A Fractal Comparison of
Real and Austrian Business Cycle Models
Robert F. Mulligan, Ph.D.
Department of Business Computer Information
Systems & Economics
Phone: 828-227-3329
Fax: 828-227-7414
Email: mulligan@wcu.edu
Acknowledgements
Robert F. Mulligan is associate professor of economics in
the Department of Business Computer Information Systems and Economics at
Western Carolina University College of Business and a research associate of the
State University of New York at
Abstract
This
paper examines macromonetary data for fractal character and stochastic
dependence. Lo's modified R/S test for
stochastic dependence is used to explore the data's fractal properties along
with five fractal analysis techniques for estimating the Hurst exponent,
Mandelbrot-Lévy characteristic exponent, and fractal dimension. Techniques employed are rescaled-range
analysis, power-spectral density analysis, roughness-length analysis, the
variogram or structure function method, and wavelet analysis. Monthly data is analyzed over the 1959-2006
range and a quarterly dataset extending from 1947-2006 is also examined. Strong evidence is found for stochastic
dependence in the monetary aggregates, which suggests an activist monetary
policy, and for nominal consumption expenditures, supporting Austrian business
cycle theory. No evidence for stochastic
dependence in investment productivity is found, and this finding is invariant
to whether productivity ratios are adjusted to account for credit expansion.
Introduction
This paper
examines the distribution of changes in a vector of macromonetary data. Statistical tests focusing on five
alternative methods for estimating
Mandelbrot's
(1972a, 1975, 1977) and Mandelbrot and Wallis's (1969) R/S or rescaled range
analysis characterizes time series as one of four types: 1.) dependent or
autocorrelated series, 2.) persistent, trend-reinforcing series, also called
biased random walks, random walks with drift, or fractional Brownian
motion, 3.) random walks, or 4.)
anti-persistent, ergodic, or mean-reverting series. Mandelbrot-Lévy distributions are a general
class of probability distributions derived from the generalized central limit
theorem, and include the normal or Gaussian and Cauchy as limiting cases (Lévy
1925; Gnedenko and Kolmolgorov 1954).
Samuelson (1982) popularized the term Mandelbrot-Lévy. The reciprocal of the Mandelbrot-Lévy characteristic exponent α is the Hurst exponent H, and estimates
of H indicate the probability distribution underlying a time series. H = 1/α
= 1/2 for normally-distributed or Gaussian processes. H = 1 for Cauchy-distributed processes. H = 2 for the Lévy distribution governing tosses of a fair coin. H is also related to the fractal dimension D
by the relationship D = 2 - H. Series
with different fractal statistics exhibit different properties as described in
Table 1.
|
Table 1 Fractal Taxonomy of
Time Series |
||||
|
Term |
'Color' |
(H) |
Fractal dimension (D) |
Characteristic
exponent (α) |
|
Antipersistent, Ergodic, Mean-reverting, Negative serial correlation, 1/f noise |
Pink noise |
0 ≤ H < ½ |
0 ≤ D < 1.50 |
2.00 ≤ α
< ∞ |
|
Gaussian process, Normal distribution |
White noise |
H º ½ |
D º 1.50 |
α º
2.00 |
|
Brownian motion, Wiener process |
Brown noise |
H º ½ |
D º 1.50 |
α º
2.00 |
|
Persistent, Trend-reinforcing, Hurst process |
Black noise |
½ < H < 1 |
1 < D < 1.50 |
1.00 < α
< 2.00 |
|
Cauchy process, Cauchy distribution |
Cauchy noise |
H º 1 |
D º 1 |
α º 1 |
|
Note: Brown noise
or Brownian motion is the cumulative sum of a normally-distributed
white-noise process. The changes in,
or returns on, a Brownian motion, are white noise. Fractal statistics are the same for brown
and white noise because the brown-noise process is differenced as part of the
estimation process, yielding white noise. |
||||
Literature review
The search for long memory in capital markets has been a fixture in the literature applying fractal geometry and chaos theory to economics since Mandelbrot (1963b) shifted his attention from income distribution to speculative prices. Fractal analysis has been applied extensively to equities (Green and Fielitz 1977; Lo 1991; Barkoulas and Baum 1996; Peters 1994, 1996; Koppl et al 1997; Kraemer and Runde 1997; Barkoulas and Travlos 1998; Koppl and Nardone 2001; Mulligan 2004, and Mulligan and Lombardo 2004), interest rates (Duan and Jacobs 1996; Barkoulas and Baum 1997a, 1997b; and Hays, Schreiber, et al 2000), commodities (Barkoulas, Baum, and Ogutz 1998), exchange rates (Cheung 1993; Byers and Peel 1996; Koppl and Yeager 1996; Barkoulas and Baum 1997c; Chou and Shih 1997; Andersen and Bollerslev 1997; Koppl and Broussard 1999, and Mulligan 2000a), and derivatives (Fang, Lai, and Lai 1994; Barcoulas, Labys, and Onochie 1997; and Corazza, Malliaris, and Nardelli 1997). Fractal analysis has also been applied to income distribution (Mandelbrot 1963a) and macroeconomic data (Peters 1994, 1996).
