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Procedia Economics and Finance 3 (2012) 119 - 124

Emerging Markets Queries in Finance and Business

Statistical evaluations of business cycle phases

Silvia Palascaa*

a'Alexandru loan Cuza' University, B-dul CarolInr. 22, Iasi, 700505, Romania

Abstract

This paper's goal is to fit the evolution of the business cycle into a simple statistical model. The price of gold was chosen as an economical indicator due to increased stability at political and accidental changes. Statistical testing using chi square test was employed to find the right distribution for each phase of the economical cycle, opposed to the idea that the normal distribution is universally valid. The reasearch proved that the chosen variable follows a different statistical distribution for each phase of the business cycle. Data analysis was performed using EasyFit software.

© 2012 The Authors. Published by Elsevier Ltd.

Selection and peer review under responsibility of Emerging Markets Queries in Finance and Business local organization. Keywords: business cycle; crisis; gold price; normal distribution.

1. Introduction

The attempt of scientists to encompass the phenomena of the surrounding world into formulas has led to quantitative or qualitative models, simple or complex, deterministic or stochastic, all representing a necessary truncation of reality. Any mathematical model will therefore, be an interpretation, an approximation of an actual fact, its accuracy given only by the precision of the estimated effect. Numerous references to the assumption of normality of the studied variables occur in the literature. Thus, statistic tests such as ANOVA require that the variable's residues follow a normal distribution, while other studies Xu and Wirjanto, 2010 use the assumption of a normal distribution of the tracked feature. In descriptions of financial market elements Costa, Cavaliere, and Iezzi, 2005, the assumption of a Gaussian distribution is often used to price financial derivatives. However, although extremely convenient in terms of computation as shown in Limpert and Stahel, 2001 this distribution does not always emphasize the important properties of the analyzed phenomen, as shown in numerous studies such as Antoniou, Ivanov, Ivanov, and Zrelov, 2004 that deny the normality assumption in favor of real analysis in terms of the distribution followed by data.

The current economic crisis is a moment worthy of interest to both economists and academics, as the origin and the solution to this negative outcome are topics worthy of research. Concerning the econometric modeling of the phenomenon, the question regards the proper tools, to ensure coverage of the essence and the possibility of replication, for future analysis. This paper aims at analyzing the correlation between business cycle phases and the best distributions

* Corresponding author. Tel.: +40-756-884-339.

E-mail address: silvia_palasca@yahoo.com

2212-6716 © 2012 The Authors. Published by Elsevier Ltd.

Selection and peer review under responsibility of Emerging Markets Queries in Finance and Business local organization. doi: 10.1016/S2212-5671 (12)00129-3

of the underlying variable, denouncing the universal Gaussian distribution in order to find a finer modeling using other distributions, which better approximate the reality. The usefulness of such a study finds motivation in developing more accurate models.

2. Business cycle theory

Modern economic activity experienced various fluctuations throughout its existence. Attempts to identify repetitive patterns in these fluctuations led to different definitions of the economic cycle with respect to duration and intensity. These were synthesized in Schumpeter's work Schumpeter, 1954, and are named after the originators of these theories: the short Kitchin cycle, the Juglar cycle with a duration of 7-11 years (sometimes abusively called the business cycle) and the Kondratiev cycle (the technological cycle). Starting from the Juglar cycle, Schumpeter, identified four phases: the recovery (characterized by increased productivity, lower prices and a low interest rate), the economic prosperity, the recession (higher interest rates, rising prices) and the crisis (multiple bankruptcies, unability to support stock markets).

In 1946, economists Arthur Burns and Wesley Mitchell provided in their paper Burns and Mitchell, 1946 a definition of business cycles: "Economic cycles are a type of fluctuation found in aggregate activity of nations that organize their work mainly in enterprises: a cycle consists of the simultaneous expansion of many industries, followed by their general recession, followed by a new phase of expansion, corresponding to a new cycle. The length of these fluctuations includes a period extending from 1 to 12 years without being divided into sub-units with similar behavior". Initially, it was considered that economic cycles have a certain regularity that helped predict them, but this concept is currently considered obsolete, the possibilities for determining the current phase of economic cycle, especially that of the turning point will likely be limited to the analysis of exterior economic events. The aim of this work revolves around identifying the appropriate distribution of the studied variable for each phase of economic cycle, previously defined, opposed to the classical hypothesis of this following a normal distribution. The use of such an approach resides in a more accurate prediction of the business cycle phases, because knowing the distribution law of the indicator is the first step in constructing a parametric model in the form of a Markov chain as described in Chauvet and Hamilton, 2005; this application is beyond the goal of the current paper.

3. Data and method

Economic evolution of countries / regions can be assessed by various indicators such as aggregate indicators like GDP, growth rates, investments, employment rate, etc. In order to compare different entities in space and time is necessary to find a variable that transcends accidental variations, reflecting actual changes in the economic structure of the observation unit. In this respect, one sought an indicator that is less influenced by the political system to achieve the widest possible range of applicability of the model, without involving any aspect of government in economic development. According to Ricardo's theory Ricardo, 1816, a useful indicator in this respect will be the price of gold expressed in the currency of the analyzed country / region. The price of gold is measured at interbank exchange rate because this quotation is closest to the real value, exempt from fees and speculation.

