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Seasonal Cointegration in Macroeconomic Systems: Case Studies for Small and Large EuropeanCountriesAuthor(s): Robert M. KunstSource: The Review of Economics and Statistics, Vol. 75, No. 2, (May, 1993), pp. 325-330Published by: The MIT PressStable URL: http://www.jstor.org/stable/2109439Accessed: 11/04/2008 12:51

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NOTES 325

SEASONAL COINTEGRATION IN MACROECONOMIC SYSTEMS: CASE STUDIES FOR SMALL AND LARGE EUROPEAN COUNTRIES

Robert M. Kunst*

Abstract-Stochastic seasonality in vector autoregressions draws attention to seasonal cointegrating vectors. Based upon the assumption of stochastic seasonality, seasonal cointegra- tion is found in a six-dimensional VAR of quarterly macroe- conomic series which were not seasonally adjusted. The same experiment is performed on parallel data from four European economies: Austria, Finland, Germany, and the United King- dom. Univariate and multivariate statistical evidence supports stochastic seasonality in Finland and Germany whereas deter- ministic cycles dominate in Austria and the United Kingdom. Eventual correspondences of seasonal structures across coun- tries are also analyzed.

I. Introduction

The strong seasonal patterns in many raw (non- adjusted) quarterly economic time series invalidate the assumption of stationary first differences. The question whether to eliminate these seasonal patterns by regres- sion on seasonal dummies ("deterministic" model) or to treat them by seasonal differencing, assuming addi- tional unit roots on the unit circle ("stochastic" model), parallels the discussion of deterministic versus stochas- tic trend models.

This discrimination becomes even more important when several series are treated together in a vector autoregression (VAR). If the stochastic model holds, seasonal cointegration (SC) deserves attention, a feature analogous to cointegration (see Engle and Granger, 1987). SC has been introduced by Engle, Granger, and Hallman (1989) and Hylleberg, Engle, Granger, and Yoo (1990, HEGY). SC means that, although individual series display stochastic seasonality reflected by unit roots (e.g., at -1 for semi-annual cycles or frequency v), there is a linear combination which is free from that kind of seasonality but might yet have unit roots at 1 or at other seasonal frequen- cies (e.g., vr/2 or annual cycles). Theory has it that non-cointegration at seasonal frequencies is equivalent to rank restrictions which would make SC the typical feature. However, empirical evidence on SC has been rather scarce so far (compare Engle et al., 1993).

All series which do not have seasonal unit roots are trivially cointegrated as the corresponding unit vectors cointegrate. Moreover, absence of SC entails that there are as many sources of seasonality as series, which is counter-intuitive if residuals from definitional relations are non-seasonal-such as output minus consumption minus investment. Hence, SC should not be uncommon in larger systems-whereas perhaps harder to find in two-variable relations.

This paper searches for SC structures in macroeco- nomic VARs known to cointegrate at frequency zero and investigates whether such structures are stable across countries. In detail, data from four European economies (Austria, Finland, Federal Republic of Ger- many, United Kingdom) are used, six quarterly series for each economy: gross domestic (or national) product Y; private consumption C; gross fixed investment I; goods (if unavailable, total) exports X; real interest rate on bonds R; real wage W (per capita wages deflated by the Y deflator). This system has been adopted from Kunst and Neusser (1990) who motivate their specification by the fact that neoclassical growth theory imposes several steady-state relations among the variables. Hence, the system provides an appropri- ate starting point for investigating multivariate cointe- gration in an international comparison.

The four national accounts series Y, C, I, X are in constant prices, so the system is in real terms. Sample periods are as follows: Austria 1964-1990, Finland 1972-1988, Germany 1960-1988, United Kingdom 1963-1988. Results on the Finnish data may suffer from low test power, as this sample is rather small. All series except R are in logarithms. Data sources are DIW (Germany), the database of the Austrian Institute for Economic Research, and the OECD quarterly na- tional accounts database. Detailed data are available upon request from the author.

