Objectives

A Fractal Comparison of Real and Austrian Business Cycle Models

Robert F. Mulligan, Ph.D.

Department of Business Computer Information Systems & Economics

Western Carolina University

College of Business

Cullowhee, North Carolina 28723

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 Binghamton. Thanks are due to Patrick Hays of the Department of Accounting, Finance, and Entrepreneurship for the program performing Lo's modified R/S test for stochastic dependence. Helpful comments from Patrick Hays, Steve Henson, Steven Miller, and Michael Smith, are gratefully acknowledged. The author retains responsibility for any errors or omissions.

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 Hurst (1951) exponent, fractal dimension, and Mandelbrot-Lévy characteristic exponent (Lévy 1925) are used. Lo's (1991) modified rescaled-range test for stochastic dependence is also used (Hays, Schreiber, et al 2000; Mulligan 2000a). The paper is organized as follows. A literature review is provided in the second section. The data are documented in the third section. Methodology and results are presented in the fourth and fifth sections. The conclusions are provided in the sixth section.

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'	Hurst exponent (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 Hurst exponent H. The Hurst exponent was introduced in the hydrological study of the Nile valley and is the reciprocal of the characteristic exponent alpha (Hurst 1951). Some series are persistent or black noise processes with (0.50 < H < 1.00). These less noisy series exhibit clearer trends and more persistence the closer H is to one. However, Hs very close to one indicate high risk of large, abrupt changes, e.g., H = 1.00 for the Cauchy distribution, the basis for the characteristic exponent test. This research used the approach of estimating the Hurst exponent for each series over the whole sample period by five alternative techniques, then testing for Gaussian character, and finally testing for stability of the Hurst exponent over two subsamples by R/S to examine whether the behavior of the data processes changed during the time studied.

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 Hurst exponent H, thus alpha = 1/H. The fractal dimension D = 2 – H.