The choice of this indicator is supported by the work of Mises, 2006, who believes that"under the gold standard, the value of the monetary unit is not directly influenced by the government". Following the definition stated above, this paper uses data covering 13 years, to ensure that at least one full business cycle is covered in the study (the analyzed period is 1st of January 1999- 31st of December 2011). As mentioned, the indicator of choice is the price of gold (XAU), expressed in the following currencies: U.S. Dollars (USD), Euro (EUR) and the Romanian new leu (RON) retrieved online from Forex trading (OANDA). Due to the large volume of data, is is more convenient to consider the average value:

__ ^ iv-.

the arithmetic mean of the recorded values of the working days (thus the average of five values), which, in turn, represent the average daily gold quotations. Using the weekly mean value has the advantage of diminishing accidental fluctuations, thus plays the role of a filter. In our model we consider the study variable (xj]) previously mentioned.The

goal of this study is to associate the most appropriate probability distribution to each set of data. Of course, most of the values could be treated as following a normal distribution, with a certain p-value, but this is a reductive approach. Our intention is to find the distribution with the largest p-value out of a certain set of distributions and we put forward the idea that each phase of the business cycle is fully characterized by e certain distribution.

A notable difference from existing literature is that we consider a business cycle consisting of four phases: prosperity, recession, crisis and recovery, as opposed to the mainstream literature Chauvet and Hamilton, 2005; Schumpeter, 1954 which only deals mainly with two phases: recession (encompassing recession and crisis) and expansion (including recovery and prosperity). As a simplifying assumption of this model, we consider that each year is entirely dominated by a particular phase of the business cycle. The values used in the study are average values, thus discrete, but they can be fitted to continuous distributions by a statistical hypothesis test.In this regard, the chi-square test from the Easy Fit software is employed to find the most appropriate continuous distribution, thus the first step is purely empiric and produces the results from Table 4.1.

4. Results

4.1. Empiric tests

Business cycle dating is performed by national or international organizations based on the fluctuation of aggregate indicators. The activity of such organizations focusses mainly on recessions and pays less importance to the other phases of the business cycle. For exemple, in the U.S., the NBER has stated that, during the considered period of time (1999-2011), there have been two recessions, namely March 2001-November 2001 and December 2007-June 2009, while in Europe, the Euro Area Business Cycle Dating Committee has also marked only one recession: 2008 quarter 1 to 2009 quarter 2. In Table 4.1, there have been recorded the respective distributions of the gold price in USD, EUR and RON for each year between 1999 and 2011. The years corresponding to the official dates of the recession have been marked in bold and we consider this information as our starting point.

We notice that a recession is characterized by a Frechet or a Cauchy distribution which is preceded by a log-normal distribution and followed by another Cauchy distribution of the gold price, while periods of economic prosperity are flagged by a Weibull or a normal distribution of the gold price. One may even notice some years when a Pareto distribution occurs, these are the expansion years, the time of a booming economy. Thus, we propose the following correspondence between business cycle phases and the distributions of the gold price, which will be discussed in further sections:

• Prosperity phase: Normal distribution and/or Weibull distribution;

• Recession phase: Log-normal distribution or Frechet distribution;

• Crisis phase: Frechet distribution or Cauchy distribution;

• Recovery phase: Pareto distribution.

For Romania there is no available official dating system, only some announcements of the National Bank. From Table 4.1 it can be inferred that, while until around 2004, Romania's business cycle was greatly influenced by the US business cycle, due to a high importance of the RON/USD exchange rate as a benchmark target, afterward the euro begun to gain more importance, mainly because of Romania's preparation and eventual joining of the European Union. In this respect, the year 2007 was exempt from the study for Romania due to inconsistent results caused by the adhering.

Table 1. Gold price distribution by year

Year XAU/USD XAU/EUR XAU/RON Year XAU/USD XAU/EUR XAU/R(

1999 Pareto Frechet/Cauchy Cauchy 2006 Weibull normal/Weibull Weibull

2000 Pareto Frechet/Cauchy Cauchy 2007 Log-N/ Frechet Log-N/Frechet

2001 Frechet Cauchy Pareto/Weibull 2008 Cauchy/Frechet Cauchy Frechet

2002 Weibull Log-N Weibull 2009 Cauchy Frechet Frechet

2003 Weibull Log-N? Weibull 2010 Cauchy Cauchy Cauchy

2004 Weibull Normal Normal 2011 Cauchy Log-N Frechet

2005 Weibull Pareto Pareto

4.2. Prosperity phase - Normal or Weibull distribution

A random variable following a normal distribution of mean ^ and variance a2 has the density function:

nx) = —L=emz-iL?*eR. (1)

a v 2k

The Weibull distribution is a continuous distribution with the density function:

fw(x,a,l3,y) = j(^)a-1e-{rC)a, a,¡5 > 0,r< * < (2)

The prosperity phase is the stage of the economic cycle closest to the assumption that the normal distribution is the best model for the price of gold. At this moment, we can talk about a stagnation of prices, relatively stable interest rates, an equilibrium on the market is reached. In the theory of economic equilibrium, prices vary freely until an optimum is found, called the equilibrium point Samuelson, 1947 and located at the intersection of the supply and demand curves. Statistically, this optimum represents the average of the random variable price and if the price follows a normal distribution law. The Weibull distribution which is skewed compared to the normal distribution, indicates a deviation from the average value. If the movement is towards a higher value, it is announcing an upward trend of gold prices as a sign of high demand. When this situation occurs, it is a possible indicator of the business cycle moving towards the recession phase, because in recession, economic agents prefer investing in safer assets (like gold) at the expense of cash. If the movement is towards a lower average value, it implies that the prosperity phase will extend for a period of time.

