Clustering Macroeconomic Time Series

Augustyński, Iwo; Laskoś-Grabowski, Paweł Article — Accepted Manuscript (Postprint) Clustering Macroeconomic Time Series ECONOMETRICS. EKONOMETRIA: Advances in Applied Data Analysis Suggested Citation: Augustyński, Iwo; Laskoś-Grabowski, Paweł (2018) : Clustering Macroeconomic Time Series, ECONOMETRICS. EKONOMETRIA: Advances in Applied Data Analysis, ISSN 2449-9994, Wydawnictwo Uniwersytetu Ekonomicznego we Wrocławiu, Wrocław, Vol. 22, Iss. 2, pp. 74-88, https://doi.org/10.15611/eada.2018.2.06 , http://www.dbc.wroc.pl/dlibra/docmetadata?id=41165 This Version is available at: https://hdl.handle.net/10419/180670 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-nc-nd/3.0/ Clustering Macroeconomic Time Series∗ Iwo Augusty´ nski†Paweł Lasko´ s-Grabowski‡ Abstract The data mining technique of time series clustering is well established in many fields. However, as an unsupervised learning method, it requires making choices that are nontrivially influenced by the nature of the data involved. The aim of this paper is to verify usefulness of the time series clustering method for macroeconomics research, and to develop the most suitable methodology. By extensively testing various possibilities, we arrive at a choice of a dissimilarity measure (compression-based dissimilarity measure, or CDM) which is particularly suitable for clustering macroeconomic variables. We check that the results are stable in time and reflect large-scale phenomena such as crises. We also successfully apply our findings to analysis of national economies, specifically to identifying their structural relations. JEL: E00, C18, C63 Keywords: time series clustering, similarity, cluster analysis, GDP 1 Introduction The algorithms for clustering similar time series or, more generally, similar highdimensional sequences, are important in areas as diverse as biomedicine, computational biology, electronic manufacturing, physics, seismology and speech recognition. Econometrics could also benefit from a vast research effort made in these and other areas. For example, according to Focardi & Fabozzi (2004), clustering of economic and financial time series includes the following areas of application: •identifying areas or sectors for policy-making purposes; •identifying structural similarities in economic processes for economic forecasting; •identifying stable dependencies for risk management and investment management. For studies in macroeconomics, one of the most promising advantages of time series clustering is its ability to identify structural similarities in the processes that generate time series at different points in time and space. This method also allows for presentation of results in the form of easy-to-understand dendrograms. ∗Published in Econometrics. Ekonometria. Advances in Applied Data Analysis, 22 (2018) 2, pp. 74–88, DOI: 10.15611/eada.2018.2.06 †Wrocław University of Economics, Wrocław, Poland; e-mail: iwo.august[email protected] ‡Institute of Theoretical Physics, University of Wrocław, Poland 1 The aim of this paper is to find the most appropriate dissimilarity measure for macroeconomic clustering analysis. The problem of time series similarity could be boiled down to measurement of the co-movement of macroeconomic aggregates. The most popular approaches to this issue are (Croux et al., 2001; Haan et al., 2008): •correlation; •cointegration, that is, the existence of a linear combination of the two processes that is stationary; •codependence, which refers to linear combinations of correlated processes that are of lower autoregressive order than others; •common features, that is, linear combinations that are unpredictable with respect to past information, and common cycles which are defined as common features in first differences for processes that are cointegrated. According to Croux et al. (2001), these concepts pose several problems. First, correlations might be detected where no correlation is present. Second, high crosscorrelation neither implies nor is implied by cointegration, common cycles, or common features. Third, these three measures are binary. For example, two processes are either cointegrated or not, but different degrees of association cannot be established. Tests such as the Johansen test, performed by commercial econometric packages, consist of fitting empirical data to cointegrated models such as Error Correcting Models (ECM). In the case of a large number of time series, cointegration is a rather cumbersome exploratory methodology. The contribution of the present paper to the literature is as follows: •to our knowledge, it is the first comprehensive analysis of the usefulness