Table 3 Lo Test for Stochastic Dependence on Macromonetary Monthly Data Processes 1959-2006
		Autoregressive Order
	N	1	2	3	4	5	6	7	8	9	10	11	12
GMB	576	1.62955	1.74704	1.87159	1.97637	2.02415	1.9856	1.93953	1.91353	1.91913	1.99047	2.02572	1.92456
		0.09502 *	0.05006 **	0.02359 **	0.01184 **	0.00850 **	0.01112 **	0.01518 **	0.01801 **	0.01737 **	0.01075 **	0.00841 **	0.01676 **
GM1	575	1.84767	2.21345	2.2386	2.25967	2.34268	2.28693	2.30709	2.3193	2.24153	2.30942	2.41049	2.22362
		0.02742 **	0.00207***	0.00169***	0.00143***	0.00072***	0.00114***	0.00097***	0.00087***	0.00165***	0.00095***	0.00040***	0.00191***
GM2	575	3.08061	3.21314	2.95373	2.79783	2.72089	2.60229	2.53809	2.4797	2.38055	2.33232	2.30495	2.21712
		0.00000***	0.00000***	0.00000***	0.00001***	0.00002***	0.00007***	0.00013***	0.00022***	0.00052***	0.00078***	0.00098***	0.00201***
GM3	565	3.4289	3.27621	2.97688	2.7754	2.64299	2.51502	2.42511	2.34378	2.25247	2.18724	2.13304	2.0593
		0.00000***	0.00000***	0.00000***	0.00001***	0.00005***	0.00016***	0.00035***	0.00071***	0.00151***	0.00254***	0.00384***	0.00662 **
GMZM	575	1.82378	1.74877	1.62799	1.55318	1.51153	1.4679	1.44394	1.42206	1.38932	1.37295	1.36474	1.33765
		0.03177 **	0.04957 **	0.09579 *	0.13889	0.1687	0.20481	0.22685	0.24838	0.28309	0.30154	0.31107	0.34378
GCUR	575	1.99419	2.18628	2.3859	2.45972	2.42729	2.43622	2.39082	2.32109	2.30198	2.37666	2.43066	2.27421
		0.01048 **	0.00256***	0.00049***	0.00026***	0.00034***	0.00032***	0.00047***	0.00086***	0.00101***	0.00054***	0.00033***	0.00127***
GRR	576	1.28968	1.43671	1.43007	1.43589	1.47097	1.4434	1.45324	1.46212	1.43545	1.45752	1.48967	1.43397
		0.40616	0.23382	0.24035	0.23462	0.2021	0.22737	0.21811	0.20999	0.23505	0.21416	0.18615	0.2365
GER	576	0.988	1.15463	1.24377	1.36766	1.56416	1.62141	1.71392	1.80396	1.93727	2.09225	2.25272	2.19475
		0.83648	0.60324	0.47046	0.30766	0.13175	0.09908 *	0.06039 *	0.03583 **	0.01541 **	0.00521 **	0.00151***	0.00239***
GCD	575	1.89645	2.28756	2.24288	2.24416	2.31028	2.24438	2.25419	2.26823	2.17607	2.20939	2.28661	2.12456
		0.02013 **	0.00114***	0.00163***	0.00162***	0.00094***	0.00161***	0.00149***	0.00133***	0.00277***	0.00213***	0.00114***	0.00410***
GTD	565	4.09756	3.54408	3.18204	2.9273	2.72514	2.56345	2.4313	2.32089	2.2253	2.14348	2.07106	2.00318
		0.00000***	0.00000***	0.00000***	0.00000***	0.00002***	0.00010***	0.00033***	0.00086***	0.00188***	0.00355***	0.00608 **	0.00984 **
GR10Y	576	1.54439	1.52775	1.53391	1.53711	1.52375	1.51733	1.52429	1.52307	1.51385	1.49982	1.48114	1.47024
		0.14481	0.15656	0.15213	0.14987	0.15949	0.16428	0.1591	0.16	0.16692	0.17789	0.19331	0.20274
GR3M	576	1.274	1.24959	1.25874	1.2697	1.27032	1.29647	1.34068	1.36586	1.36067	1.34775	1.33903	1.34247
		0.42768	0.4621	0.44908	0.43367	0.4328	0.39702	0.34002	0.30976	0.31586	0.33137	0.34208	0.33782
GR	576	0.71535	0.71662	0.71911	0.72688	0.74001	0.76082	0.78301	0.79361	0.78949	0.7868	0.78948	0.80006
		0.99562	0.99549	0.99523	0.99434	0.99255	0.98885	0.98354	0.98042	0.98168	0.98247	0.98168	0.97833
GIIP	575	1.80936	1.65349	1.53971	1.46127	1.40639	1.36101	1.32679	1.29502	1.26586	1.24042	1.21979	1.20227
		0.03468 **	0.08385 *	0.14805	0.21075	0.26462	0.31546	0.35742	0.39896	0.43903	0.4753	0.50545	0.53145
GCN	575	3.28662	3.34738	3.32592	3.28583	3.2224	3.11188	3.00656	2.90835	2.82195	2.75631	2.68461	2.62199
		0.00000***	0.00000***	0.00000***	0.00000***	0.00000***	0.00000***	0.00000***	0.00000***	0.00001***	0.00001***	0.00003***	0.00006***
GCR	575	1.2466	1.30352	1.32011	1.3356	1.34475	1.32818	1.30218	1.27093	1.24581	1.23116	1.21097	1.19413
		0.46639	0.38762	0.36596	0.34633	0.33503	0.35566	0.38939	0.43194	0.46753	0.48876	0.5185	0.54362
GCP	575	4.63613	4.15359	3.81082	3.53646	3.30485	3.10431	2.93944	2.80386	2.68482	2.57655	2.47743	2.39149
		0.00000***	0.00000***	0.00000***	0.00000***	0.00000***	0.00000***	0.00000***	0.00001***	0.00003***	0.00009***	0.00022***	0.00047***
Note: The null hypothesis is no stochastic dependence. Probability levels given below the Lo test statistics. Statistical significance at the 10% level indicated by , at the 5% level , at the 1% level by **.

This section discusses and interprets the results of five alternative fractal analysis methods for measuring the Hurst exponent H presented in Table 2. All data are converted to first differences, losing one observation. Standard errors are given in parentheses. H is estimated first for monthly data over the 1959-2006 sample.

