Empirica 27: 283–293, 2000.© 2001Kluwer Academic Publishers. Printed in the Netherlands.

283

An International Perspective on Asymmetries inConsumers’ Expenditure?

STEVEN COOKDepartment of Economics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, Wales

Abstract. Data on real consumers’ expenditure for seven major OECD economies (Australia,Canada, France, Germany, Japan, the UK and the US) are subjected to Sichel’s (1993) tests ofbusiness cycle asymmetry. The results derived show that the high cross-country correlations in con-sumers’ expenditure mask very different underlying cyclical properties. Interestingly, while Franceand Germany fail to exhibit any type of asymmetric adjustment, asymmetry of some form and tosome degree is present for all of the other economies considered. The implications of these findingsare discussed.

Key words: Asymmetry; consumers’ expenditure.

JEL Classification: C10; E21.

I. Introduction

The notion of macroeconomic time series exhibiting some form of asymmetry hasa long history stretching back at least as far as the work of Mitchell (1913) andKeynes (1936). In recent years the analysis of asymmetric behaviour has developedrapidly, with a major contribution provided by Sichel’s (1993) introduction of‘deepness’ and ‘steepness’ tests of business cycle asymmetry. Based upon originalwork by DeLong and Summers (1986), these univariate tests consider whether theheight of the recovery boom differs from the depth of the recessionary trough(deepness), and whether the speeds at which booms and troughs are approacheddiffer (steepness). A distinction can therefore be made between alternative forms ofasymmetry, which may be present either individually or simultaneously. In an earlyapplication of these tests, Holly and Stannett (1995) found evidence of asymmetricadjustment in UK consumers’ expenditure in the form of heightened booms. Incontrast to this, Cook (2000a) in a recent examination of Australian consumers’expenditure found that asymmetric behaviour existed in the form of troughs beingapproached more rapidly than booms. Given these differing results, the present

? I am indebted to Sean Holly for many discussions on related work. I am also grateful to theeditor for many useful comments which have helped improve the presentation and content of thepaper. The usual disclaimer applies.

284 STEVEN COOK

study will examine the asymmetrical properties of consumers’ expenditure in sevenmajor OECD economies (Australia, Canada, France, Germany, Japan, UK, US)over a common sample period to examine potential differences in the cyclicalvariation they exhibit. The seminal research of Holly and Stannett (1995) is there-fore extended by considering a range of economies to examine the relevance ofasymmetric behaviour over more recent data series. In addition to this, the exactform of asymmetric behaviour displayed will be focused upon via examination ofthe signs of the calculated test statistics.

The detection of asymmetric behaviour is of particular interest and importancefor a number of reasons. First, it may be that variation in the asymmetric propertiesof consumers’ expenditure reflects differing underlying expectations and attitudestowards risk amongst agents. Second, asymmetry is important as economic vari-ables adjusting and evolving at different speeds during different phases of thebusiness cycle has major implications for econometric modelling and economicpolicy analysis. An example of this is provided by the work of Arden et al. (2000)where the usually employed assumptions of symmetry and linearity are relaxed.Arden et al. (2000) find that when asymmetric specifications are incorporatedwithin a large scale macroeconomic model, very different model simulations arederived with significantly different policy implications.

To achieve its aim this paper will proceed as follows. In the following section thetests of asymmetry to be employed will be outlined. Section III will present the res-ults of applying the tests of asymmetry to the seven countries under consideration.Section IV concludes.

II. Testing for Asymmetric Behaviour

In this paper quarterly seasonally adjusted data on real consumers’ expenditure inAustralia, Canada, France, Germany, Japan, the UK and the US will be examinedover the period 1967(1) to 1997(2).1 The immediately striking features of the dataare their strong trends and the high degree of correlation between the series. Table1 shows the correlation coefficients for the economies considered. However, viathe use of univariate tests of asymmetry it will be seen that these high correlationsmask some underlying differences in the cyclical nature of consumers’ behaviourin the countries examined.

To test for the presence of asymmetric adjustment, the ‘deepness’ and ‘steep-ness’ tests of Sichel (1993) will be employed. These tests relate to the skewness of atime series, with the deepness test considering the relative depth of the recessionarytroughs below trend compared to height of the recovery booms above it. While anegative deepness statistic indicates that troughs are deeper than booms are high, apositive statistic denotes the opposite, which may then be referred to as ‘highness’.In contrast, the steepness test does not consider the levels of the booms and troughs,but rather the speeds at which they are approached. Again the derived statisticcan be positive or negative, indicating that booms or troughs are approached more