of the different dissimilarity measures for the macroeconomic research, •it offers ready-to-use methodology, •it offers tools to present data in easy-to-understand dendrograms, •it is provided with code and web application for easy use∗. The remainder of this paper is organized as follows: in Section 2 we evaluate available dissimilarity measures and propose CDM (compression-based dissimilarity measure) as a solution for clustering of the macroeconomic time series. In Section 3 we check the robustness of the presented methodology by applying the proposed clustering method and comparing created clusters with the literature. Finally, in Section 4 we present concluding remarks. All figures are the results of own calculations based on R package TSclust and Eurostat data ( namq_10_gdp dataset). 2 Experimental Evaluation of Dissimilarity Measures A well-known data mining technique, clustering is an example of unsupervised learning: a clustering algorithm creates clusters as a function of its internal rules (whereas ∗ http://iwoaugustynski.ue.wroc.pl/apps/TSclustering/ 2 in supervised learning the algorithm learns from “known” examples). The objective of clustering is to create groups of objects that are close to each other and distant from other groups of objects. If distance corresponds to similarity, clustering forms groups of objects that are maximally similar. 2.1 Design of the Experimental Approach A dissimilarity measure suitable for our applications could be informally described as attaining low values for pairs of time series that exhibit a causal relationship. This property is hard, if not outright impossible, to formulate in rigorous terms, but it can be approximated well enough by demanding that the measure is insensitive to translating the series in time. Importantly, it should be sensitive to other time transforms, including warping (acceleration), whether uniform or not. Conversely, other kinds of transforms, such as scaling the values by a constant, adding a constant to the values, or adding noise to the values, should not, in principle, affect the measure much. We can apply this observation to design an experiment to identify prospective measures. For a given time series, we can compute the distance separating it from its delayed copy (i.e. a series with the same data, but shifted in time), as well as the distance separating it from its warped copy (i.e. a series with the same data, but squeezed in time). The ratio of these distances is a measure of how well a dissimilarity measure performs. We are looking for a measure that performs well for as many as possible time series, delays, and warp factors. Furthermore, this should hold true even if the delayed copy of the series is additionally subjected to perturbations such as scaling, shifting, noise, or a combination thereof. To formalize the idea, we fix a time series X={Xi}T i=1, dissimilarity measure M, delay δ∈N, and warp factor α∈R. We call the subseries B[X;α] = {Xi}bT/αc i=1and D[X;α,δ] = {Xi}bT/αc+δ i=1+δrespectively the base and delayed series. We also obtain from Xthe warped series W[X;α] = {¯ X(α) i}bT/αc i=1by taking averages of every αconsecutive elements, trivially extending the notion for non-integer α. Strictly speaking, ¯ X(α) i=1 α∑T j=1w(α) i j Xj, where w(α) i j =            αif j−1<(i−1)α<iα<j j−(i−1)αif j−1<(i−1)α<j<iα 1 if (i−1)α<j−1<j<iα iα−(j−1)if (i−1)α<j−1<iα<j 0 otherwise. Let dM(·,·)be the distance between two time series under the measure M. We will use the ratio R(M,X,δ,α) = dM(B[X;α],˜ D[X;α,δ]) dM(B[X;α],W[X;α]) as a quality indicator, where ˜ D[X;α,δ] is D[X;α,δ]with perturbation applied. Note that Xis truncated to its prefix B[X;α]for calculating Rto fulfill a requirement of many of the measures that the series compared be of the same length. Note that while the values of distances under different dissimilarity measures are not directly comparable to one another (as different measures may e.g. attain values in different ranges), ratios such as Rare. Note also that if a measure is good, that is, in general it separates warped series more than the delayed series, its values of Rwill be in general (or, ideally, always) less than 1. Note finally that a dissimilarity measure is better than another if its values of Rare in general smaller. 