irwin/mcgraw-hill andrew f. siegel, 1997 and 2000 14-1 l chapter 14 l time series: understanding...

Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000

14-1

l Chapter 14 l

Time Series: Understanding

Changes over Time


14-2

Time Series AnalysisGoals

Understand the past Forecast the future

Different from Cross-Sectional Data Time-series data are not independent of each other

Not a random sample Does not satisfy the random-sample assumption for confidence intervals (in

Chapter 9) or hypothesis testing (in Chapter 10) New methods are needed to take account of the interdependence


14-3

Cross-Sectional and Time-SeriesCross-Sectional Data

Expect nextobservation to beabout S awayfrom

Time-Series Data Next will probably

not be about S away from(not a random sample)

X

X

XSS

S XS


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ForecastingUse a model

A system of equations that can produce data that “look like” your time series data

Estimate the model Your forecast will be the expected (mean) value of the future

behavior of the model The forecast limits are the confidence limits for your forecast (if your

model can produce them) Computed from the appropriate standard error If model is correct, the future observation has a 95% chance of being

within the forecast limits


14-5

Trend-Seasonal and Box-JenkinsTrend-Seasonal Analysis

Direct and intuitive, with four components: (1) Long-term Trend, (2) repeating Seasonal, (3) medium-term wandering

Cyclic, and (4) random Irregular Forecast comes from extending the Trend and Seasonal

Box-Jenkins ARIMA Process Flexible, but complex, probability models for how current value of

the series depends upon Past values, past randomness, and new randomness

A better way to describe the Cyclic component Forecast is

Expectation of random future behavior, given past data


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Example: Computer RevenuesSteady growth

not perfectly smooth Nonlinear (curved)

Suggests constant growth rateLogarithm of revenues

Log plot looks linearif constant growth rate

Can use regression tomodel relationship Points appear randomly

distributed about the line,so serial correlation is not a problem

Fig 14.1.4, 6

0

10

20

30

40

50

1986 1988 1990 1992 1994 1996

Rev

enue

s (bi

llion

s)

2

3

4

1986 1988 1990 1992 1994 1996

Log

of R

even

ues


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Example: Retail SalesU.S. Retail Sales (Monthly)

Growth Repeating seasonal variation

High in December Low in January, February

Seasonally-Adjusted Sales Growth Seasonal pattern removed

Shows how sales went up(or down) relative to whatyou expect for time of year

Fig 14.1.7, 8

150

200

250

1995 1996 1997 1998

Sale

s ($b

illio

ns)

150

200

250

1995 1996 1997 1998

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s ($b

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Example: Interest RatesU.S. Treasury Bills, Yearly

Generally rising

Substantial variation

Cyclic pattern Rising and falling Increasing magnitude Not perfectly repeating Not expected to continue rising indefinitely!

Fig 14.1.9

0%

5%

10%

15%

1960 1970 1980 1990 2000In

tere

st ra

te


14-9

Trend-Seasonal AnalysisDecompose a Time Series into Four Components

Data = Trend Seasonal Cyclic Irregular Trend

Long-term behavior (often straight line or exponential growth) Seasonal

Repeating effects of time-of-year Cyclic

Gradual ups and downs, not repeating each year, not purely random Irregular

Short-term, random, nonsystematic noise


14-10

Ratio-to-Moving-Average MethodMoving Average Represents Trend and Cyclic

Eliminates Seasonal and Irregular by averaging a yearDivide Data by Moving Average

Represents Seasonal and Irregular Group by season, then average, to obtain Seasonal

Seasonal Adjustment: Divide Data by SeasonalRegress Seasonally-Adjusted Series vs. Time

Represents TrendForecast by Seasonalizing the Trend

Multiply (future predicted Trend) by (Seasonal index)


14-11

Example: Ford Motor CompanyTime-series Plot

Quarterly data with strong Seasonal pattern Revenues typically highest in second quarter Does not repeat perfectly (due to Cyclic and Irregular)

Fig 14.2.1

$0

$10

$20

$30

1990 1995 2000

Sale

s ($

billi

ons)


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Example: Moving AverageAverages one year of data

2 quarters before to 2 quarters after each data value Smooths the data, eliminating Seasonal and Irregular Shows you Trend and Cyclic