Not only is the nature of seasonal structures of considerable intrinsic interest; it also has profound implications for economic short-run modeling and forecasting. SC invalidates some popular modeling strategies: individual seasonal adjustment by Census X-11; individual seasonal filtering by four-quarter mov- ing averages; VAR modeling with seasonal dummies included. Eventual misspecification of seasonality can severely influence the evidence on cointegration at zero frequency. The effects of seasonal adjustment on estimation and testing of cointegration structures are an open field for research. Jaeger and Kunst (1990)

Received for publication April 24, 1991. Revision accepted for publication December 17, 1991.

* Institute for Advanced Studies. The author would like to thank S0ren Johansen, Marius

Ooms, Pierre Siklos, and the referees for helpful comments on previous versions of this paper. He also thanks Klaus Neusser for the German data and the basic selection of the economic variables, and once more Pierre Siklos for the algorithm for the SC analysis.

Copyright ? 1993

326 THE REVIEW OF ECONOMICS AND STATISTICS

have established strong effects of Census X-11 on univariate measures of persistence. The consequences for multivariate analysis may be even more trouble- some.

The paper is organized as follows. Section II reviews the stochastic and the deterministic model of seasonal- ity and deals with univariate properties of the data. In section III, SC structures in the multivariate systems are identified and analyzed. Section IV concludes.

II. Univariate Characteristics of the Data Series

The traditional view of seasonality is that raw quar- terly series can be decomposed into a possibly deter- ministic seasonal component and into a remainder comprising "trend" and "cycles." Any further model- ing concerns the residual which is tacitly assumed to contain all economically relevant information.

This approach parallels the classic solution to the detrending problem. With respect to seasonality, how- ever, components modeling has advanced further and, in many countries for many variables, raw series are not even available. All seasonal information is de- stroyed carefully by specialists of statistical offices, often on a low level of aggregation, who usually base this "seasonal adjustment" on Census X-11 or similar procedures. Only recently, interest in raw series has increased. Within Europe, the availability of unad- justed series has confined this investigation to four countries.

Once original data are available, many researchers use simple deterministics, such as quarterly dummies, together with linear time series analysis on the stochas- tic part of the process or system. Accepting the unit root trend model, this deterministic seasonal model becomes

4

F(B)AXt = E aiDit + O(B)Et. (1) i=l

(1) recalls the modeling of trending behavior via linear functions of time. The seasonal counterpart of differ- encing trending processes is seasonal differencing (A4 = 1 - B4):

D(B)A4Xt = b + O(B)Et. (2)

If Xt is in logarithms, A 4Xt represents an annual growth rate. The filter A4 consists of the factors 1 - B and 1 + B + B2 + B3, the former one removing the trend and the latter one removing all seasonal struc- ture. (2) belongs to the SARIMA (seasonal integrated ARMA) class defined by Box and Jenkins (1976). Ac- cording to HEGY, the process is "integrated at the frequencies 0, i,-/2, il-," as these are the frequencies of the spectral poles implied by unit roots at + 1, i, -1, i.e., the roots of the A4 operator.

The two models generate different seasonal behav- ior. (1) describes a series governed by four alternating linear trends with identical slopes. The "average" sea- sonal pattern remains constant. With I'(D) * (-) 1, (2) yields four circularly merged random walks with identical drift, implying persistent changes in the sea- sonal pattern although the best prediction of its future shape is always its present one.

Whereas, with respect to detrending, analysts of macroeconomic series have been preferring the stochastic unit root model since discriminatory tests have been made available, a similar "revolution" did not occur with respect to seasonality. One reason is that reported evidence is mixed. Many authors, e.g., Ghysels (1990), Miron (1990), Franses (1991), support the dummy model; Osborn (1990), however, finds sea- sonal unit roots in some U.K. series and Lee and Siklos (1991) even detect them in seasonally adjusted Cana- dian data. Changing seasonal patterns are evident in some of the series under investigation.