AN INTERNATIONAL PERSPECTIVE ON ASYMMETRIES 285

Table I. Cross-country correlations

Australia Canada France Germany Japan UK US

Australia 1

Canada 0.989 1

France 0.985 0.994 1

Germany 0.974 0.956 0.957 1

Japan 0.994 0.994 0.994 0.973 1

UK 0.994 0.979 0.969 0.952 0.983 1

US 0.980 0.993 0.987 0.964 0.995 0.989 1

rapidly. An important feature of these tests is that they apply to the cyclical elementof a time series. Considering a time seriesxt (expressed in natural logarithms):

xt = τt + ct + ξt , (1)

where τt is the non-stationary trend component,ct is the stationary cyclicalcomponent andξt is the irregular component which isNID(0, σ 2

ξ ), the tests ofasymmetry are performed upon the cyclical component,ct . To isolate the cyclicalelement a method of trend extraction is required, and here the familiar Hodrick–Prescott (HP) (1997) filter is employed. While the choice of an appropriate methodof detrending is far from uncontroversial (see Harvey and Jaeger, 1993; Cogley andNason, 1995), the HP filter has several factors in its favour. Due to the linear struc-ture of the HP filter, it can not induce asymmetry when none exists, and although itmay accentuate booms and troughs, this would be of benefit in the present contextas it will allow any asymmetry present to be detected more easily. The HP filter alsohas some advantages over alternative approaches such as the structural time seriesmodelling (STM) of Harvey and Jaeger (1993) and the closely related exponentialsmoothing filter (see King and Rebelo, 1993). Although the STM approach hasbeen employed in previous applications of Sichel’s tests, it has been found to havelittle impact on the conclusions drawn at conventional levels of significance (seeSpeight and MacMillan, 1998). It should also be noted that alternative methods ofseasonal adjustment can have an impact on the outcomes of deepness and steepnesstests (see Cook, 2000b). However in the present study all the series considered areofficially seasonally adjusted.

Formally, the HP filter proposes a trend which is the solution to the minimiza-tion problem:

minT∑t=1

{(xt − τt )2+ λ

[(1− L)2 τt

]2}, (2)

whereL is the lag operator andλ is the smoothing parameter. Theoretically,λ canbe set at any value, with values ofλ other than zero causing the last term in (2) to

286 STEVEN COOK

smooth out the trend by penalizing the rate at which the slope of the trend changes.The extreme values of{0,∞} lead to the HP trend coinciding with the originalseries and a linear trend respectively. A value ofλ = 1600 is conventionallyimposed for quarterly data to remove low frequency components with a periodicityof more than 32 quarters, and this value has been shown to be optimal according tothe transfer function derived by Harvey and Jaeger (1993). It is this value which isadopted here.2

With the cyclical element,ct , of a series derived, the tests of asymmetry can beconstructed. The deepness test is provided by the coefficient of skewness, and isgiven as below:

D(c) =[T −1∑T

t=1 (ct − c)3]

σ (c)3, (3)

wherec is the mean ofct , σ (c) is the standard deviation ofct , andT is the samplesize. To test the significance of this statistic, a variablezt is constructed, and aregression run usingzt as the dependent variable and a constant term as the soleregressor:

zt = (ct − c)3σ (c)3

. (4)

The estimated value of the constant therefore provides an estimate ofD(c).Similarly, the significance of the estimated constant denotes the significance ofD(c).

The steepness statistic relates to the speed at which booms and troughs areapproached, and therefore uses the first difference of the cyclical element:

ST (1c) =[T −1

∑Tt=1

(1ct −1c

)3]σ (1c)3

(5)

where1c is the mean of1ct , σ (1c) is the standard deviation of1ct , and Tis the sample size. To test the significance ofST (1c) a constructed variable,z1t ,is regressed upon a constant, with the significance of the constant showing thesignificance ofST (1c):

z1t =(1ct −1c

)3σ (1c)3

. (6)

In testing the significance ofD(c) andST (1c) via the constructed variableszt andz1t , the presence of serial correlation means that OLS standard errors willbe invalid. A serial correlation consistent variance-covariance matrix estimator istherefore required to assess the significance of the constant terms in the regressions

AN INTERNATIONAL PERSPECTIVE ON ASYMMETRIES 287

using the constructed variables. In this paper Newey–West (1987) standard errorsare employed. As Newey–West standard errors use a serial correlation correc-tion based upon estimated autocovariances, decisions therefore have to be madeover the appropriate bandwidths and kernels to employ.3 With no overwhelmingevidence in the literature suggesting any particular kernel to be ‘best’, results arereported here for three alternative kernels. These kernels are the Parzen, Bartlettand Tukey kernels, which use, respectively, quadratic, declining and trigonomet-ric weighting schemes for the autocovariances. The choice of an appropriatebandwidth is dependent upon the degree of autocorrelation present in the seriesunder investigation. As this is typically unknown, this is again not well defined.Here bandwidths corresponding to approximately one quarter and one third of thesample size, giving alternative bandwidths of 30 and 40 for the present sample of122 observations, will be employed. The choice of three kernels and two band-widths means that six standard errors will be presented for each of the deepnessand steepness statistics for each of the economies considered.

