Tracking the business cycle of the Euro area: a multivariate model-based band-pass filter

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1 Tracking the business cycle of the Euro area: a multivariate model-based band-pass filter João Valle e Azevedo, Siem Jan Koopman and António Rua Abstract This paper proposes a multivariate band-pass filter that is based on the trend plus cycle decomposition model. The underlying multivariate dynamic factor model relies on specific formulations for trend and cycle components and produces smooth business cycle indicators with band-pass filter properties. Furthermore, cycle shifts for individual time series are incorporated as part of the multivariate model and estimated simultaneously with the remaining parameters. The inclusion of leading, coincident and lagging variables for the measurement of the business cycle is therefore possible without a prior analysis of lead-lag relationships between economic variables. This method also permits the inclusion of time series recorded with mixed frequencies. For example, quarterly and monthly time series can be considered simultaneously without ad hoc interpolations. The multivariate approach leads to a business cycle indicator that is less subject to revisions than the ones produced by univariate filters. The reduction of revisions is a key feature in the real-time assessment of the economy. Finally, the proposed method computes a growth indicator as a by-product. The new approach of tracking business cycle and growth indicators is illustrated in detail for the Euro area. The analysis is based on nine key economic time series. Keywords: Band-pass filter; Business cycle; Dynamic factor model; Kalman filter; Unobserved components time series model; Phase shift; Revisions. JEL classification: C13, C32, E32. Corresponding author: Department of Econometrics, Vrije Universiteit Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands. s:

2 1 Introduction Undertaking fiscal and monetary policies requires information about the state of the economy. Given the importance of this knowledge for the policymaker, a clear economic picture needs to be present at any time. However, the assessment of the economic situation can be a challenging task when one faces noisy data giving mixed signals about the overall state of the economy. In order to avoid a solely judgemental based procedure in the economic analysis one should be able to extract the relevant information through a statistical rigorous method capable of providing a clear signal regarding current economic developments. A business cycle indicator aims to present deviations from a long term trend in economic activity. Such deviations typically last between 1.5 and 8 years. The time series of real Gross Domestic Product (GDP) is undoubtedly one of the most important variables for business cycle assessment. For example, Stock and Watson (1999) consider that... fluctuations in aggregate output are at the core of the business cycle so the cyclical component of real GDP is a useful proxy for the overall business cycle.... Common practice is to signal extract the cycle from real GDP using nonparametric filters such as the ones of Hodrick and Prescott (1997) and Baxter and King (1999). These filters are based on symmetric weighting kernels and provide consistent methods of extracting trends and cycles in the middle of the time series but encounter problems at the beginning and end of the series. The Baxter and King (1999) filter is referred to as a band-pass filter since it aims to extract the dynamics that are strictly associated with the business cycle frequencies 2π/p with p ranging from 1.5 to 8 years. The optimality of band-pass filters is evaluated in the frequency domain by investigating gain functions that should only affect the business cycle frequencies. Recently, Christiano and Fitzgerald (2003) have developed an almost optimal filter in this respect together with a solution to the endpoints problem. The signal extraction of a business cycle can also be based on formulating an unobserved components time series models for real GDP as in Clark (1987) and Harvey and Jaeger (1993). Recently, Harvey and Trimbur (2003) have presented a new class of model-based filters for extracting trends and cycles in economic time series. The main feature of these model-based filters is that it is possible to extract a smooth cycle from economic time series. It is shown that the implied filters from this class of models possess band-pass filter properties similar 2

3 to those of the Baxter and King (1999) filter. Further, model-based filters provide a mutually consistent way to extract trends and cycles even at the beginning and end of the series. However, the univariate approaches of band-pass and model-based filters do not exploit the information stemming from several economic time series. Recently, the work of Forni, Hallin, Lippi, and Reichlin (2000) and Stock and Watson (2002) have shown that business cycle and growth indicators can be obtained from hundreds of economic time series. Such an approach is in sharp contrast with the common practice of using univariate band-pass filters and univariate model-based methods for obtaining business cycle indicators. In this paper we aim to take a position between the all or one strategy by considering a moderate set of time series instead of considering real GDP only or considering a vast amount of time series. Our approach requires the maintenance of a moderate database of, say, nine time series. When considering trends and cycles in multivariate time series, we also need to focus on possible lead-lag relationships between variables and the business cycle. We therefore incorporate the phase shift mechanism of Rünstler (2004) for individual cycles with respect to the base cycle that is modeled as the generalised cycle component of Harvey and Trimbur (2003). This development enables the simultaneous modeling of leading, coincident and lagging indicators for the business cycle without their a priori classification. As a result, phase shifts are estimated as parameters of the model. This contrasts with the multivariate model-based analysis of Stock and Watson (1989) where the coincident indicator is defined as the common factor of a small set of coincident variables which are identified prior to estimation. The treatment of phase shifts in a model-based approach allows the exploitation of information contained in both lags and leads of variables for computing the business cycle indicator and it overcomes the restriction of only using coincident variables. Economic and financial policy decision makers require a business cycle indicator at a high frequency despite the fact that time series information is usually not available at the desired frequency. On the other hand, information recorded at a lower frequency must not be discarded. The crucial example refers to the time series of real GDP that is disregarded by Stock and Watson (1989) for the construction of its composite indicator presumably because it is only available on a quarterly basis. Others, such as Altissimo, Bassanetti, Cristadoro, Forni, Lippi, Reichlin, and Veronese (2001), handled this shortcoming by considering a monthly interpolated 3