3 2.2 Sampled Measures, Series, Parameters, and Perturbations In our computations, we consider all dissimilarity measures provided by the R package TSdist, which, for the sake of simplicity and avoiding potential cognitive bias, may be calculated without supplying extra parameters. This is possible either thanks to their absence, or default values, or heuristics. These 24 measures are (referred to by names used within TSdist): euclidean , manhattan , infnorm , ccor , sts , dtw , fourier , acf , pacf , ar.lpc.ceps , ar.mah.statistic , ar.pic , cdm , cid , cor , cort , wav , int.per , per , ncd , spec.glk , spec.isd , spec.llr , and pdc . Furthermore, we consider warp factors αbetween 1.4 and 3.0 inclusive in steps of 0.2, and delays ∆of 2 through 10 quarters. Note that for series with quarterly data, the delay in terms of data points is δ=∆, while for monthly data it is δ=3∆. We also consider separately two different sets of time series, along with perturbations specific for each set. The first set contains 52 time series of absolute-valued data: •Eurostat quarterly GDP values for 28 EU Member States, •Eurostat quarterly GDP component values for 11 components of the UK GDP (with total GDP “component” omitted, as it is present in the preceding category), •three series obtained from the FRED quarterly UK GDP values by reversing the sign as well as concatenating prefixes and suffixes of this and the original series, •FRED monthly long-term (ten-year) government bond yields for USA, Germany, and France, •FRED monthly short-term (three-month) certificates of deposit yields for USA and interbank rates for Germany and France, •four artificial series: sine and triangular waves of three periods (each of 100 “months” for the sake of compatibility), with either constant or linearly diverging extrema. The following perturbations were applied to the delayed copies of the series from this set in distinct computation runs: •multiplying the values by a constant, •adding a constant (proportional to the standard deviation of the original series) to the values, •adding random noise (also proportional to the standard deviation) to the values, •all of the above, •none of the above. The second set contains 45 time series of quarter-to-quarter percentage changes of the same quantities as in the first set, except the three artificial modifications of the UK GDP, as well as four of the UK GDP components for which the percentage data is not available (i.e. compensation of employees, taxes on production and imports less subsidies, changes in inventories and acquisitions less disposals of valuables, and operating surplus and mixed income, gross). Also, the four artificial (sine and triangular wave) series are not converted to percentage changes, but (with different value ranges) 4 treated as percentage change series in their own right. Given the different nature of the data in this set, we deem the perturbations of scaling and shifting inapplicable here, and computations are performed in two runs (without perturbations and with random noise, not adjusted for standard deviation in this case). In each of the seven runs, ratios Rare computed for all possible combinations of measures, warp factors, delays, and time series. 2.3 Evaluation Results Examining the results of all seven computation runs, we have primarily focused on the following quantities: •maxX,∆,αRfor every measure M, that is, the maximum value of ratio Rachieved across all time series, delays, and warp factors; •the count of how often (out of 81 possible combinations of delay ∆and warp factor α) does measure Mrank either first or in top five (of 24 measures considered) when ordered by maxXR(for given M,∆,α). Based on this, we arrive at the conclusion that cdm is the overall best performing measure. Specifically (see also Table 1): •Without perturbations, both for absoluteand percentage-valued series, cdm is the only measure except for ncd that never exceeds 1 (and thus is good in the sense outlined above). Note that in these runs ncd performs better than cdm , with lower global maximum and more frequent appearances in the top spot or top five spots. In fact, cdm never ranks first in these cases, although only three measures ever do ( ncd , dtw , and, only once for absoluteand five out of 81 times for percentage-valued series, pdc ), and ranks third most often in the top five spots (behind ncd and dtw again). •In the remaining five runs, cdm has the lowest global maximum, which additionally is always less than ca. 4 3. For two runs (absolute-valued series with scaling and with all perturbations at once), it is the only measure with a maximum less than 2, while additionally for absolute-valued series with shifting ncd is the only other measure with a maximum less than 2. •For absolute-valued series with all perturbations at once, cdm ranks most often in the top spot, and, importantly, it does so for more than half of the possible combinations of warp factor and delay. Additionally, it ranks second most often in the top spot for absolute-valued series