Business and housing market cycles in the euro area: a multivariate unobserved component approach

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1 Business and housing market cycles in the euro area: a multivariate unobserved component approach Laurent Ferrara (a) and Siem Jan Koopman (b) (a) Banque de France (b) VU University Amsterdam, Department of Econometrics The Macroeconomics of Housing markets. Paris, 3-4 December 2009 Ferrara Koopman : Business and housing market cycles in the euro area p. 1

2 The macroeconomy in the euro area Quarterly time series, , GDP in volumes, for countries (i) France, (ii) Germany, (iii) Italy and (iv) Spain (i) 13.2 (ii) (iv) 12.7 (iii) Ferrara Koopman : Business and housing market cycles in the euro area p. 2

3 Trend-cycle decompositions Nonparametric filtering: Hodrick-Prescott filter (high-pass filter) Baxter-King filter (band-pass filter) Christiano-Fitzgerald filter (band-pass filter) Bandpass refers to frequency domain properties of filters for trend and cycles: trend captures the low-frequencies, cycle the mid-frequencies and irregular the high-frequencies. Ferrara Koopman : Business and housing market cycles in the euro area p. 3

4 Ideal band-pass filter properties TREND CYCLE IRREGULAR Ferrara Koopman : Business and housing market cycles in the euro area p. 4

5 Nonparametric filtering Hodrick-Prescott filter (high-pass filter) Baxter-King filter (band-pass filter) Christiano-Fitzgerald filter (band-pass filter) Drawbacks of nonparametric filtering: Forecasting not possible. Confidence intervals for extracted components not available. Extraction of spurious cycles. We therefore take a model-based approach. In particular, we adopt unobserved components time series models. Ferrara Koopman : Business and housing market cycles in the euro area p. 5

6 Univariate UC Trend-Cycle Decomposition Trend µ t : µ t+1 = µ t + β t + η t ; Irregular ε t : White Noise; y t = µ t + ψ t + ε t β t+1 = β t + ξ t Cycle ψ t : ARMA with complex roots in AR, see Clark (1987), or stochastic trigonometric functions, see Harvey (1985,1989); Trigonometric specification, enforces complex roots in AR: ( ψ t+1 ψ t+1 + ) [ ]( cos λ sin λ = φ sinλ cosλ ψ t ψ t + ) ( + κ t κ + t ), We can also have two cycles: κ t, κ + t NID(0, σ 2 κ). y t = µ t + ψ (1) t + ψ (2) t + ε t Ferrara Koopman : Business and housing market cycles in the euro area p. 6

7 The macroeconomy in the euro area Quarterly time series, , GDP in volumes, for countries (i) France, (ii) Germany, (iii) Italy and (iv) Spain (i) 13.2 (ii) (iv) 12.7 (iii) Ferrara Koopman : Business and housing market cycles in the euro area p. 7

8 The housing market in the euro area Quarterly time series, , real house prices (HP), for countries (i) France, (ii) Germany, (iii) Italy and (iv) Spain. (i) 0.3 (ii) (iii) (iv) Ferrara Koopman : Business and housing market cycles in the euro area p. 8

9 Any (common) cyclical dynamics in the data? Autocorrelograms and sample spectra, based on first differences... 1 (i) GDP Correlogram 0.4 GDP Spectrum 1 HP Correlogram 0.4 HP Spectrum (ii) (iii) (iv) Ferrara Koopman : Business and housing market cycles in the euro area p. 9

10 The basic multivariate model Multiple set of M economic time series, y it, is collected in y t = (y 1t,...,y Mt ) and model is given by y t = µ t + ψ (1) t + ψ (2) t + ε t, where the disturbance driving each vector component is a vector too, with a variance matrix. The structure of the variance matrix determines the dynamic interrelationships between the M time series. For example, if trend component µ t follows the random walk, µ t+1 = µ t + η t with disturbance vector η t, with variance matrix Σ η : diagonal Σ η, independent trends; rank(σ η ) = p < M, common trends (cointegration); rank(σ η ) = 1, single underlying trend; Σ η is zero matrix, constant. Similar considerations apply to other components. Ferrara Koopman : Business and housing market cycles in the euro area p. 10