4 GDP series. In the multivariate analysis of this paper observations of different frequencies can be incorporated since missing observations can be treated by the implemented state space methods. The ability to deal with missing observations also ensures the business cycle indicator to be up-to-date even when long delays in data releases are present. This feature is important given the significant data release delays in the Euro area. In this paper we develop a feasible procedure for the computation of business cycle indicators. The resulting business cycle filter is based on a multivariate dynamic factor model for a vector of economic and monetary time series recorded at quarterly and monthly frequencies. The dynamic specification of the model allows the latent factors to be interpreted as trend and cycle components. The common cycle component is formulated as in Harvey and Trimbur (2003) and can be shifted for individual variables as in Rünstler (2004). The signal extraction of the cycle component can be regarded as the result of a multivariate model-based approximate band-pass filter. The multivariate procedure is shown to produce a cycle indicator that is subject to smaller revisions compared to univariate band-pass filters and univariate model-based filters. Hence the real-time signal extraction has been improved against commonly used filters in business cycle analysis. This key feature is important for macroeconomic stabilisation policy decisions since reliable estimates of the business cycle position in real-time are crucial in this respect. We implement the multivariate model-based band-pass filter for the business cycle tracking of the Euro area. By pooling information from nine key economic variables (including GDP, industrial production, retail sales, confidence indicators, among others) it obtains a business cycle indicator in line with common wisdom regarding Euro area business cycle developments. The business cycle indicator can be regarded as a monthly proxy for the output gap. A growth indicator can also be constructed from our analysis as a by-product and is compared with the EuroCOIN indicator. This growth indicator of Altissimo, Bassanetti, Cristadoro, Forni, Lippi, Reichlin, and Veronese (2001) is based on a generalised dynamic factor model with almost a thousand economic series as input. The parsimonious approach proposed in this paper leads to a similar growth indicator and overcomes some drawbacks of EuroCOIN. The remaining of the paper is organised as follows. In section 2 we present the underlying dynamic factor model and its extensions. In section 3 we proceed to the empirical application 4

5 for the Euro area. Section 4 concludes. 2 Model for multivariate band-pass filtering Assume that a panel of N economic time series are collected in the N 1 vector y t and that we observe n data points over time, that is t = 1,...,n. The i-th element of the observation vector at time t is denoted by y it. 2.1 The basic trend-cycle decomposition model The basic unobserved components time series model for measuring the business cycle from a panel of economic time series is based on the decomposition model y it = µ it + δ i ψ t + ε it, i = 1,...,N, t = 1,...,n, (1) where µ it is the individual trend component for the ith time series and the business cycle component (common to all time series) is denoted by ψ t. For each series the contribution of the cycle is measured by the coefficient δ i for i = 1,..., N. The idiosyncratic disturbance ε it is assumed normal and independent from ε js for i j and/or t s. The variance of the irregular disturbance varies for the different individual series, that is ε it i.i.d. N(0, σ 2 i,ε), i = 1,..., N. (2) Each trend component from the panel time series is specified as the local linear trend component (see Harvey, 1989), that is i.i.d. µ i,t+1 = µ it + β it + η it, η it N(0, σi,η), 2 (3) i.i.d. β i,t+1 = β it + ζ it, ζ it N(0, σi,ζ 2 ), for i = 1,...,N. The disturbances η it and ζ it are serially and mutually independent of each other, and other disturbances, at all time points and across the N equations. The trends are therefore independent of each other within the panel. The cycle component is common to all time series in the panel and can be specified as an autoregressive model with polynomial 5

6 coefficients that have complex roots. Specifically, we enforce this restriction by formulating the cycle time series process as a trigonometric process, that is ψ t+1 = φ cosλ sin λ ψ t + κ t, ψ t+1 + sin λ cosλ ψ t + κ + t where κ t and κ + t κ t κ + t i.i.d. N(0, σ 2 κ ), i.i.d. N(0, σ 2 κ), are white noise disturbances and mutually independent of each other at all time points for t = 1,...,n. The damping term 0 < φ 1 ensures that the stochastic process ψ t is stationary or fixed as the cycle variance is specified as σ 2 κ = (1 φ 2 )σ 2 ψ where σ2 ψ (4) is the variance of the cycle. The period of the stochastic cycle is defined as 2π/λ where 0 λ π is approximately the frequency at which the spectrum of the stochastic cycle has the largest mass. The initial distribution of the cycle is given by ψ 1 i.i.d. N(0, σ 2 ψ ), ψ+ 1 i.i.d. N(0, σψ 2 ), (5) where σ 2 ψ = σ2 κ /(1 φ2 ). The dynamic properties of the cycle can be characteristed via the autocovariance function which is given by γ(τ) = E(ψ t ψ t τ ) = σ 2 κ φτ (1 φ 2 ) 1 cos(λτ), for τ = 0, 1, 2, Signal extraction and approximate band-pass filter properties The basic trend-cycle decomposition model falls within the class of multivariate unobserved components time series models or multivariate structural time series models which can be casted in state space form. The details of this general class of multivariate models are discussed by Harvey and Koopman (1997) including its relations with other multivariate approaches such as the common feature models of Engle and Kozicki (1993). Further, Harvey and Koopman (2000) discuss vector autoregressive model representations of unobserved component models which are related to the canonical correlation analyses of Tiao and Tsay (1985) and the multiple time series analyses of Ahn and Reinsel (1988). The unknown parameters such as variances associated with the disturbances of the model and loading factor coefficients δ i for the common cycle component are estimated by numerically maximising the likelihood function that is evaluated 6