Gilanshah and Koppl (2005) advance the thesis that postwar money demand and monetary policy behavior were mostly stable from 1945-1970, but that instability emerged during the seventies as the Federal Reserve System adopted more activist policies and procedures. If the Federal Reserve System becomes an active market player at times, the influence of this one “big player” would be to reduce the stability of money demand, as the many smaller players attempt to react to, as well as anticipate, big player moves. The smaller players' behavior should exhibit herding if it is difficult to anticipate or observe big player behavior, or if that behavior changes abruptly at the big player’s discretion, and if it is relatively easy to observe behavior of other small players. If the Federal Reserve System is a big player acting in accordance with discretion as opposed to rules, the many little players would not appear to be following any coherent behavior, even if little players developed and followed consistent and rational strategic responses. Monetary aggregates would exhibit stochastic dependence over time, and this statistical behavior could spill over into other macroeconomic data processes. Even if the little players respond according to set rules, because the big player acts unpredictably through discretion, the little players’ behavior seems incoherent. If this reading is correct, the instability in money demand is not a statistical artifact of specification error, and cannot be removed by adding variables to conventional money demand models.
As the Federal Reserve System began to adopt more activist policy measures during the 1970s, estimates generated by standard money demand specifications began to show sizable prediction errors. If activist monetary policy does indeed impose instability, this implies that the Federal Reserve System should abandon discretion and pursue money supply targets according to fixed rules. This implication runs counter to a prevailing inference presented in the literature on money demand instability. Mishkin’s (1995:572) view is representative: “because the money demand function has become unstable, velocity is now harder to predict, and setting rigid money supply targets in order to control aggregate spending in the economy may not be an effective way to conduct monetary policy.” But as Gilanshah and Koppl (2005) argue, since the money demand instability results from Federal Reserve activism, the situation calls for less discretion, not more. In their view, one mechanism introducing herding or bandwagon effects in money demand is cash managers’ attempts to enhance their reputations, which enhances their job security and earning potential. Cash managers seek to enhance their reputations in a manner similar to, and for the same reasons as, portfolio managers (Scharfstein and Stein 1990). Cash managers achieve and maintain reputation through conformity with industry practice, a global criterion, and through conduct appropriate to the unique circumstances of their business enterprise, a local criterion. Pursuit of the global criterion imposes herding behavior or bandwagon effects. If cash managers act as others do and things go well, their reputation is assured. If they act as others do and things go badly, the blame is shared throughout the profession. If cash managers defy prevailing practice in their profession and things go badly, their reputation is ruined. Sharfstein and Stein (1990:466) call this incentive to imitate standard practices the “sharing-the-blame effect.” Similar behavior can be observed among the majority of less-innovative entrepreneurial planners.
If entrepreneurial planners defy the common wisdom and things go well, their reputation is strongly enhanced and they enjoy improved income prospects and job security. This is a powerful counter-incentive to herding. Not all entrepreneurial planners are constitutionally capable of acting independently of their peers, and some may require the security of the herd. Some entrepreneurs will herd; others will not. Big player conduct affects the fraction that herds. Activist monetary policy, particularly credit expansion, impairs the value of local information which could be exploited by the more independent entrepreneurs. Thus discretionary conduct by the monetary authorities promotes herding and introduces more volatility into macromonetary data.
Data
The data are monthly-observed monetary aggregates, macroeconomic expenditures, indexes, and ratios over the 1959-2006 range. Macroeconomic data, specifically output measures and interest rates, are also examined over the same period to determine if their behavior appears significantly driven by the monetary data. Quarterly macromonetary processes were examined over the 1947-2006 period. Monetary series were extended from 1947-1958 from Gordon (1986:805-806).
