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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
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.
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.
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
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
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
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
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
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
Non-seasonal trendNon-seasonal trend
Seasonal trendSeasonal trend
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.
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
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
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
A look at the years 1929-A look at the years 1929-1931.1931.
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
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
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
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.
What happened in 1937-What happened in 1937-1938?1938?
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
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
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
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.
What about World War II?What about World War II?
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
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.
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
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
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
What about today?What about today?
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
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?
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.
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.
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%)
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.
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.
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.
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