11 Dynamic factor representations We can formulate the multivariate unobserved components model also by y t = µ + A η µ t + A (1) κ ψ (1) t + A (2) κ ψ (2) t + A ε ε t, where, for the trend component, for example, the loading matrix A η is such that Σ η = A η A η, and, similarly, loading matrices are defined for the other variance matrices of disturbances that drive the components. Here the dynamic factors or unobserved components µ t, ψ (1) t, ψ (2) t ε t are "normalised". and Ferrara Koopman : Business and housing market cycles in the euro area p. 11

12 STAMP Model is effectively a state space model: Kalman filter methods can be applied for maximum likelihood estimation of parameters (such as the loading matrices). Kalman filter methods are employed for the evaluation of the likelihood function and score vector. Kalman filter and smoothing methods are employed for signal extraction or the estimation of the unobserved components. User-friendly software is available for state space analysis. We have used S T A M P for this research project: a multi-platform, user-friendly software: econometrics, time series and forecasting by clicking. It can treat multivariate unobserved components time series models... Ferrara Koopman : Business and housing market cycles in the euro area p. 12

13 Motivation of our study Evidence of any relationship between housing prices and GDP in the euro area. Focus on more recent developments... We prefer to model the time series jointly and establish interrelationships between the time series Focus on cyclical dynamics, long-term and short-term Emphasis on real housing prices: relevant for the monetary policy We also like to discuss synchronisation of housing markets in euro area Empirical results are based on our data-set with two variables: GDP and real house prices (HP); and for four euro area countries: France, Germany, Italy and Spain. Ferrara Koopman : Business and housing market cycles in the euro area p. 13

14 Relevant literature Unobserved components model: Harvey (1989) State space methods: Durbin and Koopman (2001) Multivariate unobserved components: Harvey and Koopman (1997), Azevedo, Koopman and Rua (2006); Parametric approaches for house prices: Probit regressions: Borio and McGuire, 2004, van den Noord, 2006; Dynamic Factor models: Terrones, 2004, DelNegro and Otrok, 2007, Stock and Watson, 2008; VAR: Vargas-Silva, 2008, Goodheart and Hofmann, Presentations at this conference! Ferrara Koopman : Business and housing market cycles in the euro area p. 14

15 Univariate analysis Objectives: Verify the trend-cycle decomposition for each series Verify whether possible restrictions are realistic Results for GDP: two short cycles in France and Italy are detected (<6 years); Germany and Spain contain both a short cycle (5.42 and 3.62 years, resp.) and a long cycle (13.5 and 9.11 years) Various cycles are deterministic (fixed sine-cosine wave) Results for HP: Results are quite different for each series Two cycles for Germany (5.4 and 13.5 years) Two short cycles for Italy (3.0 and 5.5 years) and France (3.1 and 5.8 years) For Spain a cycle reduces to an AR(1) process Ferrara Koopman : Business and housing market cycles in the euro area p. 15

16 Univariate results for GDP France Germany Italy Spain GDP R R R R Trend var Cycle 1 var Cycle 1 ρ Cycle 1 p Cycle 2 var Cycle 2 ρ Cycle 2 p Irreg var N Q R Ferrara Koopman : Business and housing market cycles in the euro area p. 16

17 Univariate results for HP France Germany Italy Spain RHP R R R R Trend var Cycle 1 var Cycle 1 ρ Cycle 1 p Cycle 2 var Cycle 2 ρ Cycle 2 p Irreg var N Q R Ferrara Koopman : Business and housing market cycles in the euro area p. 17

18 Cycle correlations from univariate analysis Correlations for combined cycles (ψ (1) t + ψ (2) t ): Strong correlations between GDP of four countries (correlations range from 0.52 to 0.94) The correlations with German GDP are the lowest Correlations between HP of four countries range from 0.42 to 0.94 The highest correlation is between Spain and France HP s Correlation on combined cycle are mostly due to long-term cycle, not to the short-term cycle Correlations between GDP and HP for each country range from 0.06 for Germany to 0.76 for Spain Overall low cross-correlations between GDP of one country and HP of another country Ferrara Koopman : Business and housing market cycles in the euro area p. 18

