## Abstract

In this paper we examine the empirical features of both the business and the financial cycle in Italy. We employ univariate and multivariate trend-cycle decompositions based on unobserved component models. Univariate estimates highlight different cyclical properties (persistence, duration and amplitude) of real GDP and real credit to the private sector. Multivariate estimates uncover the presence of feedback effects between the real and the financial cycle. In addition, in the most recent period (2015–2016) the multivariate approach highlights a wider output gap than that estimated by the univariate models considered in this paper.

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## Notes

- 1.
Financial indicators do seem to have forecasting power for real activity, as shown in Borio and Lowe (2004), English et al. (2005), Hatzius et al. (2010), and Ng (2011). Furthermore, it has been argued that the estimation of potential output can benefit from the inclusion of information about the financial cycle, as captured by the behaviour of credit and property prices (Borio et al. 2013, 2014).

- 2.
- 3.
With regard to the Italian real cycle, Zizza (2006) estimated the Italian potential output using both univariate and multivariate unobserved components models, Bassanetti et al. (2010) provided a comparison of univariate structural time series models, while Busetti and Caivano (2016) estimated a bivariate output and Phillips curve model for the Italian economy.

- 4.
Adapted by Harding and Pagan (2002) to deal with series sampled at the quarterly frequency.

- 5.
Note that in the LLT the long-run growth rate (\(\Delta \tau _t\)) is driven by the shock to the slope component \(\zeta _t\) as well as by the first differences of the shock to the level component (\(\Delta \eta _t\)).

- 6.
This specification is commonly referred to as “Seemingly Unrelated Time Series Equations” (SUTSE); see Harvey (1989, Chap. 8).

- 7.
Under the restriction \(\rho _i=\rho \) and \(\lambda _i=\lambda \)\(\forall i\), the model features “similar cycles” (Harvey and Koopman 1997).

- 8.
We thank an anonymous reviewer for pointing out this issue.

- 9.
In our sample, results indicate that the IRW is preferred to the LLT specification. As we will show in the sequel of this section, in order to obtain a longer cycle and a smoother slope component in line with Rünstler and Vlekke (2018), it is necessary to impose restrictions on the slope variance; see Fig. 3 and the related discussion.

- 10.
We also performed a rolling window exercise whereby at each iteration a new data point is added and the oldest data point discarded so as to maintain the length of the estimation window fixed. While this set-up tends to magnify time variation, we reached very similar conclusions to those reported in the main text. The only exception are the estimates based on sample windows including the period between the early 80’s and 2008, when the likelihood function shows two maxima, one local at \(\lambda =0.48\) (period \(\approx 13\) quarters) and one global at \(\lambda =0.15\) (period \(\approx 40\) quarters), confirming Busetti and Caivano’s (2016) findings.

- 11.
See, e.g., Giorno et al. (1995), De Masi (1997) and Havik et al. (2014) for the methodologies employed by OECD, IMF and European Commission, respectively. The concept of a smooth potential output derives from the belief that most of the variance at business cycle frequencies is due to demand shocks while most of the variance at lower frequencies is due to supply shocks, as in Blanchard and Quah (1989). Nonetheless, this traditional view has been challenged by models in which demand shocks can have a long-lasting impact on output, as in Blanchard and Summers (1986), and by empirical analyses showing that large falls in output can be mostly driven by permanent shocks, as in Cerra and Saxena (2008).

- 12.
We tried different calibrations and finally chose the one that, in our view, delivers an economically plausible long-run growth rate of real GDP.

- 13.
See Banca d’Italia (2016).

- 14.
Conversely, working with credit-to-GDP quarterly data ranging from 1974 to 2014, Galati et al. (2016) estimated a single financial cycle with a periodicity of 16.94 years (Table 1, p. 85).

- 15.
This is mainly due to the fact that the volatility of \(\psi _{2,t}\) (measured by the signal-to-noise ratio) is much higher than the volatility of \(\psi _{1,t}\).

- 16.
Similar conclusions can be reached by leaving out longer portions of real credit data at the end of the sample period. Results are available from the authors upon request.

- 17.
Lastly, by jointly inspecting the cyclical estimates in Figs. 10 and 11, we note that when removing real credit data at the end of the sample, the financial gap tends to close; on the other hand, when removing real GDP data, the output gap tends to widen. These findings seem to suggest that the last years of data surprised to the downside with respect to real credit, while they surprised on the upside with respect to real output.

- 18.
Producing the minimum mean squared linear estimator (MMSLE) of the state vector.

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We would like to thank the Editor-in-Chief Tommaso Proietti and two anonymous reviewers, as well as Marco Bernardini, Fabio Busetti, Simone Emiliozzi, Lucia Esposito, Giuseppe Grande, Stefano Neri, Gerhard Rünstler, Stefano Siviero, Fabrizio Venditti, participants at the CFE 2017 Conference (London, 16–18 December 2017) for their constructive and valuable suggestions. We are also grateful to Gerhard Rünstler and Marente Vlekke for sharing their Matlab codes in the framework of the WGEM expert team on “Links between real and financial cycles”. Any remaining errors are ours. The views expressed herein are those of the authors and do not necessarily reflect those of the Institution they represent.

