University of Amsterdam
Msc in Economics
Monetary Policy and Banking
Master thesis under the supervision of prof.dr. Ward Romp
Second reader: prof.dr. Christian Stoltenberg
Nikolaos Apokoritis
(11390026)
MEASURING AND COMPARING
BUSINESS CYCLES AND FINANCIAL
CYCLES:
EMPIRICAL EVIDENCE FOR THE EURO
AREA AND THE UK
Statement of Originality
This document is written by Nikolaos Apokoritis who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Abstract
This thesis investigates the statistical properties of business cycles and financial cycles, as well as the possible relation between the two phenomena. This is done by applying the Kalman filter on a multivariate unobserved components model, in order to extract common cyclical components from a panel of macroeconomic and financial time series respectively. The study is conducted for five large economies of the Euro area and the United Kingdom, using data from 1976 to 2016. My analysis shows that the financial cycle can be identified as a medium-term phenomenon, mainly driven by cyclical fluctuations in property prices, while the dynamics of the business cycle are better captured over the short-term. My empirical findings also indicate that financial cycles exhibit more ample fluctuations compared to business cycles. Furthermore, the business cycles in the Euro area and the UK appear to be significantly synchronized, while their respective financial cycles exhibit substantial heterogeneity in that respect. Moreover, I find evidence of a positive correlation between cyclical movements in property prices and the medium term cyclical component of GDP. Finally, the results regarding the timing of financial crisis and economic recessions are mixed, since both phenomena are observed to precede one another through time and across different countries.
Contents
1. Introduction……….. 2. Literature Review………. 3. Methodology……… 3.1. Univariate Unobserved Components Time Series Model……… 3.2. Multivariate Unobserved Components Time Series Model………. 3.3. State Space representation and estimation method of the model………. 4. Data……….. 5. Empirical Results………. 5.1. Estimates of the multivariate model-based filter………. 5.2. Comparison of detrended time series with the common cycle component……….. 5.3. Analysis of Turning Points……….. 5.4. Correlation between the Business Cycle and the Financial Cycle………... 6. Robustness of Results……….. 7. Conclusions………. References……… Appendix……….. Intervention variables (Business Cycles)………... Intervention variables (Financial Cycles)……….. Tables of turning points………. Time series of data (seasonally-adjusted and log-transformed)...
1 4 9 9 11 13 16 17 17 22 26 34 36 45 46 49 49 52 55 58
1
1. Introduction
Financial instability, or ‘systemic risk’ by the technical term, is endemic to financial systems due to a fundamental externality in financial intermediation known in economic literature as the fallacy of composition; what might be prudent from the perspective of an individual financial entity may be imprudent for the soundness of the system as a whole. According to the post-Keynesian economist Hyman Minsky (1986, 1992), economic agents do not fully internalize the effect of their borrowing decisions on asset prices, leading to large upswings in asset prices as a result, fueled be an expansion in credit and leverage, and the build-up of financial imbalances (e.g speculative borrowing bubbles, over-accumulation of debt). The unwinding of these imbalances can ultimately lead to financial busts. This behavior, manifested in financial crises, as well as in cyclical fluctuations of financial time series beyond the range of business cycle frequencies and amplitudes, reflects the endogenous and time-varying nature of systemic risk and gave rise to the notion of the financial cycle. This concept refers to the existence of systematic boom- bust patterns in the financial system that can potentially interact with the real economy. These patterns are generated, not only due to the dynamics of the business cycle, but also through the mechanism described by Minsky that is inherent in the financial system. After the 2008 financial crisis and the subsequent European sovereign debt crisis the possible existence and dynamics of a financial cycle, as well as its interaction with macroeconomic cycles gained much popularity.
Intuitively, financial cycles could be linked to macro cycles in the following way. As mentioned, during financial booms imbalances tend to accumulate, meaning a sharp increase in credit, asset prices, indebtedness, risk taking etc. There are three main factors that can generate financial booms:
(i) Structural changes in the financial sector (e.g financial innovation, creation of new “exotic” products and procedures, which may lead to excess risk taking and a more complex financial system)
(ii) A euphoric upturn in the real economy (e.g robust growth, low inflation) could foster this increased risk tolerance
(iii) An accommodative monetary policy (e.g low interest rates) may boost credit growth even more.
When the indebtedness of economic agents is high, they become more sensitive to shocks. Even a small shock (trigger) could be detrimental to their creditworthiness, especially in the presence of asymmetric information. For instance, drops in income and sales reduce the borrowers’ net wealth, which in turn increase their probability of default dramatically, if debt is very high. As a consequence, lenders will either increase their interest rate on debt or stop funding the borrowers. In any case, this will ultimately result in fire sales, deleveraging, large asset price drops and a decrease in consumption and investment, meaning a decrease in aggregate demand. On the other hand, if an adverse shock were to occur at a bank’s balance sheet, it could lead to sharp contractions in credit and affect negatively the real economy as a consequence. If the downturn is bad enough, a deficient demand, high unemployment and a recession in the macroeconomy could be the outcome1.
1
2 Identifying the statistical properties of the business and financial cycles, and disentangling their possible relationship is very important for policy makers. If the former are characterized as two ‘different’, but interacting, phenomena, then monetary and fiscal policies are imperfect instruments for promoting financial stability and the case of macroprudential policy as a third stabilization policy is reinforced. One of the main targets of macroprudential policies is to build tools aimed at addressing the time-dimension of systemic risk, in other words, ‘dampening’ the financial cycle. An example of such a tool is the countercyclical capital buffer requirement for financial institutions introduced in Basel III, which is based on the idea of building up buffers during the expansionary phase of the financial cycle, in order to be able to draw them down in periods of financial distress. Since these policy instruments are constructed based on the properties of the financial cycle, the creation of a robust measure of the financial cycle, which allows for comparisons with the business cycle, is required.
The purpose of this thesis is to empirically investigate the statistical properties of business cycles and financial cycles, as well as examine their possible correlation and the synchronicity of their turning points. This is done by applying the Kalman filter to a multivariate unobserved components model as formulated by Durbin and Koopman (2001), in order to extract common cyclical components from a set of macroeconomic and financial variables respectively, which correspond to the desired business and financial cycle indicators. Time series on real GDP, industrial production and retail trade sales are used to extract the common business cycle indicator, while the financial cycle indicator is derived from the variables credit, credit-to-GDP and property prices. As part of the analysis of the turning points of the extracted cyclical components, I propose two simple measures, the Conditional Significance and the Unconditional Significance measures, which function as indicators of the size of a turning point conditional to the business/financial cycle of a specific country, as well as compared to the business/financial cycles of other countries. The correlation between the estimated business and financial cycles is approximated by applying a bivariate unobserved components model with ‘similar’ cycles on real GDP and property prices. The study is conducted for five countries of the Euro area, namely Finland, France, Italy, Spain and the Netherlands, as well as for the United Kingdom over the period 1976Q1 to 2016Q2. All computations were done in the econometric software package Oxmetrics 6 by utilizing the module STAMP 8.2, which is suitable for the analysis of time series within the State-Space framework.
