Marcus Scheiblecker

# **The Austrian Business Cycle in the European Context**

Marcus Scheiblecker

## **The Austrian Business Cycle in the European Context**

Dating business cycle turning points is still an important task for economic policy decisions. This study does this for the Austrian economy for the period between 1976 and 2005, using only quarterly national accounts data of Austria, Germany and the euro area. Three different filtering methods are applied: first-order differences, the Hodrick-Prescott filter, and the Baxter-King filter. To all of them, two different methods of determining the business cycle are applied: the ad-hoc determination of the business cycle and a dynamic factor model, taking into account the common variations of Austria, the euro area and the German business cycle movements. The results of both methods are dated by the Bry-Boschan algorithm in order to locate peaks and troughs of the cycle. The results are interpreted and compared to already existing studies on the euro area and the Austrian business cycle.

Marcus Scheiblecker, born in 1967, studied Economics at the Vienna University of Economics and Business Administration. After working for the Austrian Statistical Office he joined the research staff of the Austrian Institute of Economic Research (WIFO) in 1998. His current fields of research cover the analysis of international and national economic developments, short-term economic forecasting, business cycle analysis and quarterly national accounts issues. He is lecturer of international economics at the University of Applied Studies in Vienna since 2000.

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The Austrian Business Cycle in the European Context

## **Forschungsergebnisse der Wirtschaftsuniversitat Wien**

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Marcus Scheiblecker

# **The Austrian Business Cycle in the European Context**

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## **Zusammenfassung**

Die Datierung konjunktureller Wendepunkte stellt nach wie vor eine wichtige Information tor wirtschaftspolitische Entscheidungen dar. Die vorliegende Studie ermittelt den osterreichischen Konjunkturverlauf tor die Zeit zwischen 197 6 und 2005 auf Basis von Oaten der vierteljahrlichen volkswirtschaftlichen Gesamtrechung Osterreichs, Deutschlands und des Euro-Raums. Drei unterschiedliche Filtermethoden kommen tor die Extraktion konjunktureller Schwankungen zur Anwendung: Differenzen erster Ordnung, der Hodrick-Prescott-Filter und der Baxter-King-Filter. Basierend auf diesen Ergebnissen wird der Konjunkturverlauf anhand zweier Verfahren ermittelt: der Ad-hoc-Bestimmung der Konjunkturkomponente und der Erstellung eines dynamischen Faktormodells, welches die gemeinsamen zyklischen Schwankungen in Osterreich, im Euro-Raum und in Deutschland berOcksichtigt. Die Datierung des Konjunkturzyklus erfolgt danach durch den Bry-Boschan-Algorythmus. Die Ergebnisse werden interpretiert und mit bereits bestehenden Studien Ober Konjunkturschwankungen im Euro-Raum und in Osterreich verglichen.

## **Abstract**

The dating of business cycle turning points is still an important basis for economic policy decisions. The present study sets out to trace the Austrian business cycle for the Austrian economy for the period between 1976 und 2005, using quarterly national accounts data of Austria, Germany and the euro area. Three different filtering methods are applied: first-order differences, the Hodrick-Prescott filter and the Baxter-King filter. To all of them, two different methods of determining the business cycle are applied: the ad-hoc identification of the cyclical component and a dynamic factor model, taking into account the common cyclical variations for Austria, the euro area and the German economy. Subsequently, the Bry-Boschan algorithm serves to identify peaks and troughs of the cycle. Finally, the results are interpreted and compared with those of earlier studies on the euro area and the Austrian business cycle.

**Table of contents**

## **List of figures and tables**

XVI

#### XVII

## **List of abbreviations**

XVIII

## **List of variables**

## **Variable names are composed of two parts**

The first part depicts the geographical region:

ger Germany

eur euro area

The second part refers to the sectoral aggregate:

## **1. Research motivation and overview**

The dating of business cycle turning points is still an important basis for economic policy decisions. Turning points mark the point of time at which booms end and recessions are overcome. Hence, they provide important information for initiating and terminating counter-cyclical policy action. Fiscal policy interventions suffer from recognition, legislation and execution time lags and also monetary actions take time to work their way through the economy. In order to reap the benefits of stabilisation policy1, it is necessary to forecast turning points or at least to identify them timely. For both tasks, investigation into the history of turning points - with an exact dating scheme - is crucial, as it can serve as a benchmark for setting up real time based models.

Regular revisions of the economic data set, changes in methodology for measuring total economic output (like the introduction of the ESA regulation for European countries), the adoption of new statistical methods for price adjustments and further innovations make a regular update of business cycle analysis necessary also for past periods. Furthermore, economic research frequently brings forth new sophisticated methods for business cycle analysis, often supported by enhanced computational possibilities.

Despite the fact that business cycle variations are mostly understood to be a demand-based phenomenon there is merit in studying them also on a sectoral basis. According to the classical definition of the business cycle by *Burns* - *Mitchell* (l 946), business cycles are a type of fluctuation found in aggregate economic activity. This does not necessarily mean that the object of observation has

1 Examples for recent estimations of the costs of business cycle fluctuations are *Reis*  (2005) and *Del/as* (2003). *Barlevy* (2004) gives a good overview about methods and empirical results. Marcus Scheiblecker - 978-3-631-75458-0

to be an indicator of aggregate economic activity like GDP2• A comovement of such fluctuations in other economic time series whether interrelated or not -would also fulfil this criterion.

*Long* - P/osser ( 1983) stressed the comovements of sectoral output fluctuations as being one of the important features of business cycles. *Hornstein* (2000) found, that these sectoral links show up in several time series like gross output. value added and materials and energy use. *Rebelo* (2005) supplied evidence for the strong correlation between hours employed by industry and total hours employed by the private sector. This strong comovement in economic time series like sectoral output3 probably induced *Lucas*  ( 1977) to argue that business cycles were driven by aggregate shocks and not by sector-specific ones.

Despite the widely accepted view of the existence of only one business cycle driven by aggregated shocks, these variations can show up in sectoral time series with different amplitudes and with some lagging or leading characteristics. Indeed, the prominent NBER approach for detecting business cycles is based on sorting the different time series by their leading and lagging properties.

Beside the view that supports the existence of only one business cycle, studies based on band pass filters and spectral analysis methods reveal several cycles, all of which represent frequencies qualifying them for being business cycles. Apart from this, there are also statistical reasons for observing business cycle movements at the detailed sectoral level. It is quite easy to understand that if a

2 In fact, most studies on business cycles concentrate on GDP or industrial production as reference series.

3 The European System of National Accounts uses the term "sectors" only for subdividing the economy by the sectors "government", "private households", "enterprises" and the "external sector". Nevertheless, the English literature uses the term sector for different branches which is followed in the underlying study, too. In the terminology of the European System of National Accounts these are described as "kind of activities". Marcus Scheiblecker - 978-3-631-75458-0

small open economy gets export impulses from different economic areas (e.g. the US and Europe) these can show up in different economic sectors and interfere with domestic, idiosyncratic cycles4 *5 •* Confounding both cycles by aggregating the underlying time series to a higher total (like the GDP) could hamper a proper identification of the business cycle and therefore give rise to misleading conclusions about timing and size of fluctuations, and thereby to suboptimal or even wrong economic policy reactions.

Observing business cycles at the sectoral level has the great advantage that it sheds light on economic transmission mechanisms, whereas for stabilisation policy purposes its benefits are not so clear. This is based on the notion that the business cycle is mainly driven by fluctuations in demand that can be smoothed by interventions targeting certain demand components 6• Opposite to this, supply side measures are understood to aim at the trend component only. Therefore classical stabilisation policy i.e. fiscal and monetary policy intends to act on output via demand aggregates, but not on the long-run growth (trend) path. This view has somewhat changed today in that economic policy tries to generate demand by directly targeting the long term growth path7• Examples for this are investment premia, public spending for education or subsidies for research and development. Looking at sectoral cycles, apart from giving an insight into transmission mecha-

*4*An example can be the international business cycle interfering with a political one of the type mentioned by *Nordhous* ( 1975).

5 The idea of the existence of several independent driving forces has been taken up methodologically by the dynamic common component modelling approach, where several common cyclical factors represent the business cycle.

*6*Prominent exceptions to this ore Real Business Cycle models, brought forward by *Kydlond* - *Prescott* ( 1982). *Blanchard* - *Quoh* ( 1989) criticise this as lumping together supply and demand shocks whereas only the latter refer to what economists usually regard as business cycles. For a good overview about the actual development and the future relevance of the Real Business Cycle theory see *Rebelo*  (2005).

nisms, could guide economic policy by suggesting branch-specific measures, although these could be difficult to implement.

The present study intends not only to date the Austrian business cycle, but also to give information about the leading and lagging properties for several economic branches. Furthermore, their interrelation with corresponding sectors for Germany and to the euro area as a whole will be analysed. This may give insight into the transmission mechanisms from the international business cycle to the Austrian economy. In order to check the robustness of results, several procedures for isolating and analysing the business cycle will be applied that have become popular in recent time. Additionally, the results of dating the Austrian business cycle are compared with earlier studies8• Several contributions have recently been published which focus on the correlation between business cycles in different countries, but there is a lack of recent studies for Austria9 due to its small economic weight. Such analysis has been carried out for GDP as a whole10 or for industrial production only, but not for different Austrian branches 11 •

The focus of this study is on detecting business cycle comovements between countries on a branch level and dating the respective turning points. A timely identification of turning points at

11*ROnstler* (1994) is an example for a sectoral study of the Austrian economy, but focuses more on the long-run impact of foreign shocks. Marcus Scheiblecker - 978-3-631-75458-0

a *Breuss* ( 1984) and *Hahn* - *Walterskirchen* ( 1992) carried out the last thorough investigations of business cycles for Austria. *Brandner* - *Neusser* ( 1992) and *Cheung* - *Westermann* ( 1999) investigated into the impact of the German business cycle on the Austrian industrial production.

*9* Exceptions are the rather new studies of *Vijselaar -Albers* (2001) and *Artis -Krol*zig - *Toro* (2004) where only industrial production is used for determining the Austrian business cycle.

10 According to *Harding -Pagan* (2002), the use of many series in the approach of *Burns* - *Mitchell* (1946) in order to gain a synthetic indicator of the business cycle only indicates, that " ... these were surrogates for a single series, GDP, as that was unavailable to them".

the margin is not the aim of this study, however. This would require the use of business survey data with leading properties and either detrending methods based on non-symmetrical filters or series forecasts based on real-time data in order to circumvent the endpoint problem of symmetrical filters. Furthermore, variables which can explain shocks to business cycle variations (for instance oil price and exchange rate variations) are not considered explicitly, as they are assumed to affect all countries observed in the same direction12 or they feed into the business cycle in the end.

Diagram 1 gives an overview over the typical methodological steps to be taken in business cycle analysis. It presents a stylised flow of several procedures. However, this does not mean that these steps have to be followed consecutively, as some methods cover several of them at the same time. Following this sequence, the underlying study is organised accordingly. The first chapter presents the data and the procedures applied to prepare them for this study. The second deals with the various methods for extracting the business cycle frequencies from the underlying data. Related to this, a literature survey of the various methods with an examination of their theoretical properties and empirical problems is provided. As different methods can produce substantially different results, not one single is chosen for this study but three different ones. In the next chapter several approaches for the identification of the business cycle from the transformed data are presented. Again, different methods are applied in order to check the robustness of results. The fourth chapter addresses the problem of dating and analysing the business cycle for Austria. In the following chapter, the findings on dating, together with some stylised facts, are compared with the results from other studies. The last chapter offers the conclusions.

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12This assumption seems to be justified if the economies considered have similar structures, as it is plausible to assume. Marcus Scheiblecker - 978-3-631-75458-0

Source: Own illustration.

## **2. The data**

In order to study business cycles on a sectoral basis, only national accounts data broken down by sectors are used here. This is not at odds with the definition of Burns - *Mitchel* ( 1946) requiring the use of several time series of different origin for business cycle analysis. According to *Harding* - *Pagan* (2002), this requirement by Burns - *Mitchel* ( 1946) was due to a historical lack of economic data at frequency lower than a year, rather than a methodological condition.

The following quarterly time series are considered:

13 In this study, the word "real" refers to chained values with the year 2000 as the reference year.

14 For time series of older vintage, GVA does not include indirectly measured financial intermediation services (FISIM), which is discussed below.

15 This does not mean that they develop smoothly. But even if their infra-annual variability is higher than that of GDP, it is to be expected that this lies in part outside

velopments than by weather conditions. *Boehm* ( 1998) provides clear empirical evidence for the better suitability of production series adjusted for this primary sector component.