Moreover, the basic model (D( ) - ( ) 1, started from a flat seasonal pattern, generates too many changes in seasonality, "summer becoming winter" too easily. However, if the interquarterly correlation of annual growth rates is properly accounted for and a highly volatile starting pattern is used, few summer peaks will turn into winter peaks in simulations. One should not be too reluctant, anyway, to attach non-zero probability, e.g., to the event of shifting the main feast from Christmas to summer solstice after a time span of one or two centuries.

The seasonal models can be discriminated by tests developed by HEGY. HEGY show that any autore- gressive structure with deterministic component dt can be represented as

4 p

A4Xt = dt + E aiYi,t-1 + E biA4Xt-i + Et (3) i=1 i=1

Under the null hypothesis of stochastic seasonality, all ai are zero. The Yi t are obtained from Xt via the filter factors (1 + B)(1 + B2), (1 - BX1 + B2), B(1 - B) (1 + B) and (1 - BX1 + B). Each rejection of a zero restriction has its own interpretation. If only a1 # 0, (1 + B)(1 + B2) still divides the 1(D) polynomial but there will not be a unit root at one. Similarly, a2 # 0 indicates absence of the semi-annual cycle and of the root -1, whereas both a3 and a4 relate to the annual cycle with the complex roots ?i. The tests parallel the test by Dickey and Fuller (1979) which discriminates between the deterministic and stochastic trend model. HEGY give significance points for various combina- tions of deterministics included in (3). If dt comprises seasonal dummies and a linear trend, all deterministic alternatives of seasonality and trends are encompassed. All interesting test specifications were applied to the 24 data series.

NOTES 327

These HEGY tests were unable to reject unit roots for most series.1 Evidence against stochastic seasonal- ity is strongest in the United Kingdom where only C passed the test. In contrast, most German series dis- play stochastic seasonal cycles. R is seasonal as it has been deflated via the GDP deflator. In the United Kingdom, R is non-seasonal due to a non-seasonal deflator. Non-robustness of this result with respect to the inclusion of seasonal dummies directly indicates deterministic seasonality. Fixed cycles dominate in I-climatic reasons impair construction investment in winter-and also in British GDP. In the remaining series, seasonality appears to be either absent or stochastic.

Additionally, conventional unit root tests have been conducted on the original, the seasonally adjusted, and the differenced series. In summary, first-order integra- tion at frequency zero is consistent with all series except for German R which may be stationary, whereas higher-order integration is not supported.

In some series, particularly the Finnish interest rate and most U.K. variables, significant leptokurtosis oc- curred which could affect all test procedures reported in this paper. The influence of the failure of moment conditions on unit root tests has been highlighted by Phillips (1990). Retaining moment conditions, lep- tokurtosis tends to bias the size of the reported statis- tics downward in finite samples.

III. Multivariate Cointegration Analysis

In univariate analysis, working with adjusted data can be compatible with (2) as many seasonal adjust- ment filters implicitly assume seasonal unit roots. This reconciliation of (2) with seasonal adjustment, how- ever, breaks down in a multivariate framework due to SC.

A solution to the (Gaussian) maximum likelihood problem with SC restrictions on seasonally integrated variables is due to Lee (1992). It applies the trans- formed representation (3) to an n-dimensional VAR.

4 p

A4Xt=,+ EAiYi,1+ E riA4Xti +Et- (4) i=l i=l

The vectors Yit (i = 1, . . ., 4) correspond to the scalar variables of the univariate HEGY test. If A2 = A3 =

A4 = 0, (4) reduces to the usual cointegrated VAR with the "impact matrix" A1 = an'. Only in this case, (1 '+ BX1 + B2) cancels from the lag polynomial and an application of this filter to all individual series and VAR modeling of the filtered data is acceptable. This assumes n independent sources of seasonal behavior in

the system, both at the semi-annual and the annual frequency.