Multivariate unobserved components models, or else structural time series models offer a number of advantages compared to other cycle measurement/signal extraction techniques. First, this class of models provides the possibility of using information stemming from many economic time series as opposed to frequency-based filters (e.g Baxter King (1999), Christiano and Fitzgerald (2003)) which are univariate. Second, in contrast to frequency-based filters and the turning point analysis, structural time series models do not impose restrictions and ad-hoc assumptions regarding the range of cycle frequencies. These parameters are model-driven and they are estimated via the Maximum Likelihood method. For the scope of this thesis, this is particularly important, since the goal is to identify the statistical properties of the business cycles and the financial cycles. However, structural time series models are able to extract smooth cycle estimates and it is shown that they possess band-pass filter properties (Azevedo et al. (2006)). Therefore, one could say that they function as approximate, multivariate, model-based filters. Third, the model-based nature of the filter allows the researcher to perform diagnostic tests in order to check the fit of the
3 model to the data and the accuracy of its estimates. Fourth, the Kalman filter is able to deal with non-normal data with ease, a feature that is quite useful in this case, since financial data are known for having distributions with fat tails.
Nevertheless, structural time series models should be handled with caution. Imposing a common cyclical component on a number of economic time series may be too restrictive, if such a component is rather weak or even non-existent. Furthermore, the stochastic cycle model of Harvey (1989), often used to specify the unobserved cyclical component in structural time series models, implies that the cycle has a time-invariant period. Although the length of business cycles is usually constrained within a certain band, assuming that their period is fixed may sometimes be restrictive, since business cycles can have unpredictable periodicities. These issues could result in producing spurious cycles. Therefore, a robustness check of the estimation results of such models is recommended.
Similar studies on the statistical properties of the business and financial cycles found in economic literature, usually focus solely on either creating robust business cycle measures or on estimating financial cycles and comparing them merely to cyclical fluctuations in GDP. Furthermore, although many empirical papers employ multivariate analyses for the extraction of common business cycle indicators from a set of macroeconomic time series, financial cycles are usually estimated by extracting individual cycles from various credit-related variables. However, in this study I give particular emphasis on creating robust measures for both the business cycle and the financial cycle using a common methodology, which is based on the underlying idea of extracting common cyclical components from multiple economic time series. This approach allows for making more meaningful comparisons between the two types of cycles, since their statistical properties and characteristics are identified under the same methodological framework. To be more specific, in my thesis I employ a methodology similar to Azevedo et al. (2006), who signal extract a common business cycle indicator from multiple macroeconomic time series, and extend it by applying it to a set of financial time series, in order to measure the financial cycle. The financial variables I use, were also employed by Galati et al. (2016), but in their case univariate cycles are estimated from each time series and not a common financial cycle indicator. In addition, this thesis adds further to empirical literature, by utilizing an updated dataset and including new countries, such as Finland and the United Kingdom that were not analyzed in previous similar studies.
The rest of the thesis is structured in the following way. Section 2 presents a review of the existing empirical literature on the relationship of business and financial cycles. Theoretical definitions of the two phenomena and early theoretical works on these concepts are also mentioned briefly in the beginning of this section. Section 3 develops the model that is used for the measurement of the business cycles and the financial cycles. Section 4 describes the dataset that was used for the purposes of this thesis. Section 5 analyzes the empirical findings of the study. Section 6 compares the results of this study with the ones produced by an HP filter (Hodrick and Prescott (1997)) and the chronology of the business cycle turning points provided by the OECD. Section 7 concludes.
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2. Literature Review
The concept of the ‘business cycle’ has been well-established in economic literature. Economic activity can be characterized by inherent fluctuations that occur systematically, exhibit considerable cyclicality (although usually not predictable periodicity) and may vary in duration and intensity. These fluctuations interchange over time between periods (phases) of rapid economic growth and expansion, and periods of stagnation and contraction. Until the early 1800’s the dominant theory was the theory of general equilibrium and classical economists would deny the existence of business cycles. Economic fluctuations were usually attributed to external factors, such as war. It was in 1819, where Jean Charles Léonard de Sismondi laid the foundations for the concept of business cycles, by investigating the existence of periodic economic crises in his paper Nouveaux Principes d’économie politique. Sismondi supported his argument of the existence of economic cycles, by examining the ‘Panic of 1825’, which was undoubtedly the first international economic crisis that occurred during peacetime.
The standard contemporary definition of business cycles, based on the idea that economic systems function under recurring fluctuations associated with a number of variables, was given by Burns and Mitchel (1946) in their book Measuring Business Cycles.
According to the authors,
“Business cycles are a type of fluctuation found in the aggregate economic activity of nations that organize 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 character with amplitudes approximating their own”
Although Burns and Mitchel (1946) state that the upper bound of the duration of business cycle fluctuations could reach up to 10 or 12 years, a generally accepted business cycle length in economic literature is short-term fluctuations ranging from 1.5 to 8 years. This implies that movements of period longer than 8 years could be attributed to structural reasons than to conjunctural ones (Pelagatti (2004)). The strength of a business cycle is usually determined by the upward and downward movements of real gross domestic product (GDP) around its long-term trend.
The notion of the ‘financial cycle’ is used to describe the existence of systematic boom-bust patterns in the financial system that can potentially have important consequences to the macroeconomy. It is closely related to the idea of the ‘credit cycle’, which captures the expansion and contraction of access to credit over time. The concept of cyclical patterns in the financial system has been analyzed historically in the writings of Minsky (1986), Minsky (1992), Fisher (1933) and Kindleberger (1978). When credit growth is rapid and strong, asset prices tend to inflate by those who have access to leveraged capital. Leverage also increases, and excessive risk- taking and significant underpricing of risk take place. During this credit boom, which is usually accompanied by financial innovations and a euphoric macroeconomic environment, financial imbalances tend to accumulate (e.g speculative borrowing bubbles, excessive build-up of debt). At that point (documented in the literature as the ‘Minsky
5 moment’), potential triggers can cause the sudden unwinding of financial imbalances leading to a financial bust. During the crisis period, credit growth drops dramatically, fire sales and deleveraging occur at a rapid rate, driving asset prices down. The resulting effect could be a subsequent recession in the real economy. Financial crises are considered to be a manifestation of financial cycles. According to Minsky, reflecting the view of the post-Keynesian school of economic thought, boom-bust cycles in the financial system are an integral part of the process driving the business cycle.