Original data

Moving average

Fig 14.2.4

$0

$10

$20

$30

1990 1995 2000

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billi

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Example: Seasonal IndexFig 14.2.6

Average “Ratio-to-Moving-Average” by Quarter Seasonal index for each quarter, repeating each year

Shows how much larger (or smaller) this quarter is compared to a typical period throughout the year

0.8

0.9

1.0

1.1

1990 1995 2000

Seas

onal

inde

x


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Example: Seasonal AdjustmentDivide Data by Seasonal Index

To get Seasonally Adjusted Value Eliminates the expected seasonal component Shows changes that are not due to expected seasonal effects

Original data

Seasonallyadjusted series

Fig 14.2.7

$0

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$30

1990 1995 2000

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s ($

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Example: Trend LineRegress Seasonally-Adjusted Data vs. time

The resulting line can be extended into the future This gives a Seasonally-Adjusted Forecast

Fig 14.2.8

$0

$10

$20

$30

$40

1990 1995 2000

Sale

s ($b

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Seasonallyadjustedforecast

Seasonallyadjusted series

Trend line


14-16

Example: ForecastSeasonalize the Trend

Multiply Trend by Seasonal Index Can be extended into the future

Use future predicted Trend with quarterly Seasonal index

Fig 14.2.9

$0

$10

$20

$30

$40

1990 1995 2000

Sale

s ($b

illio

ns)

Original data

Seasonalized trendForecast


14-17

Box-Jenkins ARIMA ProcessesA Collection of Linear Statistical Models

Can describe many different kinds of time-series Including medium-term “cyclic” behavior

Compared to trend-seasonal analysis, Box-Jenkins Has a more solid statistical foundation Is more flexible Is somewhat less intuitive

Outline of the steps involved Choose a type of model and estimate it using your data Forecast using average future random behavior of this model Find standard error (variability in this future behavior) Find forecast limits, to include 95% of future behavior


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Random Noise ProcessA Random Sample, with No Memory

Data = (Mean value) + (Random Noise) Yt = + t

The long-term mean of Y is

Mean


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Autoregressive (AR) ProcessRemembers the Past, Adds Random Noise

Data = + (Previous value) + (Random Noise) Yt = + Yt–1 + t

The long-term mean value of Y is –

Mean


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Moving-Average (MA) ProcessRemembers Previous Noise, Adds New Noise

Data = + (Random Noise) – (Previous Noise) Yt = + t – t–1

The long-term mean value of Y is

Mean


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ARMA ProcessAutoregressive Moving Average Process

Remembers the Past, Previous Noise, Adds New NoiseData = + (Previous value) + (Noise) – (Previous Noise) Yt = + Yt–1 + t – t–1

The long-term mean value of Y is –

Mean


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Example: Unemployment

Estimated ARMA Process for this Time SeriesYt = + Yt–1 + t + t–1

where random noise has standard deviation 0.907

0%

5%

10%

1960 1970 1980 1990 2000

Une

mpl

oym

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Example (continued)Random Simulations from Estimated Process

Look similar to actual unemployment rate history Because of estimation using actual data Looking at “what might have happened instead”

0%

5%

10%

1960 1970 1980 1990 2000

Une

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Example (continued)Forecast and 95% Forecast Limits (10 years ahead)

Using the average of random future possibilities And their lower and upper 95% limits

0%

5%

10%

1960 1970 1980 1990 2000 2010

Une

mpl

oym

ent r

ate

Forecast


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Example (continued)Three Simulations of the Future

With forecast and 95% Forecast Limits To see how forecast represents future possibilities

0%

5%

10%

1960 1970 1980 1990 2000 2010

Une

mpl

oym

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14-26

Pure Integrated (I) ProcessA Random Walk from the Previous Value

Data = + (Previous value) + (Random Noise) Yt = + Yt–1 + t

Over time, Y is not expected to stay close to any long-term mean value


14-27

ARIMA ProcessAutoregressive Integrated Moving AverageRemembers its Changes

The differences, Yt – Yt–1, follow an ARMA process Over time, Y is not expected to stay close to any long-term mean

value

irwin/mcgraw-hill andrew f. siegel, 1997 and 2000 14-1 l chapter 14 l time series: understanding...

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