Five techniques for estimating the Hurst exponent are reported in this paper: 1.) Mandelbrot's (1972a) AR1 rescaled-range or R/S analysis; 2.) power spectral-density analysis; 3.) roughness-length relationship analysis; 4.) variogram analysis; and 5.) wavelet analysis:

1.) Rescaled-range or R/S analysis: R/S analysis is the traditional technique introduced by Mandelbrot (1972a). Hs estimated by this method are generally far from 0.50, suggesting non-Gaussian processes. The difference between estimated Hs and 0.50 is statistically significant over the whole sample range and both subsamples for each series examined. Hs are always below 0.50, indicating ergodicity or antipersistence, e.g., negative serial correlation meaning the data processes persistently overcorrect. This measurable antipersistence or ergodicity demonstrates policy makers habitually overreact to new information, and never learn not to.

Hs different from 0.50 demonstrate the data series have not been random walks; nevertheless, this finding may be due to short-term dependence still present after taking AR1 residuals, or systematic bias due to information asymmetries, or both.

2.) Power spectral density analysis: Hs estimated by this technique also fall in the persistent range (H < 0.50). Spectral density often provides very large standard errors for H.

3.) Roughness-length relationship method: All Hs are significantly less than 0.50, indicating antipersistence.

4.) Variogram analysis: Variogram analysis supports antipersistence for all series.

5.) Wavelet analysis: This method was developed by Daubechies (1990), Beylkin (1992), and Coifman et al (1992). Wavelet H estimates indicate antipersistence or ergodicity (H < 0.50) for time deposits, the 10 year and 3 month interest rates, the index of industrial production, and the consumption price deflator. Results suggest trend-reinforcing, persistent character for all other variables, in contrast with the other techniques.

6.) Lo's (1991) modified R/S analysis: Hypothesis tests are reported in Table 3. Lo's technique does not provide an estimate for H and the null hypothesis of no stochastic dependence is necessary to lend any credence to long memory suggested by the other five methods. Strong evidence of stochastic dependence extending at least one year is found for the monetary base, M1, M2, M3, MZM, currency, checkable deposits, time deposits, nominal consumption, and the consumption deflator. The Index of Industrial Production appears to be stochastically dependent, but only with a two-month lag.

	Table 4 Fractal Analyses of Macromonetary Quarterly Data Processes 1947-2006 Estimated Hurst Exponent H, Various Methods (Standard Errors in Parentheses)
variable		R/S	Power Spectrum	Roughness-length	Variogram	Wavelet
GDI		0.162	-0.5885	0.099	-0.032	0.351
		(0.0038666)	(4.31637)	(0.0001122)	(0.0258613)
GCN		0.087	-0.499	0.127	-0.036	0.311
		(0.0013857)	(15.4324)	(0.0003282)	(0.0789028)
GC		0.144	-0.4225	0.145	-0.003	0.229
		(0.0020723)	(3.91107)	(0.0000446)	(0.0504925)
GIN		0.066	-0.6565	0.178	-0.023	0.402
		(0.0044745)	(5.94697)	(0.0000314)	(0.0809408)
GI		0.078	-0.769	0.171	-0.018	0.336
		(0.0035645)	(7.35241)	(0.0000105)	(0.0726327)
GP		0.418	0.248	0.395	0.245	0.243
		(0.0013298)	(4.95443)	(0.0018726)	(0.0106749)
GM1		0.018	-0.0585	0.188	0.124	0.504
		(0.0139340)	(3.96447)	(0.0007898)	(0.0496912)
GM2		0.137	-0.046	0.304	0.111	0.104
		(0.0093461)	(4.71824)	(0.0000533)	(0.0270818)
GMB		0.177	-0.2795	0.206	0.018	0.083
		(0.0052624)	(6.16468)	(0.0000216)	(0.0176325)
RP		0.301	0.2655	0.372	0.225	0.199
		(0.0281422)	(3.9015610)	(0.0011170)	(0.1853207)
AP1		0.307	0.2575	0.362	0.219	0.234
		(0.0238342)	(4.8460107)	(0.0010277)	(0.1989056)
AP2		0.305	0.2665	0.383	0.252	0.328
		(0.0199283)	(8.0910254)	(0.0001840)	(0.1033507)
APB		0.295	0.249	0.376	0.211	0.191
		(0.0279218)	(4.3645687)	(0.0011405)	(0.2017907)
	Note: The Mandelbrot-Lévy characteristic exponent alpha is the reciprocal of the Hurst exponent H, thus alpha = 1/H. The fractal dimension D = 2 – H.

