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Phase-Plane Plotting the Phase-Plane Plotting the Nondurable Goods Index Nondurable Goods Index 1920 1930 1940 1950 1960 1970 1980 1990 2000 0 20 40 60 80 100 120 Year N ondurable G oods Index

Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

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Page 1: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Phase-Plane Plotting the Phase-Plane Plotting the Nondurable Goods IndexNondurable Goods Index

1920 1930 1940 1950 1960 1970 1980 1990 20000

20

40

60

80

100

120

Year

Non

dura

ble

Goo

ds In

dex

Page 2: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Nondurable goods last less than two Nondurable goods last less than two years: Food, clothing, cigarettes, years: Food, clothing, cigarettes, alcohol, but not personal computers!!alcohol, but not personal computers!!

The nondurable goods manufacturing The nondurable goods manufacturing index is an indicator of the economics index is an indicator of the economics of everyday life.of everyday life.

The index has been published The index has been published monthly by the US Federal Reserve monthly by the US Federal Reserve Board since 1919.Board since 1919.

It complements the durable goods It complements the durable goods manufacturing index.manufacturing index.

Page 3: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

What we want to doWhat we want to do

Look at important events.Look at important events. Examine the overall trend in the Examine the overall trend in the

index.index. Have a look at the annual or seasonal Have a look at the annual or seasonal

behavior of the index.behavior of the index. Understand how the seasonal Understand how the seasonal

behavior changes over the years and behavior changes over the years and with specific events.with specific events.

Page 4: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

The log nondurable goods The log nondurable goods indexindex

1920 1930 1940 1950 1960 1970 1980 1990 2000

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Year

Log

10 N

ondu

rabl

e G

oods

Inde

x

Stock Market crash

Treasure Board closes money supply

Vietnam War ends

World War II begins

Page 5: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Events and TrendsEvents and Trends

Short term:Short term: 1929 stock market crash1929 stock market crash 1937 restriction of money supply1937 restriction of money supply 1974 end of Vietnam war, OPEC oil crisis1974 end of Vietnam war, OPEC oil crisis

Medium term:Medium term: DepressionDepression World War IIWorld War II Unusually rapid growth 1960-1974Unusually rapid growth 1960-1974 Unusually slow growth 1990 to presentUnusually slow growth 1990 to present

Long term increase of 1.5% per yearLong term increase of 1.5% per year

Page 6: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

The evolution of seasonal The evolution of seasonal trendtrend

We focus on the years 1948 to 1999We focus on the years 1948 to 1999 We estimate long- and medium-term We estimate long- and medium-term

trend by spline smoothing, but with trend by spline smoothing, but with knots too far apart to capture knots too far apart to capture seasonal trendseasonal trend

We subtract this smooth trend to We subtract this smooth trend to leave only seasonal trendleave only seasonal trend

Page 7: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Smoothing the dataSmoothing the data

We want to represent the data yj by a smooth curve x(t). The curve should have at least two smooth derivatives. We use spline smoothing, penalizing the size of the 4th derivative.

2 2000 2

1948

4( | )n

j

F x dtjj x x ty t D

A function Pspline in S-PLUS is available by ftp from

ego.psych.mcgill.ca/pub/ramsay/FDAfuns

Page 8: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Year

Lo

g1

0 N

on

du

rab

les G

oo

ds I

nd

ex

1964.0 1965.0 1966.0 1967.0

1.6

21

.64

1.6

61

.68

1.7

01

.72

Three years of typical trend: Three years of typical trend: 1964-19661964-1966

Page 9: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Seasonal TrendSeasonal Trend

Typically three peaks per yearTypically three peaks per year The largest is in the fall, peaking at The largest is in the fall, peaking at

the beginning of Octoberthe beginning of October The low point is mid-DecemberThe low point is mid-December

Page 10: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Non-seasonal trendNon-seasonal trend

Page 11: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Seasonal trendSeasonal trend

Page 12: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Phase-Plane PlotsPhase-Plane Plots

Looking at seasonal trend itself does not Looking at seasonal trend itself does not reveal as much as looking at the interplay reveal as much as looking at the interplay between: between: VelocityVelocity or its first derivative, reflecting or its first derivative, reflecting kinetic kinetic

energyenergy in the system. in the system. AccelerationAcceleration or its second derivative, reflecting or its second derivative, reflecting

potential energypotential energy.. The The phase-plane diagramphase-plane diagram plots acceleration plots acceleration

against velocity. against velocity. For purely sinusoidal trend, the plot would For purely sinusoidal trend, the plot would

be an ellipse. be an ellipse.

