Computer Science : VT Group : Mike Stannett : Asymmetry of Economic Time Series

(Transferred from NoiseFactory Research Archives)

## Asymmetry of Economic Time Series
This survey was carried out for its own sake, but also on behalf of - and with the support of - Sean Holly,
while I was employed as a researcher at the DAE in Cambridge,
and is related to certain early aspects of what is now the
Small UK Model. My
especial thanks go to the
Marshall Librarian at Cambridge University, as it would have been impossible
to compile the data concerning nineteenth century theories without the
exceptional access to rare German and French resources granted me by himself and his staff. Parts of this survey have
appeared in summarised form as part of the paper [Sean Holly & Mike Stannett (1995) "Are there asymmetries
in UK consumption? A time series analysis" |

Introduction | References | Sources of asymmetry |

Business Cycle Theories: 1825-1913 | ||

Theories from the nineteenth century | Early twentieth century theories | Recent investigations (as at 1993) |

Introduction

A popular quote introducing recent papers on nonlinearities in economic times series is taken from Keynes (1936:314),

- There is, however, another characteristic of what we call the trade cycle
which our explanation must cover if it is to be adequate; namely the
phenomenon of the

The theory of business cycles appears to have arisen with contemporary discussions of the economic crisis of 1825 [Mitchell (1913)], and many of the arguments raised at this time are still aired in today's journals. The intensity of discussion was such that the topic developed rapidly, and by the end of the nineteenth century histories of the theory of crises began to appear [von Bergmann (1895); Jones (1900); Lescure (1907:433-522)].

Many conflicting theories arose as to the cause of economic crises. We discuss them briefly here, and will consider their implications for the asymmetry debate below.

By 1900, a certain confluence of opinion had emerged. Crises occurred too regularly to be seen as abnormal catastrophes in an otherwise orderly economy; instead, they came to be seen as part of a three-phase business cycle, encompassing prosperity, crisis, and depression. The emphasis moved, therefore, from one on explaining crises, to one on explaining the cycle itself. Viewed from the perspective of nineteenth century theory, this change has important consequences; if crises are to be seen as part of the normal course of events, we must reject the view that they are the consequence simply of abnormal events which temporarily confuse decision making. Instead, explanations were sought which explained the necessity of crisis in terms of economic organisation. There was no shortage of candidate theories.

May (1902) observed that continued growth in production required steadily rising wages accompanied by falling prices, since otherwise markets could not grow sufficiently quickly to enable production to be cleared promptly. Traditionally, however, times of prosperity have been times of rising prices, so that this condition for continuous growth is violated. A glut occurs, and prices are forced down until the markets can clear. An alternative explanation was proposed by Beveridge (1912), who argued that business cycles result from "the simple and well nigh universal fact of industrial competition"; it is in the nature of a competitive economy that when potential demand is seen not to be satisfied, several competing producers will seek to fill the gap. Inevitably overproduction occurs, and the goods cannot all be sold at a profit. The crisis now occurs as prices necessarily fall, and depression ensues until demand has once again outstripped supply, when the cycle begins again.

Bouniatian (1908) suggested a similar theory, based on the notion of over-capitalisation. If prices rise in some market, this will tend to concentrate purchasing power, and eventually capital, in the hands of those who own the goods sold in this market. This greater concentration of capital eventually leads to greater investment in related production, and so concentrates purchasing power in the hands of those who control the means of production. Thus, no matter which market initiates price rises, capital will eventually accumulate in those industries which produce industrial equipment. But this reduces the capital available for general consumption, so that the growth of productive capacity outstrips the growth of potential consumption, and prices must fall, thereby dispersing capital once more. Meanwhile, investment demand falls, and the rate of interest is driven down. Eventually, therefore, the restoration of balanced demand is achieved, at the same time that low cost loans are available for new investment in productive capacity. Any price rise in any sector now triggers the cycle to start again.

There are clearly a number of common threads running through most of these theories. First, we see a concern with the distribution of capital and incomes, and the relationship between these distributions and potential mismatches between the growth of supply and demand. Second, we see much emphasis being placed on the identification of a specific industrial sector - in this case producers of industrial equipment - whose fortunes are deemed to reflect, or even drive, those of the economy at large. The economic relationship between this sector and others within the economy are then seen as paramount to an understanding of the cycle as a whole. This is, perhaps, not surprising given the context in which authors were writing - a context in which progress in the development of machinery was seen as the ultimate force driving economic expansion.