Table 5 Lo Test for Stochastic Dependence on Macromonetary Quarterly Data Processes 1947-2006 (Probability Levels in Parentheses)
	Autoregressive Order
	1	2	3	4	5	6	7	8
GDI	1.40673	1.38759	1.37816	1.41393	1.46503	1.49573	1.50748	1.52146
	0.26426	0.285	0.29558	0.25672	0.20737	0.18118	0.17184	0.16119
GNC	3.07276	2.72132	2.53354	2.42323	2.32137	2.2517	2.17212	2.10373
	0.00000***	0.00002***	0.00013***	0.00036***	0.00086***	0.00152***	0.00285***	0.00478***
GC	1.17211	1.07542	1.03438	1.02329	1.01993	1.03555	1.04176	1.06216
	0.5768	0.72103	0.77803	0.79266	0.79702	0.77647	0.7681	0.73988
GNI	0.79473	0.75217	0.74163	0.76791	0.81151	0.87051	0.93236	0.99374
	0.98007	0.99053	0.9923	0.98732	0.97423	0.94429	0.89547	0.82966
GI	0.79473	0.75217	0.74163	0.76791	0.81151	0.87051	0.93236	0.99374
	0.98007	0.99053	0.9923	0.98732	0.97423	0.94429	0.89547	0.82966
GP	4.20451	3.48924	3.05428	2.75439	2.53518	2.36609	2.23081	2.11905
	0.00000***	0.00000***	0.00000***	0.00002***	0.00013***	0.00059***	0.00180***	0.00427***
GM1	3.32571	2.89285	2.60312	2.4221	2.27724	2.15938	2.07015	1.99938
	0.00000***	0.00000***	0.00007***	0.00036***	0.00124***	0.00315***	0.00612 **	0.01011 **
GM2	3.46758	2.99392	2.69202	2.48458	2.32315	2.19482	2.09177	2.00902
	0.00000***	0.00000***	0.00003***	0.00021***	0.00084***	0.00239***	0.00522 **	0.00945 **
GMB	3.27019	2.8996	2.66144	2.48498	2.34316	2.23141	2.13708	2.05403
	0.00000***	0.00000***	0.00004***	0.00021***	0.00071***	0.00179***	0.00373***	0.00687 **
RBCPROD	2.7654	2.333	2.09299	1.94231	1.83737	1.75876	1.6953	1.64139
	0.00001***	0.00078***	0.00518 **	0.01490 **	0.02923 **	0.04679 **	0.06694 *	0.08935 *
ABCPROD(M1)	3.00105	2.52607	2.25997	2.09153	1.97312	1.88391	1.81197	1.75144
	0.00000***	0.00014***	0.00142***	0.00523 **	0.01211 **	0.02182 **	0.03414 **	0.04881 **
ABCPROD(M2)	3.44684	2.88252	2.55647	2.34426	2.19134	2.07343	1.97766	1.89731
	0.00000***	0.00000***	0.00011***	0.00071***	0.00246***	0.00597 **	0.01174 **	0.02002 **
ABCPROD(MB)	2.79195	2.35869	2.1195	1.97094	1.86826	1.79163	1.72937	1.67584
	0.00001***	0.00063***	0.00425***	0.01229 **	0.02410 **	0.03857 **	0.05537 *	0.07442 *
Note: The null hypothesis is no stochastic dependence. Probability levels given below the Lo test statistics. Statistical significance at the 10% level indicated by , at the 5% level , at the 1% level by **.

Five estimates of the Hurst exponent for quarterly variables over the 1947-2006 range are presented in Table 4. R/S analysis suggests antipersistence for all series. Power spectral density functions indicate the GDP deflator, M1, and M2 are persistent, but that all other series are antipersistent, though standard errors are too high to allow much confidence in these estimates. Roughness-length and variogram estimates indicate antipersistence for all series. Wavelets suggest all series are antipersistent except M1which is either nearly normal or slightly persistent. Lo tests are reported in Table 5 with up to eight quarters of autoregressive processes removed. Results indicate stochastic persistence for nominal consumption, M1, M2, the monetary base, the GDP deflator, and for all productivity ratios, for up to eight quarters.

