Phase-Plane Plotting the Nondurable Goods Index

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Phase-Plane Plotting the Nondurable Goods Index
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• Nondurable goods last less than two years Food,
  clothing, cigarettes, alcohol, but not personal
  computers!!
• The nondurable goods manufacturing index is an
  indicator of the economics of everyday life.
• The index has been published monthly by the US
  Federal Reserve Board since 1919.
• It complements the durable goods manufacturing
  index.

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What we want to do

• Look at important events.
• Examine the overall trend in the index.
• Have a look at the annual or seasonal behavior of
  the index.
• Understand how the seasonal behavior changes over
  the years and with specific events.

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The log nondurable goods index
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Events and Trends

• Short term
• 1929 stock market crash
• 1937 restriction of money supply
• 1974 end of Vietnam war, OPEC oil crisis
• Medium term
• Depression
• World War II
• Unusually rapid growth 1960-1974
• Unusually slow growth 1990 to present
• Long term increase of 1.5 per year

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The evolution of seasonal trend

• We focus on the years 1948 to 1999
• We estimate long- and medium-term trend by spline
  smoothing, but with knots too far apart to
  capture seasonal trend
• We subtract this smooth trend to leave only
  seasonal trend

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Smoothing 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.
A function Pspline in S-PLUS is available by ftp
from ego.psych.mcgill.ca/pub/ramsay/FDAfuns
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Three years of typical trend 1964-1966
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Seasonal Trend

• Typically three peaks per year
• The largest is in the fall, peaking at the
  beginning of October
• The low point is mid-December

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Non-seasonal trend
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Seasonal trend
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Phase-Plane Plots

• Looking at seasonal trend itself does not reveal
  as much as looking at the interplay between
• Velocity or its first derivative, reflecting
  kinetic energy in the system.
• Acceleration or its second derivative, reflecting
  potential energy.
• The phase-plane diagram plots acceleration
  against velocity.
• For purely sinusoidal trend, the plot would be an
  ellipse.

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Position of a swinging pendulum
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Phase-plane plot for pendulum
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Phase-plane plot for 1964

• There are three large loops separated by two
  small loops or cusps
• Spring cycle mid-January into April
• Summer cycle May through August
• Fall cycle October through December

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A look at the years 1929-1931.
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1929 through 1931

• The stock market crash shows up as a large
  negative surge in velocity.
• Subsequent years nearly lose the fall production
  cycle, as people tighten their belts and spend
  less at Christmas.

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What happened in 1937-1938?
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1937 and 1938

• The Treasury Board, fearing that the economy was
  becoming overheated again, clamped down on the
  money supply. The effect was catastrophic, and
  nearly wiped out the fall cycle.
• This new crash was even more dramatic than that
  of 1929, but was forgotten because of the
  outbreak of World War II.

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What about World War II?
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• During World War II, the seasonal cycle became
  very small, since the war, and the production
  that fed it, lasted all year long.
• Now look at three pivotal years, 1974 to 1976,
  when the Vietnam War ended and the OPEC oil
  crisis happened. Watch the shrinking of the fall
  cycle.

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What about today?
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These days

• Over the last ten years the size of all three
  cycles have become much smaller.
• Why?
• Is variation now smoothed out by information
  technology?
• Are the aging baby boomers spending less?
• Are personal computers, video games, and other
  electronic goods really durable?
• Has manufacturing now moved off shore?

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Conclusions

• We can separate long- and medium-term trends from
  seasonal trends by smoothing.
• Phase-plane plots are great ways to inspect
  seasonality.
• Derivatives were used in two ways to penalize
  roughness, and to reflect the dynamics of
  manufacturing.

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Trends in Seasonality

• We see by inspection that seasonal trends change
  systematically over time, and can also change
  abruptly.
• We first estimate the principal components of
  seasonal variation, using a version of principal
  components analysis adapted to functional data,
  and sensitive only to effects periodic over one
  year.

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The Components

1. Relative sizes of spring and summer cycles (53)
2. Joint size of spring and summer cycles (25)
3. Size of fall cycle (11)

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Plotting Component Scores

• We can compute scores at each year for these
  three principal components, sometimes called
  empirical orthogonal functions.
• Plotting the evolution of these scores over the
  51 years shows some interesting structural
  changes in the economics of everyday life.

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Wrap-up

• Phase-plane plots are good for inspecting
  seasonal quasi-harmonic trends
• Principal components analysis reveals main
  components of variation in seasonal trend.
• Plotting component scores shows how trend has
  evolved.

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• This was joint with work with James B. Ramsey,
  Dept. of Economics, New York University, and is
  reported in
• Ramsay, J. O. and Ramsey, J. B. (2001) Functional
  data analysis of the dynamics of the monthly
  index of non-durable goods production. Journal
  of Econometrics, 107, 327-344.