7 by the Kalman filter. The signal extraction of unobserved trend and cycle components is carried out by the Kalman filter and associated smoothing methods. An introduction to such techniques is found in, for example, Harvey (1989) and Durbin and Koopman (2001). The parameters are identified It may be of interest to know how observations are weighted when, for example, the business cycle estimate at time t is constructed. Such weights can be obtained as discussed in Harvey and Koopman (2000). The implications of the filter in the frequency domain can be assessed via the spectral gain function, see Harvey (1993, Chapter 6). A filter is said to have band-pass filter properties when the gain function has sharp cut-offs at both ends of the range of business cycle frequencies. 2.3 Generalised components The basic trend plus cycle decomposition model (1) can be modified in such a way that the model-based filtering and smoothing method possesses band-pass filter properties, see Gomez (2001) and Harvey and Trimbur (2003). For example, the m-th order stochastic trend for series i can be considered as given by µ it = µ (m) it where m µ (m) i,t+1 = ζ it, ζ it i.i.d. N(0, σ 2 i,ζ), (6) or µ (j) i,t+1 = µ(j) it + µ (j 1) it, j = m, m 1,...,1, (7) with µ (0) it i.i.d. = ζ it (0, σi,ζ 2 ) for i = 1,..., N and t = 1,..., n. The trend specification (6) can be taken as the model representation of the class of Butterworth filters that are commonly used in engineering, see Gomez (2001). The special case of (6) with m = 2 reduces to the local linear trend component (3) with σi,η 2 = 0 from which it follows that β it = µ (1) it. The effect of a higher value for m is pronounced in the frequency domain since the low-pass gain function will have a sharper cut-off downwards at a certain low frequency point. In the time domain the effect is evident since m µ (m) i,t+1 = ζ it suggests that the time series y it implied by model (1) and (6) needs to be differenced m times to become stationary. From this perspective, the choice of a relatively large choice for m may therefore not be realistic for many economic time series. 7

8 ψ (k) t The generalised specification for the k-th order common cycle component is given by ψ t = where ψ(j) t+1 ψ +(j) t+1 = φ cosλ sin λ sin λ cosλ ψ(j) t ψ +(j) t + ψ(j 1) t ψ +(j 1) t, j = 1,..., k, (8) with ψ 0 t = κ t i.i.d. N(0, σκ 2 ), ψ+0 t = κ + t i.i.d. N(0, σ 2 κ ), for t = 1,...,n. A higher value for k leads to more pronounced cut-offs of the band-pass gain function at both ends of the range of business cycle frequencies centered at λ. For example, the model-based filter with k = 6 leads to a similar gain function as the one of Baxter and King (1999), see Harvey and Trimbur (2003). However, by increasing k, there is a trade-off between the sharpness of the implicit band-pass filter and its behaviour at the endpoints of the sample. A sharper cut-off implies larger revisions of the estimates near the endpoints since more weight is given to the (implicitly forecasted) observations in the far future. In the empirical study of this paper, a good compromise is found at k = 6 although smaller values of k did not lead to completely different estimates of the key parameters in the model. Different variations within this class of generalised cycles are discussed in Harvey and Trimbur (2003) who refer to specification (8) as the balanced cycle model. They prefer this specification since the timedomain properties of cycle ψ t are more straightforward to derive and it tends to give a slightly better fit to a selection of U.S. economic time series. The number of unknown parameters remain three for cycle specification (4), that is 0 < φ 1, 0 λ π and σκ 2 > 0, for any order k. 2.4 Phase shifts of business cycle In the multivariate case, with N > 1, multiple time series are considered for the construction of a business cycle indicator. The contribution of the ith time series to cycle ψ t is determined by loading δ i, for i = 1,...,N. In the current model (1) with generalised trend and cycle components only economic time series with coincident cycles are viable candidates to be included in the model. Prior analysis may determine whether leads or lags of economic variables are appropriate for its inclusion. To avoid such a prior analysis, the model is modified to allow 8