16 NACE (Nomenclature d'Activites des Communautes Europeennes) is the harmonised framework for classifying branches contributing to GDP in EU member states. Marcus Scheiblecker - 978-3-631-75458-0

should develop somewhat more smoothly and volatility rising in NACE K, instead.

All other economic activities like public administration, social services (containing education and health services) and other community, social and personal services (including culture, sports, etc.) are not considered separately as the production of these services is not assumed to be subject to business cycle variations. Nevertheless, such production is included in GVA and GDP figures.

According to the general view, the typical business cycle lasts from around 2 to 8 years 17• This calls for the use of economic time series covering some multiples of that time span in order to isolate cyclical frequencies and to identify a representative number of turning points for analytical purposes. Moreover, these series should be consistent so as to measure output over a long period in a similar way. Unfortunately, there have been several events which complicated the search for long and consistent time series, especially for euro area aggregates.

The process of European integration - moving towards a common economic policy in the Union - required a harmonisation of the national accounts framework. Member States have been obliged to compile their GDP data according to the ESA95 regulation. Several further regulations have been passed, focussing particularly on national accounts aggregates. This was necessary because gross national income constitutes the tax base of Member States' contributions to the Union budget. Furthermore, certain development programmes are related to the income of regions or countries and the Stability and Growth Pact is explicitly linked to the development of real GDP.

17Burns *-Mitchell* (1946) defined it as phases between 6 and 32 quarters, *Hording* - *Pagon* (2002) between 5 and 32 quarters and there are several studies which take a time span between 8 and 32 quarters as a reference. Marcus Scheiblecker - 978-3-631-75458-0

The ESA95 itself - being the European version of the United Nations SNA93 - marks a massive change in the interpretation of what has to be recorded as output. Most prominently, computer software was included into output, whereas it was not covered by the definition of production before.

More generally, national accounting is still far from being a static framework of recording production. Recently, all EU Member States had to change over from a fixed base year for price adjustment in national accounts to one of taking the previous year as the base18• The re-allocation of indirectly measured financial services (FISIM) was a further big change. These services, provided mainly by banks, are implicitly paid for by the difference in interest rates between loans and deposits. Until recently, they were by convention treated entirely as intermediate consumption by enterprises and therefore not included in GDP. Nowadays, FISIM have to be split up into components of private, government and intermediate consumption, exports and imports.

Revisions - introducing all the statistical innovations referred to undermined a backward calculation of national accounts data far enough for business cycle analysis. In order to generate longer time series, data sets had to be chained backward.

For **Germany,** consistent seasonally adjusted national accounts time series on a quarterly, as well as on a yearly basis, only go back until 1991. In order to restore data for the period before reunification, seasonally unadjusted data covering only West-Germany but describing a similar sectoral disaggregation have been used for chaining19• Thereby, it is implicitly assumed, that the

18 The result is a host of time series of two years' length. In order to construct longer time series, growth rates are calculated and chained up using a specific period as reference in order to get long time series in absolute values. Therefore, this method is called "chaining" or "chain-linking".

19 These data have been downloaded form the Federal German Statistical Office website, see www.destatis.de. Marcus Scheiblecker - 978-3-631-75458-0

cyclical variations for West Germany before 1991 exhibit the same turning points as for the new re-united German economy. Whereas this assumption seems warranted for the turning points, it is probably less so for the amplitudes of these variations. Since planned economies, like the former East Germany, are expected to show smaller cyclical variations, the coefficient of variation for the sum of both areas should be lower than the one produced by chaining. This has to be borne in mind for an appropriate interpretation of business cycle variations. Nevertheless, data have been chained backward as has been done by Fagan - *Henry* - *Mestre*  (200 l) for the whole euro area, after a seasonal adjustment procedure.

Due to the German reunification, data for the whole **euro area**  also dated back only until 199 l. Unlike for Germany, no data - neither annual nor quarterly - were available on a sectoral basis for the euro area. Only a seasonally-adjusted series for total GDP, made available by the Euro Area Business Cycle Network (EABCN) 20and reaching from to beginning of 1970 till the fourth quarter of 2003, entered the data set. Using the latest euro area GDP data, this series has been extended until the second quarter of 2005.

For **Austria** three data sets have been used to construct quarterly sectoral time series reaching back to the beginning of 197 6. The first are annual national accounts data starting 197 6, the second quarterly national accounts series reaching back to the first quarter of 1988 which are consistent with the respective annual figures. For chaining back quarterly figures before 1988, national accounts data of an earlier vintage - showing a similar sectoral definition have been used as a third data set. In order to retain business cycle variations of the annual time series, the process of chaining quarters took account of these annual totals. The method applied

20See www.eabcn.org. The method of construction of this series was published by Fagan *-Henry -Mestre* (2001). Marcus Scheiblecker - 978-3-631-75458-0

is based on the Chow-Un approach21 , where the earlier quarters were taken as an indicator for distributing annual totals. With this method, sectoral quarterly time series summing up to annual totals have been produced.

The restriction to only sectoral national accounts data does not provide insight into the sources of the business cycle phenomenon, which lies outside this study. Instead, the focus is on transmission mechanisms between sectors i.e. their lead and lag structure, their cyclical variation and their dependence on euro area and German aggregates. Shocks which can potentially explain business cycle variations (like changes in oil prices or exchange rates) are not considered as they are assumed to hit all observed regions or their effect will be transmitted by the cyclical variation from one region to the other over time22.

21See *Chow-Un* (1971).

22*Artis* - *Zhang* ( 1999) found empirical evidence that there is a connection between exchange rates and the transmission of business cycles across national frontiers. Marcus Scheiblecker - 978-3-631-75458-0

## **3. Methods of extracting business cycle characteristics**

Before one can study the business cycle, the latter has to be extracted from the underlying time series. For that purpose, at least a vague idea of the nature of the business cycle is necessary.

## **3.1 Defining the business cycle**

From studying the literature it becomes apparent, that there is no consensus about the nature and definition of the business cycle. This lack of a precise definition results first of all from the absence of a widely accepted theory of the business cycle phenomenon. The most prominent approaches relate to demand-side imbalances caused by information asymmetries between economic agents or to supply-side shocks due to technical progress (real business cycle theories}. These problems make theory a poor guide for measuring business cycles. Secondly, the empirical approach for discriminating between competing theories is problematic, too. Looking at the behaviour of economic time series, one can see smooth up and down changes (especially if one imagines some kind of simple trend} without any regularities concerning the length or amplitude of a cycle. As a consequence, there exists a plethora of statistical tools and methods for extracting this kind of movements.

### *3. 1. 1 The classical business cycle definition*

Despite a long tradition in business cycle analysis, starting from the middle of the last century by *Burns -Mitchell* (1946), their definition still forms the basis of one strand of business cycle studies:

"Business cycles are a type of fluctuations found in the aggregate economic activity of nations that organise their work mainly in business enterprises: a cycle consists of expansions occurring at about the same time in many economic activities, followed by similarly general recessions, contractions and revivals which merge into the expansion phase of the next cycle; this sequence of changes is recurrent but not periodic; in duration business cycles vary from more than one year to ten or twelve years; they are not divisible into shorter cycles of similar characters with amplitudes approximating their own."

As such a definition is far too imprecise for being a working base for a study; it has been the subject of several refinements and extensions23. This definition, also called the classical definition of the business cycle, has been criticised not only for being imprecise concerning e.g. what constitutes the aggregate economic activity and the pattern of these movements24, but also for being "measurement without theory"25. In order to study the classical business cycle, several economic time series deemed representing the aggregate economic activity are analysed, with trended and stationary time series (e.g. interest rates) both being observed within the same framework. It is not required to separate stationary movements from a trend, only some prior adjustment for working days and seasonal variations is recommended. But, without breaking down changes in economic activity into trend growth and cyclical movements it may be difficult to interpret the cycle, with problematic consequences for economic policy interventions.

Apart from shortcomings in the theoretical definition, this method does not distinguish between different sources of economic growth and has lost some of its popularity26. If this definition of

23Examples for this are *Zarnowitz* ( 1992), and *Zarnowitz -Moore* ( 1982).

2, See *Harding* -Pagan (2003).

2s See *Koopmans* ( 1947).

26Whereas business cycle analysis with level data lost its scientific importance in the last decades, there are some recent studies by *Harding* - *Pagan* (2002), Hess - *Iwata* ( 1997) and *Clements* - *Krolzig* (2004) giving evidence for a kind of revival of such approaches. Marcus Scheiblecker - 978-3-631-75458-0

business cycle is applied where the focus lies on developments in the level of time series, economies showing a strong trend growth will experience only few recessions.

## *3. 1.2 The deviation cycle definition*

As business cycles seem to be recurrent but not periodic, as *Burns*  - *Mitchell* ( 1946) pointed out, *Mintz* ( 1969) suggested separating the trend from time series in order to study them properly. This was quite in line with *Lucas'* ( 1977) suggestion of defining the business cycle as the deviations of aggregate real output from trend, which puts the definition on a more theoretical basis. The *Solow*  ( 1970) growth model, describing production as following a secular trend - mostly driven by technological progress - with demanddriven imbalances causing temporary departures (the business cycle) from it, serves here as the theoretical framework. This is probably the definition most people have in mind, when they talk about the business cycle.

If the business cycle is defined as departure from a trend it is called the "deviation cycle" or "growth cycle" approach. As the term "growth" usually refers to growth rates or first order differences in logs - which represents a special method for de-trending time series and therefore a kind of deviation cycle approach- the term "deviation cycle" will always be used hencef orth27•

The choice of the appropriate definition of business cycles - classical or deviation cycle - may depend on the specific research topic. If the purpose is to give guidance for business cycle intervention (as deviations from the steady state growth path imply welfare losses) then the deviation cycle is more appropriate. This is based on the fact, that the business cycle defined by the devia-

27 Sometimes "deviation cycles" and "growth cycles" are used as synonyms; in this case, calculating first differences for extracting the trend is labelled the "growth rate cycle" approach in order to distinguish between them. Marcus Scheiblecker - 978-3-631-75458-0

tion concept shows a lead in time vis-a-vis the one interpreted in the classical way. Furthermore, it has to be kept in mind that applying the classical approach yields fewer and shorter contraction phases than the deviation method as documented in *Harding* - *Pagan* (2002). This may lead to a substantially different business cycle stabilisation policy. Especially the lead property makes deviation cycle approaches more appropriate for economic policy issues. For all those reasons, the following analysis concentrates exclusively on the deviation cycle approach.

## **3.2 Isolation of business cycle frequencies**

Following *Zarnowitz* ( 1992), economic time series measuring output may be represented in the following form2a

$$\{I\} \qquad \qquad Y\_t = X\_t + C\_t + S\_t + \mathbf{e}\_t$$

where *x,* represents the trend component, c, the business cycle variation and s, other components like seasonal, working day or weather effects. *e,* represents an error term capturing all residuals like special events and measurement errors. In earlier approaches, all components have been modelled as being deterministic, whereas modern ones understand all or some of them as stochastic processes29.

28Frequently this model is specified in logarithms where the underlying model is multiplicative *r,* = *X,* • *c,* • s, • *e,* .

*29*Examples for this kind of models are structural time series models following the approach of *Harvey* ( 1989) where the trend is modelled as a stochastic process. Also the method of using just growth rates (out of first order differencing process) which implies random walk behaviour of the trend component can be regarded as a stochastic approach. Marcus Scheiblecker - 978-3-631-75458-0

Even the assumption of orthogonality of trend and cycle has been relaxed by some studies30, which is in line with *Zarnowitz'* ( 1992) consideration that trend and cycle are possibly influencing each other.

As theory specifies neither the trend nor the business cycle precisely, there is a large dissent in business cycle research, how the business cycle can be extracted from the underlying time series. As a consequence, several methods for isolating the cyclical component have been proposed.

Based on the component structure given in *Figure 1,* business cycle extraction methods can be classified mainly by three none mutually exclusive characteristics

Direct approaches *try* to extract directly only cyclical variations (according to a specified definition) out of one or more time series, without prior adjustment of other components like trend, seasonality or noise. This class comprises univariate filter techniques like band-pass filters as well as models assuming a special structure for all components31 • The latter can be constructed in a multi- or univariate manner.

Indirect approaches by contrast, first try to single out other information not belonging to the business cycle in order to obtain the business cycle as the residual. As the data used here are already cleared for seasonal and working day effects, this is mainly a task of separating a trend from the series, eventually followed by some

30*Dellos* (2003) estimated a business cycle model where trend and cycle interact in a non-trivial way.

smoothing procedure to get rid of high frequency noise variation. The subtraction of deterministic time trends, first order differencing, the Hodrick-Prescott filter, the Christiano-Fitzgerald filter and some ad-hoc moving averages belong to this class. In most cases, several steps have to be combined in order to single out the business cycle.