More recent writings in the financial cycle literature, relate that phenomenon to the concept of the procyclicality of the financial system and the endogenous nature of systemic risk (see e.g Borio et al. (2001), Brunnermeier et al. (2009), Adrian and Shin (2010)). According to Kauffman and Scott (2003), systemic risk refers to the probability of collapse of an entire financial system, as opposed to the collapse of individual parts of it, and is characterized by comovements between most or all of these parts. Systemic risk is considered to be endemic in the financial system and consists of two dimensions; the structural dimension, which is related to the interconnectedness of the individual financial institutions and their exposure to common risk factors, and the time dimension which is associated with the evolution of systemic risk over time and the transition of a financial system from stability to fragility. Financial cycles are believed to capture the patterns of systemic risk over time, or else the procyclicality of the financial system, which are closely linked to cyclical movements in real economic activity.
Although a formal definition of the financial cycle is yet to be established in economic literature, according to Borio (2014), “the financial cycle can be defined as self-reinforcing interactions between perceptions of value and risk, attitudes towards risk and financing constraints, which translate into sequences of financial booms followed by busts” (see also Galati et al. (2016)).
There is a long tradition in empirical literature on measuring business cycles and identifying their statistical properties. On the other hand, the empirical investigation of the characteristics of financial cycles and their interconnection with the business cycle is still in its infancy. Three different strands of methodologies have mainly been employed in economic literature for measuring business and financial cycles.
The first methodology is the traditional turning point analysis, introduced by Burns and Mitchel (1946), which relies on identifying business cycles by dating peaks and troughs in observed economic time series. A commonly used algorithm to detect turning points in the log-level of a series is the one developed by Bry and Boschan (1971), which is based on searching for local minima and maxima in the series that meet certain censoring rules, regarding the length and the phase of the cycle. This approach is still employed by the NBER and the Euro area business cycle dating committees. Claessens et al. (2011) and Claessens et al. (2012) use this method to measure financial cycles for a large set of countries. In particular, they detect peaks and troughs in credit, equity prices and property prices- financial time series that are thought to be associated with the financial cycle. One of their main findings is that cycles in these series tend to be longer and more ample compared to business cycles. Furthermore, their results indicate that cyclical fluctuations in credit and housing prices are highly synchronized (however, this is not the case for equity prices). Moreover, they find evidence that business and financial cycles are strongly interrelated.
6 The second methodology is based on the spectral analysis of economic time series by utilizing statistical, ‘non-parametric’, frequency-based filters. Such filters include the well-known Hodrick and Prescott filter (1997) and the bandpass filters of Baxter and King (1999) and Christiano and Fitzgerald (2003). The main function of bandpass filters is to isolate the component of a time series that lies within a certain band of frequencies. Therefore, such filters are used to separate the trend from the cyclical dynamics in a time series by pre-specifying the range of cycle frequencies to be extracted. Aikman et al. (2015) use the Christiano and Fitzgerald filter in order to investigate the relationship between the credit cycle and the business cycle. Their analysis is focused on estimating spectral densities of real and financial variables, namely real GDP growth, real money aggregates, real bank loan growth and real bank asset growth. Their sample consists of 12 developed countries and long time series ranging from 1880 to 2008. They find that the credit cycle is a well-defined empirical regularity. Medium term fluctuations (about 8 to 20 years) account for the most part of the overall variation of credit-related variables, compared to short-term fluctuations consistent with business cycle frequencies (2 to 8 years), which play a negligible role to the total variation of these variables. Furthermore, the cyclical components of credit-related variables have amplitude that is roughly five times that of fluctuations in real GDP over the short term. As a result, the credit cycle can be characterized as a phenomenon distinct from the business cycle, in terms of amplitude and frequency. Moreover, the authors conclude that credit booms usually foreshadow banking crises.
In a prominent study, Drehmann et al. (2012) combine both approaches in an attempt to reach sharper and more robust conclusions. They apply both the turning point analysis and the bandpass filter of Christiano and Fitzgerald on a broad range of variables that potentially capture the financial cycle, namely credit, credit-to-GDP ratio, property prices, equity prices and an aggregate asset price index, which combines property and equity prices. Their sample consists of seven developed countries over the period 1960-2011. They identify four main empirical findings. First, the financial cycle can be adequately captured by medium-term cyclical fluctuations in credit and property prices, which are highly correlated and explain most of the total variation in these variables compared to short-term components. On the other hand, equity prices and aggregate asset prices co-move much less with the former variables. Second, the average length of the financial cycle is around 16 years. However, financial cycles seem to have become longer and more ample after the mid-1980’s. Third, systemic banking crises are closely linked to the financial cycle, since all of them occur at, or close to the peak of the financial cycle. Fourth, the business and the financial cycle are two different, but interacting phenomena. In particular, economic recessions are more severe when they coincide with the contraction phase of the financial cycle.
Mendoza and Terrones (2008) also analyze interactions between credit and economic cycles by combining the Hodrick and Prescott filter with their proposed ‘threshold method’, for a large number of countries over the period 1960-2006. According to this approach, after the trend and cyclical components of credit-related variables have been extracted with the HP filter, a credit boom is identified as an event in which credit exceeds its long-term trend by more than a given ‘boom’ threshold. This threshold is proportional to the standard deviation of credit over the business cycle. The authors find that credit booms exhibit a procyclical behavior, meaning that the build-up phase of a credit boom is associated with periods of economic expansion, while the contraction phase is linked to economic recessions.
7 The third approach of measuring business and financial cycles employs structural time series models to extract cyclical components from macroeconomic and financial time series. The underlying philosophy of this methodology involves the decomposition of a time series into unobserved trend, cyclical and irregular components, which are explicitly modeled according to appropriate stochastic processes. These models can be easily casted into state space form, and therefore, filtration techniques, such as the Kalman filter, can be used to estimate the unobserved components of the time series. Furthermore, the parameters of the model can be estimated through the Maximum Likelihood method, where the likelihood function is computed via the Kalman filter. Azevedo et al. (2006), as well as Creal et al. (2010) create business cycle indicators by utilizing a multivariate unobserved components model, in order to extract a common cyclical component (with a fixed frequency) from a large set of macroeconomic time series. Both papers incorporate the phase shifting mechanism of
Rünstler (2004), which allows for modelling potential lead/lag relationships among
the economic time series.