## Appendices

### Appendix

### Estimation, filtering and smoothing

Consider the following linear Gaussian state space model:

where \(\varvec{y}_t\) is the \(N\times 1\) vector of observed variables, \(\varvec{\varepsilon _t}\) is the \(N\times 1\) vector of measurement errors, \(\varvec{\alpha }_t\) is the \(m\times 1\) vector of state variables and \(\varvec{\eta }_t\) is the corresponding \(m\times 1\) vector of innovations. The two innovation vectors are assumed to be Gaussian distributed and uncorrelated for all time periods, that is, \(E(\varvec{\varepsilon }_t \varvec{\eta }_s')=\varvec{0} \)\(\forall t,s\); this assumption can be relaxed at the cost of a slight complication in some of the filtering formulae. The initial value of the state vector is also assumed to be Gaussian \(\alpha _1 \sim {\mathcal {N}}(\varvec{a}_1,\varvec{P}_1)\) and uncorrelated \(\forall t\) with \(\varvec{\varepsilon _t}\) and \(\varvec{\eta _t}\).

Following Harvey (1989), conditional on the information set \(\varvec{Y}_{t-1}=\{\varvec{y}_{t-1},...,\varvec{y}_1\}\) and on the vector of parameters \(\varvec{\theta }\), the observations and the state vector are Gaussian, i.e., \(\varvec{y}_t|(\varvec{Y}_{t-1};\varvec{\theta }) \sim {\mathcal {N}} (\varvec{Z} \varvec{a}_t, \varvec{F}_t)\) and \(\varvec{\alpha }_t|(\varvec{Y}_{t-1};\varvec{\theta }) \sim {\mathcal {N}}(\varvec{a}_t, \varvec{P}_t)\). Therefore, it follows that the log-likelihood function for the full vector of observations \(\varvec{y}=(\varvec{y}_1',\dots ,\varvec{y}_n')'\) can be expressed by the *prediction error decomposition*:

The prediction error \(\varvec{v}_t\), its covariance matrix \(\varvec{F}_t\), the state vector conditional mean \(\varvec{a}_t\), and its mean square error (MSE) matrix \(\varvec{P}_t\) are estimated optimally^{Footnote 18} by means of the Kalman Filter (KF):

with initial values \(\varvec{a}_1\) and \(\varvec{P}_1\). Regarding the initialization of the state vector, a diffuse initial condition is used for the trend component, while the cyclical component is initialized with the unconditional moments. Once the vector of parameters \(\varvec{\theta }\) is estimated by ML, the unobserved components can be extracted from the observations using the KF and the associated Kalman smoother. Specifically, \(\varvec{a}_{t}=E(\varvec{\alpha }_{t}|\varvec{Y}_{t-1},\varvec{\theta })\) is the so-called *predictive* filter and \(\varvec{P}_{t}=E[(\varvec{a}_{t}-\varvec{\alpha }_{t})(\varvec{a}_{t}-\varvec{\alpha }_{t})']\) is the associated MSE, while the *real-time* filter is \(\varvec{a}_{t|t}=E(\varvec{\alpha }_t|\varvec{Y}_t, \varvec{\theta })\) and its MSE is \(\varvec{P}_{t|t}=E[(\varvec{a}_{t|t}-\varvec{\alpha }_t)(\varvec{a}_{t|t}-\varvec{\alpha }_t)']\). It is worth stressing that in this linear model the MSEs are independent from the observations, thus they are also the unconditional covariance matrices associated with the conditional mean estimators; see Harvey (1989, Sect. 3.2.3).

The *smoother* algorithm allows us to estimate the state vector given all the available information, namely \(\varvec{a}_{t|n}=E(\varvec{\alpha }_{t}|\varvec{Y}_n,\varvec{\theta })\) and the associated MSE \(\varvec{P}_{t|n}=E[(\varvec{a}_{t|n}-\varvec{\alpha }_{t})(\varvec{a}_{t|n}-\varvec{\alpha }_{t})']\). It is a backward recursion:

with initial values \(\varvec{r}_n=\varvec{0}\) and \(\varvec{N}_n=\varvec{0}\). For more details see Harvey (1989, Sect. 3.6) and Durbin and Koopman (2012, Sect. 4.4).

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Bulligan, G., Burlon, L., Delle Monache, D. *et al.* Real and financial cycles: estimates using unobserved component models for the Italian economy.
*Stat Methods Appl* **28, **541–569 (2019). https://doi.org/10.1007/s10260-019-00453-1

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### Keywords

- Business cycle
- Financial cycle
- Unobserved components
- Model-based filters

### JEL Classification

- C32
- E32
- E44