Conclusion

The logarithmic differences of macroeconomic data for a stable and growing economy should have Hurst exponents approximately equal to 0.50, indicating these series change in a purely random, normally-distributed manner. Series with long-term trends and non-periodic cycles should display time persistence with H > 0.50, unless economic efficiency imposes randomness and normality anyway. Much of the macroeconomic data in this study yield strong evidence of antipersistence, ergodicity, or negative serial correlation. The conclusion suggested is that decision makers are incapable of correctly evaluating economic data, persistently overreact to the arrival of new information, and never learn not to overreact, and this behavioral shortcoming applies to monetary policy makers, money managers, and entrepreneurial planners.

A scenario rendering this finding more intuitive is that information relevant to a nation's macroeconomic performance arrives frequently, in infinitesimal increments, and seemingly at random. Decision makers habitually ignore the vast majority of this information, because the vast majority is unimportant or irrelevant, until it accumulates a critical mass they must finally recognize. Then, perceiving they have ignored a body of relevant information which they have allowed to accumulate, they attempt to compensate for their history of informational sloth by overreacting. The expression "informational sloth" can just as validly be characterized as "filtering out noise."

References

Andersen, Torben G.; Bollerslev, Tim. 1997. "Heterogeneous Information Arrivals and Return Volatility Dynamics: Uncovering the Long-run in High Frequency Returns." Journal of Finance, 52(3): 975-1005.

Barkoulas, John T.; Baum, Christopher F. 1996. "Long-term Dependence in Stock Returns." Economics Letters 53: 253-259.

_____; _____. 1997a. "Fractional Differencing Modeling and Forecasting of Eurocurrency Deposit Rates." Journal of Financial Research 20(3): 355-372.

_____; _____. 1997b. "Long Memory and Forecasting in Euroyen Deposit Rates." Financial Engineering and the Japanese Markets 4(3): 189-201.

_____; _____. 1997c."A Re-examination of the Fragility of Evidence from Cointegration-based Tests of Foreign Exchange Market Efficiency." Applied Financial Economics 7: 635-643.

Barkoulas, John T.; Baum, Christopher F.; Oguz, Gurkan S. 1998. "Stochastic Long Memory in Traded Goods Prices." Applied Economics Letters 5: 135-138.

Barkoulas, John T.; Labys, Walter C.; Onochie, Joseph. 1997. "Fractional Dynamics in International Commodity Prices" Journal of Futures Markets 17(2): 161-189.

Barkoulas, John T.; Travlos, Nickolaos. 1998. "Chaos in an Emerging Capital Market? The Case of the Athens Stock Exchange" Applied Financial Economics 8: 231-243.

Beylkin, Gregory. 1992. "On the Representation of Operators in Bases of Compactly Supported Wavelets." SIAM Journal on Numerical Analysis 29(6): 1716-1740.

Black, Fisher; Scholes, Myron. 1972. "The Valuation of Option Contracts and a Test of Market Efficiency." Journal of Finance 27: 399-418.

_____; _____. 1973. "The Pricing of Options and Corporate Liabilities." Journal of Political Economy 81: 637-418.

Byers, J.D.; Peel, D.A. 1996. "Long-memory Risk Premia in Exchange Rates." Manchester School of Economic and Social Studies 64(4): 421-438.

Calvet, Laurent; Fisher, Adlai; Mandelbrot, Benoit B. 1997. "Large Deviations and the Distribution of Price Changes," Cowles Foundation Discussion Paper no. 1165. Yale University.

Cheung, Yin-Wong. 1993. "Tests for Fractional Integration: a Monte Carlo Investigation." Journal of Time Series Analysis 14: 331-345.

Cheung, Yin-Wong; Lai, Kon S. 1993. "Do Gold Market Returns Have Long Memory?" The Financial Review 28(3): 181-202.

Chou, W.L.; Shih, Y.C. 1997. "Long-run Purchasing Power Parity and Long-term Memory: Evidence from Asian Newly-industrialized Countries." Applied Economics Letters 4: 575-578.

Coifman, Ronald; Ruskai, Mary Beth; Beylkin, Gregory; Daubechies, Ingrid; Mallat, Stephane; Meyer, Yves; Raphael, Louise, eds. 1992. Wavelets and Their Applications. Sudbury, Massachusetts: Jones and Bartlett Publishers.

Corazza, Marco; Malliaris, A.G.; Nardelli, Carla. 1997. "Searching for Fractal Structure in Agricultural Futures Markets." Journal of Futures Markets 71(4): 433-473.

Daubechies, Ingrid. 1990. "The Wavelet Transform, Time-frequency Localization and Signal Analysis." IEEE Transactions on Information Theory 36: 961-1005.