19 Correlations between combined cycles for eight series Combined cycle (ψ (1) t + ψ (2) t ) F GDP F HP G GDP G HP I GDP I HP S GDP S HP F GDP F HP G GDP G HP I GDP I HP S GDP S HP Ferrara Koopman : Business and housing market cycles in the euro area p. 19

20 Correlations between short cycle for eight series Short cycle ψ (1) t F GDP F HP G GDP G HP I GDP I HP S GDP S HP F GDP F HP G GDP G HP I GDP I HP S GDP S HP Ferrara Koopman : Business and housing market cycles in the euro area p. 20

21 Correlations between long cycle for eight series Long cycle ψ (2) t F GDP F HP G GDP G HP I GDP I HP S GDP S HP F GDP F HP G GDP G HP I GDP I HP S GDP S HP Ferrara Koopman : Business and housing market cycles in the euro area p. 21

22 Bivariate analysis For each country, we carry out a bivariate analysis between GDP and RHP: y t = µ t + ψ (1) t + ψ (2) t + ε t, where y t is a 2 1 vector for two series: GDP and HP. We can conclude that highest correlation is found for cycle components (except Italy) for France, high correlation for medium-term cycle (8 years) but no dependence for long-term cycle (15.6 years) for Spain, strong correlations for both medium-term (8.2 years) and long-term (14.4 years) for Germany, correlations for both cycles, but with low periods (4.3 and 7 years) Ferrara Koopman : Business and housing market cycles in the euro area p. 22

23 Bivariate results for GDP and HP GDP RHP corr per ρ diag GDP RHP FRA trend N cyc Q cyc R irreg GER trend N cyc Q cyc R irreg Ferrara Koopman : Business and housing market cycles in the euro area p. 23

24 Bivariate results for GDP and HP GDP RHP corr per ρ diag GDP RHP ITA trend N cyc Q cyc R irreg SPN trend N cyc Q cyc R irreg Ferrara Koopman : Business and housing market cycles in the euro area p. 24

25 Four-variate cross-country analysis of GDP and RHP Now we incorporate earlier findings and impose a strict short- and long-term cycle decomposition for our analysis. In particular, we have an independent trend µ t (i.e. diagonal variance matrix Σ η for disturbance vectors of µ t+1 = µ y + η t ) similarly, an independent irregular component ε t (i.e. diagonal variance matrix Σ ε ) a two-cycle parametrization with restricted periods of 5 and 12 years the rank of the 4 4 cycle variance matrices Σ κ is 2: common cyles... we load the two "independent" cycles on France and Germany, i.e. cyclical dynamics of Spain and Italy are obtained as linear functions of the two times two (short and long) cyclical factors Ferrara Koopman : Business and housing market cycles in the euro area p. 25

26 Four-variate decomposition for GDP, cross-country LFRA_GDP Level LFRA_GDP Cycle 1 LFRA_GDP Cycle 2 LFRA_GDP Irregular LGER_GDP Level LGER_GDP Cycle 1 LGER_GDP Cycle 2 LGER_GDP Irregular LITA_GDP Level 0.02 LITA_GDP Cycle LITA_GDP Cycle LITA_GDP Irregular LSPA_GDP Level LSPA_GDP Cycle LSPA_GDP Cycle LSPA_GDP Irregular Ferrara Koopman : Business and housing market cycles in the euro area p. 26

27 Four-variate results for cross-country: GDP Fra Ger Ita Spn Fra Ger Cycle short (cov 10 6 ) factor loadings Fra Ger Ita Spn Cycle long (cov 10 6 ) Fra Ger Ita Spn Ferrara Koopman : Business and housing market cycles in the euro area p. 27

28 Four-variate results for cross-country: GDP Diagnostic statistics are satisfactory Strong correlation France-Germany for long-term cycle Business cycles for Italy and Spain are closely connected with the one for France (however, negative??? marginal correlation Fra-Spa for short-term cycle) German cycles strongly affect business cycles in Italy and Spain (however, their marginal correlations for longer cycle are negative) Ferrara Koopman : Business and housing market cycles in the euro area p. 28