9 the base cycle ψ t to be shifted for each time series. For this purpose, we adopt the phase shift mechanism for a multivariate cycle specification as proposed by Rünstler (2004). For a given cycle component given by (4), it follows from a standard trigonometric identity that the autocovariance function of cycle ψ t is shifted ξ 0 time periods to the right (when ξ 0 > 0) or to the left (when ξ 0 < 0) by considering cos(ξ 0 λ)ψ t + sin(ξ 0 λ)ψ + t, t = 1,...,n. The shift ξ 0 is measured in real-time so that ξ 0 λ is measured in radians and due to the periodicity of trigonometric functions the parameter space of ξ 0 is restricted within the range 1 2 π < ξ 0λ < 1π. For example, it follows that when a cosine is shifted by xλ, its correlation with the base 2 cosine is the same as its correlation with a cosine that is shifted by (π x)λ but with an opposite sign, for 1 π < xλ < π. The shifted cycle is introduced in the decomposition model by 2 replacing equation (1) with the equation { } y it = µ (m) it + δ i cos(ξ i λ)ψ (k) t + sin(ξ i λ)ψ +(k) t + ε it, i = 1,..., N, t = 1,..., n, (9) where the specifications in (6), (8) and (2) remain applicable. Rünstler (2004) shows that the phase shift appears directly as a shift in the autocovariance function of the stochastic cycle. This remains to hold for model specification (9), see Appendix A for more details. We further restrict attention to a common cycle specification. It is assumed that the first equation identifies the base cycle and therefore ξ 1 = 0 and δ 1 = 1 so that y 1t = µ 1t + ψ t + ε 1t, t = 1,...,n. The common cycle ψ t coincides contemporaneously with the first variable in the vector y t. The restriction δ 1 = 1 can be replaced by the constraint of a fixed value for σκ. 2 The restriction ξ 1 = 0 allows the cycles of other variables to shift with respect to the cycle of the first variable. The general multivariate trend-cycle decomposition model can be formulated in state space with partially diffuse initial conditions. The details are given in Appendix B. 2.5 Some further issues The multivariate trend-cycle decomposition model contains a common cycle that is specified as a generalised cycle and is allowed to shift for every individual series with respect to a base cycle. 9

10 The generalised cycle specification is taken from Harvey and Trimbur (2003) and the phase shift mechanism is taken from Rünstler (2004). The trend and irregular components in the model are idiosyncratic factors that account for the dynamics in the non-cyclical range of frequencies in the time series. The model is casted in the state space framework. Therefore, variables with different observation frequencies in the panel can be handled straightforwardly using the Kalman filter, see Durbin and Koopman (2001, Section 4.6). For example, the treatment of quarterly time series using a monthly time index in the state space framework amounts to having two consecutive monthly observations as missing followed by the quarterly observation. There are alternative ways to account for mixed frequencies, see, for example, in Mariano and Murasawa (2003). Such solutions require nonlinear techniques since macroeconomic time series are often modeled in logs. Our main interest is to capture the cycle and growth dynamics. Therefore, the different solutions will have minor impact on the extracted cycle. The specification of the dynamic factor model (9) can be extended to a more general setting. Nevertheless, the key features of business cycle analysis are incorporated in the model. The generalised cycle enables us to interpret the extracted cycle as the result of an approximate band-pass filter to the reference series in the spirit of Harvey and Trimbur (2003). As a result, the model can be viewed partly as a filtering device and partly as a specification of the true data generating process. When we take the cycle of a particular series as the base cycle and we enforce the base cycle on the other series (adjusted for scale and phase) as in the proposed model specification, we effectively use information from the cyclical fluctuations of the other series only to the extent that they match those of the series that are originally attached to the base cycle. Subject to phase shift adjustments, the cyclical components of the various series are perfectly correlated. The band-pass filter interpretation is therefore warranted. We should note that cross-correlations (adjusted for phase) can only be equal to unity if the phase shift implies an integer shift in the cross-correlation function, see Appendix A. It is unlikely that an integer shift occurs in practice since the shift parameter is a typical non-integer value and needs to be estimated. These insights of the model indicate that the model is appropriate for the purpose of the analysis: to obtain a business cycle indicator that is close to the one obtained from a univariate band-pass filter but one that has better properties in terms of revision and availability at a higher frequency. 10

11 The identification of the parameters in the model is guaranteed as they all appear in the autocovariance function of the model in its reduced form. These results are standard for the multivariate unobserved components time series model of which our model without cycle phase shifts is a special case, see Harvey and Koopman (1997). The identification of parameters associated with cycle shifts is discussed by Rünstler (2004, Proposition 2) and requires an unique decomposition for the autocovariance function of the cycle component. The specification that we have adopted is a special and simplified case of the Choleski decomposition. 3 Empirical results for the Euro Area 3.1 Data The multivariate framework presented in the previous section allows the simultaneous modeling of leading, coincident and lagging indicators for the business cycle with phase shifts being estimated as an integrated part of the modeling process. However, as a preliminary step, one has to define the base cycle. We follow Stock and Watson (1999) who consider the cyclical component of real GDP as a useful proxy for the overall business cycle. Therefore, we impose a unit common cycle loading and a zero phase shift for Euro area real GDP. The remaining series included in the multivariate analysis are the most commonly used monthly indicators on Euro area economic monitoring. This includes industrial production, retail sales, unemployment, several confidence indicators (consumers, industrial, retail trade and construction) and interest rate spread. As it is well known, industrial production is highly correlated with the business cycle and it is coincident or even slightly leading. On the other hand, retail sales provide information about private consumption, which accounts for a large proportion of GDP. Unemployment is a countercyclical and lagging variable but it also displays a high correlation with the business cycle. Confidence indicators provide information regarding economic agents assessment of current and future economic situation so they are expected to present coincident or leading properties. Finally, the interest rate spread has been widely recognised as a forwardlooking indicator for real activity. These time series are available for the period between 1986 and 2002 and graphically presented in figure 1. The resulting dataset covers different economic 11