Page 13: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Position of a swinging Position of a swinging pendulumpendulum

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

Hor

izon

tal P

ositi

on

Large velocity, big kinetic energy

Large acceleration, big potential energy

Page 14: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Phase-plane plot for Phase-plane plot for pendulumpendulum

-8 -6 -4 -2 0 2 4 6 8-40

-30

-20

-10

0

10

20

30

40

Velocity

Acc

eler

atio

n

Page 15: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Velocity

Acc

ele

ratio

n

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

-10

-50

51

0

1964

j

F

m

A

M

J

J

A

S

O

N

D

j

Phase-plane plot for 1964Phase-plane plot for 1964

There are three large There are three large loops separated loops separated by two small loops by two small loops or cusps:or cusps:

1.1. Spring cycleSpring cycle: mid-: mid-January into AprilJanuary into April

2.2. Summer cycleSummer cycle: : May through May through AugustAugust

3.3. Fall cycleFall cycle: October : October through Decemberthrough December

Page 16: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

A look at the years 1929-A look at the years 1929-1931.1931.

Page 17: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Velocity

Acc

ele

ratio

n

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

-10

-50

51

0

1929

jF

m

A

M

J

J

A

SO

N

D

j

Page 18: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Velocity

Acc

ele

ratio

n

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

-10

-50

51

0

1930

j

F

m

A

M

J

J

A

S

ON

D

j

Page 19: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Velocity

Acc

ele

ratio

n

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

-10

-50

51

0

1931

j

F

m

AMJ

J

A

S

O

N

D

j

Page 20: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

1929 through 19311929 through 1931

The stock market crash shows up as The stock market crash shows up as a large negative surge in velocity.a large negative surge in velocity.

Subsequent years nearly lose the fall Subsequent years nearly lose the fall production cycle, as people tighten production cycle, as people tighten their belts and spend less at their belts and spend less at Christmas.Christmas.

Page 21: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

What happened in 1937-What happened in 1937-1938?1938?

Page 22: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Velocity

Acc

ele

ratio

n

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

-10

-50

51

0

1937

j

FmA

MJ

J

A

S

O

N

D

j

Page 23: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Velocity

Acc

ele

ratio

n

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

-10

-50

51

0

1938

j

F

m

AM

JJ

A

S

O

N

D j

Page 24: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Year

Lo

g1

0 I

nd

ex

1920 1940 1960 1980 2000

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Page 25: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

1937 and 19381937 and 1938

The Treasury Board, fearing that the The Treasury Board, fearing that the economy was becoming overheated economy was becoming overheated again, clamped down on the money again, clamped down on the money supply. The effect was catastrophic, supply. The effect was catastrophic, and nearly wiped out the fall cycle. and nearly wiped out the fall cycle.

This new crash was even more dramatic This new crash was even more dramatic than that of 1929, but was forgotten than that of 1929, but was forgotten because of the outbreak of World War II.because of the outbreak of World War II.

Page 26: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

What about World War II?What about World War II?

Page 27: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Velocity

Acc

ele

ratio

n

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

-10

-50

51

0

1943

j

F

mAMJ

J

AS

O

N

D

j

Page 28: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

During World War II, the seasonal cycle During World War II, the seasonal cycle became very small, since the war, and became very small, since the war, and the production that fed it, lasted all the production that fed it, lasted all year long.year long.

Now look at three pivotal years, 1974 to Now look at three pivotal years, 1974 to 1976, when the Vietnam War ended 1976, when the Vietnam War ended and the OPEC oil crisis happened. and the OPEC oil crisis happened. Watch the shrinking of the fall cycle.Watch the shrinking of the fall cycle.