III. Results

Before presenting the results of applying the deepness and steepness tests to theseven economies under consideration, the cyclical elements of consumers’ ex-penditure are presented in Figures One to Four. Figure Four provides two graphsof the cyclical element of consumers’ expenditure for Germany. The first graphshows the derived cyclical element over the whole sample, while the second showsthe cyclical element up to 1990(4). This reflects the effect of unification from1991(1) onwards. Prior to this point the data refer to West Germany, thereafterthey refer to the unified Germany. Figure Four illustrates this with a sharp spike atthe point of unification. As this overshadows other fluctuations, the second graphwas constructed to allow smaller variations to be seen.

The results of applying the deepness and steepness tests are presented in TablesTwo and Three, with the calculated statistics given along with their asymptoticstandard errors and asymptotic marginal significance levels (p-values). These p-values indicate the significance of the relevant test statistic. Following conventionalpractice, significance is considered at the 5% level, requiring p-values≤ 0.05 toindicate a significant deepness or steepness statistic. In addition to this p-valuesbetween 0.05 and 0.1 are taken to indicate weak evidence of asymmetry beingpresent. Due to the potential effects of unification on the outcomes of the tests ofasymmetry for Germany, two sets of results are presented corresponding to thewhole sample period (denoted Germany) and for the period up to 1990(4) (denotedGermany∗). Turning to the results for the deepness tests, negative statistics werederived for Canada, France and the US. These statistics indicate troughs which aredeeper than booms are tall. However, the statistics are not, in any instance, signi-ficant at conventional levels. For Australia, Germany, Japan and the UK positivestatistics were found, which suggest highness, or the height of booms exceeding

288 STEVEN COOK

the depth of troughs. These results raise some interesting issues. In accordancewith the findings of Holly and Stannett (1995) over a less recent sample period,the UK displays significant highness, at up to even the 1% level. None of the othercountries have highness which is significant at conventional levels, although thereis evidence of significance for Japan at the 10% level. The size of the statisticfor Japan is also noteworthy, being approximately 65% larger than that for theUK, suggesting a great degree of highness. The results for Germany also meritdiscussion, as the impact of unification leads to the deepness statistic changingfrom 0.0582 to 0.8707. Although in neither the pre- or post-unification phase is thedeepness test significant, the p-values post-unification are significantly reduced.Along with the increased size of the statistic, this suggests that the effect of uni-fication is to bias the test towards detecting asymmetry in the form of highness, aswould be expected.

The results for the steepness tests are again mixed, with both positive and neg-ative statistics. France and post-unification Germany both have positive statistics,suggesting that expenditure is increased more sharply than it is cut. However,in neither case are the statistics significant, and the results for post-unificationGermany are in any case biased for the reasons given above. Table Three alsoshows that negative steepness statistics were derived for Australia, Canada, pre-unification Germany, Japan, the UK and the US. For Australia and Canada thesestatistics are of moderate size and are significant at the 5% level for Australia,and beyond that for Canada. This indicates that consumption is cut more sharplythan it is increased. For post-unification Germany and the UK the statistics are notsignificant. For Japan and the US the steepness statistics are larger in absolute valuethan those for Australia and Canada, especially so in the case of Japan, and thereis evidence of their significance at the 10% level.

Combining the results derived, Australia and Canada display asymmetric ad-justment in consumers’ expenditure in the form of reductions being much sharperthan increases, which is supported at high levels of significance. France andGermany display no evidence of asymmetric behaviour, although the impact ofunification is to introduce a bias towards significant highness and expansionarysteepness. The results for Japan are interesting as the statistics are very large in ab-solute value for both forms of asymmetry, indicating that booms very much higherthan troughs are deep, and that troughs are approached very much quicker than thebooms. This suggests that the cyclical peaks in the level of consumers’ expenditureare approached very steadily in Japan, although the statistics for both tests weresignificant only at the 10% level. Finally, both the UK and the US display someform of asymmetry in consumers’ expenditure, with the UK exhibiting highnessbeyond the 1% level of significance, and the US showing recessionary steepness ata lower level of significance.