12 sectors and it is based on both quantitative and qualitative data as well as on both real and monetary variables. GDP is available on a quarterly basis whereas the other indicators are monthly. Appendix C provides a detailed description of the data GDP IPI Interest rate spread Construction confidence indicator Consumer confidence indicator Retail sales Unemployment (inverted sign) Industrial confidence indicator Retail trade confidence indicator Figure 1: Nine key economic variables for the Euro area between 1986 and Details of model for multivariate band-pass filtering The empirical study of the Euro area business cycle will be based on the multivariate model (9) with generalised trend µ it, given by specification (6) with m = 2, and common cycle ψ t, given by specification (8) with k = 6. The first time series in the panel is real GDP. The time index refers to months and applies to all series in the panel except for GDP that is observed every quarter. This implies that the monthly series of GDP has two consecutive monthly observations as missing followed by its quarterly observation. A preliminary analysis of the panel of time series using band-pass filters reveals that cyclical correlations of the various series with GDP ranges from 0.70 to This range of correlation 12

13 values is relatively high and we do not expect the model to capture this perfectly. However, it justifies, to some extent, the decomposition model with a common base cycle and with trend and irregular components that account for the other dynamics in the series. Harvey and Trimbur (2003) show that the choice of m = 2 and k = 6 leads to estimated trend and cycle components with band-pass filtering properties similar to the ones of Baxter and King (1999). The role of the trend component µ it in model (9) is to capture the persistence of the time series excluding the cycle. The order m of the trend (6) applies to all time series in the panel. The frequency of the common cycle component is fixed to assure that the cycle captures the typical business cycle period in the time series. The frequency is therefore chosen as λ = which implies a period of 96 months (8 years) and is consistent with Stock and Watson (1999) for the U.S. and the ECB (2001) for the Euro area. The frequency parameter is fixed to guarantee the extraction of typical business cycle fluctuations. When applying bandpass filters to a typical integrated series, the peak in the spectrum of the filtered series is reached at the lowest frequency of the band of interest. The value of λ is therefore chosen to correspond approximately with the location of the peak in the spectrum. The choice is not based on minimizing the difference of the gain function between an ideal filter and a filter that is implicitly given by the model. Further, we note that δ 1 and ξ 1, the load and shift parameters associated with the base series, GDP, are restricted to one and zero, respectively. Apart from these restrictions, the number of parameters to estimate for each equation is four (σ 2 i,ζ, δ i, ξ i and σ 2 i,ε) and for the common cycle is two (φ and σκ 2 ). The total number therefore is 4N (4 for each equation minus two restrictions plus two for the common cycle). The maximum likelihood estimates of the parameters are obtained by maximising the loglikelihood function that is evaluated through the Kalman filter. This is a standard exercise although computational complexities may arise because the parameter dimension rises quickly as N increases. For example, with N = 9 the number of parameters to be estimated is 36. In principle, there is no difficulty of being able to empirically identify the parameters from the data since the number of observations increases accordingly with N. It can however be computationally and numerically cumbersome to obtain the maximum value of the log-likelihood function in 13

14 a 36 dimensional space. A practical and feasible approach to overcome some numerical difficulties is to obtain starting values from univariate and bivariate models. The model dimension is then increased step by step and for each step realistic starting values will be available from the earlier low-dimensional models. This approach can be fully automated and different orderings in the panel of time series can be considered in order to be assured (at least, to some extent) that the likelihood maximum is obtained. Further, the maximisation process is stopped when specific parameter values become unrealistic. The maximisation is restarted at a different set of (realistic) values in order to safeguard the estimation process from producing unsettling model estimates. It should be noted that this has occurred a few times when the dimension of the model increases. Once the model parameters are estimated, the trends and the common business cycle can be extracted from the data using the Kalman filter and associating smoothing algorithms. The Kalman filter is sometimes known to be slow for multivariate models, especially when specific modifications are considered for the diffuse initialisations of nonstationary components such as trends. However, we would like to stress that the implementation of the Kalman filter for multivariate models as documented in Koopman and Durbin (2000) and implemented using the library of state space functions SsfPack was not slow nor numerically unstable, see Appendix B for further details. 3.3 The business cycle indicator The estimated components for the Euro area real GDP series are presented in figure 2. Several interesting features become apparent. First, although GDP is recorded on a quarterly basis the estimated components are monthly. These components can be seen as the outcome of the underlying monthly GDP decomposition which can be recovered resorting to the information contained in the remaining dataset. Second, by analysing the trend it seems that Euro area potential growth has declined after the major recession that occurred in This feature becomes clear by plotting the trend slope against the time axis. Before 1993 recession monthly potential growth was around 0.3 per cent (3.7 per cent in annualised terms) while afterwards it was closer to 0.2 per cent. This is 2.4 per cent in annualised terms and falls within the 14