The former study was conducted for the Euro area and the unknown parameters of the model where estimated via the Maximum likelihood method. The latter study was conducted for the U.S and the parameters of the model were estimated through Bayesian methods. In addition, Creal et al. (2010) incorporate stochastic volatility processes and mixture distributions for the disturbances of the irregular and cyclical components. Stock and Watson (1989, 1999), Kim and Nelson (1998) and Diebold and Rudebusch (1996) follow similar procedures by constructing business cycle indicators based on factor models that contain a common cycle component. Forni et al. (2000), Forni et al. (2001) and Altissimo et al. (2007) combine factor models with the principal component analysis to build business cycle indicators.Koopman and Lucas (2005) were among the first researchers that applied structural time series models on financial variables. In particular, they estimate a tri-variate unobserved components model using U.S data on real GDP, credit spreads and business failure rates from 1933 to 1997, in order to identify correlations between the credit cycle and the business cycle. In their model, they distinguish between two types of cyclical components, one with a shorter period of around 6 years and one with a longer period of approximately 11 years. They identify a negative correlation between the cycles in GDP and the aforementioned financial variables mainly at the lower frequencies. At the higher frequencies, the co-cyclicality between these variables is less well-defined. Similarly, Galati et al. (2016) apply a multivariate unobserved components model to a set of financial variables, namely credit, credit-to-GDP and property prices, in order to measure financial cycles for the U.S and the Euro area over the period 1970 to 2014. They identify the financial cycle as a medium-term phenomenon with a period ranging from 8 to 25 years, which exhibits more ample fluctuations compared to the business cycle. Furthermore, they find evidence of heterogeneity within a financial cycle over time (e.g significant differences in the amplitude of U.S cycles over time), but also between financial cycles of different countries. Moreover, they identify that the cyclical components in credit, credit-to-GDP and property prices share some common statistical properties (e.g persistence and period), therefore they characterize these cycles as ‘similar’. Rünstler and Vlekke (2016) employ a multivariate structural time series model to extract cyclical components from GDP, credit volumes and housing prices (allowing for possible phase shifts between the time series), in an attempt to investigate the relationship between the business and the financial cycle. The study was conducted for the U.S and the Euro area over the period 1973 to 2014. The authors find that the financial time series under analysis exhibit low frequency cycles with large amplitudes. As in Galati et al. (2016),
8 heterogeneity across the financial cycles of different countries is also detected. Furthermore, they provide evidence that cycles in financial variables are highly correlated with medium-term cycles in GDP. Koopman et al.(2016) also use a multivariate model-based filter in order to extract two common cyclical components from a set of economic and financial variables, aimed to capture business cycle fluctuations and financial cycle dynamics respectively. Their empirical findings indicate the existence of a financial cycle which length is approximately twice the length of a regular business cycle. Furthermore, they find that credit-related variables load heavily on the financial cycle, while property prices follow their own cyclical dynamics and do not depend significantly on neither the business nor the financial cycle.
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3. Methodology
In this section, I will describe the unobserved components time series model (UCM), as formulated in the works of Durbin and Koopman (2001) and Koopman and Ooms (2010). This is the core model I employ in my thesis to extract cyclical components from a set of macroeconomic and financial variables that will be discussed in detail in the following section. The main characteristic of this class of models is that they are based on the decomposition of a time series into trend, seasonal, cycle, irregular and other relevant components, which are considered to be unobserved. All the variables used in the analysis have been seasonally adjusted first, therefore I do not include seasonal components in the model. The evolution of each of these components over time is modeled separately using appropriate stochastic processes. The trend represents a low frequency component of the time series, describing long term developments, while the cycle represents a high frequency component, which, in economic time series, can be associated with the movement of the business cycle. The irregular component constitutes the random noise in the signal. Such a decomposition of a time series allows for better understanding of the dynamic features of the time series and how these features change over time. These structural time series models are in sharp contrast to the philosophy underlying the Box-Jenkins ARIMA models, which require removing these components by differencing the time series, before the analysis takes place. The effectiveness of UCM compared to ARIMA models is discussed by Harvey, Koopman and Penzer (1999). Particularly, they emphasize that a UCM is preferred to an ARIMA model, when messy features are present in the time series, such as missing values, mixed frequencies, outliers, structural breaks and non-linear, non-Gaussian aspects.
3.1. Univariate Unobserved Components Time Series Model
First, I will introduce the univariate unobserved components model. This is to ensure that there is a clear picture of the underlying mechanics of this methodology and that the transition to the multivariate case will be a lot smoother. A UCM can be formulated in various ways. A specification that is relevant in this case is
𝑦𝑡 = 𝜇𝑡+ 𝜓𝑡+ 𝜀𝑡, 𝜀𝑡~𝑁𝐼𝐷(0, 𝜎𝜀2), 𝑡 = 1,2, … … , 𝑛 (1)
where 𝑦𝑡, 𝜇𝑡, 𝜓𝑡, 𝜀𝑡 represent the time series, trend, cycle and irregular components
respectively. The number of observations is n and the disturbance terms are normally distributed and serially uncorrelated.
The trend component is usually modeled in empirical literature as a random walk process with a stochastic drift, reflecting the fact that most macroeconomic time series usually have a maximum order of integration equal to 2. The specification of the trend component is given by the equations
𝜇𝑡+1 = 𝜇𝑡+ 𝛽𝑡+ 𝜉𝑡, 𝜉𝑡~𝑁𝐼𝐷(0, 𝜎𝜉2) (2)
10 where the error terms 𝜉𝑡 and 𝜁𝑡 are normally distributed, serially uncorrelated and mutually
independent of each other and all the disturbances present in equation (1). The slope component 𝛽𝑡 is a random walk (stochastic drift) and the trend is specified as an I(2) process.
The initial variables 𝜇1, 𝛽1 are considered to be unknown parameters that need to be
estimated together with the unknown variances 𝜎𝜉2 and 𝜎𝜁2. If we impose the restriction
𝜎𝜁 = 0 then the trend component follows a random walk with a constant drift. If one would
want to enforce further smoothness to the trend, this could be done by setting 𝜎𝜉= 0, in
which case the trend is modeled as an integrated random walk. If both variances of the disturbances are set to zero, then we obtain a linear deterministic time trend. Even further smoothness can be achieved by modeling the trend as a higher order stochastic process I(k) for k>2 by using the specification 𝛥𝑘𝜇𝑡+𝑘= 𝜁𝑡, where 𝛥𝑘 is the 𝑘𝑡ℎ order difference operator.
