Short-term GDP forecasting with a mixed frequency dynamic factor model with stochastic volatility Massimiliano Marcellino 1 , Mario Porqueddu 2 and Fabrizio Venditti 2 1 EUI and Bocconi University 2 Banca d’Italia - Research Department section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 1 / 36
Plan of the talk Motivation The model Estimation strategy An empirical application: forecasting euro area GDP Full sample results Daily business: some bayesian tools for nowcasting Out of sample: point and density forecast evaluation Some concluding remarks section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 2 / 36
Motivation 1 Interest in policy making and forecasting in probability distributions around a central forecast Fan charts: Bank of England, Bank of Canada, Norges Bank, SA Re- serve and Sveriges Riksbank, more recently Bank of Italy and also the US Fed (2008) Density forecasts are sensitive to shifts in the parameters of the model: Jore, Mitchell, and Vahey (2010), Clark (2011) Discrete breaks far away in the past: use sample split More recently: increasing interest in modeling small continuous breaks (time varying parameter models, Cogley and Sargent, 2003, Primiceri, 2005, large literature following) The Great Recession: spot light on volatility breaks (end to the Great Moderation?) section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 3 / 36
Motivation 2 Most of the work on density forecast/time varying models falls in the medium/long term forecasting literature Nowcasting/Short term forecasting is a world of its own mixed frequency data ragged edge data different timeliness (soft/hard data) There are existing tools that deal with the above issues but No applications on density forecasts Time constant parameters (some allow for discrete random breaks in the mean: MS models) Galvao (2009), STAR-MIDAS Guerin Marcellino (2011) MS-MIDAS Carriero, Clark, Marcellino (2012) U-MIDAS with stochastic volatility in the context of Nowcasting/Short term forecasting section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 4 / 36
Our contribution Extend the mixed frequency factor model by Mariano and Murasawa (2003) to account for continuous shifts in volatility Derive some interesting tools: Density forecasts / fan charts for GDP short term forecasts Probability distributions of the news content of indicator releases section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 5 / 36
Empirical application: forecasting euro area GDP We document a dramatic increase in both common and idiosyncratic business cycle volatility in the euro area in the past few years Evaluate the contribution to forecast accuracy of stochastic volatility in terms of: Point forecast accuracy (RMSE) S-vol lowers uniformly but marginally RMSE Ability to produce normalized forecast errors (computed via PITS) which are close to normal The model produces good pits with and without S-vol Interval forecast accuracy (coverage rates) S-vol improves significantly the coverage rates section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 6 / 36
The model � y ⋆ � u 1 , t � µ ⋆ � � � 1 t 1 = + β f t + y 2 t u 2 , t µ 2 y 1 t = 1 1 , t + 2 1 , t − 2 + 2 1 , t − 3 + 1 3 y ⋆ 3 y ⋆ 1 , t − 1 + y ⋆ 3 y ⋆ 3 y ⋆ 1 , t − 4 � y 1 t � µ 1 � β 1 ( 1 3 f t + 2 3 f t − 1 + f t − 2 + 2 3 f t − 3 + 1 � � � 3 f t − 4 ) = + + y 2 t µ 2 β 2 f t � 1 3 u 1 , t + 2 3 u 1 , t − 1 + u 1 , t − 2 + 2 3 u 1 , t − 3 + 1 � 3 u 1 , t − 4 u 2 , t section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 7 / 36
Idiosyncratic shocks: the baseline model � p f j = 1 φ f j f t − j + ǫ f ǫ f f t = t ∼ N ( 0 , σ f ) t � p 1 j = 1 φ 1 j u 1 , t − j + ǫ 1 ǫ 1 u 1 , t = t ∼ N ( 0 , σ 1 ) t � p 2 j = 1 φ 2 j u 2 , t − j + ǫ 2 ǫ 2 u 2 , t = t ∼ N ( 0 , σ 2 ) t section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 8 / 36
Stochastic volatility � p f j = 1 φ f j f t − j + ǫ f t ( λ f , t ) 0 . 5 ǫ f f t = t ∼ N ( 0 , 1 ) � p 1 j = 1 φ 1 j u 1 , t − j + ǫ 1 t ( λ 1 , t ) 0 . 5 ǫ 1 u 1 , t = t ∼ N ( 0 , σ 1 ) � p 2 j = 1 φ 2 j u 2 , t − j + ǫ 2 t ( λ 2 , t ) 0 . 5 ǫ 2 u 2 , t = t ∼ N ( 0 , σ 2 ) We let the log-stochastic volatility components follow a random walk without drift log ( λ i , t ) = log ( λ i , t − 1 ) + η i , t η i , t ∼ N ( 0 , σ η, i ) section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 9 / 36
