Time Series Understanding Changes Over Time

8/3/2019 Time Series Understanding Changes Over Time

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Chapter 14

Time Series: Understanding

Changes over Time


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Time Series Analysis

Goals

Understand the past

Forecast the future

Different from Cross-Sectional

Data

Time-series data are not independentof 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


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Cross-Sectional and Time-Series

Cross-Sectional Data

Expect next

observation to be

about S away

from

Time-Series Data

Next will probably

notbe about S

away from

(not a random sample)

X

X

XS

S

S

XS


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Trend-Seasonal and Box-Jenkins

Trend-Seasonal Analysis

Direct and intuitive, with four components:

(1) Long-term Trend, (2) repeatingSeasonal, (3) medium-term

wandering Cyclic, and (4) randomIrregular

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: Radio, TV, Computer Stores

Steady growth

not perfectly smooth

Nonlinear (curved)

Suggests constant growth rate

Logarithm of revenues

Log plot looks linear

if constant growth rate

Can use regression to

model relationship

Points are not randomly

distributed about the line,

soserial correlation is present

Fig 14.1.4, 6

0

10

20

30

40

50

60

70

1980 1985 1990 1995 2000Year

Sales(billions)

2

3

4

5

1980 1985 1990 1995 2000

Year

Sales(logarithm)


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Example: Retail Sales

U.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 what

you expectfor time of year

Fig 14.1.7, 8

$150

$200

$250

$300

$350

1997 1998 1999 2000 2001

Year

Sales(billions)

$150

$200

$250

$300

$350

1997 1998 1999 2000 2001

Year

Sales(billions)


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Example: Interest Rates

U.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 2000

Year

Interestrate


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Trend-Seasonal Analysis

Decompose a Time Series into Four Components

Data = Trend v Seasonal v Cyclic v Irregular

Trend

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

Seasonal

Repeating effects of time-of-year

Cyclic

Gradual ups and downs, notrepeating each year, notpurely

random

Irregular

Short-term, random, nonsystematic noise


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Ratio-to-Moving-Average Method

Moving Average Represents Trend and Cyclic

Eliminates Seasonal and Irregularby averaging a year

Divide Databy Moving Average

Represents Seasonal andIrregular

Group by season, then average, to obtain Seasonal

Seasonal Adjustment: Divide Databy Seasonal

Regress Seasonally-Adjusted Series vs. Time

Represents Trend

ForecastbySeasonalizing the Trend

Multiply (future predicted Trend) by (Seasonal index)


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Example: Ford Motor Company

Time-series Plot

Quarterly data with strong Seasonalpattern

Sales typically highest in second quarter

Does not repeat perfectly (due to Cyclic and Irregular)

Fig 14.2.1

$20

$25

$30

$35

$40

1994 1995 1996 1997 1998 1999 2000 2001

Year

FordSa

les(billions)


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Example: Moving Average

Averages 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

Fig 14.2.4

$20

$25

$30

$35

$40

1994 1995 1996 1997 1998 1999 2000 2001

Year

Fordsales(billions)

Original data

Moving average

1994 1995 1996 1997 1998 1999 2000 2001

Year

Original data


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Example: Seasonal Index

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

Fig 14.2.6

1994 1995 1996 1997 1998 1999 2000 2001

Year

Se

asonalindex

0.8

0.9

1.0

1.1

1994 1995 1996 1997 1998 1999 2000 2001

Year


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Example: Seasonal Adjustment

Divide Data by Seasonal Index

To get Seasonally Adjusted Value

Eliminates the expected seasonal component

Shows changes that are not due to expected seasonal effects

Fig 14.2.7

1994 1995 1996 1997 1998 1999 2000 2001

Year

Seasonallyadjusted

Original data

$20

$25

$30

$35

$40

1994 1995 1996 1997 1998 1999 2000 2001

Year

Fordsales(billions)


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Example: Trend Line

Regress Seasonally-Adjusted Data vs. time

The resulting line can be extended into the future

This gives a Seasonally-Adjusted Forecast

Fig 14.2.8

$20

$25

$30

$35

$40

Fordsal

es(billions)

Seasonallyadjusted series

Trend line

seasonally

adjustedforecast

1995 2000 2005

Year


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Example: Forecast

Seasonalize the Trend

Multiply Trendby Seasonal Index

Can be extended into the future

Use future predicted Trend with quarterly Seasonal index

Fig 14.2.9

2005

Year

Fordsales(billions)

$20

$25

$30

$35

$40

$45

1995 2000 2005

Year

Forecast

Original data

Seasonalized trend


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Box-Jenkins ARIMA Processes

A 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 Process

A Random Sample, with No Memory

Data = (Mean value) + (Random Noise)

Yt = Q+ It

The long-term mean ofY

is Q

Mean


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Autoregressive (AR) Process

Remembers the Past, Adds Random Noise

Data = H + N(Previous value) + (Random Noise)

Yt = H + NYt1 + It

The long-term mean value of Y is H N

Mean


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Moving-Average (MA) Process

Remembers Previous Noise, Adds New Noise

Data = Q + (Random Noise) U(Previous Noise)

Yt = Q + ItUIt1

The long-term mean value of Y is Q

Mean


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ARMA Process

Autoregressive Moving Average Process

Remembers the Past, Previous Noise, Adds New Noise

Data = H + N(Previous value) + (Noise) U(Previous Noise)

Yt = H + NYt1 + ItUIt1

The long-term mean value of Y is H N

Mean


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Example: Unemployment

Estimated ARMA Process for this Time SeriesYt = + Yt1 + It+ It1

where random noise has standard deviation 0.907

0%

5%

10%

1960 1970 1980 1990 2000

Unem

ploymentrate


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0%

5%

10%

1960 1970 1980 1990 2000

Unemp

loymentrate

Example (continued)

Random Simulations from Estimated Process

Look similar to actualunemployment rate history

Because of estimation using actual data

Looking at what might have happened instead


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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 upper95% limits

0%

5%

10%

1960 1970 1980 1990 2000 2010

Une

mploymentrate

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

Unemp

loymentrate


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Pure Integrated (I) Process

A Random Walk from the Previous Value

Data = H + (Previous value) + (Random Noise)

Yt = H + Yt1 + It

Over time, Y is notexpected to stay close to any long-

term mean value


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ARIMA Process

Autoregressive Integrated Moving Average

Remembers its Changes

The differences, YtYt1, follow an ARMA process

Over time,Y

is notexpected to stay close to any long-term mean value

Time Series Understanding Changes Over Time

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