4.3. Recession phase - Log-normal or Frechet distribution

Recession is defined as a contraction of the economic cycle, a general slowdown of activity in a period of time. Recessions are rooted in a general reduction of costs. Governments counteract recessions through macroeconomic policies, which include expansion of the monetary mass, increased public spending, lower taxes, etc. If a random

variable follows a log-normal distribution, it has the following density function:

m = (3)

xa v 2n

while a Frechet distribution is characterized by the density function:

fF(x,a,P,y) = , a,¡5 >0,y<x< (4)

The Log-normal distribution, respectively the Frechet distribution are in close correlation with normal thr distribution, and the Weibull distribution. Using the logarithmic transformation:

ex+y = ex ■ ey & ln(ex+y) = ln(ex) + ln(ey).

justifies the observation that log-normal and Frechet distributions show the accumulation of multiplicative effects, opposed to the normal distribution, which through the central limit theorem designates the accumulation of additive effects.

4.4. Crisis phase - Cauchy distribution

A random variable following a Cauchy distribution has the density function:

where ^ is the location parameter and a is the scale parameter.

Note that this distribution does not allow an average and a dispersion, which, in practical terms, is equivalent to the fact that regardless of the number of observations, it will never reach a value close to the average of the random variable considered. In economics, the crisis is a sharp recession phase. It is characterized by a prolonged duration, abnormal growth of unemployment, a decrease of credit amid a banking / financial crisis, lower investment, numerous bankruptcies, decrease of commercial activities, increase of currency fluctuations due to high depreciation. This period is characterized by increased investor caution, and a rise of strategic reserves. During a crisis, investors will steer funds to purchase the precious metal because it is a sound investment that will increase its value in time because, unlike currency, is not subject to sharp devaluation. In these conditions, the gold price will experience an explosive growth due to extremely high demand. This is illustrated by the Cauchy distribution. The presence of this distribution which does not allow an average (and is therefore impossible to predict), indicates a serious market imbalance, an excess of demand that causes uncontrolled growth of money in a very short time, with no resources of money or time to reach the equilibrium situation illustrated by the normal distribution. The Cauchy distribution is still present on the market even after the acute crisis is over because the effects of the crisis continue for years and the disequilibrium situation is difficult to overcome.

4.5. Recovery phase-Pareto distribution

The Pareto distribution is described by the density function:

f(x,a,ß) = -^rp 0 < ß <x< +o°,a > 0. (6)

A useful characterization of the Pareto distribution of a random variable X is given by the mean excess function

P(X > u)

ex(u) = E(X - u\x > u) = —-- / (x- u)f(x)dx, (7)

which, for the Pareto distribution has the form:

t \ k + U /o^

a — 1

During periods of recovery, economic agents prefer to own goods, not precious metals. The supply of gold on the market will increase, leading to a decrease of its price. Transactions on the market will be priced lower than those established by the equilibrium point due to highly abundant supply.

Adapting Revankar's model Revankar, Hartley, and Pagano, 1974 we have:

• m e R the minimum acceptable for any transaction;

• X - random variable defining the gold price set by market;

• Y - random variable defining the accepted gold price (Y < X);

• U = X — Y - random variable, 0 < U < max(0,X — m). The average of the price difference is proportional to X — m:

E(U |X = x) = b(x — m) = a + bx (9)

Revankar's Theorem Revankar, Hartley, and Pagano, 1974 shows that in the previous conditions, for

E(U|X > y) = a + by (10)

it is necessary and sufficient that X has a Pareto distribution with a finite mean.

5. Conclusions

This empiric study is a starting point to a new approach of the business cycle. It has proven that this elusive phenomenon may be statistically modeled using a financial indicator such as the price of gold in various currencies. Although the assumptions of this particular model may be considered coarse, the idea of this study can find multiple

applications for analyzing the business cycle of any country, due to a high availability of necessary data and an easy application of the method: fitting the data to distributions using the highest the p-value. The limits of the research include political regulations of the gold price opposed to free prices, change of phases earlier then a year and even the selection of distributions, which may be changed according to the number of phases under study, for example, for two phases the normal and log-normal distributions would suffice. Further study includes a refinement of the assumptions, especially concerning the length of each phase of the business cycle, the ultimate goal is determining the exact turning point of each phase of the business cycle.

Acknowledgements

The author is grateful to Professor Jaba Elisabeta and Professor Stoleriu Iulian for many valuable suggestions. References

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