The Ai admit interpretations analogous to the zero- frequency case treated by Johansen (1988). If A2 has a rank deficiency, it can be represented as a2i'. The columns of f62 contain vectors which remove the root at -1 from the resulting series. ,' X, trends but does not exhibit semi-annual seasonality while 3'2Y2t is sta- tionary. Hence, the columns of /62 cointegrate at frequency vr. The columns of f63 in A3 = a3f3'3 cointe- grate at frequency vr/2 if A4 = 0.2 If A4 # O, "dy- namic cointegration vectors" (see Engle et al., 1993) can complicate the analysis.

Lee (1992) shows that maximum-likelihood estimates of pi (i = 1,2,3) are obtained from the canonical vectors on Yi 1t- with respect to A4Xt corresponding to the non-zero canonical roots. Canonical analysis is performed conditional on the other Y1,t-l elements (e.g., the Y1, t- to A4Xt correlations are calculated conditional on Y2,t-1, Y3,t-1 Y4,t-1) and on p lags of A4Xt. In the variant used for this paper, p3 was estimated under the assumption A4 = 0, thus only "static" SC vectors were admitted. The algorithm gen- eralizes Johansen's (1988) cointegration procedure to the seasonal case.

The rank of Ai is determined by likelihood-ratio (LR) statistics which are cumulated sums

n

LRi(k) =-T E log(1 - ri) (5) j=n-k+1

over the k smallest canonical roots of the correspond- ing problem. Under the null hypothesis, k roots are zero. If LRi(k) becomes significant, rn-k+ 1 is taken to be non-zero and rank Ai > n-k. Lee (1992) gives the distribution of LRi(k) in homogeneous systems. Al- though intercepts change significance points in finite samples, these modifications only entail important ef- fects for LRj(k) (see Johansen and Juselius, 1990). Constants generate linear trends in Y1t but not in the seasonal variates Yit, i > 1.

For LR2(k) and LR3(k), Lee's fractiles tables were extended to higher dimensions by simulation. For LR,(k), table Al of Johansen and Juselius was used. GAUSS program codes by Siklos (1989, 1990) helped to simplify the calculations. Table 1 gives LR statistics for the four six-variable country systems. Values signif- icant at approximately 5% are in bold face. For all countries, evidence on SC is substantial and compara- ble to that on frequency-zero cointegration.

Lag orders were specified by Akaike's Information Criterion AIC. Although information criteria should be applied with care for VARs (Nickelsburg, 1985), the

1All test results mentioned are available on request from the author.

2One could also restrict A3 = 0 and investigate cointegra- tion by A4. The choice A4 = 0 amounts to preferring syn- chronous seasonal cycles to those with a phase shift.

328 THE REVIEW OF ECONOMICS AND STATISTICS

TABLE 1.-SEASONAL COINTEGRATION TEST LR STATISTICS

Austria Finland Germany United Kingdom Frequency (p = 0) (p = 0) (p = 1) (p = 0) Rank

Co = 0 220.190 105.001 121.237 100.949 0 Co = 7r 124.143 100.291 97.432 141.645 Co = v/2 144.190 120.317 111.200 171.860 co = 0 105.971 63.094 72.065 61.463 1 Co = 7r 79.584 61.811 46.637 96.749 co = v/2 70.567 72.436 44.132 92.792 co = 0 59.897 38.433 42.933 27.701 2 Co = 7r 53.216 35.549 22.410 53.082 Co = v/2 32.884 43.267 22.677 59.538 co = 0 30.530 17.397 24.883 12.816 3 Co = 7r 31.067 19.461 10.392 23.251 Co = v/2 14.043 24.261 9.947 30.662 co = 0 9.370 4.013 8.282 5.371 4 Co = 7r 11.279 6.590 4.970 11.063 co = v/2 5.355 8.252 1.275 10.241 co = 0 2.283 0.580 2.631 0.725 5 co = 7r 0.184 0.085 0.254 2.727 Co = v/2 0.701 0.216 0.086 2.859

Note: "Rank" indicates the rank of the Ai matrix (A1 in the first line of each block etc.) which is tested for. Rejection of "rank A2 = 0," e.g., means there is (at least) one vector cointegrating at 7r. Values significant at approximately 5% are in boldface.

quest for parsimony suggests AIC in a situation where none of the lag orders can be regarded as the correct one. All models left substantial residual autocorrela- tion in at least one of the residuals. Parsimonious models are justified as too many conditioning lags in the VAR would distort the evidence on cointegrating structures. AIC on VARs in levels recommended lag orders 3, 1, 5, 4 for the four economies. Hence, only for Germany, one lag of A4X1 was included in (4).

