James B. Ramsey

The Decomposition of Economic Relationships by Time Scale Using Wavelets: Expenditure and Income

Economists have long known that time scale matters, in that the structure of decisions as to the relevant time horizon, degree of time aggregation, strength of relationship, and even the relevant variables differ by time scale. Unfortunately, until recently it was difficult to decompose economic time series into orthogonal time-scale components except for the short and long run, in which the former is dominated by noise. This paper uses wavelets to produce an orthogonal decomposition of some economic variables by time scale over six different time scales. The relationship of interest is the permanent income hypothesis. We confirm that time-scale decomposition is very important for analyzing economic relationships and that a number of anomalies previously noted in the literature are explained by these means. The analysis indicates the importance of recognizing variations in phase between variables when investigating the economic relationships.

Time Irreversibility and Business Cycle Asymmetry

The problem of business-cycle symmetry is addressed within the context of time reversibility. To this effect, the authors introduce a time domain test of time reversibility, the TR test. In an application, they show that time irreversibility is the rule rather than the exception for two well-known representative macroeconomic data sets. This shows that many components of the business cycle have asymmetric fluctuations. The characterization of asymmetry provided by the TR test shows that many series exhibit steepness asymmetry. A few series appear to be either deep or sharp. Copyright 1996 by Ohio State University Press.

The Statistical Properties of Dimension Calculations Using Small Data Sets

The statistical properties of estimates of pointwise dimension and their errors have been investigated for some maps and for some random variables. Substantial bias in the estimates is detected and modelled as a function of the sample size and the embedding dimension. The usual methods for calculating error bars are shown to underestimate the actual error bars by factors of ten and more. Procedures to improve the estimation of dimension in the cases studied are discussed, as are methods to improve the ability to distinguish noise from an attractor when using small data sets. Some idea of small is given when the attractors are similar to the ones studied.

The Statistical Properties of Dimension Calculations Using Small Data Sets: Some Economic Applications

Several recent attempts have been made to test for chaos in economic time series through dimension calculations. Relative to the large data sets used in the natural sciences, economic time series are small. Using a procedure developed by J. B. Ramsey and H. Yuan, the authors show that, with the techniques available to date and for the time series examined so far, there is virtually no evidence for the presence of simple chaotic attractors. Copyright 1990 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Bias and error bars in dimension calculations and their evaluation in some simple models

Abstract With respect to two simple models and various distributions of noise, it is shown that there is substantial bias in dimension calculations with small data sets. Some of the practical difficulties involved in dimension estimation are briefly discussed. It is shown that estimated error bars are a small fraction of the actual error bars. Two empirical equations are presented to model these phenomena. One of these equations can be used to make some modest improvements in estimating dimension with small data sets The second equation indicates a procedure to obtain a more accurate assessment of error bars. Two statistical tests for deciding whether a desired time series is from an attractor or not are aslo suggested.

An Analysis of U.S. Stock Price Behavior Using Wavelets

Using wavelets we re-examine the U.S. stock market price index for any evidence of self-similarity or order that might be revealed at different scales. The wavelet transform localized in time can be used to indicate how the power of the projection of the signal onto the kernel varies with the scale of observation. By comparing how the local power scales vary over time much information about the structure of the data can be obtained. Such evidence is not at all evident from standard analyses of untransformed data, including projections onto a Fourier basis. Wavelets can detect structures in data that are highly localized in time and therefore non-detectable by Fourier transforms. The main conclusion is that while the data are clearly complex, there seems to be some evidence of non-randomness in the data. There is also some limited evidence of quasi-periodicity in the occurrence of large amplitude shocks to the system.

Functional data analysis of the dynamics of the monthly index of nondurable goods production

Functional data analysis techniques pioneered by Ramsay and Bernard Silverman are used to analyze the dynamics of a monthly nonseasonally adjusted index of production. After determining the order (three) of the ordinary differential equation that describes the data and estimating its coefficients; the time variation of each vector of coefficients is examined in detail to determine the evolutionary dynamics of each series over a 70-year span. The dynamical properties of the solution paths are also analyzed for their qualitative dynamics. The resulting models are used to examine the interaction between seasonal dynamics, dynamics at business cycle frequencies, and long-term growth.

The US Wage Phillips Curve across Frequencies and over Time

Although widely used in many areas of applied sciences, wavelet analysis has not fully entered the economic discipline yet. In this article we apply wavelet analysis to one of the most investigated relationships is in empirical macroeconomics: the relationship between wage inflation and unemployment. Using US postwar data we find a frequency-dependent relationship of a sort that is consistent with Phillips’ original insights. It also turns out that this relationship is remarkably stable over the 1948–93 period, but not in the aftermath, as a consequence of a process of adaption of the wage formation process to a low inflation environment.

The Application of Wave Form Dictionaries to Stock Market Index Data

A matching pursuit algorithm is used to implement the application of wave form dictionaries to decompose the signal in the stock market (Standard and Poor’s 500) index. A wave form dictionary is a class of transforms that generalizes both windowed Fourier transforms and wavelets. Each wave form is parametrized by location, frequency, and scale. Such transforms can analyze signals that have highly localized structures in either time or frequency space as well as broad band structures.The Standard and Poor’s 500 stock market index is found to be highly complex, but not a random walk. There are bursts of high energy that arise suddenly with very localized energy and die out equally quickly. In addition there is evidence of Dirac delta functions representing impulses, or shocks, to the system that seem to cluster more than would be expected under an hypothesis of random variation. It would appear that the energy of the system is largely internally generated, rather than the result of external forcing. Finally, there is apparently some evidence for a quasi-periodic occurrence of oscillations that are well localized in time, but that involve almost all frequencies.

Regression over Timescale Decompositions: A Sampling Analysis of Distributional Properties

In two previous papers, Ramsey and Lampart demonstrated that regression analyses between timescale decompositions provided important insight into the properties of economic relationships. The idea in those papers was that the relationship between any two variables, say consumption and income, was the union of the individual relationships between consumption and income at each timescale and that the regression relationship might, differ across timescales. This paper is dedicated to discovering the approximate distributional properties of the regression estimators and of the residuals in the context of such models. Sampling procedures are used to verify the distributional properties of the regression estimators at each timescale and those of the residuals. This analysis is necessary to provide the appropriate distributional information required to specify tests of hypotheses and confidence intervals.