GMB is the logarithmic first difference of the monetary base.
GM1 is the logarithmic first difference of M1.
GM2 is the logarithmic first difference of M2.
GM3 is the logarithmic first difference of M3.
GMZM is the logarithmic first difference of MZM.
GCUR is the logarithmic first difference of currency.
GRR is the logarithmic first difference of required reserves.
GER is the logarithmic first difference of excess reserves.
GCD is the logarithmic first difference of checkable deposits.
GTD is the logarithmic first difference of time deposits.
GR10Y is the first difference of the 10 year Treasury bond rate.
GR3M is the first difference of the 3 month Treasury bill secondary market rate.
GR is the first difference of the term spread, the 10 year rate minus the 3 month rate.
GIIP is the first difference of the index of industrial production.
GCN is the logarithmic first difference of nominal consumption expenditures.
GC is the logarithmic first difference of real consumption expenditures.
GCP is the first difference of the consumption expenditures deflator.
GDI is the logarithmic first difference of real disposable income.
GIN is the logarithmic first difference of nominal investment expenditures.
GI is the logarithmic first difference of real investment expenditures.
GP is the first difference of the GDP deflator.
RP is a productivity index computed as the ratio of nominal consumption expenditures divided by nominal investment expenditures. The productivity indices are difference-stationary so are not differenced for fractal analysis.
AP1 an ABC-adjusted productivity index computed as the ratio of nominal consumption expenditures minus the increase in M1 from the last period divided by nominal investment expenditures plus the increase in M1 from the last period. As with RP, AP1 is difference-stationary.
AP2 is the first difference of a productivity index computed as above, but using M2 in place of M1.
APB is a productivity index computed as above, but using the monetary base in place of M1.
Time series which were already represented as interest rates, percentages, or ratios, were simply first differenced without taking logarithms. Productivity measures were not differenced because they were already difference-stationary.
Methodology
Long memory series exhibit non-periodic long cycles, or persistent dependence between observations far apart in time; i.e., observable patterns which tend to repeat. Long memory or persistent series tend to reverse themselves less often than a purely random series. Thus, they display a trend, and are also called black noise, in contrast to purely random white noise. Persistent series have long memory in that events are correlated over long time periods. In contrast, short-term dependent time series include standard autoregressive moving average and Markov processes, and have the property that observations far apart exhibit little or no statistical dependence. R/S or rescaled range analysis distinguishes random from non-random or deterministic series. The rescaled range is the range divided (rescaled) by the standard deviation. Seemingly random time series may be deterministic chaos, fractional Brownian motion (FBM), or a mixture of random and non-random components.
Conventional statistical techniques lack power to distinguish unpredictable random components from highly predictable deterministic components. R/S analysis evolved to address this difficulty. R/S analysis exploits the structure of dependence in time series irrespective of their marginal distributions, statistically identifying non-periodic cyclic long run dependence as distinguished from short dependence or Markov character and periodic variation (Mandelbot 1972a: 259-260). The difference between long-memory processes, also called non-periodic long cycles, and short-term dependence, is that each observation in long memory processes has a persistent effect, on average, on all subsequent observations, up to some horizon after which memory is lost, whereas in contrast, short-term dependent processes display little or no memory of the past, and what short-term dependence can be observed often diminishes with the square of the time elapsed. For equity prices, long memory can be observed when a stock follows a trend or repeats a cyclical movement, even though the cycles can have time-varying frequencies. Short-term dependence is indicated when there are no observable trends or patterns beyond a very short time span, and the impact of any outliers or extreme values diminishes rapidly over time.