State space representation The model has a time varying state space representation y t = F µ t = H µ t − 1 + η t η t ∼ N ( 0 , Q t ) µ t Λ t = Λ t − 1 + ζ t ζ t ∼ N ( 0 , Ξ) y t collects both quarterly and monthly variables, µ t includes the unobserved factor and idiosyncratic shocks Q t collects the drifting volatilities σ i λ i , t section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 10 / 36
6 blocks of parameters: 6 steps Metropolis Hastings within Gibbs algorithm y t = F µ t = H µ t − 1 + η t η t ∼ N ( 0 , Q t ) µ t Λ t = Λ t − 1 + ζ t ζ t ∼ N ( 0 , Ξ) 1 Elements of F ( β ) 2 Elements of H ( φ ) 3 Time constant elements of Q t ( σ i ) 4 Time varying elements of Q t ( λ i , t ) 5 Variances of s-vol ( σ η, i ) 6 The unobserved state vector ( µ t ) Uncorrelated disturbances: estimation can be performed equation by equation section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 11 / 36
Steps 1-2: β i and σ i Take a measurement equation y i , t = β i f t + u i , t Autocorrelated 1 − φ ( L ) and heteroscedastic λ 0 . 5 i , t residuals Filter with 1 − φ ( L ) and divide by λ 0 . 5 i , t f t and all other parameters can be treated as known This is a standard regression Normal-gamma conjugate prior → Normal-gamma posterior section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 12 / 36
Step 3: φ i Take a transition equation p i � µ i , t = φ i µ i , t − j + η i , t j = 1 heteroscedastic λ 0 . 5 i , t residuals divide by λ 0 . 5 i , t This is a standard regression Normal conjugate prior → Normal posterior Discard explosive roots section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 13 / 36
Step 4-5: λ i , t , σ η, i Use block-by-block Jacquier-Polson-Rossi algorithm Involves drawing from a candidate density (log-normal) Metropolis acceptance step section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 14 / 36
Step 6: µ t Conditional on F ( β ) , H ( φ ) , Q t ( σ i , λ i , t ) use state space representation Durbin and Koopman disturbance smoother gives draws of µ t Missing values in GDP equation are treated as in Mariano/Murasawa (skip the filtering step) section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 15 / 36
Priors Estimate with OLS on a training sample of τ initial observations Normal prior means are set at OLS estimates, variances at 10 3 the OLS variances Gamma degrees of freedom set to τ + 1 for time constant variances Gamma degrees of freedom for the variance of λ i , t set to 1 and scale parameter to 0 . 025 (in line with Clark, 2011) We set τ to 36 (first three years of data). section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 16 / 36
Empirical application: Euro area GDP forecasting - indicators Table: Variable selection summary Indicator Country Quarterly series GDP Euro Area Monthly series Industrial Production Euro Area Industrial Production - Pulp/paper Euro Area Business Climate - IFO Germany Economic Sentiment Indicator Euro Area PMI composite Euro Area Exchange rate US-Euro 10y spread US-Euro Michigan Consumer Sentiment US section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 17 / 36
FULL SAMPLE RESULTS section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 18 / 36
Factor Loadings - posterior estimates Table: Factor Loadings - posterior estimates Percentiles 25th 50th 75th GDP 0.27 0.38 0.54 IP 0.40 0.49 0.60 IP-PULP 0.23 0.29 0.36 IFO 0.10 0.12 0.13 ESI 0.10 0.12 0.14 PMI 0.12 0.13 0.15 US $ TO EURO -0.08 -0.05 -0.02 US-spread -0.06 -0.04 -0.02 Michigan Consumer 0.04 0.06 0.08 section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 19 / 36
GDP: Median monthly estimate and Eurocoin 1.5 1 0.5 0 −0.5 −1 −1.5 −2 GDP−monthly estimate (median) −2.5 GDP quarterly ECOIN −3 2000 2002 2004 2006 2008 2010 section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 20 / 36
Stochastic volatilities - factor and selected indicators Factor GDP 1.6 2 1.4 1.2 1.5 1 0.8 1 0.6 0.5 0.4 1994 1996 1998 2000 2002 2004 2006 2008 2010 1994 1996 1998 2000 2002 2004 2006 2008 2010 3 IP US−spread 2 2.5 1.5 2 1.5 1 1 0.5 0.5 1994 1996 1998 2000 2002 2004 2006 2008 2010 1994 1996 1998 2000 2002 2004 2006 2008 2010 section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 21 / 36
NEWS AND FORECASTS section Marcellino-Porqueddu-Venditti (2012) Bundesbank-IFO Workshop June 2012 22 / 36
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