It is not immediately clear that the mechanisms proposed would necessarily, of themselves, result in asymmetry. While it is natural to concern ourselves with the transition from prosperity to depression, we may ask whether the corresponding transition back to prosperity might not simply reflect the dynamics of the crisis in reverse. One explanation of asymmetry is particularly simple. Clearly, during a crisis, the first to suffer are the reckless. The existence of regular crises therefore serves to increase the proportion of cautious individuals and organisations involved in business. But it is in the nature of caution to avoid risks as soon as they are evident, and to proceed only slowly when fortunes appear once more to be picking up. Accordingly, it is to be expected that the withdrawal of economic activity that accompanies a crisis should occur more readily than the reinvestment that accompanies the transition from depression to prosperity.

Another possibility is that asymmetry is introduced by asymmetries in the information available to participants market mechanisms. Gul & Postlewaite (1992) consider the cumulative effects of asymmetric information in large exchang economies. They observe that asymmetry of information is probably characteristic of economic situations, since agents usually have information about their own utility functions which are not known to all other agents. In order to move from microlevel information to macromodels, it is necessary to assume that the informational relevance of each agent is small - the advantages that accrue to them from possessing non-common knowledge are assumed to be insignificant to the economy as a whole. They explain, however, that while this is an attractive notion, it suffers both at a technical and a practical level. First, it may be that, far from cancelling each other out, the effects of informational asymmetry at the agent level accumulate when we aggregate to the economy as a whole. And secondly, it is far from obvious how the 'informational smallness' of an agent may be established empirically. Gul & Postlewaite consider conditions under which asymmetries will necessarily cancel out, and point out that situations exist when their effects are indeed cumulative. Indeed, Akerlof (1970) demonstrated a model economy in which a large number of identical agents had private information about a car they wished to sell. Although each agent is intuitively 'informationally small' the market outcome was not as it would have been had the asymmetry not existed.

Burgess
(1992) suggests another mechanism centred on the nature of search costs
involved in locating a trading partner. "In the nature of matching models, this
will depend on the 'tightness' of the market for [their good] *x*, and
hence on the amount of *x* traded. So the adjustment costs and hence the
speed of adjustment of *x* will depend on the tightness of the market for
*x*. This gives rise to nonlinear dynamics ... [such that] in the downswing
*x* chases a falling target quite rapidly, while in the upswing it chases a
rising target more slowly".

Although there has clearly been much interest in the business cycle, the analytical investigation of general asymmetries in the economy appears to have begun only recently, the first papers coinciding to some extent with a general explosion of interest in nonlinearity theory in the early 1980s.

It is convenient to take as our starting point Neftçi's (1984) paper in which he investigates whether economic time series might be asymmetric over the business cycle.

Neftçi argues that it is inappropriate to test for asymmetry by splitting data into two groups - one representing contractions, the other expansions - and then performing separate analyses upon the two sets of data to determine whether significant dynamic differences in structure can be identified. Such a procedure, he argues, will bias the results in favour of asymmetry. Accordingly, he rejects the use of such strategies, and suggests instead that the statistical theory of finite-state Markov processes should be used.

Specifically, suppose that {*X _{t}*} is some series of interest.
Let {

so that {*I _{t}*} is generally positive during upswings in
{

where

- p
_{0}is the probability of the initial state - l
_{10}=*P*{*I*= +1 |_{t}*I*= +1 and_{t-1}*I*= -1 }_{t-2} - l
_{01}=*P*{*I*= -1 |_{t}*I*= -1 and_{t-1}*I*= +1 }_{t-2} - the parameters
*N*and_{ij}*T*denote the number of occurrences of the states implied by the associated transition probabilities._{ij}

Once the l's have been estimated by maximising the
log-likelihood function, a confidence ellipse for l_{00} and l_{11}
can be constructed by solving the quadratic equation

where

**H**is the 2×2 matrix of second partial derivatives of*L*with respect to l_{00}and l_{11}evaluated at- a is the desired confidence level

We reject the hypothesis of asymmetry (*i.e.* l_{11} > l_{00}
for a procyclical variable) if any part of this ellipse falls on or below the
45° line, with l_{00} measured on the
horizontal axis.