The next important step is to specify an appropriate process for the cycle component 𝜓𝑡. In its
simplest form, 𝜓𝑡 has a deterministic nature given by a pure sine wave function of the form
𝜓𝑡 = 𝜓̃ cos 𝜆𝑡 + 𝜓̃ sin 𝜆𝑡 ∗
where λ is the frequency of the cycle measured in radians, 2π/λ is the period of the cycle and 𝜓̃ , 𝜓̃ are constants. However, cycles in their pure form rarely, if ever, capture economic ∗
phenomena adequately. Therefore, more economically meaningful formulations need to be considered. Writing the simple deterministic cycle in recursive form, adding random noise and introducing a damping factor for additional modeling flexibility, yields a stochastic generalization of the cyclical component 𝜓𝑡, which has a fixed period but time-varying
amplitude and phase. The stochastic cycle model was first introduced by Harvey (1989) and is formulated as [𝜓𝑡+1 𝜓𝑡+1∗] = 𝜑 [ cos 𝜆 sin 𝜆 − sin 𝜆 cos 𝜆] [ 𝜓𝑡 𝜓𝑡∗] + [ 𝜅𝑡 𝜅𝑡∗] (4)
where φ is the persistence parameter or ‘damping factor’ which is restricted to be 0 < φ < 1 so that the cycle 𝜓𝑡 is modeled as a stationary process. The parameter λ is restricted in the closed
interval [0,π]. This is based on the essential frequency-domain concept of time series analysis that a stationary process (𝜓𝑡 in this case) can be decomposed into random components that
occur at frequencies in the interval [0,π], where each of these components is weighted according to its relative contribution to the composite signal. Thus, λ represents the central frequency at which the random components are distributed around. The persistence parameter φ describes the dispersion of the random components around the frequency λ. When φ is close to 0 then there is no clustering of the random components around the frequency λ, whereas when φ converges to 1, there is significant clustering around the central frequency, meaning a more persistent cycle. The stochastic cycle reduces to an AR(1) process if λ=0 or λ=π. The disturbance terms are serially uncorrelated and mutually independent with variances given by [𝜅𝜅𝑡
𝑡∗] ~𝑁𝐼𝐷(0, 𝜎𝜅 2𝐼
2). The distributions of the initial conditions are [
𝜓1
𝜓1∗] ~𝑁𝐼𝐷(0, 𝜎𝜅2 1−𝛷2𝐼2)
where 𝐼2 is a 2x2 identity matrix2. The variance of the cyclical component is 𝜎𝜓2 = 𝜎𝜅2 1−𝛷2. The
coefficients φ, λ and 𝜎𝜅2 are unknown and need to be estimated along with all the other
2
It is apparent that by restricting the damping factor φ between 0 and 1, it is ensured that the variance of the cycle 𝜎𝜅
2
11 unknown parameters. The variable 𝜓𝑡∗ is an auxiliary process that helps in the formulation of
𝜓𝑡.
Intervention effects can also be easily included in the UCM if appropriate. Intervention variables are usually introduced in the model in order to capture outliers or level breaks in the observed time series. Outliers are abnormally large innovations in the irregular component, while a level break is detected when there is a large disturbance in the trend or slope component. Level breaks and outliers can be detected by employing normality tests on the residuals of the level, slope and irregular components. Harvey (2006), Harvey et. al (2009) and Mendelssohn (2011) discuss about the merits of incorporating intervention variables for improving the performance of unobserved components models. Suppose that we would like to introduce an intervention at time τ due to an observed irregularity in the time series under scrutiny.
Trend breaks can be captured by defining the variable 𝑤𝑡= {1, 𝑡 ≥ 𝜏0 , 𝑡 < 𝜏 (5)
As regards outliers the intervention variable 𝑤𝑡 is defined as
𝑤𝑡 = {1, 𝑡 = 𝜏 0 , 𝑡 < 𝜏 , 𝑡 > 𝜏 (6)
For capturing changes in the slope component, 𝑤𝑡 is defined as
𝑤𝑡 = { 𝑡 − 𝜏, 𝑡 > 𝜏0 , 𝑡 ≤ 𝜏 (7)
Therefore, the model in equation (1) takes the form
𝑦𝑡 = 𝜇𝑡+ 𝜓𝑡+ 𝜃𝑡𝑤𝑡+ 𝜀𝑡, 𝜀𝑡~𝑁𝐼𝐷(0, 𝜎𝜀2), 𝑡 = 1,2, … … , 𝑛 (8)
where 𝜃𝑡 measures the effect of the intervention at time τ. Since the coefficient 𝜃𝑡 is time
invariant, it is specified by the equation
𝜃𝑡+1= 𝜃𝑡
Under the UCM framework multiple intervention variables can be introduced in the model to capture level breaks and outliers at different points in time.
3.2. Multivariate Unobserved Components Time Series Model
The univariate unobserved components time series model as developed in this section, can be generalized to a multivariate setting to allow for simultaneous analysis of multiple time series. Let’s consider again the structural model that consists of the equations (1), (2), (3) and (4). Its multivariate analog has the same formulation with the exception that 𝒚𝒕 represents a px1
vector of p different time series, while 𝝁𝒕, 𝜷𝒕, 𝝍𝒕, 𝝍𝒕∗ are also px1 vectors containing the
unobserved components of each time series. Furthermore, we have that 𝜺𝒕~𝑁(𝟎, 𝜮𝜺), , 𝝃𝒕~𝑁(𝟎, 𝜮𝝃), , 𝜻𝒕~𝑁(𝟎, 𝜮𝜻), [
𝜿𝒕
12 where 𝜺𝒕, 𝝃𝒕, 𝜻𝒕, 𝜿𝒕, 𝜿∗𝒕 are px1 vectors containing the error terms and 𝜮𝜺, 𝜮𝝃, 𝜮𝜻, 𝜮𝜿 are pxp
covariance matrices. The key feature of this multivariate version of the model, also called the seemingly unrelated time series equations model, is that each of the time series in the vector 𝑦𝑡 is modeled as in the univariate case, however, the disturbances driving the unobserved
components are allowed to be instantaneously correlated across the different time series, therefore creating a connection between them.
The dynamic properties of the multivariate model clearly depend on the specification of the covariance matrices. A specification particularly suitable for the analysis of many economic phenomena is when the covariance matrices of the disturbances do not have full rank, meaning 𝑟𝑎𝑛𝑘 (𝜮𝒊) = 𝑟𝑖 < 𝑝, 𝑤𝑖𝑡ℎ 𝑖 = 𝜉, 𝜁, 𝜅. In this case, the model contains only 𝑟𝑖
underlying components (e.g trends and/or cycles) instead of p, which are called common components. Since the purpose of the current study is to create a multivariate model-based filter in order to extract a common cycle component from a set of economic variables, I will only impose common cycles in the model and in particular one common cycle for the p time series under scrutiny. In other words, I restrict the covariance matrix 𝜮𝒌 to have a rank
𝑟𝑘 = 1 (see Carvalho, Harvey and Timbur (2007)). Following a methodology similar to
Azevedo, Koopman and Rua (2006), who signal extract a business cycle indicator from a panel of macroeconomic time series, the multivariate model-based filter employed in this study is formulated as
𝑦𝑖𝑡𝑐 = 𝜇𝑖𝑡 + 𝛿𝑖𝜓𝑡+ 𝜃𝑖𝑡𝑤𝑖𝑡+ 𝜀𝑖𝑡, 𝑖 = 1, … … , 𝑝 , 𝑡 = 1, … … , 𝑛 (10)
where 𝑦𝑖𝑡𝑐 is the 𝑖𝑡ℎ time series included in the 𝒚𝒕𝑐 vector of country c of the sample, 𝜇𝑖𝑡 is the
individual trend component for the 𝑖𝑡ℎ time series, 𝜓
𝑡 is the common cycle component
(independent of the index i) for all the time series and 𝑤𝑖𝑡 is an intervention variable
introduced in the 𝑖𝑡ℎ time series. For simplicity of notation I only illustrate the inclusion of one intervention variable in the model. However, more than one interventions were included in the model in order to account for level breaks and outliers in the time series. The variable 𝑤𝑖𝑡 can be defined according to equations (5), (6) or (7), depending on the type of
intervention I choose to introduce. The coefficient 𝛿𝑖 measures the contribution of the cycle
component to each one of the time series. Furthermore, 𝛿𝑖 provides a useful and intuitive
interpretation for the extracted cyclical components. As I will explain in detail in the following section (Data section), all the macroeconomic and financial variables included in the sample are expressed in logarithms or percentages. Therefore, the common cyclical component captures the percentage deviation of the level of each variable 𝑖 from its long-term trend, adjusted for its cycle coefficient 𝛿𝑖. In other words, if the common cyclical component
is multiplied by the cycle coefficient of the variable 𝑖, then the resulting cyclical component measures the percentage deviation of that particular variable from its trend.