Diebold, Francis X.; Inoue, A. 2000. "Long Memory and Regime Switching." National Bureau of Economic Research technical working paper no. 264.

Duan, Jin-Chuan; Jacobs, Kris. 1996. "A Simple Long-memory Equilibrium Interest Rate Model." Economics Letters 53: 317-321.

Fama, Eugene; Fisher, L.; Jensen, M., and Roll, R. 1969. " The Adjustment of Stock Prices to New Information" International Economic Review 10: 1-21.

Fang, H.; Lai, Kon S.; Lai, M. 1994. "Fractal Structure in Currency Futures Prices." Journal of Futures Markets 14: 169-181.

Fisher, Adlai; Calvet, Laurent; Mandelbrot, Benoit B. 1997. "Multifractality of Deutschemark/US Dollar Exchange Rates." Cowles Foundation Discussion Paper no. 1166. Yale University.

Gilanshah, Catherine B.; Koppl, Roger. 2005. "Big Players and Money Demand." In: Backhouse, Jürgen. (ed.) Modern Applications of Austrian Economics. New York: Routledge.

Gnedenko, Boris Vladimirovich; Kolmogorov, Andrei Nikolaevich. 1954. Limit Distributions for Sums of Random Variables. Reading, Massachusetts: Addison-Wesley.

Gordon, Robert J. 1986. The American Business Cycle: Continuity and Change. Chicago and London: University of Chicago Press.

Granger, C.W.J. 1989. Forecasting in Business and Economics (2nd ed.) Boston: Academic Press.

Granger, C.W.J.; Hyung, N. 1999. "Occasional Structural Breaks and Long Memory." Discussion paper 99-14. University of California at San Diego.

Greene, M.T.; Fielitz, B.D. 1977. "Long-term Dependence in Common Stock Returns." Journal of Financial Economics 5: 339-349.

Hays, Patrick; Schreiber, Max; Payne, James E.; Ewing, Bradley T.; Piette, Michael J. 2000. "Are Net Discount Ratios Stationary? Evidence of Mean-Reversion and Persistence." Journal of Risk and Insurance 67(3): 439-449.

Heiner, Ronald A. 1980. "The Origin of Predictable Behavior." American Economic Review 73(3): 560-595.

Hurst, H. Edwin. 1951. "Long-term Storage Capacity of Reservoirs." Transactions of the American Society of Civil Engineers 116: 770-799.

Hurst, H. Edwin; et al. 1965. Long-term Storage, an Experimental Study. London: Constable.

Jarque, C.M., and A.K. Bera (1980). "Efficient Tests for Normality, Homoskedasticity, and Serial Independence of Regression Residuals." Economics Letters 6: 255-259.

Kaen, Fred R.; Rosenman, Robert E. 1986. "Predictable Behavior in Financial Markets: Some Evidence in Support of Heiner's Hypothesis." American Economic Review 76(1): 212-220.

Kraemer, Walter; Runde, Ralf. 1997. "Chaos and the Compass Rose." Economics Letters 54: 113-118.

Koppl, Roger; Ahmed, Ehsan; Rosser, J. Barkley; White, Mark V. 1997. "Complex Bubble Persistence in Closed-End Country Funds." Journal of Economic Behavior and Organization 32(1): 19-37.

Koppl, Roger; Broussard, John. 1999. "Big Players and the Russian Ruble: Explaining Volatility Dynamics." Managerial Finance 25(1): 49-63.

Koppl, Roger; Nardone, Carlo. 2001. "The Angular Distribution of Asset Returns in Delay Space." Discrete Dynamics in Nature and Society 6: 101-120.

Koppl, Roger; Yeager, Leland, 1996. "Big Players and Herding in Asset Markets: The Case of the Russian Ruble," Explorations in Economic History, 33(3): 367-383.

Lévy, Paul. 1925. Calcul des Probibilités. Paris: Gauthier Villars.

Lo, Andrew W. 1991. "Long-term Memory in Stock Market Prices." Econometrica 59(3): 1279-1313.

Lo, Andrew W.; MacKinlay, A. 1988. "Stock Market Prices Do Not Follow a Random Walk: Evidence from a Simple Specification Test." Review of Financial Studies 1: 41-66.