Neftçi applied this techniques to three US unemployment rate series: the seasonally adjusted rate, the rate for 15 weeks and over, and the rate for insured workers. In each case, he found that the diagonal remained outside the 80% confidence ellipse.

Subsequently, Falk (1986) applied Neftçi's methods in an analysis of the behaviour of real GNP, investment, and productivity in the US, as well as non-US industrial production. He found the claim for asymmetry to be uncompelling: the null hypothesis that recessions in GNP and investment do not tend to last as long as recoveries was rejected at the 80% confidence level. In no case did he find evidence in support of the asymmetry thesis.

Sichel (1989) attempted to replicate Neftçi's results, using data from the same sources, and discovered a probable error in Neftçi's empirical work that reverses the significance of his evidence for the asymmetry of the unemployment rate. On calculating the confidence ellipse, Sichel obtained confidence levels below the 80% levels cited by Neftçi. However, he notes (p. 1259)

Given the potential sensitivity to noise or measurement error, it is interesting to note that there is some evidence of asymmetry in annual data in which, presumably, more of the noise is averaged out. In particular, there is strong evidence of asymmetry in annual unemployment in both a 1949-81 and 1949-87 sample. For example, for 1949:1-1987:4, the quarterly datat-statisticfor [ l_{11}-l_{00}=0 ] is 0.88 while the annual datat-statisticis 2.0 ...... the annual data evidence and the power analysis suggest that Neftçi's test applied to quarterly data may not identify asymmetry that is, in fact, present and has been identified by other researchers.

Despite these shortcomings, Neftçi's work has triggered subsequent promising investigations which continue to suggest the presence of nonlinearities in economic time series data, although it is not always apparent that these nonlinearities are connected with movements through the business cycle. A key paper is that of Hamilton (1989), in which he applies Markov methods to derive a series of useful results under the assumption of switching in the dynamics of series under study.

Suppose that {X_{t}} is some particular series, which may or may not
display non-zero trend. Either way, let {n_{t}} denote the trend
component of {X_{t}}. Then {n_{t}} obeys a *Markov trend in
levels* if

where the a's are constants, and s_{t} takes
the value 0 or 1, and denotes some unobserved state of the system under
observation. The switching variable {s_{t}} is assumed to be governed by
a first order Markov process of the form

P{ s_{t} = 1 | s_{t-1} = 1 } = p |
P{ s_{t} = 0 | s_{t-1} = 1 } = 1-p |

P{ s_{t} = 0 | s_{t-1} = 0 } = q |
P{ st = 1 | s_{t-1} = 0 } = 1-q |

Hamilton demonstrates that, as one would expect, information about the
initial state of the economy has no effect on the long-run growth rate {Dn_{t}}, but a permanent effect on the level
{n_{t}}, since

where l = (p+q) - 1.

Since {n_{t}} represents only the trend component of {X_{t}},
we need to examine combinations of {n_{t}} with other stochastic
processes. Hamilton considers the process

where

- the f
_{i}are constants; and - the {e
_{t}} are IID(0,s²) and independent of {n_{t+j}} for all j These specifications imply a model of the form DX_{t}= a_{1}s_{t}+ a_{0}+ Dz_{t}, where

Hamilton provides extensive details (pp 367-71) of how probabilistic
inferences about {s_{t}} may be drawn from knowledge of {DX_{t}}, on the assumption that {DX_{t}} can be observed, but that this is true for
neither {s_{t}} nor {Dz_{t}}.

Hamilton applied his technique to US post-war data on real GNP, using

for the period 1951:2 to 1984:4, and found evidence of negative growth (-0.4%) during state 0, and positive growth (+1.2%) during state 1, which he interpreted as indicating true cycles, rather than switches between two positive growth rates. Furthermore, he estimated dates for peaks and troughs for the post-war business cycles, and obtains results in close agreement with NBER estimates.

Hamilton's highly detailed paper concludes with an analysis of the behaviour of consumers, and asks how unanticipated increases to current income affect estimates of permanent income. His results suggest that knowledge of the state of the economy is relevant: "the certain knowledge that the economy has gone into a recession is associated with a 3% drop in permanent income".

as suggesting a process which draws from two normal distributions which differ only in their means. In order to allow for more extensive asymmetry, Kähler & Marnet extend Hamilton's model to allow for differing variances as well, and apply his techniques to GDP data for four countries, Canada, the UK, the USA, and West Germany. They find that variance is smaller in recessions than in booms for Canada and West Germany, but smaller in booms for the UK and the USA. But perhaps the most surprising results concern the transition probabilities between booms and recessions, and the expected duration of recessions.