As regards the extraction of the business cycle indicator, the unobserved trend components are specified according to the integrated random walk model, similarly to the univariate case
𝜇𝑖,𝑡+1= 𝜇𝑖𝑡 + 𝛽𝑖𝑡 (11)
13 As far as the extraction of the financial cycle indicator is concerned, the unobserved trend components are modeled as 4th order trends in order to capture the considerably smooth trend that is observed in the financial time series under analysis, given by the equation
𝛥4𝜇
𝑖,𝑡+4= 𝜁𝑖𝑡 (13)
The above definition of higher order trends in time series is discussed by Harvey and Trimbur (2003) as well as Galati et al. (2016). Defining the smoothness of the trend determines how much fluctuation in the variable is assigned to it, compared to the cycle. Therefore, if a variable exhibits smooth patterns (as housing prices do for example) then a higher order trend is suggested to adequately capture the cyclical dynamics.
The coefficient of the intervention variable is time invariant, therefore it is modeled by the equation
𝜃𝑖,𝑡+1 = 𝜃𝑖𝑡 (14)
The cycle component, concerning the extraction of both business and financial cycle indicators, is modeled following the standard approach by Harvey (1989) as a time-varying stochastic trigonometric process. Its specification is exactly the same as in equation (4) described previously, along with all the relevant assumptions mentioned. As a reminder for the reader, the stochastic cyclical component is specified as
[𝜓𝑡+1 𝜓𝑡+1∗ ] = 𝜑 [ cos 𝜆 sin 𝜆 − sin 𝜆 cos 𝜆] [ 𝜓𝑡 𝜓𝑡∗ ] + [𝜅𝜅𝑡 𝑡∗] (4)
As regards the error terms of the model, they are formulated as in (9) with the exception that the covariance matrices 𝜮𝜺 and 𝜮𝜻 contain non-zero values only in the main diagonal where
the variances of the disturbances are located. This is derived from the fact that the idiosyncratic errors 𝜀𝑖𝑡, as well as the disturbances associated with the slope component
𝜁𝑖𝑡 are assumed to be serially and mutually independent of each other. In addition, 𝜮𝝃= 𝟎,
where 𝟎 is a pxp matrix with zero elements, since no disturbances are introduced in the trend equation (11). Furthermore, 𝜮𝜿= 𝜎𝜅2 since the cycle component is common across the p time
series (the covariance matrix 𝜮𝜿 reduces to a scalar). The multivariate model with common
cycles as described here, can be interpreted as a dynamic factor model, where the latent factors can be interpreted as trend and cycle components due to their dynamic specification.
3.3. State Space representation and estimation method of the model
The unobserved components time series model belongs in a general class of models that can be represented in state space form. The general linear Gaussian state space form consists of the observation equation and the state equation. Its mathematical formulation is as follows
𝒚𝒕 = 𝒁𝒕𝜶𝒕+ 𝜺𝒕, 𝜺𝒕~𝑁(𝟎, 𝑯𝒕) (𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛)
𝜶𝒕+𝟏= 𝑻𝒕𝜶𝒕+ 𝑹𝒕𝜼𝒕, 𝜼𝒕~𝑁(𝟎, 𝑸𝒕), 𝑡 = 1, … … 𝑛 (𝑠𝑡𝑎𝑡𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛)
where 𝒚𝒕 is a px1 vector containing the observed time series and 𝜶𝒕 is an unobserved mx1
vector called the state vector. The main philosophy underlying state space models is that, the state of a system at a certain point in time is defined by the state vector, while the evolution of the system over time is governed by the state equation. However, because the state vector is
14 unobserved, we must base our inference regarding the evolution of the system by means of the observation vector. The matrix 𝒁𝒕 is a linear transformation that maps the state vector to
the observation vector. The observation equation can be seen alternatively as a linear regression equation where the coefficients vector 𝜶𝒕 is time variant. The matrix 𝑻𝒕 is called
the transition matrix and it describes the linear fashion in which the system moves from one state to the other. The Markovian nature of the state space models becomes apparent by interpreting the state equation as a first order vector autoregressive model. The vectors 𝜺𝒕 and
𝜼𝒕 contain the error terms which are considered to be serially independent. The error terms
between the observation and state equations respectively are mutually independent at all points in time. The matrices 𝑯𝒕 and 𝑸𝒕 are covariance matrices, while the matrix 𝑹𝒕 is usually
an identity matrix.
The multivariate unobserved components model as defined previously, regarding the extraction of the business cycle indicator of each country included in the sample, can be put in state space form in the following way3:
t t t t t t t t t t t t t t t t t t t tw
w
w
y
y
y
3 2 1 3 2 1 3 2 1 3 2 1 3 3 2 2 1 1 3 2 1*
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
, with 𝑯𝒕= 𝜮𝜺 = [ 𝜎𝜺𝟐𝟏 0 0 0 𝜎𝜺𝟐𝟐 0 0 0 𝜎𝜺𝟑𝟐 ] (Observation equation) 3As it will be discussed in the Data section, the dimension of the observation vector is p=3, since 3 macroeconomic time series are used for the extraction of the business cycle indicator. Therefore, the state vector 𝒂𝑡 has a dimension m=11.