Model-based approaches estimate all or some components by assuming some specific structure for them. Further model-based approaches concentrate on time series models. As they do not assume a specific structure, they can also be regarded as filter techniques. This goes for instance for the Beveridge-Nelson decomposition where the economic time series is represented as a time series model which is factorised after identification in order to extract the business cycle.

## *3.2.1 Outliers*

A proper identification of the business cycle requires a consistent data base, adjusted for disturbances caused by outliers. This goes for all approaches, whether they are model- or filter-based, direct or indirect methods. According to the decomposition possibilities outlined above, these outliers are contained in most cases in the error term *et* of ( 1 ) , together with other high frequency noise.

For an initial cleaning of the underlying time series, three different types of outliers had been considered:

Additive outliers appear at one point in time and vanish thereafter without having any lasting effect on the further development of the series. In a graphical representation of a time series they are marking a one-time spike in the plot32•

Level shifts are innovations which mark a break in the time series, shifting the successive mean by a constant value. They appear as an upward or downward step in the time series.

Transitory components are outliers lasting only for a limited time and vanishing thereafter. They are either fading out in successive time periods (decay) or they evolve slowly over time and vanish suddenly (ramp).

Some of these outliers can be explained by variables outside this model. Ignored working day variables or weather conditions - influencing the economic output - are examples. Furthermore, every unusual strong movement in the history of a time series may appear as an outlier. In this special case here, this goes for sharp recessions, too. It is a difficult task to detect outliers and one has to define first of all what should be considered an outlier and by which value it has to be replaced.

For the present study, outlier detection has been carried out by modelling all time series as seasonal ARIMA models with regression effects (sometimes also called Reg-SARIMA models for short), which took care of the varying number of working days of different countries by regressing on them. Several steps of estimation are carried out in order to check whether an elimination of the significant outliers (in this study dummy variables with a t-value higher then 3.8) improves the following model33• A purely me-

32 It has to be borne in mind that in a first-order differenced time series an additive outlier marks a spike with a counter-reaction in the next period. If no counterreaction follows, it appears as a level shift in the original series.

33 This procedure was assisted by the software package TRAMO developed by *Gomez* - *Marava/1* ( 19921. It is widely used also in business cycle studies like for example in *Artis -Krolzig -Toro* (2004), *Artis -Marcellino -Proietti* (2004) and *Altissimo eta/.* (2001}. Marcus Scheiblecker - 978-3-631-75458-0

chanical detection procedure according to some statistical criterion can substantially influence the result of a study. In order to avoid these pitfalls, outliers have been discussed with experts and considered in the light of economic history.

## 3.2.2 *Calendar effects*

Apart from the business cycle, the varying number of working days influences economic output, too. Therefore, time series have to be cleaned for this influence. As far as they have not already been derived from the data base in adjusted form, this adjustment was made by the same SARIMA model combined with a regression on working day information as it was used for seasonal adjustment. As not only the varying number of working days constitutes the calendar effect, but also Easter and the leap year effectJ-', their influence had been considered, too.

Apart from testing for a significant Easter and leap year effect, two specifications for modelling the working day effect have been applied. One tests for the significance of the varying number of working days in total and the other checks for all working days separately. Thus, the regression includes the number of Mondays, Tuesdays, and so on. The discrimination between the two specifications was done according to their t-values and auto-correlation properties of the residuals from the SARIMA model estimation.

In most applications, as in this study, calendar effects are cleaned from the series under the assumption that they are independent from other components like the trend or the cycle. This orthogonality assumption can be challenged by considerations of possibly larger working day effects in times of high capacity utilisation. During recessions a large number of employees have to work part time and their productivity is probably quite low, so that an addi-

3' Other holidays like Christmas have not to be considered separately, as they occur always in the same quarter of the year. Marcus Scheiblecker - 978-3-631-75458-0

tional working day due to specific calendar constellations will not yield the same value added as in a boom. As a consequence, the business cycle reaction of calendar effects shows up wrongly in the business cycle itself, if calendar effects are estimated as being orthogonal. But this should bias only the amplitude of the business cycle, but not interfere with the dating of the cycle.

## 3.2.3 *Seasonal variations*

These high frequency movements can interfere with the dating of the overall peaks and troughs. *Sims* (1993) and Hansen - *Sargent*  ( 1993) have shown that seasonal pre-filtering is essential in business cycle analysis in order to avoid a distortion of the business cycle pattern.

Therefore, the largest part of studies is based on the analysis of seasonally adjusted time series, but there are also good arguments for considering unadjusted series as well. *Cubadda* ( 1999) found that in the case of existence of seasonal co-integration between analysed time series, the results for unadjusted and adjusted time series can be different. Despite the theoretically difficult economic implication of having a steady state trend showing seasonal variations, he argues in favour of using unadjusted time series and clearing them for seasonal variations within the same framework used for business cycle analysis. In a similar direction points the study by *Joger* - *Kunst* ( 1990) who showed that the process of seasonal adjustment spuriously amplifies the persistence of variables.

There are several approaches of clearing for seasonal variations which estimate the trend and the cycle within one framework, as proposed by *Cubadda* ( 1999)35• In unobserved components

*35*It has to be noted, that the application of band-pass filters for frequencies of the business cycle not only cancels out the trend component but also seasonal variations. It is unclear whether this is conforming to a separation process within one

models, this is done either by assuming a fixed trigonometric movement of seasonality36 or by regressing on dummy variables. In both cases, seasonality is modelled as being stable over the whole time span of the series, which seems to be a rather strong assumption. So the advantage of isolating the seasonal variation within the same framework used for identification of the business cycle has to be weighed against the advantage of allowing for a flexible trend for seasonal variations.

The most popular methods for univariate seasonal extraction are X-12 from the US-Census Bureau and TRAMO-SEATS developed by the Bank of Spain37• Both procedures allow - in contrast to unobserved component models with a basic structure - a variation of the seasonal component over time. This captures the possibility of moving seasonal patterns due to technical progress or changing economic circumstances38 which could be very important for cleaning longer time series like the ones used for business cycle analysis.

Weighing the pros and cons of the different approaches, it was decided to separate the seasonal component externally. This was done using the TRAMO-SEATS software. *Fiorentini* - *Planas* (2003) have shown that business cycle dating is not sensitive to the use of either X-12 or TR AMO-SEATS. This can be seen in *Figure* 2.

Nevertheless, both approaches, the unobserved component model and the univariate time series model, assume orthogonality between seasonal variations and the business cycle, which should be borne in mind when interpreting the respective results.

36*Artis* - *Marcellino* - *Proietti* (2004) or *Hahn* - *Walterskirchen* ( 1992) are examples for this approach.

37 See Gomez - *Marava/1* ( 1992).

38Reasons for that could be new processing techniques in the construction sector stabilising output during winter periods or Marcus Scheiblecker - 978-3-631-75458-0 changing seasonal patterns in tourism.

#### *3.2.4 The trend*

The deviation cycle approach requires a separation of the business cycle fluctuations from the trend component of a series. There is no definition of the properties that such a trend should possibly have. There is consensus only about the residual part of time series after de-trending. Cyclical variations should be second order stationary. show some autocorrelation characteristics and are expected to have a cycle frequency of some years. According to several authorsJ9, this type of cycle is best mirrored in the capacity utilisation of enterprises, which varies auto-regressively without any trend. Therefore. the analysis of business cycles is in most cases based on figures showing these patterns of changing economic activity.

As there is no consensus concerning the shape of the trend, there exists a plethora of approaches to de-trending. Unfortunately, according to Canova ( 1998). these different methods of de-trending or direct business cycle extraction can give substantially different results for dating the business cycle. Considering this, various methods of de-trending will be used in the present study. The selection of these techniques will be based on their popularity as well as their appropriateness for analysing the Austrian business cycle.

In order to judge on the latter. a theoretical evaluation together with reported empirical features of the different approaches will be presented.

#### **<IP_ADDRESS> Deterministic trend**

Due to its simplicity, this approach was historically very popular in business cycle analysis. A linear (or log-linear) trend - in most cases obtained by fitting a regression line - was deducted from the time

39 See e.g. *Tichy* (1994) *orBreuss* (1984). Marcus Scheiblecker - 978-3-631-75458-0 Downloaded from PubFactory at 01/11/2019 04:38:46AM via free access

series. The residual is taken to represent the business cycle (and the remaining components). Instead of a linear trend, other polynomial functions of time are possible. With the study of *Nelson* - *Plosser* ( 1982), this approach has lost most of its popularity. For the U.S., they found that a large part of economic time series are allegedly difference stationary, where the trend cannot be removed by linear de-trending.

For this reason, this method is not state-of-the-art and should not be considered as an option further on in the present study.

#### **<IP_ADDRESS> Phase-average trend**

This is a heuristic method of determining the trend of non-stationary time series, usually applied in the classical business cycle framework for estimating the amplitudes of business cycle variations of trended time series. Basically, it consists of the following steps40: After identifying the turning points in the trended time series, it is split up into segments between two consecutive turning points (called "phases"). For each of these segments the mean is calculated (this is the "phase-average"). In order to obtain the trend for the whole series, these averages are combined with a smoothing three-term moving average. As this method of trend extraction requires an ex-ante determination of the turning points, it is not suitable for this study41 •

#### **<IP_ADDRESS> First-order differencing**

Here, de-trending is made by deducting the value at time *t* - 1 from *t* in order to get stationary differences {if they are first-order difference stationary which is in most cases assumed or tested). If

"° A detailed description of the full procedure can be found in *Boschan* - *Ebanks*  (1978).

41*Breuss* ( 1984) has used this approach for isolating the Austrian business cycle. Marcus Scheiblecker - 978-3-631-75458-0

this method is applied to a series for which the logarithm has been taken, the outcome can be interpreted as growth rates.

Therefore the basic assumptions behind the method of applying first-order differences are that the trend component of the series corresponds to a random walk process without drift, that the cyclical component is stationary and that the two components are uncorrelated42• The original time series *y,* is assumed to have a unit root, which is entirely due to the trend component of the series43• This assumption of a random walk trend gives anything but a smooth profile of the trend component, which is for example the case in the Solow growth model. Every shock to this trend will never decay in impact in the future. Nevertheless, this assumption is still popular in empirical economics since the pioneering work of *Nelson* - *Plosser* ( 1982) who found unit roots in 13 out of 14 longterm annual US macro series, including real GDP44•

In the case of a time series model based on seasonally adjusted data, this behaviour can be represented as

*(2) y 1 =x1 +c1 +&1*

*y,* is our trended aggregate, *c,* the cycle to be extracted and *e,*  some white noise process where *e,* ~ N(O,cre). The assumption of a random walk trend implies that *x,* = *y,* \_ 1 which, entered into *(2),*  gives

42 See *Canova* ( 1998).

43 There are also multivariate forms of this approach. Cheung - *Westermann* ( 1999) estimated the trend of industrial net production as the co-integration relation between Germany and Austria. Business cycle effects are captured by the residual short- run relationship not included in the error correction term.

44Since then, several authors (e.g. *Rudebusch,* 1993 and *Diebold -Senhadji,* 1996) have challenged these findings by showing that the testing procedure has not enough power against economically Marcus Scheiblecker - 978-3-631-75458-0 relevant trend-stationary alternatives.

$$\langle \mathbf{3} \rangle \qquad \mathbf{y}\_t = \mathbf{y}\_{t-1} + \mathbf{c}\_t + \mathbf{s}\_t$$

Simple arithmetics isolates the cycle as

$$\{\mathbf{4}\}\qquad\qquad\mathbf{c}\_{t}=\mathcal{y}\_{t}-\mathcal{y}\_{t-1}+\mathbf{c}\_{t}$$

It can be seen, that the cyclical component includes the error term, which makes it very erratic and does not show up in the smoothness one would expect from a pure cyclical variation. Furthermore, forming first-order differences does not correspond to a symmetric filter as it generates only differences to past values. This leads to a phase shift of the cyclical component and therefore yields a dating output different from the case of using symmetric filters.

In order to look at the properties of this first-order difference method at the frequency domain, *Figure* 2 depicts the gain function when filtering data that are already seasonally adjusted. The grey shaded area marks the frequency band of the typical business cycle between ,r/3 (which corresponds to cycles of 6 quarters length) and 1r/16 (corresponding to a length of 32 quarters). The bold red line reveals that the first-order difference filter indeed cancels out frequencies that are located close to zero and therefore can be regarded as variations of the trend component. But it also wipes out mistakenly some spectral mass of the business cycle frequencies in the shaded area. At ,r/2, which corresponds to cycles of 4 quarters) there is no spectral mass at all, as the data have been seasonally adjusted beforehand.

