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 years: Food, clothing, cigarettes, alcohol, but not personal computers!! The nondurable goods manufacturing index is an indicator of the economics of everyday life. 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 The index has been published monthly by the US Federal Reserve Board since It complements the durable goods manufacturing index. It complements the durable goods manufacturing index.

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

The log nondurable goods index

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

The evolution of seasonal trend We focus on the years 1948 to 1999 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 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 We subtract this smooth trend to leave only seasonal trend

Smoothing the data We want to represent the data y j by a smooth curve x(t). The curve should have at least two smooth derivatives. We use spline smoothing, penalizing the size of the 4 th derivative. A function Pspline in S-PLUS is available by ftp from ego.psych.mcgill.ca/pub/ramsay/FDAfuns

Three years of typical trend:

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

Non-seasonal trend

Seasonal trend

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

Position of a swinging pendulum

Phase-plane plot for pendulum

Phase-plane plot for 1964 There are three large loops separated by two small loops or cusps: 1. Spring cycle: mid- January into April 2. Summer cycle: May through August 3. Fall cycle: October through December

A look at the years

1929 through 1931 The stock market crash shows up as a large negative surge in velocity. 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. Subsequent years nearly lose the fall production cycle, as people tighten their belts and spend less at Christmas.

What happened in ?

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.

What about World War II?

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.

What about today?

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? Is variation now smoothed out by information technology? Are the aging baby boomers spending less? Are the aging baby boomers spending less? Are personal computers, video games, and other electronic goods really durable? Are personal computers, video games, and other electronic goods really durable? Has manufacturing now moved off shore? Has manufacturing now moved off shore?

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

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.

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%)

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.

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

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,