15
0
0
0
*
0
0
0
*
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
cos
sin
0
0
0
0
0
0
0
0
0
sin
cos
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
1
0
0
1
*
3 2 1 3 2 1 3 2 1 3 2 1 1 , 3 1 , 2 1 , 1 1 1 1 , 3 1 , 2 1 , 1 1 , 3 1 , 2 1 , 1 t t t t t t t t t t t t t t t t t t t t t t t t t t t tR
,with 𝑸𝒕= 𝑑𝑖𝑎𝑔(𝟎3𝑥3, 𝜮𝜻, 𝜎𝜿𝟐𝜤𝟐, 𝟎𝟑𝒙𝟑) (State equation)
where 𝑑𝑖𝑎𝑔 refers to a block diagonal matrix and 𝑅𝑡 is an identity matrix.
The multivariate unobserved components model associated with the extraction of the financial cycle indicator can be casted into state space form in a similar fashion. The only difference is that the trend component has a slightly different specification, as given by equation (13). We can see that the state vector 𝜶𝒕 contains all the unobserved components of the model4.
The advantage of using the state space form representation is that we can apply filtration techniques, such as the Kalman filter in order to estimate the unobserved components of the model. According to Harvey (2004), once the unobserved components model has been casted into state space form, the Kalman filter yields estimators of the components based on current and past observations. However, the signal extraction relies on all the information included in the sample. Specifically, the estimation of the components is the product of the forward iteration of the Kalman filter recursions and the backward iteration of relevant smoothing recursions, which run backwards from the last observation. Furthermore, the Kalman filter can be used to compute the likelihood function. Through the maximization of the log-likelihood function (Maximum Likelihood estimation method) we can obtain estimates for the unknown parameters of the model, namely the variances of the disturbances, the frequency and damping factor of the cyclical component, the cycle coefficients and the initial conditions for the state equations. A more thorough explanation and formal proofs of the Kalman filter can be found in Harvey (1989) and Durbin and Koopman (2001).The maximization of the log-likelihood function with respect to a vector γ that contains the unknown parameters of the model, can be done using numerical methods, such as the Broyden-Fletcher-Goldfarb-Shannon (BFGS) method, which is generally regarded as efficient in terms of convergence speed and numerical stability. More information about the Maximum Likelihood estimation method can be found in Durbin and Koopman (2001).
4
The coefficients of the interventions variables are also included in the state vector 𝒂𝒕, whereas the
16
4. Data
According to empirical literature, real GDP is one of the most commonly used variables for capturing the overall business cycle (see Stock and Watson (1999)). However, since the nature of the model I employ in this thesis is multivariate, it allows for a simultaneous analysis of multiple macroeconomic time series, to extract the desired business cycle indicator. I create a coincident business cycle indicator, using three coincident macroeconomic variables commonly used in Euro area economic monitoring, namely the real GDP, industrial production and retail trade sales5. Ideally, one would want to use long time series from a large set of variables, including leading and lagging variables as well, such as unemployment, interest rates, inflation, several confidence indicators (construction confidence indicator, consumer confidence indicator etc, see Azevedo, Koopman and Rua (2006), Creal, Koopman and Zivot (2010)), in order to extract a more complete measure for the business cycle. Nevertheless, data limitations constrain the length of the time series, the number of variables that can be used as well as the amount of countries that can be included in the analysis. As a result, the use of the three aforementioned coincident variables seems to be the best option for this study, in terms of homogeneity in the length of the time series, in their frequencies, in their nature as economic indicators (coincident) and in the number of countries for which these variables are available. Following the same logic, I approximate the financial cycle by using credit, credit-to-GDP and housing prices. These variables have been widely used in the literature on financial cycle measurement (see Galati, Hindrayanto, Koopman and Vlekke (2016), Koopman, Lit and Lucas (2016)), although there is still not a clear consensus amongst academics on which variables are the best proxies for the financial cycle. The variable ‘credit’ measures the outstanding amount of credit at the end of each quarter and it covers core debt (e.g loans), debt securities, currency and deposits. The longest available time series on credit that I eventually used for the purpose of my research, refers to lending directed by all sectors of the economy (i.e banks, non-financial corporations etc) to the private non-financial sector. The ‘housing prices’ variable is comprised of residential property prices (indexed to the year 1995). The analysis is carried out for five of the largest economies in the Euro area, namely Finland, France, Italy, Spain and the Netherlands, as well as the United Kingdom. The set of countries was selected according to the size of their respective economies, their geographical position in Europe, in order to assemble a diverse and representative sample and data availability. As regards the sources of the business cycle variables, the real GDP time series was extracted from the Quarterly National Accounts database of the OECD, the industrial production index from the IMF International Financial Statistics database and the retail trade sales index from the Federal Reserve Bank of St. Louis (FRED) website. As far as the financial cycle variables are concerned, all of them were obtained from the Bank of International Settlements (BIS) macroeconomic database. All variables are documented in a quarterly frequency and have been seasonally adjusted using the X-12-ARIMA filter. Furthermore, they are expressed in logs, so as to remove (potential) exponential growth patterns and to approximately linearize the series, except for the credit-GDP ratio, which is already expressed as a percentage. The sample period for all countries is from 1976Q1 to 2016Q2.
5 The total retail trade sales index variable was not available for Italy and Spain for the time period 1976Q1-2016Q2. Therefore, I used the variable retail trade sales: passenger car registrations index, for which data were available.
17 Given all the information provided in the Methodology and Data sections, it is now apparent that the observation vector 𝒚𝒕𝒄 regarding the estimation of the business cycle indicator has a
dimension p=3 and is defined as 𝒚𝒕𝒄= [
ln (𝐼𝑃𝐼) ln (𝐺𝐷𝑃) ln (𝑅𝑇𝑆𝐼)
]. The corresponding observation vector related to the extraction of the financial cycle indicator also has a dimension p=3 and is defined as 𝒚𝒕𝒄= [
ln (𝐶𝑟𝑒𝑑𝑖𝑡) 𝐶𝑟𝑒𝑑𝑖𝑡/𝐺𝐷𝑃
ln (𝐻𝑃𝐼)
]. IPI, RTSI and HPI stand for Industrial Production Index, Retail Trade Sales Index and Housing Prices Index respectively. The country superscript c takes the following values
𝑐 = 𝐹𝑖𝑛𝑙𝑎𝑛𝑑, 𝐹𝑟𝑎𝑛𝑐𝑒, 𝐼𝑡𝑎𝑙𝑦, 𝑆𝑝𝑎𝑖𝑛, 𝑡ℎ𝑒 𝑁𝑒𝑡ℎ𝑒𝑟𝑙𝑎𝑛𝑑𝑠, 𝑈𝑛𝑖𝑡𝑒𝑑 𝐾𝑖𝑛𝑔𝑑𝑜𝑚
5. Empirical Results
5.1. Estimates of the multivariate model-based filter
In this section I will analyze in depth and interpret the empirical findings of this study. As a first step, I will present the main Maximum Likelihood estimates of the multivariate unobserved components models employed in this study, as well as graphs of the extracted business and financial cycle indicators. An analysis of this first group of results will follow directly after. The estimated cyclical components are scaled with respect to the cycle coefficients of log(GDP) and log(HPI). This means that the extracted business cycle indicators measure the percentage deviation of the level of GDP from its long-term trend, while the financial cycle indicators measure the percentage deviation of the level of HPI from its long-term trend. Furthermore, the intervention variables that were included in the model, along with their estimated coefficients and their standard errors are presented in the appendix. The software STAMP 8.2 is able to automatically select interventions for a given model. First, it identifies a set of potential outliers and breaks based on diagnostic checking and normality tests (Bowman Shelton) performed on the residuals of the multivariate model. Nextly, the software chooses to include in the model the interventions that are sufficiently significant.