A remarkable property of this filter is that it superimposes all higher frequencies, raising the gain for very high frequencies above 1 . This explains the very erratic output of the filter as one can see from looking at growth rate series.

Applying first-order differences to processes which are either trend stationary or show a higher degree of integration biases the outcome for business cycle interpretation. In the first case, this would Marcus Scheiblecker - 978-3-631-75458-0 correspond to over-differencing which biases the cyclicality to the high frequency spectrum, whereas in the second case underdifferencing shifts it to the low frequency area45•

Due to these deficiencies, the first-order difference filter has lost most of his popularity in empirical studies on business cycles. If it is used at all, it is mostly to check for robustness of the output across several methods or to illustrate its deficiencies. Despite these undesired properties which limit the use of this filter for business cycle analysis, it will be considered in this study due to the popularity of growth-rate-based interpretations as a rule-of-thumb method46•

#### **<IP_ADDRESS> Hodrick-Prescott filter**

The use of the business cycle filter proposed by *Hodrick* - *Prescott (* 1980) is very popular among business cycle researchers. It is a flexible tool that is capable of removing non-stationary components that are integrated of order four or less47• Effectively, the trend implicitly fitted by the HP filter amounts to a process of curve fitting. It results from constructing a trend as smooth as possible, with penalizing all squared deviations from this trend from the original time series. The precise formula is

$$\langle \mathcal{G} \rangle = \min\_{\{\mathbf{g}\_t\}\_{t=1}^T \in \mathcal{I}^{\omega+1}} \sum\_{\iota=1}^T [(\mathbf{y}\_t - \mathbf{g}\_t)^2 + \lambda \left[ (\mathbf{g}\_{t+1} - \mathbf{g}\_t) - (\mathbf{g}\_t - \mathbf{g}\_{t-1}) \right]^2] \quad \lambda \ge 0$$

where *y,* is the original trended time series and *g,* is the trend to be estimated and subtracted. J is acting as the signal-to-noise ration, being the weight for penalizing all deviations from trend, which has to be fixed by the user. If J = 0, there is no difference between the trend and the original series, if J approaches infinity

*' 7*See *Baxter -King* (1995). Marcus Scheiblecker - 978-3-631-75458-0 Downloaded from PubFactory at 01/11/2019 04:38:46AM via free access

*'5*See e.g. *Ritschl -Uebele* (2006).

*'6*Like it is done by the CEPR Business Cycle Dating Committee.

the trend component becomes linear. Typically a value of 1600 is chosen for applications using quarterly data48• The HP filter extracts a trend which can be stochastic but moves smoothly over time and is uncorrelated with the cyclical component49• A further property is its symmetry, so that no phase shift is introduced. In its ideal representation (its infinite sample version), it approximately places zero weight at the zero frequency (the trend) and close to unit weight at high frequency. This last property leads to a nontrended output but which carries high-frequency information like the noise component *e,* . *Figure* 2 depicts the gain function of this filter. It wipes out rather clear-cut all frequencies lower than n/16 (i.e. cycles longer than 32 quarters) and leaves all other frequencies unchanged (again, the loss of spectral mass at *n/2* is due to the preceding seasonal adjustment process).

In order to overcome the disadvantage of residual high frequencies, sometimes a kind of low-term moving average is applied after the HP filter for smoothing the output. *Artis* - *Marcellino* - *Proietti* (2004) used a band-pass version of the HP filter. They combined two HP filters - which both are virtual approximations of high-pass filters - in order to get a smoother output, corresponding to the business cycle frequency band.

The trend resulting from the minimisation process of (5) can also be represented as a linear symmetric filter. Like all symmetric filters, the HP filter is - in its ideal presentation - subjected to the endpoint problem, which means that there is a loss of several observations at the beginning and the end of the series. There exists a trade-off relation concerning symmetric filters, i.e. the more observations one is willing to lose, the more exactly the filtering procedure works. In the case of the practically applied HP filter, there is no such obvious end point problem as there is a kind of built-in me-

*48*It can be shown that J = 1600 corresponds to a cut-off of frequencies lower than 32 quarters. See e.g. *Prescott* ( 1986) or *Baxter -King* ( 1995).

chanical forecast feature which allows de-trending even at the endpoints50. The HP filter calculates the trend component and identifies the cyclical component as the difference between the original series and the trend component51 • The end-point problem is therefore concentrated on changes in the trend component. This makes the HP filter very attractive for practical purposes, but the end-point problem is solved at the cost of the accuracy of the filter. Especially at the end points, the filter works imprecisely in that it shows a stronger leakage. This means that some frequencies belonging to the trend can pass, whereas fluctuations of cyclical nature (especially those of low frequency order) are filtered out.

*Cogley* - Nason ( 1995) and *Canova* ( 1998) pointed out that despite the HP filter's ability to de-trend difference stationary time series, it distorts the frequency spectrum so that business cycle extraction could be problematic.

### **<IP_ADDRESS> The Baxter-King filter**

Band-pass filters are understood as frequency filters, which give in their ideal representation - zero weight to the frequency band to be filtered out and unit weight to the rest52. *Baxter -King* (1995) proposed such a filter for business cycle filtering, which is an optimal linear approximation to an ideal band-pass filter. They constructed a band-pass filter by starting from a low-pass filter, which allows all frequencies *w* (below or equal a certain threshold .Q?) to pass. This requires that all frequencies get a unit weight *p* if they are above this threshold and zero otherwise:

50See *Baxter -King* ( 1995).

51See *Kranendonk* - *Bonenl<omp* - *Verbruggen* (2004).

52In a wider sense, high-pass filters, which filter out all frequencies below a certain frequency threshold (e.g. the trend) as well as low-poss filters, which let pass only frequencies below a certain threshold, can be regarded as band-pass filters. But what is meant here are only the filters which capture the band between two

$$\langle \mathsf{A} \rangle \qquad \mathcal{J}(\mathsf{a}) = \begin{cases} \mathsf{l} & \text{if } |\mathsf{a}| \le \underline{\underline{\underline{\alpha}}} \\ \mathsf{0} & \text{if } |\mathsf{a}| > \underline{\underline{\underline{\alpha}}} \end{cases}$$

With the help of these frequency weights p(m) it is possible to derive the time series filter weights *b{h)* by applying the inverse Fourier transformation to the frequency response function

$$\left(\nabla\right)\_{\mathbf{k}} \quad \quad b\_{\mathbf{k}} = \int\_{-\pi}^{\pi} \beta\left(\alpha\right) e^{i\omega h} \, d\alpha$$

The time series filter weights *b(h)* can be used for constructing the ideal low-pass filter in the time domain

$$\begin{array}{cc} \langle \mathsf{B} \rangle & b \ (L) = \sum\_{-\infty}^{\infty} b\_k \ L^k \end{array}$$

This ideal low-pass filter is symmetrical (as it goes from -oc to +oc} and is constructed as an infinite-order moving average. The weights in this moving average are *b0 =[!!/tr,* and *bh* = *sin{hgj/h,r* for *h*= 1, 2, ... 53• In practice, an approximation of this ideal filter will be enough, such that a shorter filter can be applied which solves the end point problem at the cost of a leakage.

From this low-pass filter, a band-pass filter can be easily derived from two consecutive low-pass filters, one working at the lower boundary *f!l* and the other atm. The approximation to this ideal band-pass filter with the weighting scheme derived this way is called the Baxter-King filter or BK filter for short. Its weights are given by

$$\langle \Theta \rangle \qquad \widetilde{\boldsymbol{b}}(L) = \frac{\sin L \overline{\boldsymbol{\alpha}} - \sin L \underline{\boldsymbol{\alpha}}}{L \pi} - \frac{\mathbf{l}}{2K + \mathbf{l}} \sum\_{L=-K}^{L=K} \frac{\sin L \overline{\boldsymbol{\alpha}} - \sin L \underline{\boldsymbol{\alpha}}}{L \pi} \mathbf{l}$$

53For details the reader is referred to *Baxter* Marcus Scheiblecker - 978-3-631-75458-0 *-King* ( 1995). Downloaded from PubFactory at 01/11/2019 04:38:46AM via free access

with again *f!l* and *m* being the lower and upper boundaries and *K*  representing the length of the filter.

*Figure 2: Gain function of stationary transformations*

Source: *Fiorentini* - *Planas* (2003).

The upper and lower boundaries (usually cyclical components between 8 and 32 quarters)54 are captured, by filtering out all frequencies above or below. Convenient properties of this BK filter are that the identification assumption requires no restriction concerning the trend to be either deterministic or stochastic and it al-

54 In the frequency domain, this corresponds to a band between ,r;/16 (= 32 quarters) and ,r;/4 (= 8 quarters). Marcus Scheiblecker - 978-3-631-75458-0

lows for changes in the trend behaviour over time as long as the changes are not too frequentss.

An advantage over the HP filter is that the BK filter cancels out higher frequencies above the cyclical variations, too, whereas the former - acting as a pure high-pass filter - still gives an erratic but de-trended output. This can be seen again in *Figure 2.* The BK filter cancels out lower frequencies dedicated to the trend frequencies below 1t/3. This allows the filter to be applied theoretically even to non-seasonally-adjusted series.

A further advantage is that the loss of frequency information due to data aggregation seems to be less of a problem for band-pass filters than for the HP filter. *Aadland* (2005) has shown that aggregated high-pass filtered data (e.g. by the HP filter) can lead to spurious cycles at the business cycle frequencies, if the disaggregated data carried strong variations in the high frequency area. This is due to the so-called "aliasing"-problem which arises when high frequency data are observed at lower frequencies.

As the BK filter is symmetric like the HP filter, it causes no phase shift. A convenient feature of the BK filter is its transparency. The user can explicitly fix the upper and lower level of the band to be filtered; thereby defining what should be understood as the business cycle. Furthermore, the degree of approximation to the ideal band-pass version can be chosen. This can be done by sacrificing observations towards either end point in order to make the filter work more exactly, i.e. to reduce its leakage. The problem of leakage arises with the approximation of the filter. As it is not possible in practice to work with infinitely long time series, shorter filters have to be applied. This has two consequences: First, frequencies can pass which should be filtered out and some are mistakenly filtered out which should pass; and second, frequencies are superimposed at the borders of the frequency band, which appears as

*55*See *Canova* (1998). Marcus Scheiblecker - 978-3-631-75458-0 Downloaded from PubFactory at 01/11/2019 04:38:46AM via free access

side-lobes. As *Woitek* (2001) pointed out, this could possibly lead to spurious results in business cycle analysis. But the leakage problem as well as the one of amplified side-lobes can both be reduced by using longer-term filters and sacrificing more observations at either end of the time series.

Applying the BK filter, one has the possibility to individually select the trade-off between the accuracy of the filter and the end-point problem. Compared with this, the HP filter properties depicted in the gain function of *Figure 2* are calculated by accepting a loss of two observations at either end of the time series. It can be seen, that the precision of the filtering process at rr/ 16 is not as exact as in the BK filter case. Furthermore, the part in the middle of the business cycle frequency band is superimposed (values above l) and there is some leakage at higher frequencies.

*Figure 3* shows how the precision of the filtering process changes if one is willing to sacrifice a higher number of observations at the start and the end of the series. *Figure 3a* shows the gain function obtained by a BK filter with a window length of 6 quarters, causing a loss of 3 quarters at either end. Compared with this, the 12 quarter-window filter (resulting in a loss of 6 quarters at both ends like it is used in our calculations) cuts out more precisely the desired frequencies, without strongly superimposing business cycle frequencies and with a lower leakage. But this precision is obtained at the cost of losing the possibility of analysing the business cycle at the margin. As this is the period for which the dating is most important in order to take timely economic policy measures, this can be regarded as a drawback.

To overcome this problem, some authors propose an extension of the time series by applying forecasting techniques. *Fiorentini* - *Planas* (2003) propose an ARIMA forecast for that purpose. But all forecasts based on univariate time series methods (without including external information like business survey data) may reduce the

end-point problem at the cost of risking forecasting errors of possibly the same size56\_

#### **<IP_ADDRESS> The Christiano-Fitzgerald filter**

A somewhat different band-pass filter has been proposed by *Christiano* - *Fitzgerald* (2003). It also represents an optimal approximation of an ideal band-pass filter, by imposing a somewhat different criterion of optimality. This criterion minimises the sum of the squared approximation errors which are weighted by their spectral density *fx ( m}* of the data being filtered57.

$$\langle f|O\rangle \Big|\_{\hat{B}\_f^{\mathfrak{P},f}, f^{\sum\_{j=-f}^{\infty}\int\_{-\infty}^{\infty}}\Big|\_{-\pi}^{\pi}\Big|B(e^{-i\omega}) - \hat{B}^{\mathfrak{P},f}\big(e^{-i\omega}\Big)\Big|^2 f\_x(\omega) \,d\omega$$

Furthermore, the filter length is allowed to vary over the time series and is not restricted to being symmetrical. Moving towards the start and the end of the time series the filter becomes more and more asymmetric, which allows circumventing the end-point problem.