29 Four-variate decomposition for HP, cross-country 5.5 LFRA_RHprice Level LFRA_RHprice Cycle LFRA_RHprice Cycle 2 LFRA_RHprice Irregular 0.3 LGER_RHprice Level LGER_RHprice Cycle LGER_RHprice Cycle e 5 0 5e 5 LGER_RHprice Irregular LITA_RHprice Level LITA_RHprice Cycle LITA_RHprice Cycle LITA_RHprice Irregular LSPA_RHprice Level 0.04 LSPA_RHprice Cycle LSPA_RHprice Cycle 2 LSPA_RHprice Irregular Ferrara Koopman : Business and housing market cycles in the euro area p. 29

30 Four-variate results for cross-country: HP Fra Ger Ita Spn Fra Ger Cycle short (cov 10 6 ) factor loadings Fra Ger Ita Spn Cycle long (cov 10 6 ) Fra Ger Ita Spn Ferrara Koopman : Business and housing market cycles in the euro area p. 30

31 Four-variate results for cross-country: HP Overall, these results seem to indicate that there is less evidence of common (cyclical) dynamics in HP series Low correlations between France and Germany Strong negative correlations for the 5-year cycle between Fra-Ita and Ger-Spa However, more commonalities for the 12-year cycle (Fra-Spa, Fra-Ita, Ger-Spa) Similarities between correlation matrices for the 12-year HP and GDP cycles, except that relationship Fra-Ger is stronger for GDP (0.79 against 0.38 for HP) Ferrara Koopman : Business and housing market cycles in the euro area p. 31

32 Eight-variate results: HP and GDP for four countries Similar restrictions apply as in four-variate analyses. We conclude that strong correlations among GDPs for short-term cycles but less evidence for long-term cycles, especially for Germany low correlations among HP series. for short-term cycle, these correlations for HP Fra-Ger is 0.65 and for HP Spa-Ger is only a few positive correlations for the long-term cycle in HP have been found: Fra-Spa (0.58) and Ger-Ita (0.57) correlations HP-GDP are only found for long-term cycle, especially for France and Spain. Ferrara Koopman : Business and housing market cycles in the euro area p. 32

33 Eight-variate results: short cycle correlations France Germany Italy Spain GDP HP GDP HP GDP HP GDP HP F-G F-H G-G G-H I-G I-H S-G S-H 1 Ferrara Koopman : Business and housing market cycles in the euro area p. 33

34 Eight-variate results: long cycle correlations France Germany Italy Spain GDP HP GDP HP GDP HP GDP HP F-G F-H G-G G-H I-G I-H S-G S-H 1 Ferrara Koopman : Business and housing market cycles in the euro area p. 34

35 Synchronization... Synchronisation of cycles may be needed, but note that... 1 (i) GDP Correlogram 0.4 GDP Spectrum 1 HP Correlogram 0.4 HP Spectrum (ii) (iii) (iv) Ferrara Koopman : Business and housing market cycles in the euro area p. 35

36 Conclusions Synchronisation of cycles may be needed... This is not a weakness in our analyses... more realistic... No evidence of common cycles for HPs, but commonalities are found in long-term cycles Strong correlation for Spain is found between HP and GDP cycles, such issues can be detected in our analysis... Multivariate UC models are powerful and general: can take shifts into account, time-varying synchronisation, forecasting... Ferrara Koopman : Business and housing market cycles in the euro area p. 36

37 So can we learn more from the data? 12.8 (i) 13.2 (ii) 5.0 (i) 0.3 (ii) (iv) 12.7 (iii) (iii) (iv) Ferrara Koopman : Business and housing market cycles in the euro area p. 37

38 Business and housing market cycles in the euro area: a multivariate unobserved component approach Laurent Ferrara (a) and Siem Jan Koopman (b) Thank you! Ferrara Koopman : Business and housing market cycles in the euro area p. 38