Main Business Cycle Estimates
FIN FR IT ES NL UK φ 0.959 [0.908, 0.983] 0.900 [0.822, 0.946] 0.932 [0.887, 0.960] 0.959 [0.904, 0.983] 0.972 [0.933, 0.988] 0.941 [0.887, 0.970] λ 0.309 0.344 0.391 0.426 0.156 0.293 Period (in years) 5.09 [4.15, 6.28] 4.57 [3.27, 6.49] 4.02 [3.46, 4.69] 3.68 [3.24, 4.20] 10.08 [8.08, 12.59] 5.36 [4.28, 6.74] 𝝈𝝍𝟐 0.000158 [6.92e-005, 0.000362] 4.91e-005 [2.42e-005, 9.93e-005] 0.000132 [7.74e-005, 0.000226] 3.11e-005 [1.38e-005, 7.01e-005] 0.000363 [0.000158, 0.000833] 0.000150 [7.35e-005, 0.000304] 𝑹𝑫𝟐 (GDP) 0.660 0.256 0.472 0.794 0.600 0.303 (IPI) 0.373 0.390 0.472 0.378 0.408 0.495 (RTSI) 0.363 0.127 0.636 0.577 0.438 0.269 Table 1
18
Main Financial Cycle Estimates
FIN FR IT ES NL UK φ 0.980 [0.957,0.991] 0.996 [0.983, 0.999] 0.989 [0.973, 0.996] 0.9988 [0.9950, 0.9997] 0.997 [0.982, 0.9996] 0.990 [0.960, 0.997] λ 0.231 0.127 0.170 0.101 0.163 0.131 Period (in years) 6.79 [6.23, 7.41] 12.41 [11.25, 13.70] 9.23 [7.75, 11] 15.54 [14.56, 16.59] 9.66 [8.17, 11.43] 11.98 [9.88, 14.55] 𝝈𝝍𝟐 0.00330 [0.00150, 0.00725] 0.00466 [0.00109, 0.0199] 0.00276 [0.00107, 0.00706] 0.0347 [0.00658, 0.183] 0.0205 [0.00332, 0.126] 0.00879 [0.00229, 0.0338] 𝑹𝑫𝟐 (Credit) 0.0577 0.174 0.183 0.226 0.266 0.375 (Credit/ GDP ) 0.433 0.391 0.624 0.703 0.520 0.493 (HPI) 0.758 0.836 0.875 0.803 0.751 0.664 Table 2
Coefficients (𝜹𝒊) for Business Cycle
FIN FR IT ES NL UK GDP 1.95** (0.0884) 1.05** (0.00791) 1.03** (0.00423) 1.96** (0.0548) 1.18** (0.0328) 1.51** (0.0527) IPI 2.84** (0.395) 2.35** (0.188) 2.19** (0.140) 3.90** (0.381) 1.48** (0.221) 1.53** (0.0820) RTSI 1.11** (0.143) 0.793** (0.124) 2.29** (0.565) 7.32** (1.51) 1.37** (0.165) 0.956** (0.130) Table 3
Coefficients (𝜹𝒊) for Financial Cycle
FIN FR IT ES NL UK Credit -0.183 (0.202) 0.0760 (0.371) -0.120 (0.335) 0.993** (0.188) 0.727** (0.251) 0.571** (0.194) Credit/GDP -19.85** (8.09) -6.22 (7.20) -3.38 (5.81) 22.13** (5.43) 0.232 (9.20) -4.99 (7.61) HPI 1.053** (0.0557) 1.054** (0.0520) 0.999** (0.0212) 1.23** (0.0350) 1.09** (0.0488) 0.993** (0.0378) Table 4
Description of parameter notation: φ refers to the damping factor, λ refers to the frequency of the cycle and 𝜎𝜓2 refers to
the variance of the cycle. 𝑅𝐷2 is the coefficient of determination with respect to the total sum of squared differences and
measures the goodness of fit of the model to the data. An 𝑅𝐷2 value is given for each equation of the multivariate
models. The numbers in blue colour within square brackets refer to 68% asymmetric confidence intervals for the parameters, as produced by the STAMP 8.2 software. The numbers in parentheses refer to the standard errors of the estimated cycle coefficients and ** is used to indicate statistical significance of a coefficient at a 5% significance level.
19 Estimated Business Cycles
-.06 -.04 -.02 .00 .02 .04 .06 1980 1985 1990 1995 2000 2005 2010 2015 FINLAND -.03 -.02 -.01 .00 .01 .02 1980 1985 1990 1995 2000 2005 2010 2015 FRANCE -.04 -.03 -.02 -.01 .00 .01 .02 .03 1980 1985 1990 1995 2000 2005 2010 2015 ITALY -.04 -.03 -.02 -.01 .00 .01 .02 .03 1980 1985 1990 1995 2000 2005 2010 2015 SPAIN -.06 -.04 -.02 .00 .02 .04 .06 1980 1985 1990 1995 2000 2005 2010 2015 NETHERLANDS -.04 -.02 .00 .02 .04 .06 1980 1985 1990 1995 2000 2005 2010 2015 UNITED_KINGDOM
20 Estimated Financial Cycles
-.20 -.15 -.10 -.05 .00 .05 .10 .15 .20 1980 1985 1990 1995 2000 2005 2010 2015 FINLAND -.15 -.10 -.05 .00 .05 .10 .15 1980 1985 1990 1995 2000 2005 2010 2015 FRANCE -.12 -.08 -.04 .00 .04 .08 .12 1980 1985 1990 1995 2000 2005 2010 2015 ITALY -.5 -.4 -.3 -.2 -.1 .0 .1 .2 .3 .4 1980 1985 1990 1995 2000 2005 2010 2015 SPAIN -.2 -.1 .0 .1 .2 .3 1980 1985 1990 1995 2000 2005 2010 2015 NETHERLANDS -.16 -.12 -.08 -.04 .00 .04 .08 .12 .16 1980 1985 1990 1995 2000 2005 2010 2015 UNITED_KINGDOM