*Christiano* - *Fitzgerald* (2003) have shown that the length of the BK filter window not only determines the degree of approximation to the ideal band-pass filter, but influences especially its capability of filtering out long-term movements. The filter proposed by them can also be adjusted to filter a specific frequency band, but - in contrast to the BK filter - its length is a result of the optimisation process and cannot be altered by the applier. Therefore, it is not possible to define explicitly the business cycle with regard to a certain frequency band.

56In order to extend the HP filter, *Kaiser* - *Morovo/1* ( 1999) proposed on IMA (2.2) time series model for forecasting.

Empirical studies of economic time series for the Netherlands carried out by *Kranendonk* - *Bonenkamp* - *Verbruggen* (2004) have shown a better performance with regard to revisions of the endpoint output than with the BK filter. Nevertheless, the use of asymmetrical type filters suffers from phase shifts or they contain an implicit forecast (based on past observations), respectively. Whether a thorough forecast combined with the BK filter yields a better performance depends on the specific case.

In the literature, modifications of this approach can be found, like in *Altissimo et al.* (2001) who applied a multivariate Christiano-Fitzgerald filter in order to study business cycles for the euro area. *Goldrian* - *Lehne* ( 1999) proposed a further band-pass filter that is not based on a minimisation process, but on the common pattern of weight matrices, instead.

#### **<IP_ADDRESS> The Beveridge-Nelson decomposition**

Instead of filtering time series for special frequency bands, *Beveridge* - *Nelson* ( 1981) proposed a time series model-based approach. The idea behind is that after fitting a time series model to the underlying data, its structure can be explored in order to separate the trend from the cycle. The fitting of an **ARIMA** model58 implicitly models the trend as a stochastic process59• The stationary **ARMA** part is assumed to be (or at least contains) the business cycle.

In order to show some interesting properties of this approach, a simple example is set up. Assuming the underlying time series can be modelled by an ARIMA (0, 1, 1) process, the time series becomes stationary after forming first order differences. This can be

58In the case of not-seasonally adjusted data, a S-ARIMA model has to be applied.

*59* In most cases it will be of order one, which implicitly yields a random walk process. Marcus Scheiblecker - 978-3-631-75458-0

represented by a moving average process of order 1 and shows at the same time the cycle:

$$\{\{\}\}\qquad\Delta\ \nu\_t = e\_t + \beta \, e\_{t-1}$$

where *fly,* is the (log) differenced time series, *e,* a white noise term and *p* represents the moving average parameter with */J* < 1 .

If this expression is solved recursively and the start values are assumed *Yo* = *e0* = o , the following expression emerges

$$\{\{\{\mathcal{Q}\}\}\qquad \mathcal{Y}\_t = \sum\_{i=1}^t e\_i + \mathcal{B} \sum\_{i=1}^{t-1} e\_i$$

or

$$\{I\{\mathbf{3}\}\}\qquad\wp\_{\iota}=\{\mathfrak{l}+\beta\}\sum\_{i=1}^{\iota}\mathbf{e}\_{i}-\beta\,\mathbf{e}\_{i}$$

In *(13)* the first term *y,* = (1 + */J* )I *e;* is the trend part which - being a random walk process - is the sum of its past shocks, and *p e,* represents our stationary cyclical part. Equation *(13)* implies one interesting feature of this kind of splitting the trend from the cycle: The secular as well as the cyclical component are both driven by the same shock at the particular point in time. This means, trend and cycle are perfectly correlated60• In order to make this somewhat clearer, we can transform *(13)* to

$$\{\{\{\mathbf{l}\}\}\}\qquad\boldsymbol{\nu}\_{t}=\left[\left(\mathbb{I}+\mathcal{J}\right)\sum\_{i=2}^{t}\boldsymbol{e}\_{i-1}+\left(\mathbb{I}+\mathcal{J}\right)\boldsymbol{e}\_{t}\right]+\boldsymbol{\beta}\,\boldsymbol{e}\_{t}$$

showing that the trend as well as the cyclical component at time *t*  both depend on shock *e.* This implication is in stark contrast to the

60See *Canova* (1998). p. 481. Marcus Scheiblecker - 978-3-631-75458-0 Downloaded from PubFactory at 01/11/2019 04:38:46AM via free access

usual assumption of orthogonality of trend and cycle. But it does not necessarily represent a deficiency, as it seems plausible sometimes to see both components influenced by one shock. Admittedly, this means - as criticised by *Blanchard* - *Quah* (1989} lumping together supply side shocks, which affect the secular component, and demand-related shocks influencing the cycle.

As the Beveridge-Nelson approach is based on ARIMA time series models, all problems linked to that sort of modelling are carried over to this trend-cycle separation approach. Foremost, applying this kind of modelling the problem is usually not to find a suitable model but to discriminate between the large amounts of feasible ones61 • The identification of the cycle therefore remains a somewhat arbitrary task and may be challenged in academic discussions. Furthermore, ARIMA models became prominent on account of their good short-run forecasting properties, whereas they perform quite poorly for longer-term projections. As a consequence, one can extract quite different trends and cyclical components from a plethora of suitable models.

#### **<IP_ADDRESS> Unobserved components models**

This type of approach intends to model all components explicitly. A special time series structure is modelled for the trend, the cycle, some regression effects like working days, (possibly the season) and even innovations like structural breaks or suspected outliers62• This method is thus very flexible and a wide variety of specifications is possible. Usually the trend is modelled as some kind of random walk process, possibly allowing for a drift. A prominent representative of this approach is *Harvey* (1989}, who proposed a so-

61*Christiano* - *Eichenbaum* ( 1992) mentioned this problem for practical business cycle analysis.

62For instance, *CaNa/ho* - *HaNey* (2004) estimated a model for the euro area with a trend evolving stochastically in its slope and level with adding a serially correlated stationary component representing the cycle. Marcus Scheiblecker - 978-3-631-75458-0

called local linear trend where the level as well as the slope is represented by stochastic processes. Depending on restrictions of the variance of these shocks, deterministic as well as random walks (with or without drift) and smoothly evolving trends can result63. This flexibility makes it difficult to judge the appropriateness of unobserved components models for business cycle analysis. In some more simple specifications they come close to the Beveridge-Nelson decomposition method.

*Morley* - *Nelson* - *Zivot* (2002) argue that in the unobserved components framework the common (but not necessary) restriction that trend and cycle are uncorrelated leads to great differences in the cyclical output. Whereas the Beveridge-Nelson decomposition implies that a stochastic trend accounts for most of the variation in output, the cyclical variation is dominant in unobserved components models for their part. Lifting the restriction of trend and cycle being uncorrelated, the two approaches can yield identical decompositions. As *Blanchard* - *Quah* ( 1989) assigned supply shocks to the trend and demand shocks to the cycle, the assumption of uncorrelatedness in isolating the cycle can have decisively different consequences for economic policy.

A possible advantage of modelling explicitly all components - as it is done in the unobserved components approach - is that this process can bear closer relation to economic theory than mechanical filtering procedures do. Whereas the purely statistical non-parametric procedures of detrending can be accused of practicing "measurement without theory", Canova ( 1998) accuses all economic-theory-based decompositions as, "at best, attempts to approximate unknown features of a series and therefore subject to specification errors". So, the explicit modelling strategy not only allows capturing theoretical aspects, but also contains several subjective assumptions.

63*Koopman et al.* (2000) give an overview over how certain combinations of these restrictions affect the trend. Marcus Scheiblecker - 978-3-631-75458-0

*Figure 3b: Gain function of* a *BK filter with* a *length of 12 quarters*

Furthermore, in practical work it sometimes turns out that the results obtained are not substantially different from less sophisticated approaches. For instance, *Hahn* - *Walterskirchen* ( 1992) pointed out that if the size of amplitudes is measured by simple variation coefficients of growth rates rather than through calculations by their model, results are largely the same.

## **4. Identifying the business cycle**

After extracting business cycle variations from the underlying time series, either directly by band-pass filters, by explicit modelling in an unobserved components model or indirectly by removing special components, it has to be decided which of the results (or which combination of them) should represent the business cycle. According to *Burns* - *Mitchell's* 11946) definition, the business cycle is a type of fluctuation found in aggregate economic activity. As they refrained from clarifying what this aggregate economic activity should be, several options are open.

Nowadays, there are ready-made statistics available, which measure some sort of aggregated economic activities. These are weighted by their economic significance in monetary units. Examples are GNI, GDP, gross value added and several aggregates of specific sectors. This variety of aggregates poses the problem to determine which of them is best suited for studying the business cycle. This decision in favour of a certain time series can be made on the basis of theoretical considerations, which is called the adhoc method in the present study.

Another possibility is to construct a measure of such an "overall economic activity" out of several economic time series as a kind of index, with a special weighting scheme applied to the time series. Doing this, one has the possibility to include those series which are considered to carry a large content of business cycle information. Furthermore, the weights for their aggregation can be chosen freely. This is of great advantage if series are to be analysed which are not measured in monetary terms, like unemployment rates or interest rates.

## **4.1 Construction of composite economic indices**

## *4. 1. 1 The empirical NBER approach*

In order to isolate and date the business cycle *Burns* - *Mitchell*  ( 1946) constructed lagging, leading and coincident composite indices out of a large amount of monthly economic time series64• The reason for considering several economic time series was partly the lack of a series covering aggregated output on a sub-annual frequency (like quarterly GDP), and partly the fact that noisy components are reduced through aggregation.

The following illustrates only the basic steps of the empirical NBER approach in order to give an impression of its properties. Good descriptions of the details are provided by *Zarnowitz* ( 1992) and Bry - *Boschan* ( 1971), to which the reader is referred.

After a possibly necessary adjustment of all time series for seasonal variations, turning points of them have to be calculated (usually based on the method proposed by *Bry -Boschan* (1971) which will be explained in chapter 6.2). Following this, a reference chronology for determining the leads and lags of each time series based on these turning points has to be set up. This has been done originally by clustering them into groups of leading, lagging and coincident ones on a judgmental basis. Furthermore, synthetic up/down turning point indices (indicating the percentage of time series which show a peak or trough at a certain point in time) or diffusion indices (indicating the percentage of time series in expansion at a certain point in time) are developed. Taking GDP or industrial production as the reference chronology may serve the

*64*This method was developed historically at the NBER and still forms the basis of this kind of analysis, carried out by the Conference Board for the US economy or the OECD. Since then, several improvements have been made with every institution having its own special version. The present study only draws on the NBER basic version and the reader is kindly referred to the special literature for variations. Marcus Scheiblecker - 978-3-631-75458-0

same purpose, but in this case the method comes close to univariate methods like the ones presented in chapter 4.2.

Using this reference chronology, several statistics can be calculated in order to classify the series as leading, lagging or coincident. The most popular statistics in the time domain are crosscorrelations and means, medians and standard deviations of leads and lags. In the frequency domain the calculation of coherences is used frequently. Other possible measures - along with their properties - are discussed in chapter 5. As a result, all time series have to be classified into separate sets of leading, lagging and coincident economic time series.

In order to construct one composite index for each class, a specific weight (ranging from O to 100) is given to each according to seven properties6s:

Subsequently, a restriction allowing only the best-performing series (according to these criteria) to enter the composite indices has to be implemented. Again, there are different methods of aggregation in order to obtain only one indicator from several time series. Basically, standardisation methods accounting for the different variabilities of the time series, detrending steps (usually clearing for

65See *Zarnowitz -Boschan* (1975). Marcus Scheiblecker - 978-3-631-75458-0 Downloaded from PubFactory at 01/11/2019 04:38:46AM via free access

phase-average-trends as presented in <IP_ADDRESS>.) and special heuristic weighting schemes are employed66.

It is easy to see that the whole process is rather time consuming and hard to track, apart from its high degree of arbitrariness. Koopmans (1947) criticised it fiercely for being "measurement without theory". For all these reasons, this method is not considered to be the right one for the present study and will not be pursued hereafter.

### *4. 1.2 Index models*

In order to deal with the more or less arbitrary procedure of constructing business cycle indicators in the line of the empirical NBER approach, several more sophisticated multivariate time series approaches have been proposed. The concept behind using index (or factor) models is the estimation of a common driving force (considered to be the business cycle) in a multivariate manner67. This multivariate framework for analysing the business cycle can be understood as weighing all time series together in order to identify their common driving force. Weights for aggregating all time series can be determined such as to maximise the variability content of the aggregate in relation to the one of the whole data set.