An approximate 5% risk level supports four cointe- grating vectors at ir for Austria, two for Finland, one for Germany, and three for the United Kingdom. At vr/2, the outcome is similar: two for Austria, three for Finland, one for Germany, four for the United King- dom. As theory backs two SC vectors-one unit vector for non-seasonal X and one relating seasonality in Y to C and I-the following cross-country comparisons assume two vectors for each country.

Let vi, i = 1, . . ., 6 denote the canonical vectors at the semi-annual frequency, with v1 corresponding to the largest root and V6 to the smallest one. Table 2a shows v,1 and v2 for each economy.

In Austria, v 1 relates seasonality in Y to I with a lesser influence from X. V2 links the remaining season- ality in output to the other two important sources, W and C. The correspondence between seasonal fluctua- tions in C and Y plays a lesser role than in the remaining countries.

In Finland, v, links seasonality in C to influences from output and wages. v2 joins seasonality in W to the accounts aggregates.

In Germany, v, relates seasonality in exports to that of consumption with a slight contribution from Y. v2

treats seasonal effects in W and I and relates them to the consumption quota C - Y. The X unit vector is not contained in the linear space generated by v1 and

3 V2, reflecting the seasonal nature of German exports.

In the United Kingdom, both vectors heavily depend on C - Y. Consumption accounts for the bulk of the seasonal fluctuations in total output. The remaining seasonality stems from W and, to a lesser extent, from I. The comparatively small influence of I mirrors the weak seasonal pattern of that variable and separates the United Kingdom from the other countries where cold winters impair investment in the construction sec- tor.

Additional to the evaluation of test statistics, which suffer from limited test power, it is recommended to look at sample spectra of the six canonical variables v1Y2t (i = 1,... , 6) for each country.4 These spectra are naturally ordered: the first one is certainly station- ary, the sixth one clearly reflects the unit root at -1 and the remaining ones are in between. Visual evi- dence tends to yield more "cointegrating" vectors than the conservative formal testing procedure. A feature enhanced by the spectra is that Germany has more independent sources of stochastic seasonality than Austria. In Austria, all seasonal structure concentrates in the last component which could accommodate a completely deterministic model of seasonality. In Ger- many, the eye recognizes three to four independent

3Scales in the series R and W have been adjusted in order to have the same magnitude as Y, C, I, X. Yet, R fails to make a big impression on the cointegrating vectors.

4 Due to space constraints, these graphs are not shown here but can be obtained on request from the author.

NOTES 329

TABLE 2.-FIRST Two SOLuTION VEcrORS TO THE SC PROBLEMS

Y C I X R W

Panel A: Seasonal Cointegration at Frequency 7-

Austria 1.000 0.013 -0.279 -0.140 -0.001 0.021a 1.000 -0.192 0.011 0.087 -0.000 - 0.481a

Finland 1.000 - 2.321 - 0.087 - 0.088 0.034 1.656a 1.000 0.704 -0.826 0.611 -0.015 - 2.585a

Germany 1.000 - 3.570 - 0.244 4.860 0.003 0.714a 1.000 -1.024 1.148 -0.288 0.001 -2.006

United Kingdom 1.000 -0.956 0.271 -0.061 -0.007 2.624a 1.000 -0.924 -0.629 -0.037 0.001 0.156a

Panel B: Seasonal Cointegration at Frequency v1/2 Austria 1.000 - 0.568 - 0.076 7.120 0.003 - 4.998a

