sales).

Our first step is to estimate thetrend component of our series.This is accomplished using a

ordinary least squares, or OLS for short.


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Let:Yt be the actual value of furniture store sales in

month t;

Let t be the trend value of furniture store salesin month t.T he trend function we are seekingsatisfies the following condition:

2

187

1)(. t t

t Y Y M IN

!


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We estimate a linear trend function wit h Excel.It is displayed on t he nextslide.


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y = 17 .62 x + 1475 .

0

1000

2000

3000

4000

5000

6000

7000

0 20 40 60 80 100 120 140 160 180 200

L inear Trend L ine Fitted toHome Furnishing Data

t

R 2 = 0 .8 3


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1000

2000

3000

4000

5000

6000

7000

92 94 96 98 00 02 04 06

Actual TREND

Year/Month

Actual and Trend Values of Hom Furniture Sales (in millions)


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Month IndexJan 0.8799Feb 0.8475Mar 0.9823

Apr 0.9004May 0.9939Jun 1.0197Jul 0.9729

Aug 1.0487

Sep 1.0042Oct 0.9962Nov 1.123Dec 1.2969

If you sum t hemont hly values anddivide by 12 , youget 1 .00 .

Later we s how asimple tec hniquefor computing a

seasonal index.

Seasonal Index


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Performing an in -sample forecast of homefurnis hing sales

A n in-sample forecast means we are forecastinghome furshing sales for those months for which wealready have data that have been used to estimate the

trend, seasonal, and other components . Comparingforecasted, or fitted values of home furnishing saleswith actual time series data gives us an idea of howwell this performs.

We will assume that the cyclical index is equal to 1(Ct = 1). This is a poor assumption since our periodcontains two business cycle contractions.


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t t t Apr S F vv!9 8

71.532,2$19 00.0]1475)7662.17[( 9 8 !vvv! A pr

Lets give an example how weuse this model to Home

furnishing sales for a particular month, say, A pril199 8 . t = 76 for this month


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1000

2000

3000

4000

5000

6000

7000

94 96 98 00 02 04 06

Multiplcative model Home furnishing sales (millions)

In-Sample Forecas t of Home Furnishing Sales Us ing Multiplicative Model

Year/Month


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-300

-200

-100

0

100

200

300

94 96 98 00 02 04 06

Res i ls f I -samp le ecas t f me ish i a les i milli s)

Recess i is shaded

Year / th275.103$! M SE


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t Yr/Mo Trend Seasonal Cyclical Forecast190 2007/Oct 4822.8 0.9962 0.999 4799.669

191 2007/Nov 4840.42 1.123 0.979 5321.64

192 2007/Dec 4858.04 1.2969 0.975 6142.882

F orecasting Using t he MultiplicativeModel

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