_____; _____. 1989. "The Size and Power of the Variance Ratio Test in Finite Samples: a Monte Carlo Investigation." Journal of Econometrics 40: 203-238.

Malkiel, Burton G. 1987. "Efficient Market Hypothesis." New Palgrave Dictionary of Economics 2: 120-122. New York: Stockton.

Mandelbrot, Benoit B. 1963a. "New Methods in Statistical Economics." Journal of Political Economy 71(5): 421-440.

_____. 1963b. "The Variation of Certain Speculative Prices." Journal of Business 36(3): 394-419.

_____. 1972a. "Statistical Methodology for Non-periodic Cycles: From the Covariance to R/S Analysis." Annals of Economic and Social Measurement 1(3): 255-290.

_____. 1972b. "Possible Refinements of the Lognormal Hypothesis Concerning the Distribution of Energy Dissipation in Intermittent Turbulence." In Rosenblatt, M.; Van Atta, C. (eds.) Statistical Models and Turbulence. New York: Springer Verlag.

_____. 1974. "Intermittent Turbulence in Self Similar Cascades: Divergence of High Moments and Dimension of the Carrier." Journal of Fluid Mechanics 62: 331-358.

_____. 1975. "Limit Theorems on the Self-normalized Range for Weakly and Strongly Dependent Processes." Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 31: 271-285.

_____. 1977. The Fractal Geometry of Nature. New York: Freeman.

Mandelbrot, Benoit B.; Fisher, Adlai; Calvet, Laurent. 1997. "A Multifractal Model of Asset Returns." Cowles Foundation Discussion Paper no. 1164. Yale University.

Mandelbrot, Benoit B.; van Ness, J.W. 1968. "Fractional Brownian Motion, Fractional Noises and Application." SIAM Review 10: 422-437.

Mandelbrot, Benoit B.; Wallis, James R. 1969. "Robustness of the Rescaled Range R/S in the Measurement of Noncyclic Long-run Statistical Dependence." Water Resources Research 5(4): 976-988.

Mulligan, Robert F. 2000a. "A Fractal Analysis of Foreign Exchange Markets." International Advances in Economic Research 6(1): 33-49.

_____. 2000b. "A Characteristic Exponent Test for the Cauchy Distribution." Atlantic Economic Journal 28(4): 491.

_____. 2004. "Fractal Analysis of Highly Volatile Markets: an Application to Technology Equities." Quarterly Review of Economics and Finance. 44(1): 155-179.

Mulligan, Robert F.; Lombardo, Gary A. 2004. "Maritime Businesses: Volatile Prices and Market Valuation Inefficiencies." Quarterly Review of Economics and Finance 44(2): 321-336.

Mussa, Michael. 1984. "The Theory of Exchange Rate Determination." In Bilson, John F.O.: Marston, Richard C. (eds.) Exchange Rate Theory and Practice (pp. 13-78.) Chicago: University of Chicago Press.

Neely, Christopher; Weller, Paul; Dittmar, Robert. 1997. "Is Technical Analysis in the Foreign Exchange Market Profitable? a Genetic Programming Approach." Journal of Financial and Quantitative Analysis 32(4): 405-426.

Osborne, M. 1959. "Brownian Motions in the Stock Market." Operations Research 7: 145-173.

Peters, Edgar E. 1994. Fractal Market Analysis. New York: Wiley.

_____. 1996. Chaos and Order in the Capital Markets: a New View of Cycles, Prices, and Market Volatility (2nd ed.) New York: Wiley.

_____. 1999. Complexity, Risk, and Financial Markets. New York: Wiley.

Samuelson, Paul A. 1982. "Paul Cootner's Reconciliation of Economic Law with Chance." In Scharpe, William F.; Cootner, Cathryn M. (eds.) Financial Economics: Essays in Honor of Paul Cootner (pp. 101-117.) Englewood Cliffs, New Jersey: Scott Foresman.

Glossary of Fractal Analysis Terms

ARFIMA process – autoregressive fractally integrated moving average process, represented by ARFIMA(p,d,q), where, as with an ARIMA(p,d,q) process, p represents the order of autoregression process, d the order of integration (i.e., the number of times the process must be differenced to impose stationarity), and q the order of the moving average process. An ARFIMA process differs from ARIMA in that the order of integration is not constrained to be an integer.