15 per cent GDP potential growth range underlying the ECB monetary policy. Third, the resulting GDP cyclical component, that is our Euro area business cycle indicator, is in line with the common wisdom regarding Euro area business cycle developments (see ECB (2002)). In particular, a major recession was recorded in 1993 (as German unification boost delayed the impact of the recessions in the United States and the United Kingdom in the early 1990s) while there was an economic slowdown in the late 1990s following the crisis in Asian emerging market economies and the financial instability in Russia. (i) (ii) (iii) (iv) Figure 2: Decomposition of GDP: (i) GDP and trend; (ii) slope; (iii) cycle; (iv) irregular. The variances of the trends and the damping factor and standard deviation of the common cycle are estimated. For the latter parameters we obtained estimates ˆφ = 0.57 and ˆσ κ = , respectively. The variances of the individual trends are also estimated and they vary from very small values (suggesting that the trend is fixed and may coincide with the fact that the trend does not exist) to relatively high values (suggesting that trend and its corresponding slope change throughout the series). The estimated business cycle component is computed by the multivariate Kalman filter and 15

16 smoother which effectively weight the observations in an optimal way for signal extraction. The associating gain function is also multivariate but is difficult to interpret from a practical point of view. Therefore, figure 3 presents the gain function of the univariate filter as in Harvey and Trimbur (2003, section IV). The gain function is for the extraction of the monthly business cycle associated with GDP and is based on coefficients that are obtained from parameter estimates of the multivariate model. The cut-off at the lower frequencies is sharp while the cut-off at the higher frequencies is more smooth. The latter is partly due to the estimate ˆφ = 0.57 as it implies a somewhat less persistent cycle component. G(λ) λ 3.0 Figure 3: Gain function of the implied univariate filter for extracting the cycle for GDP at a monthly frequency. The gain coefficients are from parameter estimates of multivariate model. The approach of this paper can be taken as an attempt to extract a business cycle that is similar to the one obtained from a filter implied by the gain function of figure 3 but that is actually based on a dataset of monthly economic indicators as well as quarterly GDP. The motivations of this approach are to circumvent the non-availability of GDP at a monthly frequency and to obtain a clearer signal at the end of the sample. These objectives are of importance 16

17 for the policy maker. It should be stressed that figure 3 gives a distorted picture of the gain function at the end of the sample. However, by using multiple sources of information at a monthly frequency, this distortion is diminished. Furthermore, it may lead to smaller revisions of the signal at the end of the sample when new observations become available. Some key estimation results are presented in table 1. It is noted that we do not aim to develop a model that captures all dynamic characteristics in the available panel of time series. Therefore we do not present a detailed discussion of the estimation results for each individual equation of the multivariate model. The table aims to provide information about the adequacy of our choice of economic variables and their roles in the construction of the multivariate bandpass filter. The individual time series are decomposed by trend, common cycle and irregular components. The relative importance of each component for a particular time series i is determined by the scalars σ i,ζ (trend), δ i σ κ (cycle) and σ i,ε (irregular). In table 1 the relative importance of the trend and the cycle is presented based on the maximum likelihood estimates of these scalars. Trend variations appear most prominently in the unemployment series and, to a lesser extent, in the series interest rate spread, consumer confidence and industrial confidence. It is interesting to observe that industrial confidence tops the list of contributors to the construction of the business cycle while GDP, industrial production, consumer confidence and construction confidence are important too. Series related to retail trade contribute relatively little to the estimated business cycle. The ability of simultaneously identifying and estimating cycle shifts within the model for a given set of time series is a key development that is important for the current analysis since it avoids the pre-selection of coincident variables, whether or not by lagging or leading the variable. The estimation results reveal that unemployment lags the business cycle by more than one year (almost 16 months) while interest rate spread clearly leads the business cycle by almost 17 months. The common cycle component for the other series displays relatively small shifts, confirming the coincident character of these series. These results are in line with what was expected and according to what was found elsewhere, see for example Altissimo, Bassanetti, Cristadoro, Forni, Lippi, Reichlin, and Veronese (2001). An indication whether the multivariate trend-cycle model fits the individual series successfully is the goodness-of-fit RD 2 17

18 Table 1: Selected estimation results time series rel trd rel cyc shift RD 2 Gross domestic product (GDP) Industrial production Unemployment (inverted sign) Industrial confidence Construction confidence Retail trade confidence Consumer confidence Retail sales Interest rate spread The following values are based on maximum likelihood estimates (denoted by hats): rel trd measures the relative importance of trend component and is computed by ˆσ i,ζ /(ˆσ i,ζ + ˆδ iˆσ κ + ˆσ i,ε ); rel cyc is for the cycle component and is computed by ˆδ iˆσ κ /(ˆσ i,ζ + ˆδ iˆσ κ + ˆσ i,ε ); shift reports estimate ˆξ i. The coefficient of determination RD 2 is with respect to the total sum of squared differences. 18