Using Hamilton's original technique, the probability of growth in the next period given growth now is very high: 98.1% for Canada, 95.2% for the UK, 95.7% for the USA, and 93.9% for West Germany. In contrast, the probability of repeated depression was considerably smaller. Accordingly, the associated unconditional probability of recession is (arguably) low: 8.6% for Canada, 18.5% for the UK, 5.8% for the USA, and 12.4% for West Germany. Such results are arguably reasonable for the UK, but Kähler & Marnet note that for the USA the average length of a recession would stand at about 1.5 quarters, in sharp contrast to conventional wisdom that USA recessions have an average duration of some 13 months.

Allowing for changing variances changes these figures considerably. 'Recession' in West Germany becomes something of a misnomer over the sample period, since even in slow-growth periods, that growth is still positive. The unconditional probability of recession in the USA now climbs to 50%, and that for the UK to 60.8%, figures which might in turn be considered unreasonable.

Why does the allowance for changing variance matter appear to have such
significant effects, and why do such counter-intuitive results emerge? Kähler
& Marnet suggest that Markov-switching models are not picking up level
changes in the D's, but rather changes in
*volatility* - an effect which is not allowed for in the unmodified
technique. They continue

Is the Markov-switching model wrong? We do not think so. Instead, we would blame the differencing of the data for the failure to identify business cycles. It is apparent ... that there is hardly any persistence in levels of these series and that variance effects are clearly dominant. It is obviously futile to search for business cycle phenomena in GDP data which are differenced since, for these data, differencing is a filter which removes the level effects ... We could show that for the study of business cycle phenomena, the differencing of GDP data has fatal effects. Within the Markov-switching model, which is designed to identify booms and recessions,

we were not able to find meaningful business cycle phenomena if we allow for asymmetries in business cycles.

- Aftalion A. 1909
*Essai d'une théorie des crises générales et périodiques*. Paris. - Akerlof G. 1970 The market for lemons.
*Quarterly Journal of Economics***89**pp 488-500. - Beveridge W.H. 1912
*Unemployment*(3rd ed). London. - Bouniatian M. 1908
*Studien zur Theorie und Geschichte der Wirtschaftskrisen*. Munich. - Burgess S.M. 1992 Asymmetric employment cycles in
Britain: evidence and an explanation.
*The Economic Journal***102**pp 279-90. - Carver T.N. 1903 A suggestion for a theory of
industrial depressions.
*Quarterly Journal of Economics*(May) pp 497-500. - Falk B. 1986 Further evidence on the asymmetric behaviour
of economic time series over the business cycle.
*Journal of Political Economy***94**pp 1096-1109. - Fisher I. 1911
*The Purchasing Power of Money*. New York. - Gul F., and Postlewaite A. 1992 Asymptotic efficiency in
large exchange economies with asymmetric information.
*Econometrica***60**(6) pp 1273-92. - Hamilton J.D. 1989 A new approach to the economic
analysis of nonstationary time series and the business cycle.
*Econometrica***57**(2) pp 357-84. - Hobson 1909
*The Industrial System*. London. - Hull G.H. 1911
*Industrial Depressions*. New York. - Jones E.D. 1900
*Economic Crises*. New York. - Kähler J., and Marnet V. 1992 International business
cycles and long-run growth: an analysis with Markov-switching and
cointegration methods.
*Recherches Economiques de Louvain***58**(3-4) pp 399-417. - Lescure J. 1907
*Des crises générales et périodiques de surproduction*. Paris. - May R.E. 1902
*Das Grundgesetz der Wirtschaftskrisen*. Berlin. - Mitchell W.C. 1913
*Business Cycles*. Berkeley: University of California Press. - Neftçi S.N. 1984 Are economic time series asymmetric
over the business cycle?
*Journal of Political Economy***92**pp 307-28. - Sichel D.E. 1989 Are business cycles asymmetric? A
correction.
*Journal of Political Economy***97**pp 1255-60. - Veblen T.B. 1904
*Theory of Business Enterprise*. New York. - von Bergmann E. 1895
*Geschichte der nationalökonomischen Krisentheorie*. Stuttgart.

Maintained by and copyright © 2000 Mike Stannett. All rights reserved.