In the last few years, factor models have become more and more popular in several areas of macro economic analysis. They have been applied for constructing economic indicators68, in forecast-

*66Green* - *Beckman* ( 1992) give a description of the method employed by the US Bureau of Economic Analysis (BEA) together with some critical remarks.

*67*The idea of extracting common factors for business cycle analysis goes back to *Geweke* (1977) and *Sargent-Sims* (1977).

68*Goyer* - *Genet* (2006) used this approach to derive a weighting scheme for the Joint Harmonized EU Programme of Business and Consumer Survey. Marcus Scheiblecker - 978-3-631-75458-0

ing, in monetary policy analysis and for business cycle analysis69• Their big advantage is that they try to extract common dynamics from a large number of different time series without running out of degrees of freedom. Even if the number of time series observations is lower than the number of variables, this does not constitute a problem. By filtering out common dynamics, the time series are cleaned for idiosyncratic shocks and cycles at the same time. Furthermore, this can be done without making prior assumptions about a special structure of the common or idiosyncratic dynamics. However, the estimation and identification of the number of factors or components is not straightforward and may lead to less than robust results in some cases.

In its simplest form, several time series are to be represented by one common factor and an idiosyncratic part for each of the series. Here *p* represents the number of time series, each ranging over the time domain 1, ... , T. The common factor *F* and an idiosyncratic part *u;,* represent each of the *p* time series *z;,* which can be formalised as

*(15)* **z;,,** *=I; F, +u;,,* i = l, ... ,p

*I;* is the so-called loading of the factor showing the contribution of the factor to the common driving force *F,* represented in time series *i.* For identification the factor is supposed to be uncorrelated with the idiosyncratic parts *u;,.* and the *u;.,-,* are also uncorrelated among each other. This single common factor approach can easily be extended to the identification of more than one common factor7o\_

69*Breitung* - *Eickmeier* (2005) give a rather complete overview of applications of factor models in recent times.

A drawback of this static approach is that one of the basic f eatures of the business cycle, its auto-regressive development, is not considered since it is assumed that the factors are identically and independently distributed. In order to overcome this drawback, *Stock* - *Watson (* 1989, 1991) proposed a method whereby a single coincident common factor is allowed to move auto-regressively of order 2 and the idiosyncratic ones are represented by **AR(])** processes. Again as in *( 15)*

*( 16) Z;,,* = *A; F,* + *U;,,* i = l, ... ,p

but now a special auto-regressive structure is demanded for the factors

$$\{\{\Gamma\}\}\qquad F\_{\mu} = \phi\_{-1}F\_{\mu-1} + \phi\_{-2}F\_{\mu-2} + \mathcal{s}\_{\mu}$$

and for the components specific to each time series

$$\{\{\mathcal{B}\}\}\qquad\omega\_{\ell,t} = \rho \,\, u\_{\ell,t-1} + \nu\_{\ell,t}$$

with *&,* and *u;.,* assumed to be white noise terms mutually independent in order to allow a proper identification of the common factors. This is usually done by writing all equations above in state space form (with *( 16)* representing the measurement equation and *( 17)* and *(18)* the respective transition equations) and generating maximum-likelihood estimators by the Kalman filter.

This approach is capable of extracting a common dynamic factor - showing business cycle AR properties - from the whole set of observed time series. It is also possible to capture more than one common factor, but in this case the identification of the first common factor influences the following. Thus the number of considered common factors is crucial for the output. A further possible drawback of this approach is "that it cannot be directly extended to analyse large data sets due to computational problems", as *Gayer* - *Genet* (2006) pointed out.

A different method of estimating factor models is the principal component approach. It was originally developed in order to reduce the variance of large data sets by singling out a common component that is capable of representing the largest part of the total variation, by constructing simple linear combinations of it. Each individual time series is represented by one or more common components and some idiosyncratic variation. In the static case, the idiosyncratic parts of all observed time series are either assumed to be mutually uncorrelated (this is called the "strict factor model") or they are allowed to be weakly cross-correlated and heteroskedastic ("approximate factor models")71• A proper identification of the common factors with relaxed assumptions - as it is done in the approximate factor model - requires that the number of time series considered exceeds the time dimension by far and theoretically goes to infinity. This allows for the existence of idiosyncratic common movements between business sectors, as long as they are not too dominant. In the latter case they would enter into the common component.

The dynamic version of this type of factor model approach seems to be ideal for business cycle analysis. It allows the common factors to move auto-regressively and the observed time series can be classified as leading, lagging or coincident according to the common component represented by them. Again, some weak cross-correlation between the idiosyncratic components is allowed and the factors are required to be uncorrelated among each other. *Forni et al.* (2000) suggested a generalised dynamic factor model where the dynamic factors are identified in the frequency domain. Instead of observing the cross-correlation matrix in order to identify the dynamic factors, they focus is on the spectral density matrix.

71 In the approximate factor model case, a weak correlation is allowed even between the factors and the idiosyncratic components. *See Breitung* - *Eickmeier*  (2005). Marcus Scheiblecker - 978-3-631-75458-0

This approach has the advantage that the identified common factors are invariant to their number being identified. The first common factor explains the maximum proportion of total variation of the *p* observed time series, with all further factors showing diminishing explanatory power, till the *p th* common component completes the total variance of the whole data set. *Forni et al.*  (2004) suggest an information criterion for determining the number of factors to be extracted, which concentrates on the amount of common variance that can be explained by the factors.

Again, each time series is represented as the sum of one or more common factors and its residual idiosyncratic component. In the dynamic formulation, the static factors common to all series can be loaded with lead and lag structures, but all are constructed as being mutually orthogonal at all leads and lags. For the idiosyncratic components it can again be assumed that either they are orthogonal to all other idiosyncratic ones (strict factor model) or that some amount of correlation is allowed.

Formally, this problem can be expressed72

where as **z,** is the *N* x 1 vector of observed second-order stationary time series at time *t.* It is represented as the sum of the vector of common components y, and the idiosyncratic *l;,i.* The common term y, itself is constructed by a vector of *q* orthogonal common components **x,** loaded by time-shifted weights *Cq (L),* where *L*  stands for the lag operator73.

72 See *Forni et al.* (2000).

73 Allowing for *a* dynamic structure in factor models - which shows up in leading and lagging properties of the loadings - leads to a loss of information at the beginning and the end of the time series. As already mentioned, this endpoint problem is not a disadvantage for the present study, because no real time dating is intended. Marcus Scheiblecker - 978-3-631-75458-0

Assuming orthogonality between the common and the idiosyncratic parts allows - in the frequency domain - representing the total spectral density matrix ~(ro) as the sum of the one responsible for the common component and the one of the idiosyncratic part

$$\text{(2O)}\qquad \sum \text{(o)} = \sum\_{\gamma} \text{(o)} + \sum\_{\zeta} \text{(o)}$$

where ro is a certain frequency within the interval [-n and +n]. This enables the common part y, to be identified by analysing the eigenvalues of the matrix ~(ro). According to the method proposed by *Forni et al.* (2000), the eigen value solutions of the covariance matrix are ranked by their magnitude, whereby the largest corresponds to the first principal component explaining the largest part of full data variation, the second principal component explains the second largest part of total variation and so on. As the common factor loadings appear with some lead or lag structure, the practical approach for solving this requires a fixing of the number of frequencies at which the spectral density matrix is to be evaluated. *Forni et al.* (2000) recommended the number of frequencies to be observed within the interval [0,2n] as the rounded square root of the time series length divided by four. This should reasonably settle the trade-off between the length of the time series and the quality of the identification output.

The identification of each of the common components is supported by demanding that their variance at several leads and lags inherent in each of the time series is maximised if it is summed up over all time series

$$\left(\mathcal{Z}\right)\Big|\_{\ell\left(\left[\dots,\infty\right)}\sum\_{j=1}^{n}\operatorname{var}\left(\mathcal{I}\_{j,\ell}^{q}\right)$$

with *n* being the number of observed time series and q is again one specific common component. The essential process of identifying one or more common components consists of looking at the eigen values and eigen vectors of the spectral density matrix of the vector containing the original, de-trended data z,, at certain frequencies? 4•

This general dynamic factor model approach has the big advantage that the analysis of leading and lagging features of all time series is not done as usual after determining the business cycle, but that instead it constitutes an essential part of the identification procedure itself.

Unlike the dynamic factor model approach proposed by *Stock* - *Watson* ( 1989, 1991), the business cycle is not modelled specifically by demanding some special auto-correlation of some specific order for the factors. This potentially allows common variations that do not belong to the business cycle to be included in the common component, if the data used are not pre-filtered for high frequency movements. An example would be a terrorist attack which influences economic growth for a short period and is possibly felt in a large part of observed time series. Whereas this appears as a common shock - and therefore influences the estimation of the common components - this cannot be regarded as a business cycle phenomenon. The same argument goes for other high-frequency common or idiosyncratic components like seasonal or working day variations.

In the context of the present study, the low number of time series could be a problem for the identification of the common factors. To obtain consistent estimates of the common factors, the number of series should be considerably higher than the number of observations and theoretically go to infinity. This allows idiosyncratic components to be separated from common variations. Recent research by *Boivin* - *Ng* (2006) shows that under certain circumstances, the number of observed variables can be reduced without losing explanatory power. The authors have shown that in real forecasting exercises 40 time series can yield better results than

*14*See *Forni et al.* (2000). Marcus Scheiblecker - 978-3-631-75458-0 Downloaded from PubFactory at 01/11/2019 04:38:46AM via free access

models based on 147 series75• This is especially the case if idiosyncratic components are not cross-correlated as it is required in the strict factor model approach. In some cases, adding more timeseries (which are probably noisier) lowers the average variation of the common component while assigning more to the idiosyncratic parts76. The findings of *Ink/oar* -Jacobs - *Romp* (2003) are a further example that a higher number of series for studying business cycles does not necessarily deliver better results.

In the present study, time series have been carefully selected with regard to their theoretical content concerning business cycle information. Therefore, series belonging to the domain of public services and agriculture and fores try have been discarded from this analysis. For theoretical reasons, they will not facilitate the identification process, but instead bring some noisy information into the data base, hampering a proper identification. Therefore, a reduction of the number of time series will probably improve the outcome. The pre-filtering of time series in order to isolate only business cycle frequencies should improve the identification process of the common component further. Low-order cross correlation of idiosyncratic components is substantially reduced by adjusting for seasonal, working day and irregular variations. The beforementioned facts warrant the expectation that the restriction to somewhat more than 1 O time series can still produce meaningful results77 •

*75*A further example is Watson (2000) who found the gain from going beyond 50 time series being only marginal.

*16*See *Boivin -Ng* (2006).

77The chapters that compare our results to ad-hoc methods and other studies confirm this expectation. Marcus Scheiblecker - 978-3-631-75458-0

## **4.2 Univariate determination of the business cycle**

This encompasses all approaches where one reference series is chosen (eventually after some kind of aggregation process) without recurring to its statistical properties but for its assumed theoretical suitability. *Harding -Pagan* (2002) emphasised that the use of several time series in the NBER approach to represent the business cycle was motivated by a historical lack of aggregated production data (like the quarterly GDP) rather than by methodological considerations. Nowadays, where such time series are widely available, their use seems to be justified as a reference for the business cycle.

First and foremost, GDP - as a compound measure of total production of goods and services - is expected to carry business cycle information. Without denying its suitability, this series includes by definition a lot of components expected to carry no business cycle variations like agriculture or public services. Theoretically, this should not be a problem if one is able to model or filter out precisely only business cycle frequencies. Unfortunately, in practical life the empirical toolbox is far from being that perfect as to distinguish precisely between different shocks and components influencing the time path of a series. Theoretical guidance for the process of selecting the reference series can be of great value.

As described in chapter 2, GDP contains some sectors whose output can reasonably be expected to carry no business cycle information. Agriculture is more dependent on weather conditions than on business conditions. *Boehm* ( 1998) gives clear empirical evidence for the better suitability of production series adjusted for this primary sector component. We follow this approach in order to avoid the danger that such variations inhibit a proper identification process. Furthermore, taxes on production - like VAT and some excise duties - are at best coincident with regard to the cycle, if not lagging78, which distorts the exact dating of the business cycle.

In view of such considerations, the gross value added in real terms excluding agriculture (GVAex) was chosen as the reference series, instead of GDP. This means that the business cycle frequencies discovered in this aggregate are assumed to be the most suitable measure of the business cycle for Austria, and dating results for all other series are interpreted in relation to this reference series. A large number of studies use the industrial production index as an ad-hoc basis for measuring the business cycle. Whereas this approach has the major advantage of offering long time series at a monthly frequency, this is not the kind of business cycle *Burns* - *Mitchell* ( 1946) had in mind. According to their definition, the business cycle shows up in aggregate economic activity and not only in industrial production.