Antipersistence – a series that reverses itself more often than a purely random series, also called pink noise, ergodicity, 1/f noise, or negative serial correlation. (Peters 1994: 306).

Black noise – a series that reverses itself less often than a purely random series, displaying trends or repetitive patterns over time, also called persistence, positive serial correlation, or autocorrelation. (Peters 1994: 183-187).

Brown noise – the cumulative sum of a normally-distributed random variable, also called Brownian motion. (Peters 1994: 183-185; Osborne 1959).

Efficient Market Hypothesis – the proposition that market prices fully and correctly reflect all relevant information. A market is described as efficient with respect to an information set if prices do not change when the information is revealed to all market participants. There are three levels of market efficiency: weak, semi-strong, and strong. (Fama et al 1969; Malkiel 1987).

Lo test – the statistical test of the null hypothesis of no stochastic dependence based on computation of Q_n, a modified rescaled range statistic (Lo 1991).

Long memory – the property that any value a series takes on generally has a long and persistent effect, e.g., extreme values that repeat at fairly regular intervals. (Peters 1994: 274).

Multifractal Model of Asset Returns (MMAR) – a very general model of asset pricing behavior allowing for long-memory and fat-tailed distributions. Instead of infinite-variance distributions such as the Mandelbrot-Lévy and Cauchy distributions, the MMAR relies on fractional combinations of random variables with non-constant mean and variance, providing many of the properties of infinite-variance distributions. (Mandelbrot, Fisher, and Calvet 1997).

Non-periodic long cycles - a characteristic of long-memory processes, i.e., of statistical processes where each value has a long and persistent impact on values that follow it, that identifiable patterns tend to repeat over similar, though irregular, cycles (non-periodic cycles.) Also called the Joseph effect. (Peters 1994: 266).

Non-stationarity – the property that a series has a systematically varying mean and variance. Any series with a trend, e.g., U.S. GDP, has a growing mean and therefore is non-stationary. Brown-noise processes are non-stationary, but white-noise processes are stationary. (Granger 1989: 58).

Persistence or persistent dependence – a series that reverses itself less often than a purely random series, and thus tends to display a trend, also called black noise. Persistent series have long memory in that events are correlated over long time periods, and thus display non-periodic long cycles. (Peters 1994: 310).

Semi-strong-form Market Efficiency – the intermediate form of the efficient market hypothesis, asserting that market prices incorporate all publicly available information, including both historical data on the prices in question, and any other relevant, publicly-available data, and thus it is impossible for any market participant to gain advantage and earn excess profits, in the absence of inside information. (Peters 1994: 308).

Short-term dependence - the property that any value a series takes on generally has a transient effect, e.g., extreme values bias the series for a certain number of observations that follow. Eventually, however, all memory of the extreme event is lost, in contrast to long-memory or the Joseph effect. Special cases include Markov processes and serial correlation. (Peters 1994: 274).

Spectral density or Power-spectral Density Analysis – a fractal analysis based on the power spectra calculated through the Fourier transform of a series. (Peters 1994: 170-171).

Stationarity – the property that a series has a constant mean and variance. White, pink, and black-noise processes are all stationary. Because it is the cumulative sum of a white-noise process, a brown-noise process is non-stationary. (Granger 1989: 58).

Stochastic dependence – the property that time series exhibit non-periodic cycles. Behavior will be persistent for an unpredictable period (the Joseph effect) but may also be washed out, reversed, changed, or memory may be lost by some randomly-occuring event (the Noah effect.)

Strong-form Market Efficiency – the most restrictive version of the efficient market hypothesis, asserting that all information known to any one market participant is fully reflected in the price, and thus insider information provides no speculative advantage and cannot offer above average returns. (Malkiel 1987: 120).

Weak-form Market Efficiency – the least restrictive version of the efficient market hypothesis, asserting that current prices fully reflect the historical sequence of past prices. One implication is that investors cannot obtain above-average returns through analyzing patterns in historical data, i.e., through technical analysis. Also referred to as the Random Walk Hypothesis. One common way of testing for weak-form efficiency is to test price series for normality, however, normality is a sufficient rather than a necessary condition. (Malkiel 1987: 120).

White noise – a perfectly random process exhibiting no serial dependence. Normal processes meet this requirement, and normality is often conflated with white noise. Normality is a sufficient condition rather than a necessary condition for white noise. (Peters 1994: 312).