with scaling and with shifting (behind per ), and for absoluteand percentage-valued series with noise (behind dtw ). •For four runs (absolute-valued series with scaling, with shifting, and with all perturbations at once, as well as percentage-valued series with noise), cdm ranks most often in top five spots (although on par with pdc for absolute-valued series with shifting). For absolute-valued series with scaling and with all perturbations at once, it always ranks in top five spots, while for percentage-valued series with noise, it fails to do so for only three combinations of warp factor and delay. For absolute-valued series with noise, cdm ranks third most often in top five spots (behind dtw and cid ). 5 Another criterion that could be sensibly used here is the count of how often the ratios for a given measure exceed 1. Admittedly, with perturbations present, cdm does not perform well in this aspect. However, precisely in the presence of perturbations this requirement can be argued to be too restrictive: the delayed and perturbed series can be informally considered as at a disadvantage compared to its warped and not perturbed counterpart (especially if perturbations are, in some sense, large). It also does not give consistent conclusions across different perturbations: pdc ranks best for absolute-valued series with scaling or shifting, dtw for absoluteand percentagevalued series with noise, and int.per for absolute-valued series with all perturbations at once. In general, for absolute-valued series, measures performing well with noise mostly perform poorly with other two perturbations, and vice versa. Finally, measures should perform well also without perturbations, and that in these cases cdm together with ncd rank best as the only good measures. CDM (compression-based dissimilarity measure) introduced by Keogh et al. (2004) and further elaborated upon in Keogh et al. (2007), has already been demonstrated (e.g. in these two references) to be immensely useful in various data mining aspects, prominently including time series clustering. It warrants mention, however, that it is not a distance measure (which is why we avoid using that phrase altogether throughout this paper), as, among other properties, by definition it attains values in the range [1 2,1], i.e. does not reach 0, even for equal arguments. This also means that the ratios Rwe compute in our experiment are always going to be within the range [1 2,2]for CDM, and it may be argued that the setup is biased in favour of this measure. However, we uphold our conclusion because no other measure emerges as a clear alternative, especially if we demand that it performs well both with noise and with other types of perturbations. As a bonus, CDM also performs well with both the absoluteand percentage-valued series, making this measure much more versatile, even though we do not explicitly require such behaviour. Also, the unambiguously good results without perturbations (i.e. ratios below 1) and aforementioned reports of CDM suitability for clustering in general support this choice. 2.4 Choice of Clustering Method The next step in the procedure was to choose the most suitable clustering method. After clustering one should obtain a figure resembling a tree with some main branches and many smaller side branches. Conversely, a structure of ascending steps would disqualify a given clustering method. We tested the following approaches: •single linkage, •complete linkage, •Ward, •average (UPGMA), •McQuitty (WPGMA), •median (WPGMC), •centroid (UPGMC). 6 Only the first five of these produced useful dendrograms. Ultimately we decided to pick the Ward method, as it is less sensitive to changes in the length of the time series and creates better separated groups. 3 Empirical Application For the purposes of this paper we employed the time series of GDP for analysis of the measure’s stability, and its direct components (investment, consumption, import, export, employment, wages, etc.) to compare the economic structures of four EU Member States: Germany, France, Italy, and Spain. The reasoning for this choice is as follows: all of them are members of the Eurozone, Germany and France are the most advanced European countries, whereas Italy and Spain are the biggest lagging-behind members of the EU. In order to establish international comparisons, we used data from the Eurostat after the introduction of the euro currency, that is, from the first quarter of 2000. We decided to use the “raw” data, that is, quarterly, not seasonally adjusted time series. We argue that using data subject to preprocessing such as seasonal adjusting or detrending could introduce artificial distortions to the time series (Hamilton, 2017; Haan et al., 2008). 