The univariate ad-hoc determination of the time series which carries most of the business cycle variations is probably the most widely used approach. Studies based on modelling techniques rely heavily on this kind of business cycle determination. There are several studies using multivariate models in business cycle analysis79, but they are only multivariate in looking at several countries to extract the business cycle by measuring common trends or common cycles directly. But for all of these countries, one type of time series is chosen ad-hoc as being most appropriate for business cycle analysis.

78 National accounts regulations and recommendations require the recording of tax revenues for GDP purposes not on a cash basis, but to adjust them at an accrual basis. If this is done properly, such kind of taxes are at best coincident as a lead seems to be rather implausible for them. In practice however, lagged cash revenues are used which potentially shift included business cycle variations in time.

*79*Examples are the multivariate unobserved component model proposed by *Artis -Marcellino* - *Proietti* (2004) or *Cheung* - *Westerman* (1999), who estimated in a multivariate manner the common trend for the industrial production of Austria and Germany. Marcus Scheiblecker - 978-3-631-75458-0

## *5.* **Analysing cyclical comovements**

After extracting the business cycle, an analysis about the leading and lagging properties of several economic time series is of great interest80• There exists a broad range of statistics in this respect, which can be classified according to their belonging to time or frequency domain statistics. It is worth mentioning that this kind of analysis already forms an implicit part of the dynamic common component approach and constitutes an essential part of the identification and extraction of the business cycle.

## **5.1 Time domain statistics for analysing comovements**

To this class all sort of statistics belong, which exhibit comovements between time series at discrete points in time or for certain intervals. They are mainly based on correlation statistics referring to second moments of time series.

Calculating cross-correlations in order to detect potential leads and lags is the simplest form of identifying comovements between the business cycle and other series. They are constructed as

$$\langle \text{22} \rangle \qquad \rho\_{b,s}(\mathbf{r}) = \frac{Cov(z\_{b,t}, z\_{s,t-\mathbf{r}})}{\sqrt{Var(z\_{b,t})Var(z\_{s,t})}}$$

with the subscript *b* denoting the business-cycle-carrying time series (i.e. the reference series). *s* = *1,* ... , *N* counts for all other *N* time series and -r determines the different leads and lags for which cross-correlations are calculated. If -r = O the synchronised comovement is observed. The classification as leading or lagging is mainly based on the delay with which the highest cross-correlation is achieved. It can only be regarded as clear-cut, if the cross-

**eo As** the dating procedure of the business cycle is so prominent, this will be handled separately in the next chapter together Marcus Scheiblecker - 978-3-631-75458-0 with some analysis on turning points.

correlation is significantly higher than for other leads and lags in close neighbourhood.

### **5.2 Frequency domain statistics for analysing comovements**

Sometimes it is easier to interpret comovements not in the time domain, with discrete time data points, but in the frequency domain. This is especially the case in business cycle analysis where special frequencies or frequency bands are the focus of interest. These statistics can be derived by the Fourier transformation of time domain statistics. So is the cross-spectra the frequency- domain-equivalent of the cross correlation *Pb.i* (.), like the spectral density function is the equivalent of the auto-correlation function. The cross-spectra of two series for a certain frequency *tu* is defined as

$$\text{(23)}\qquad \gamma\_{b.s.}\text{ (}\alpha\text{)} = \frac{1}{2\pi} \sum\_{r=-\infty}^{\infty} \rho\_{b.s.}\text{ (}\tau\text{)}\ e^{-i\omega\tau}$$

with *tu* being a frequency within [-n;.n) and *Pb.,* being the crosscorrelation as defined in *(22).* As can be seen in *(23),* the crossspectrum contains a complex part which does not cancel out because the cross-correlation function is not symmetric, i.e. *Pb.s (,)\*Pb.,(-,).* Therefore, this statistic cannot be interpreted in order to determine leads and lags directly, but has to be transformed into another statistic like the coherence, the phase spectra or the mean delay.

## *5.2. 1 Coherence81*

The coherence measures the linear relatedness of two stationary processes. It can be regarded as the frequency-domain-equivalent to the cross-correlation in the time domain. The output is defined over the interval [O, 1t] and shows the correlation of the cyclical (or at least stationary) component of the series at each frequency. It is defined by the squared cross-spectrum, divided by the product of the spectral density functions of both series

$$\text{(24)}\qquad co\ (o) = \frac{\left|\boldsymbol{\gamma}\_{b,s}\ (o)\right|^2}{\boldsymbol{\gamma}\_{b,b}\ (o)\ \boldsymbol{\gamma}\_{s,s}\ (o)}$$

Applying this quadratic transformation ensures that values are real and symmetric. It can be interpreted as the frequency-domaincounterpart to R2, the well known coefficient of determination, as it shows the proportion of variance of one series explained by the other for a given frequency *m.*

But this transformation has a substantial disadvantage. *Croux* - *Forni* - *Reich/in (* 1999) stressed that this statistic "does not measure correlation at different frequencies, because it disregards the phase difference between variables"82• Thus, only the synchronised comovement of two time series, over some specified spectrum or at a certain frequency, can be observed. Whether these frequencies are phase-aligned or not is of no influence to this measure. Related to our problem, the coherence assumes high values if both series show similar frequency gains, irrespective of whether one time series is leading or lagging.

81 In the literature, this measure is sometimes called squared coherence.

82 See *Croux* - *Forni* - *Reich/in* ( 1999), p. 4. Marcus Scheiblecker - 978-3-631-75458-0 Downloaded from PubFactory at 01/11/2019 04:38:46AM via free access

## *5.2.2 Phase spectra and* mean *delay*

In order to supplement the coherence with a statistic that informs about leading and lagging properties of time series, the phase spectrum can be calculated. This can be done by combining the real and the complex part of the cross-spectrum as defined in (23) and calculating the arcustangens of it in order to obtain the phase angle:

$$\langle \text{2.5} \rangle \qquad \gamma\_{b,s} \text{ (}\text{\textit{\textquotedblleft}}\text{\textit{\textquotedblright}}\text{)} = \frac{\text{l}}{2\pi} \sum\_{\text{r}=-\infty}^{\infty} \rho\_{b,s} \text{ (}\text{\textit{\textquotedblleft}}\text{)} \, e^{-i\,\text{\textit{\textquotedblleft}}\text{r}} \, = \left\| \frac{\text{l}}{2\pi} \sum\_{\text{r}=-\infty}^{\infty} \rho\_{b,s} \text{ (}\text{\textit{\textquotedblleft}}\text{)} \right\| e^{-i\,PH} \, (\text{\textit{\textquotedblleft}}\text{)} \, \text{r} \right\| $$

where *PH{ m)* is the phase-delay-generating function multiplied by a scalar, and further

$$\text{(2\text{\textquotesingle})}\qquad \xi(\text{o}) = \arctan \frac{\text{l}}{2\pi} \sum\_{\text{r} = -\infty}^{\text{o}} \frac{\rho\_{b,s}\left(\text{r}\right)}{PH\left(o\right)\left(\text{r}\right)}$$

~(ffi) is the phase spectrum over all frequencies, indicating how large the lead (positive numbers) or lag (negative) is. Averaging over a specified frequency band yields the mean delay for this term. In this way one can look at the leading and lagging properties only for frequencies within the business cycle boundaries or for other frequencies in the focus of interest.

### *5.2.3 Dynamic correlation*

Another way of overcoming the deficiency of the coherence of not accounting for phase shifts of frequencies has been proposed by *Croux -Forni -Reich/in* (1999). They recommended a measure that looks quite similar to its time-domain-equivalent, the crosscorrelation, shown in *(22):*

$$\left(2\mathcal{T}\right)\quad\widetilde{\rho}\_{b,s}\left(\mathcal{o}\right) = \frac{\chi\_{b,s}\left(\mathcal{o}\right)}{\sqrt{\chi\_{b,b}\left(\mathcal{o}\right)\chi\_{s,t}\left(\mathcal{o}\right)}}$$

but this time defined only for co within the interval [0, 1t). This measure is related to coherence - as it shows its real part - but instead of rendering values real by a quadratic transformation, the complex term is cancelled out by summing negative and positive waves of the same frequency. This can be seen in a different representation of

$$\langle \text{2B} \rangle \qquad \widetilde{\rho}\_{b,s} \text{ (}\!\!\/ \text{\(\mu\)} = \frac{\chi\_{b,s} \left(\begin{smallmatrix} \mu \end{smallmatrix}\right) + \chi\_{b,s} \left(\begin{smallmatrix} -\mu \end{smallmatrix}\right)}{2} \text{)}$$

This should not be a drawback as (co) and (-co) have the same periodicity, which is the focus of interest in business cycle analysis. Furthermore, the authors stressed that dynamic correlation observed at a special frequency band is theoretically fully equivalent to static correlation applied to band-pass pre-filtered data accordingly.

#### *5.2.4 Cohesion*

In order to analyse the comovement of more than one time series, Cro ux - *Forni* - Reich/in ( 1999) proposed a multivariate version of the dynamic correlation index. It is constructed by weighing together all dynamic correlation coefficients as defined in *(27)* and is called cohesion by them

$$\langle \mathcal{D}\mathcal{D} \rangle \qquad coh\_b \text{ (}\boldsymbol{\omega}\text{)} = \frac{\sum\_{i \neq j} \,\,\,\omega\_i \,\,\,\omega\_j \,\,\widetilde{\rho}\_{s\_i s\_j} \,\, (\boldsymbol{\omega})}{\sum\_{i \neq j} \,\,\,\omega\_i \,\,\,\,\omega\_j}$$

the *p\_,* being again the dynamic correlation coefficients as defined in *(27)* over all combinations of time series except their diagonal elements. The authors suggest choosing the weights according to their economic significance (e.g. by their proportion of contributing to GDP). This statistic has the advantage of allowing to capture common comovements of the underlying set of time series at all frequencies at one glance. This makes it more appropriate for certain tasks than looking at cross-correlations because Marcus Scheiblecker - 978-3-631-75458-0

high values of the latter neither imply nor are implied by cointegration, common cycles or common features8J. A practical example of application of this statistic can be found in Croux - *Forni* - *Reich/in (* 1999) and in *Partridge -Rickman* (2005).

83 See *Croux -Forni -Reich/in* (1999), p. 1. Marcus Scheiblecker - 978-3-631-75458-0 Downloaded from PubFactory at 01/11/2019 04:38:46AM via free access

## **6. Dating the business cycle**

Only part of the studies on business cycles restricts itself to comovement analysis. Especially if studies are motivated by cyclical policy intervention (rather than by the identification of structural similarities), the dating of the business cycle is most important. The process of dating can be described as looking for turning points in the business cycle84• There is a difference between the definition of turning points in the classical context of business cycle analysis where the object of study are the absolute levels of some time series - and the interpretation of cyclical deviation. *Figure 4* shows a stylised example of an economic time series - like GDP - consisting of a deterministic trend plus a cyclical variation.

*Figure 4: Turning points in the classical and the cyclical deviations approach*

Source: Own illustration.

ll-4 In business cycle analysis turning points are defined in a different way as in geometrics. Whereas the first mark local minima and maxima the latter is the point where a graph starts to change the sign of its curvature. Marcus Scheiblecker - 978-3-631-75458-0

If the cycle is defined in the classical way, as depicted by the trended bold line in the upper part of *Figure 4,* then point A represents one of the upper turning points occurring at time t A as it constitutes a local minimum. Following the deviations approach and observing only turning points of the extracted business cycle component, point B marks the upper turning point at time ts. This point is derived by analysing only the stationary, bold line in the lower part of the figure, which should represent the business cycle according to the deviations approach. It can easily be seen - as it is a well-known fact - that the turning points according to the cyclical deviations approach (bold line in the lower part) lead the ones of the classical definition (bold line in the upper part)B5•

If instead the focus of interest lies on growth rates (growth rate cycle approach), the result is again different turning points. The dotted line in the lower part of *Figure 4* represents the growth rates of the series. When interpreting the business cycle as a sine function (like it is done here with the bold line in the lower part), growth rates follow a cosine function as this is its first derivative. This series of growth rates is illustrated by the dotted line in the lower part of *Figure 4* (for graphical reasons it has been linearly transformed). It reaches its peaks earlier (e.g. at time tc) than the others86, which is in part due to its asymmetric construction.