1.000 0.151 -0.221 -0.174 0.002 -1.182a Finland 1.000 2.679 - 0.817 - 0.049 1.362 3.441a

1.000 -0.041 -0.331 0.406 -0.048 -0.114a Germany 1.000 0.305 -0.183 -1.507 0.002 -0.175a

1.000 - 0.925 0.419 - 0.088 0.007 - 0.835 United Kingdom 1.000 -0.644 -0.180 -0.426 0.006 0.849a

1.000 -0.200 -0.318 -1.089 -0.006 -2.012a

Note: For the labels of the variables, see section I. a Significant vectors.

sources of stochastic seasonality which would entail three or at least two seasonal cointegrating vectors. Deterministic seasonality also appears to prevail in the United Kingdom.

In analogy to common trends, common seasonal sources deserve attention which are found in the last components. By calculating mutual correlations among the vectors V5 and V6 across the four countries, a common source of seasonality in output, consumption, and wages can be identified which is rather similar in all four economies. A second seasonal source is com- mon both to Finland and the United Kingdom, which shows stronger relations to investment. Coincidences between v and v2 vectors are weaker, with Finland/U.K. and Austria/Germany forming loose couples.

At frequency vr/2, independent seasonal cycles are fewest in the United Kingdom and in Austria, possibly only two. Germany and Finland insinuate three to four non-stationary components. The vectors corresponding to the two largest roots are given in table 2b. There are some closer similarities among Austria and Germany and less so between these two and the United King- dom, while Finnish cointegrating vectors are entirely different. One common source of seasonality is shared approximately by Austria, Germany, and the United Kingdom. This seasonal source is rooted in wages and has less impact on the Finnish system.

One can also test for the hypothesis that the same vectors cointegrate at both seasonal frequencies. In the Austrian and Finnish systems, this hypothesis is firmly rejected, indicating that the true cointegrating vectors

might be dynamic. Germany and the United Kingdom come closer to sharing vectors at ir and vr/2.

To further assess the amount of deterministic sea- sonality in the data, a parallel experiment was per- formed with seasonal dummies included in (4). With- out additional restrictions, seasonality is granted too much freedom in such a system, as seasonal patterns are allowed to persistently change shape and to expand simultaneously. Therefore, no detailed results of this experiment are provided.

If the deterministic model is correct, the optimum VAR lag order should decrease as the stochastic sea- sonal model entails over-differencing and artificial moving-average terms. Also, all roots on the ir and ir/2 problems should be significant as the residuals from conditioning on the dummies are non-seasonal. Moreover, the roots at frequency zero should not change too much, inflated roots indicating stochastic seasonality.

Compared to the purely stochastic model, the opti- mum lag order decreases only slightly for Germany (from 5 to 4) but falls substantially for the United Kingdom (from 4 to 2). In the remaining two countries, seasonal dummies do not affect the lag length selected by AIC. In all systems, substantial residual autocorrela- tion remains.

Seasonal dummies eliminate all evidence on season- ality in the United Kingdom and leave very little in Austria. In contrast, the dummy model does not suc- ceed for Germany and Finland. Slight spill-over effects to the zero frequency reduce the number of cointegrat- ing vectors to three in Austria and increase their

330 THE REVIEW OF ECONOMICS AND STATISTICS

number to two in Finland. These findings agree well with the evidence from the analysis without dummies that Germany displays more stochastic seasonality than Austria or the United Kingdom.

IV. Summary and Conclusions

Sources and natures of seasonal fluctuations vary widely among national economies. Even the key ques- tion must be answered differently for each country: deterministic seasonality may accurately describe the Austrian and British economies but fails to do so for Germany and Finland. Multivariate SC analysis has been shown to help to clarify this problem.

Just as it is difficult if not impossible to definitely classify series as integrated versus trend-stationary on the basis of finite samples, the same goes for the classification of the seasonal patterns. Perhaps, it makes more sense to speak of vectors that reduce the correla- tion structure of the resulting series substantially, com- pared to the components series.

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