3.1 Type of Data In the following calculations we used nominal GDP in millions of euro, but other options are also available: •real (chain linked) values, •national currency, •first differences (or percentage change). Which one is the most suitable for the considered method? First, it depends on data availability. Usually the most up-to-date and the most accessible data is represented in nominal values in a given national currency. Second, when comparing the behaviour of different real variables we are usually not interested in their nominal values, but their changes. Therefore, percentage change is the one most commonly used. But it is problematic in the case of, for instance, financial data, which is often presented as a percent rate (interest rates, yields, etc.). Percentage change of a percent rate may be hard to interpret. Third, the answer to the question if real or nominal values are preferable is much more straightforward. If the dataset contains variables measured both in real terms (such as the number of unemployed) and in currency terms (such as consumption or GDP), chain-linked values should be employed. Otherwise, when all variables are represented in currency terms, distortions caused by inflation could be probably neglected. We argue that the proposed dissimilarity measure is suitable in all of the aforementioned cases. 3.2 Clustering Stability We begin empirical testing with analysis of the time stability of the CDM measure. To this end, we compare time series of the GDPs of the EU Member States, cov7 ering three periods: 2000Q1–2007Q4, 2008Q1–2017Q1, and the complete period of 2000Q1–2017Q1 (Figure 1). The first conclusion is rather obvious: the longer the time series are, the less similarity emerges. This phenomenon is indicated by the dotted line. A related remark is that the financial crisis reversed processes of economic integration within the EU (both periods are of similar length). This is consistent with the findings of Belke et al. (2017); Gächter et al. (2012); Ahlborn & Wortmann (2018), who employed different synchronization measures (respectively: correlation, panel regressions, and nonparametric regressions; correlation; and fuzzy clustering). The employed similarity method allows us to distinguish two main groups of countries (grey outer rectangles in Figure 1c): the “core”, consisting of France, Germany, the Netherlands, Austria, Belgium, and Spain; and the “periphery”. This structure was slightly different before and after the financial crisis (respectively, Figures 1a and 1b). Hierarchical representation of the distances enables changing the level of clustering (black inner rectangles). On this lower level, in the first period (2000–2007) the countries formed four quite similar groups consisting of: (a) Italy, Denmark, and Slovenia; (b) Germany, the Netherlands, Malta, and Romania; (c) France, Spain, UK, Finland, and Belgium; (d) the remaining 14 Member States. In the second period, generally speaking, after the financial crisis, countries are less similar and it is more appropriate to cluster them into three groups: (a) the “core”, consisting of France, Germany, the Netherlands, Austria, Belgium, and Spain; (b) the “semi-periphery”, consisting of Croatia, Malta, Slovakia, Sweden, Slovenia, Lithuania, Bulgaria, and Luxembourg; (c) the “periphery”, consisting of the remaining 12 countries, including Greece, Italy, Portugal, and the United Kingdom. This structure holds also for the full time series. These results are similar to (countries in common typeset in boldface): •Belke et al. (2017), who distinguished Finland,France,Germany, Austria, and the Netherlands as the core countries; •Papageorgiou et al. (2010), where the core countries group in 2000–2009 consists of Sweden, Portugal, Germany,France,Spain,Belgium, Denmark, Austria, and the Netherlands; •Ahlborn & Wortmann (2018), who grouped Austria, Belgium,Denmark,Finland,France,Germany, Hungary, Ireland, the Netherlands, Norway, Poland, Portugal, Spain, Sweden, Switzerland, and the United Kingdom as the core countries in 1996Q1–2015Q4. To validate the usefulness of the proposed method in the analysis of the structure of an economy, we have selected variables used to calculate the gross domestic product (GDP) and GDP itself. 3.3 Analysis of National Economies According to Eurostat (2013, p. 273), there are three approaches to calculating GDP: production approach GDP is the sum of the gross value added of the various institutional sectors or the various industries plus taxes and minus subsidies on products (which are not allocated to sectors and industries); 8