19 statistic as reported in table 1. The sum of squared prediction errors is compared to the sample variance of the differenced time series for each equation of the model. These statistics give evidence that the fit of the model for the individual time series is adequate to good Figure 4: Euro area business cycle indicator with ECB datings. Figure 4 presents the smoothed estimate of the business cycle indicator together with the Euro area business cycle turning points settled by the European Central Bank, see ECB (2001). Since the ECB dating was made on a quarterly basis, the quarters where presumably occured a turning point are represented by a shaded area due to the monthly nature of the time axis. One can see that the suggested business cycle indicator tracks, in general, quite well the turning points. Only at the end of the sample it seems to perform less satisfactorily. However, one should note that the ECB dating was made using data only up to In particular, this can explain why the last peak dated by ECB is not confirmed by our business cycle indicator since it suggests that the peak only occurred at the beginning of The historical minimum value of the business cycle is observed in August 1993, which falls within the most severe recession period recorded in the Euro area, while the maximum value is at January

20 H C A K R C F H P Figure 5: Comparison of four different business cycle indicators: H-P is the Hodrick-Prescott filter; C-F is the Christiano-Fitzgerald filter; H-C is the model-based decomposition of Harvey and Clark; A-K-R is our business cycle indicator. The business cycle indicators are presented at a quarterly frequency. 20

21 In Figure 5 the business cycle obtained from our multivariate filter is compared with the ones resulting from some univariate approaches to business cycle assessment based on quarterly real GDP. In particular, we consider the Hodrick and Prescott (1997) filter, the Christiano and Fitzgerald (2003) band-pass filter and the Harvey-Clark model-based filter, see Clark (1987) and Harvey (1989). The latter model-based filter is based on the univariate trend-cycle decomposition model (1) with N = 1. As it is standard in the literature for quarterly data, we set λ = 1600 for Hodrick-Prescott filter and considered cycles of periodicity between 6 and 32 quarters for Christiano-Fitzgerald filter. All cycle indicators appear to move upwards or downwards at roughly the same time. However, our multivariate filter provides a smooth business cycle similar to the one from Christiano and Fitzgerald (2003) band-pass filter while the other detrending approaches are more affected by irregular movements. The cycle indicators are presented at a quarterly frequency whereas our multivariate filter obtains business cycle estimates at the higher monthly frequency and is able to recover the latent monthly output gap which is highly relevant for the policy maker. The underlying methods used in this paper also overcome the release lags that end up delaying the assessment of the current economic situation and are noteworthy in the Euro area. Since the state space framework can deal with the missing observations problem, it ensures an up-to-date high-frequency business cycle estimate and allows for a timely analysis of current economic developments. 3.4 Revisions A major issue regarding business cycle estimates is their real-time reliability, see Orphanides and van Norden (2002). In practice, business cycle estimates are subject to revisions over time due to data revision and to re-computation when additional data becomes available. Although it is not possible to evaluate the consequences of the first source of revision for the Euro area due to lack of data vintages, the second source of revision can be assessed by comparing smoothed and filtered estimates of the business cycle. The filtered estimate is the one conditional on the information set available up to each time point. The smoothed estimate is the one based on the full set of data. Their comparison reveals the magnitude of the revision. In figure 6 final (smoothed) and real-time (filtered) estimates are presented for different filters. In line with the 21

22 H P 0.01 C F H C A K R Figure 6: Comparison of filtered (dotted line) and smoothed (solid line) estimates related to four different business cycle indicators: H-P is the Hodrick-Prescott filter; C-F is the Christiano- Fitzgerald filter; H-C is the model-based decomposition of Harvey and Clark; A-K-R is our business cycle indicator. The business cycle indicators are presented at a quarterly frequency. 22

23 Table 2: Reliability statistics for different business cycle indicators Method correlation noise-to-signal sign concordance Hodrick-Prescott Christiano-Fitzgerald Harvey-Clark Azevedo-Koopman-Rua Correlation is the contemporaneous correlation between the real time (filtered) estimates and the final (smoothed) estimates of the business cycle. Noise-to-signal ratio is the ratio of the standard deviation of the revisions against the standard deviation of the final estimates. Sign concordance is the percentage of times that the sign of the real time and final estimates are equal. The statistics are reported for the sample periods of (86-02) and (93-02). findings of Orphanides and van Norden (2002) for the U.S., the difficulty of assessing the cyclical position in real-time for the Euro area is clear. In fact, Orphanides and van Norden (2002) show that the reliability of real-time estimates of U.S. output gap is low whatever method is used. In addition to a graphical inspection, the importance of the revisions can be measured by the cross-correlation between final and real-time estimates, by the noise-to-signal ratio (the ratio of the standard deviation of the revisions to the standard deviation of the final estimate) and by the sign concordance (the proportion of time that final and real-time estimates share the same sign). These measures are presented in table 2 for the whole sample and for a more recent period. It should be beared in mind that it is extremely hard to estimate the output gap in real-time since only with the advance of time one can be more accurate about the cyclical position at a given period. From table 2, one can see that the Hodrick-Prescott filter presents the worst results while the multivariate band-pass filter method outperforms considerably all univariate filters in real-time. Therefore, it seems that a multivariate setting allied with the use of leading, coincident and lagging variables can overcome, to some extent, the shortcomings of the currently used filters for real-time analysis. 23