Page 29: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Velocity

Acc

ele

ratio

n

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

-10

-50

51

0

1974

j

Fm

AM

J

J

AS

O

N

D

j

Page 30: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Velocity

Acc

ele

ratio

n

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

-10

-50

51

0

1975

j

F

m

A

M

J

J

AS

O

N

D

j

Page 31: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Velocity

Acc

ele

ratio

n

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

-10

-50

51

0

1976

j

F

m

AM

J

J

A

S

ON

D

j

Page 32: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

What about today?What about today?

Page 33: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Velocity

Acc

ele

ratio

n

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

-10

-50

51

0

1996

j

FmA

M

J

J

AS

O

N

D

j

Page 34: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

These daysThese daysOver the last ten years the size of all three Over the last ten years the size of all three

cycles have become much smaller.cycles have become much smaller.Why?Why?

Is variation now smoothed out by information Is variation now smoothed out by information technology?technology?

Are the aging baby boomers spending less?Are the aging baby boomers spending less? Are personal computers, video games, and Are personal computers, video games, and

other electronic goods really durable?other electronic goods really durable? Has manufacturing now moved off shore?Has manufacturing now moved off shore?

Page 35: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

ConclusionsConclusions

We can separate long- and medium-We can separate long- and medium-term trends from seasonal trends by term trends from seasonal trends by smoothing.smoothing.

Phase-plane plots are great ways to Phase-plane plots are great ways to inspect seasonality.inspect seasonality.

Derivatives were used in two ways: Derivatives were used in two ways: to to penalize roughnesspenalize roughness, and to reflect , and to reflect the the dynamicsdynamics of manufacturing. of manufacturing.

Page 36: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Trends in SeasonalityTrends in Seasonality

We see by inspection that seasonal trends We see by inspection that seasonal trends change systematically over time, and change systematically over time, and can also change abruptly.can also change abruptly.

We first estimate the principal We first estimate the principal components of seasonal variation, using components of seasonal variation, using a version of principal components a version of principal components analysis adapted to functional data, and analysis adapted to functional data, and sensitive only to effects periodic over sensitive only to effects periodic over one year.one year.

Page 37: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!
Page 38: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!
Page 39: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!
Page 40: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

The ComponentsThe Components

1.1. Relative sizes of spring and summer Relative sizes of spring and summer cycles (53%)cycles (53%)

2.2. Joint size of spring and summer Joint size of spring and summer cycles (25%)cycles (25%)

3.3. Size of fall cycle (11%)Size of fall cycle (11%)

Page 41: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Plotting Component ScoresPlotting Component Scores

We can compute scores at each year We can compute scores at each year for these three principal components, for these three principal components, sometimes called empirical sometimes called empirical orthogonal functions.orthogonal functions.

Plotting the evolution of these scores Plotting the evolution of these scores over the 51 years shows some over the 51 years shows some interesting structural changes in the interesting structural changes in the economics of everyday life.economics of everyday life.

Page 42: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!
Page 43: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!
Page 44: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

Wrap-upWrap-up

Phase-plane plots are good for Phase-plane plots are good for inspecting seasonal quasi-harmonic inspecting seasonal quasi-harmonic trendstrends

Principal components analysis Principal components analysis reveals main components of reveals main components of variation in seasonal trend.variation in seasonal trend.

Plotting component scores shows Plotting component scores shows how trend has evolved.how trend has evolved.

Page 45: Phase-Plane Plotting the Nondurable Goods Index. Nondurable goods last less than two years: Food, clothing, cigarettes, alcohol, but not personal computers!!

This was joint with work with James B. This was joint with work with James B. Ramsey, Dept. of Economics, New Ramsey, Dept. of Economics, New York University, and is reported in:York University, and is reported in:

Ramsay, J. O. and Ramsey, J. B. (2001) Ramsay, J. O. and Ramsey, J. B. (2001) Functional data analysis of the Functional data analysis of the dynamics of the monthly index of dynamics of the monthly index of non‑durable goods production. non‑durable goods production. Journal of EconometricsJournal of Econometrics, 107, , 107, 327‑344.327‑344.