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Table II. Deepness tests

Asymptotic Standard Error (p-value)

D(c) Parzen Tukey Bartlett

30 40 30 40 30 40

Australia 0.2728 0.2253(0.228) 0.2014(0.178) 0.1934(0.161) 0.1862(0.146) 0.2334(0.245) 0.2142(0.205)

Canada −0.6272 0.6450(0.333) 0.5802(0.282) 0.5838(0.285) 0.4677(0.182) 0.5510(0.257) 0.4491(0.165)

France −0.9809 0.7399(0.187) 0.7129(0.171) 0.6966(0.162) 0.7107(0.170) 0.7200(0.176) 0.7373(0.186)

Germany 0.8707 0.8609(0.314) 0.8005(0.279) 0.7888(0.272) 0.7361(0.239) 0.7999(0.279) 0.7585(0.253)

Germany∗ 0.0582 0.4936(0.906) 0.3964(0.883) 0.3899(0.882) 0.2142(0.786) 0.3264(0.859) 0.2398(0.809)

Japan 1.123 0.7326(0.128) 0.6873(0.105) 0.6742(0.098) 0.6530(0.088) 0.6956(0.109) 0.6744(0.098)

UK 0.6784 0.3900(0.084) 0.2727(0.014) 0.2283(0.004) 0.1403(0.000) 0.2848(0.019) 0.2645(0.012)

US −0.2399 0.3941(0.544) 0.2795(0.392) 0.2204(0.279) 0.1958(0.223) 0.3015(0.428) 0.3126(0.444)

Germany: these results are for the period up to 1990(4) and use bandwidths of 24 and 32, rather than 30 and 40.

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Table III. Steepness tests

Asymptotic Standard Error (p-value)

ST (1c) Parzen Tukey Bartlett

30 40 30 40 30 40

Australia −0.4934 0.3089(0.113) 0.2832(0.084) 0.2814(0.082) 0.2399(0.042) 0.2990(0.101) 0.2411(0.043)

Canada −0.5858 0.2315(0.012) 0.1849(0.002) 0.1722(0.001) 0.1396(0.000) 0.2095(0.006) 0.2002(0.004)

France 1.767 1.614(0.276) 1.598(0.271) 1.598(0.271) 1.572(0.263) 1.589(0.268) 1.567(0.262)

Germany 4.688 4.252(0.272) 4.068(0.251) 4.047(0.249) 3.839(0.224) 4.076(0.252) 3.901(0.232)

Germany∗ −0.0592 0.3248(0.856) 0.2713(0.828) 0.2669(0.825) 0.1920(0.758) 0.2711(0.828) 0.2416(0.807)

Japan −2.500 1.641(0.130) 1.544(0.108) 1.537(0.107) 1.409(0.078) 1.539(0.107) 1.430(0.083)

UK −0.4277 0.3852(0.269) 0.3530(0.228) 0.3435(0.216) 0.3252(0.191) 0.3838(0.267) 0.3633(0.241)

US −0.8873 0.5539(0.112) 0.5186(0.090) 0.5025(0.080) 0.5111(0.085) 0.5676(0.121) 0.5438(0.105)

Germany∗: these results are for the period up to 1990(4) and use bandwidths of 24 and 32, rather than 30 and 40.

AN INTERNATIONAL PERSPECTIVE ON ASYMMETRIES 291

Figure 1. The cyclical elements for Australia and Canada.

Figure 2. The cyclical elements for France and Japan.

Figure 3. The cyclical elements for United Kingdom and United States.

292 STEVEN COOK

Figure 4. The cyclical elements for Germany.

IV. Conclusions

Despite interest in asymmetric adjustment over the business cycle having a longhistory in macroeconomics, only recently have formal tests become available toempirically examine this. In this paper, Sichel’s (1993) tests of deepness and steep-ness have been applied to real consumers’ expenditure for seven major OECDeconomies. The results show widespread evidence of asymmetric adjustment ofdifferent types and to differing degrees, with only France and Germany failing todisplay some form of asymmetric behaviour. The very high correlations betweenconsumers’ expenditure in the economies considered therefore mask some verydifferent underlying cyclical properties which may be indicative of differing un-derlying expectations and attitudes towards risk. The prevalence of asymmetricbehaviour contrasts with the usually employed implicitly symmetric linear spe-cification of econometric models, and also has major policy implications. As anexample, the significant highness found for the UK helps explain the failure ofmacroeconomic modellers to predict the strength of the consumption boom of thelate eighties following a relaxation of previously tight economic policy.

Notes

1. The data used were obtained from Datastream and have codes AUCONEXPD, CNCONEXPD,FRCONEXPD, BDCONEXPD, JPCONEXPD, UKCONEXPD and USCONEXPD respectively.The analysis of the data was conducted usingMicrofit 4.0 (see Pesaran and Pesaran, 1997) andPc-Give 9.0(see Hendry and Doornik, 1996).

2. Note that for annual data the choice ofλ is not well defined. Cook (2000c) experiments withalternative values finding this has a significant impact on the conclusions drawn from Sichel’stests.

3. For a more complete discussion of consistent variance-covariance matrix estimators see Andrews(1991), Andrews and Monahan (1992), Newey and West (1987,1994), and Pesaran and Pesaran

AN INTERNATIONAL PERSPECTIVE ON ASYMMETRIES 293

(1997). In this paper the Newey–West standard errors were calculated usingMicrofit 4.0(Pesaranand Pesaran, 1997).

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