As for their importance, turning points have been given special names. In the classical approach, the lower turning points are called *troughs,* which are followed by a phase of *recovery* leading into a period of *expansion* whose end is marked by the upper turning point, the *peak.* Thereafter, a *contraction* takes place followed

85The length of the lead is related positively to trend growth and negatively to the amplitude of the series. The higher trend growth, the more it is capable of delaying the downward-sloping effect of the cycle on the whole series.

86 If a high frequency error term is included, the leading property of growth rates cannot be exploited for an earlier indication of turning points. This is especially true as it has been shown that taking first-order differences superimpose high frequency error terms. Marcus Scheiblecker - 978-3-631-75458-0

by a *recession* ending with the next *trough.* which completes the cycle. The nomenclature is somewhat different in the deviation cycle approach. The local peaks of the business cycle component - *as* shown in the lower part of *Figure 4* - are often referred to *as highs* and the lower turning points are called *lows.* Phases between them are called *expansions* or *contractions.*

Having this sequential pattern of the business cycle in mind, dating the turning points has to fulfil several conditions. Firstly, every dating procedure - arbitrary or mechanical - has to ensure that turning points alternate. It must not occur that the turning point following a peak is again a peak. Secondly, the period between the turning points should not be too short in order to allow some economically meaningful process of recovery or contraction. Furthermore. turning points should represent some local minima or maxima according to certain criteria. Several methods have been developed to this end. but only the most important shall be explained heres1.

### **6.1 The expert approaches**

To this class belong all approaches which are mainly based on more or less subjective evaluations. rather than on a mechanical method. The most prominent approach in this area is the dating schedule set up by the NBER. Here, a Business Cycle Dating Committee publishes officially the turning points calendar for the US economy. According to the NBER. a recession is characterised by "a significant decline in economic activity spread across the economy. lasting more than a few months, normally visible in real GDP. real income. employment. industrial production. and wholesale-retail sales88.'' This gives insight into the database that forms

87Examples for alternative methods not reported here are proposed by *Diebold* - *Rudebush* ( 1987), *Wecker* ( 1979), *Hess* - *Iwata* ( 1997) or *Neftci* ( 1982).

the backbone for the NBER dating procedure. The term "normally visible in real GDP" is not only a hint at the subjectivity of this approach, but has something to do with the availability of GDP, i.e. usually only at a quarterly frequency.

Before this dating can be done, several steps to identify the cycle take place, which have already been described above. Despite its subjectivity, the dates resulting from this approach are not only the official ones, but serve as a benchmark for many innovative new dating methods. This is probably due to the fact, that this approach is adopted by experts who can take into account specific developments in the economy as well as in the underlying time series which could otherwise be misinterpreted if treated in a purely mechanical way. For the same reason, *Breuss* ( 1984) argues in favour of a re-evaluation of identified turning points by using external knowledge about the characteristics of time series. In case of doubt whether a turning point is dated exactly, an ad-hoc intervention should take places9•

This was probably the reason why a similar procedure has recently been adopted for the euro area. With the formation of a currency union in 1999, demand for a business cycle dating calendar has emerged with a view to giving adequate guidance to economic policy. To this end, the Centre for Economic Policy Research (CEPR) has set up a committee similar to the one of the NBER for determining the dates of turning points. Its definition of a recession is quite close to that of the NBER, but accommodates for euro area characteristics.

Unlike the NBER, the CEPR focuses on the spread between the EU Member States, whereas the former seems to focus more on sector differences. Furthermore, the CEPR refers to quarterly instead of monthly developments, which has to do with the fact that the

89The author cites as an example a higher-than-usual economic activity resulting from a front-loading of private expenditure in anticipation of a planned tax rise, which could wrongly be interpreted as an upper turning point. Marcus Scheiblecker - 978-3-631-75458-0

most reliable economic information for the euro area is only available on a quarterly basis.

Most interestingly, the CEPR concentrates on growth rates. The interpretation of growth rates makes a dating procedure not only sensitive to superimposed error terms (making such dating a poor guide for policy interventions), but also mixes up trend and cycle movements. From a theoretical point of view, it makes a decisive difference whether slow growth results from a gradually sloping trend or a sharp fall in the business cycle.

## **6.2 The Bry-Boschan routine**

The best-known methodical approach is the algorithm proposed by the NBER economists *Bry* - *Boschan* ( 1971). This non-parametric method can be applied to seasonally-adjusted trended as well as de-trended monthly time series and tracks quite well turning points for the US as set by the NBER Business Cycle Dating Committee.

This approach starts with a possibly necessary de-trending procedure and the removal of outliers according to some standard deviation criterion. Subsequently, a sequence of smoothing filters is applied, each followed by running the dating procedure. The smoothing process starts with the filter performing the highest degree of smoothness and goes down consecutively till the original (unsmoothed) time series is dated. The final turning points are dated on the basis of the original series, but must be consistent with all earlier dates according to the smoothed series. The specific points are found by checking for local minima and maxima and imposing some criteria for the minimum length of phases and cycles between them. If two or more peaks (or troughs) follow each other, then only the highest (deepest) one enters the dating output.

Disadvantages of the Bry-Boschan routine are that only the alternation of turning point signs and the phase-length impose restric-

tions for the identification of peaks and troughs. There is no possibility to take into account the amplitude of cyclical variations and, as it is a purely deterministic approach, nor the uncertainty encountered when choosing dates for the turning points.

As this procedure proposed by *Bry* - *Boschan* (l 971) was developed originally only for monthly data, some amendments have been proposed to use it analogically for quarterly series90• The version used in this study is the one implemented in the BUSY software package developed by the *Joint Research Centre of the European Commission* (1993). There, de-trended time series are smoothed by a Spencer curve, which is a symmetric *2:x7* moving average filter with the following weights

$$\begin{array}{ll} \text{(\text{\textquotedblleft}O\textquotedblright)} & \text{\textquotedblright} (L) = \frac{1}{320} \begin{bmatrix} \text{\textquotedblleft}4 + \delta\text{\textquotedblright} \left(L + L^{-1}\right) + 4\delta\left(L^{2} + L^{-2}\right) + 2\left\{(L^{3} + L^{-3}) + \\ \text{\textquotedblleft}3 + \delta\text{\textquotedblright} \left(L^{4} + L^{-4}\right) - \delta\left(L^{5} + L^{-5}\right) - \delta\left(L^{6} + L^{-6}\right) - \Im\left(L^{7} + L^{-7}\right) \end{bmatrix} \end{array}$$

In order to compensate the loss of seven data points on either end of the series, the series is extended by extrapolating the growth of the first and last four observations. This smoothed series is used, first of all, for replacing outliers in the uncorrected series, which are detected by imposing that its standard deviation be a certain multiple of the series total standard deviation, to be defined by the user.

Following this, the series corrected for outliers is filtered by a 2 x 4 term MA term in the quarterly time series case. The output of this filter and the one generated by the Spencer curve are both scanned for common turning points, which are characterised by local minima or maxima with intervals of five periods91 • Furthermore, a minimum phase length of 6 quarters is imposed, together with an alternation of the signs of the turning points.

*90Harding -Pagan* (2002) is an example in this respect.

91 The size of the interval can be altered by the user. Marcus Scheiblecker - 978-3-631-75458-0 Downloaded from PubFactory at 01/11/2019 04:38:46AM via free access

An additional, less smoothed, series is calculated by further shortening to MA order according to a parameter called QCD (quarters of cyclical dominance). This QCD yields the minimum delay for which the average of absolute deviations of growth in the Spencer cycle is larger than that in the irregular component. This series is again checked for common turning points with the series as calculated before, subject to the same requirements for cycle length and the alternation of signs. As this last short filter is able to move close to the end of the series - while for the longer filters synthetic extrapolations had to be made - turning points found in the last or first two observations are dropped.

Versions of the Bry-Boschan routine used for dating the quarterly US-GDP time series have been quite successful in tracking the turning points as published by the NBER92•

### **6.3 Hidden Markovian-switching processes**

Studies resorting to the method proposed by *Hamilton* ( 1989) try to represent the business cycle component by a Markov-switching autoregressive model. Here, the series examined is to be represented by a stochastic process that can switch between a contractive and an expansionary regime93• In its most general form, not only the mean *µ* is allowed to vary between both states, but also the AR parameters efJ of order p and the variance of the error term (with assumed zero mean) *e,* .

$$\{\mathcal{Z}\}\Big|\qquad z\_t = \mu\_{s\_t} + \sum\_{i=1}^{\varrho} \phi\_{s\_t} \ z\_{t-i} + e\_t^{s\_t}$$

92Examples for this kind of studies ore *King* - *Plosser* (1994) and *Harding* - *Pagan*  (2003).

93In *Artis* - *Krolzig* - *Toro* (2004) a third regime is considered for some countries, in order to capture the effect of a break in trend growth in the second half of the seventies. Marcus Scheiblecker - 978-3-631-75458-0

with *z* being the analysed time series and *s,* = 1 if the economy is in expansion and *s,* = 2 if it is in contraction, implying that the mean is above or below the average mean of the total series

$$\begin{array}{cc} \text{(32)} & \mu\_{s\_t} = \begin{cases} \mu\_1 > 0 & \text{if } s\_t = 1\\ \mu\_2 < 0 & \text{if } s\_t = 2 \end{cases} \end{array}$$

The probability of being in either state of the cycle evolves according to a Markov chain defined by certain transition probabilities for switching from one state to the other

$$\{\text{\{\ $3\}}\} \quad \text{\{\text{\$ not\}}\text{\{\ $}s\_t = i \mid s\_{t-1} = j\} = \mathfrak{s}\_{l,j} \in \text{(\text{\$ },l)} \quad \forall i, j = l, \mathfrak{L} \text{ \{\ $}s\_t = i \mid s\_{t-1} = j\} = \mathfrak{L} \text{ \{\$ }s\_t = i \mid s\_{t-1} = j\} \quad \forall i$$

Thus, the probability of being in expansion if the preceding period was also in expansion is (1-e1), or for a contraction following a contraction (1- s2 ), and the changes from one state to the other are s1 or s 2 , respectively.

The estimation of such Markov-switching models is quite complicated, since a global nonlinear process is combined with an unobserved component character. In practice, this is carried out with the Kalman filter technique in order to get Maximum-Likelihood estimators.

As a result, probabilities for each observation of being in either state are obtained, which can be transformed into a binary variable indicating contraction or expansion periods. The advantage of this procedure, compared with the non-parametric approaches, is that for all points in time it can be inferred whether the economy is expanding or contracting94• Furthermore, the size of deviation plays a role in the identification process, so that minor ups and downs can be excluded from the dating calendar. The advantage of obtaining probabilities for turning points may be a

9• Except for the very rare points where the probability for either regime is insignificant. Marcus Scheiblecker - 978-3-631-75458-0

disadvantage at the same time. Usually, there remains a - hopefully small - number of time series points, for which it is inconclusive in which state the economy is. As they are located between expansions and contractions, such points become the focus of interest. If they have to be identified, only a judgemental evaluation can help.

## **6.4 Threshold autoregressive models**

Threshold autoregressive models {TAR} represent a further method of a model-based dating procedure. Here, different regimes {in the case of business cycle analysis there will again be two of them: contraction and expansion} are modelled, with a threshold *r* for their identification and with a certain threshold delay *d* to which the threshold refers. The structure is basically the same as in *(31)*

$$\langle \text{Cd} \rangle \qquad z\_{\text{f}} = \mu\_{s\_{\text{f}}} + \sum\_{i=1}^{p} \phi\_{s\_{\text{f}}} \, z\_{\text{f}-i} + s\_{\text{f}}^{s\_{\text{f}}}$$

but the process of determining the state is different, in that it depends on a certain threshold {the threshold delay}, which has to be exceeded

$$\{\text{G.5}\}\qquad z\_{s\_t} = \begin{cases} \mu\_1 + \sum\_{i=1}^p \phi\_1 \ z\_{t-i} + \varepsilon\_t^{\frac{1}{t}} & \text{if } z\_{t-d} > r \\\\ \mu\_2 + \sum\_{i=1}^p \phi\_2 \ z\_{t-i} + \varepsilon\_t^{\frac{2}{t}} & \text{if } z\_{t-d} < r \end{cases}$$

This approach is rather flexible, as for various regimes different auto-correlation behaviours can be modelled as well as separate error term variances. The TAR model is a piecewise linear model which shows non-linear global behaviour. As first steps, the threshold delay *d,* the threshold *r* and the order of the AR polynoms for either state have to be estimated. After applying an identification

procedure95 of both processes, the turning points mark the dates of change from one state to the other. As in the case of the Markov-switching model, restrictions for the amplitude of deviations can be formulated and - as it is a probabilistic approach probabilities for being in a contractive or expansionary state can be generated. However, this can also be seen as problematic, as outlined in the case of the Markov-switching model.

## **7. Analysis of turning points**