24 3.5 A growth indicator The focus of this paper is on the business cycle indicator and on how the information content of an economy-wide dataset can be exploited for the assessment of the cyclical position of the economy using a multivariate approximate band-pass filter. Nevertheless, the policymaker may also be interested in tracking economic activity growth. As a by-product a growth indicator can be obtained as part of our model-based approach. This can be done by restoring the smoothed estimate of the business cycle together with the corresponding smoothed estimate of the trend component of a particular time series and by computing its growth rate. The three-monthly change of the trend restored business cycle indicator is presented in figure 7. The resulting growth indicator can be interpreted as the latent monthly measure of GDP quarterly growth. Figure 7 shows that the growth indicator tracks GDP quarterly growth quite well. Moreover, it is available on a monthly basis and avoids the erratic behaviour of GDP quarterly growth rate which complicates the assessment of the current economic situation. Figure 7 also presents our growth indicator together with EuroCOIN, see Altissimo, Bassanetti, Cristadoro, Forni, Lippi, Reichlin, and Veronese (2001). The EuroCOIN is based on the generalised dynamic factor model of Forni, Hallin, Lippi, and Reichlin (2000) and resorts to a huge dataset (almost a thousand series that refer to the six major Euro area countries). The model underlying EuroCOIN can be described briefly in the following way. Each series in the dataset is modeled as the sum of two uncorrelated components, the common and the idiosyncratic. The common component is driven by a small number of factors common to all variables but possibly loaded with different lag structures, while the idiosyncratic component is specific to each variable. EuroCOIN is the common component of the GDP growth rate after discarding its high frequency component. The EuroCOIN indicator presented in panel (ii) of figure 7 has been subject to revisions each time new observations become available. Note that the GDP monthly series used as input was obtained through the linear interpolation of quarterly figures. Although this approach allows for more general cyclical dynamics when compared with our model, it also presents additional drawbacks. For instance, it requires the detrending of the series prior to dynamic factor analysis and one cannot use simultaneously series of different periodicity. Moreover, with the estimation of only two parameters for each series, the common cycle factor load and the shift, we are able 24

25 to get a quite similar outcome (i) (ii) Figure 7: Comparisons of (i) Euro area quarterly GDP growth (dotted line) versus our growth indicator (solid line) and (ii) EuroCOIN indicator (dotted line) versus our growth indicator (solid line). 4 Conclusion In this paper we propose a multivariate approximate band-pass filter for obtaining a business cycle indicator. The filter is based on a multivariate unobserved components time series model that incorporates two recent and relevant contributions: (i) the inclusion of a particular cycle component that has properties similar to those of band-pass filtered series, see Harvey and Trimbur (2003); (ii) the incorporation of phase shifts within the cycle component, see Rünstler (2004). The resulting multivariate filter is used to extract trend and cycle components using several economic time series for the Euro area. The innovations to the cycle component of each time series are common but their contribution is derived from a model-based filter that depends 25

26 only on three parameters. One parameter is the frequency of the cycle, which we fix so as to assure that the typical business cycle frequencies are captured. Another parameter accounts for phase shifts while the third parameter accounts for the relative contribution of the cycle to the dynamics of the individual time series. The common cycle component resumes to our business cycle indicator. These generalisations lead to a smooth business cycle indicator and the suggested procedure can be seen as a multivariate band-pass filter. Besides the multivariate setting, we are not constrained to the use of only coincident variables to identify the business cycle, as in Stock and Watson (1989), since we account simultaneously for phase shifts between the cyclical components of the series. Thus, one can also exploit the information contained in both leading and lagging variables and this allows for a richer information set when obtaining the business cycle. This multivariate approach is shown to provide business cycle estimates that are less subject to revisions than the ones resulting from the commonly used filters. This feature represents a valuable improvement for policymaking since reliable estimates of the current cyclical position are required in real-time when policy decisions are made. Moreover, we are also able to reconcile information available at different frequencies. It becomes possible to obtain a high frequency business cycle indicator without discarding series such as GDP despite of its quarterly frequency as in Stock and Watson (1989). The use of quarterly GDP together with monthly indicators can be dealt with straightforwardly by exploiting the state space framework in a consistent manner. Furthermore, the ability to deal with missing obervations also allows timely business cycle estimates regardless of data release lags that can be substantial in the Euro area. This feature is also crucial for the policymaker since it permits an evaluation of the current state of the economy without delay. The new approach is applied to Euro area data and the resulting business cycle indicator appears to be in line with common wisdom regarding Euro area economic developments. As a by-product, one can also obtain a growth indicator. In particular, our more parsimonious approach leads to a growth indicator for the Euro area similar to EuroCOIN. The latter is based on a more involved approach by any standard and uses hundreds of time series from individual countries belonging to the Euro area. Furthermore, the suggested procedure also overcomes some of its drawbacks. 26

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