Mandelbrot (1963a, 1963b) demonstrated all
stationary series can be categorized in accordance with their
Results
Many macromonetary
series are anti-persistent
or ergodic, mean-reverting, or pink noise processes
with (0.00 < H < 0.50), indicating they are more volatile than a random
walk. Pink noise processes are used to
model dynamic turbulence. Ergodic or
antipersistent processes reverse themselves more often than purely random
series. Ergodicity, that is, H
significantly below 0.50, indicate policy makers persistently over-react to new
information, imposing more macroeconomic volatility than would maintain in the
absence of policy, and never learn not to over-react. This observed phenomenon is directly
analogous to Mussa's (1984) disequilibrium overshooting, in which the market process
of adjustment toward final equilbrium is unstable, and never quiets down. Hs significantly above 0.50 demonstrate
macroeconomic data series are not random walks.
|
Table 2 Fractal Analyses of Macromonetary Monthly Data Processes 1959-2006 Estimated Hurst Exponent H, Various Methods (Standard Errors in Parentheses) |
|||||
|
variable |
R/S |
Power Spectrum |
Roughness-length |
Variogram |
Wavelet |
|
|
GMB |
0.018 |
-0.392 |
-0.044 |
0.003 |
0.514 |
|
|
|
(0.0194634) |
(6.2957945) |
(0.00269) |
(1.1896007) |
|
|
|
GM1 |
-0.013 |
-0.404 |
-0.048 |
-0.006 |
0.851 |
|
|
|
(0.0128657) |
(11.1950188) |
(0.0014695) |
(2.8028963) |
|
|
|
GM2 |
0.075 |
-0.29 |
-0.030 |
0.006 |
0.767 |
|
|
|
(0.0100815) |
(13.3460836) |
(0.0008523) |
(1.3703301) |
|
|
|
GM3 |
0.049 |
-0.1605 |
-0.034 |
0.034 |
0.597 |
|
|
|
(0.0171313) |
(5.5036998) |
(0.0009275) |
(0.6806059) |
|
|
|
GMZM |
0.176 |
-0.2285 |
0.032 |
0.040 |
0.545 |
|
|
|
(0.0056194) |
(5.624958) |
(0.0005663) |
(0.3498935) |
|
|
|
GCUR |
-0.007 |
-0.4175 |
0.014 |
-0.010 |
0.819 |
|
|
|
(0.0198405) |
(12.1109514) |
(0.0004026) |
(2.8371727) |
|
|
|
GRR |
0.098 |
-0.496 |
-0.056 |
0.001 |
0.748 |
|
|
|
(0.0118314) |
(6.1537476) |
(0.0012239) |
(0.5085489) |
|
|
|
GER |
0.079 |
-1.0185 |
-0.093 |
-0.004 |
0.985 |
|
|
|
(0.0106919) |
(6.4823799) |
(0.0048483) |
(0.0637986) |
|
|
|
GCD |
-0.012 |
-0.4115 |
-0.066 |
-0.008 |
0.872 |
|
|
|
(0.0132301) |
(10.5566654) |
(0.0014590) |
(2.2587965) |
|
|
|
GTD |
0.279 |
-0.058 |
0.078 |
0.113 |
0.192 |
|
|
|
(0.0040276) |
(5.5141182) |
(0.0010699) |
(0.0619385) |
|
|
|
GR10Y |
0.191 |
-0.4425 |
0.064 |
0.021 |
0.436 |
|
|
|
(0.0154230) |
(2.7973051) |
(0.0002411) |
(0.0531330) |
|
|
|
GR3M |
0.241 |
-0.5015 |
0.119 |
0.024 |
0.361 |
|
|
|
(0.0110211) |
(5.6629744) |
(0.0013522) |
(0.0937309) |
|
|
|
GR |
0.228 |
-0.663 |
0.083 |
0.016 |
0.553 |
|
|
|
(0.0055240) |
(8.0740194) |
(0.0007747) |
(0.0593133) |
|
|
|
GIIP |
0.215 |
-0.304 |
0.089 |
0.034 |
0.255 |
|
|
|
(0.0142341) |
(4.0755129) |
(0.0001330) |
(0.0258054) |
|
|
|
GCN |
0.110 |
-0.539 |
-0.005 |
-0.008 |
0.786 |
|
|
|
(0.0040738) |
(6.8441381) |
(0.0004587) |
(0.0303925) |
|
|
|
GC |
0.132 |
-0.5175 |
0.006 |
-0.001 |
0.755 |
|
|
|
(0.0044611) |
(4.0951791) |
(0.0002475) |
(0.0305760) |
|
|
|
GCP |
0.220 |
-0.165 |
-0.006 |
0.039 |
0.341 |
|
|
|
(0.0056719) |
(6.2382107) |
(0.0004521) |
(0.0955249) |
|
|
|
Note: The
Mandelbrot-Lévy characteristic exponent alpha is the reciprocal of the |
|||||