Using Entropic Tilting to Combine BVAR Forecasts with External Nowcasts

Transcription

1 Using Entropic Tilting to Combine BVAR Forecasts with External Nowcasts Fabian Krüger (HITS), Todd Clark (FRB Cleveland), and Francesco Ravazzolo (Norges Bank & BI) Federal Reserve Bank of Cleveland June 2015 Todd Clark (FRBC) Entropic Tilting June / 18

2 Motivation Medium-term forecasting models (VARs, DSGE models) are not very good at short-term forecasting So VAR and DSGE forecasts are often combined with nowcasts from another source Judgmental forecast from survey or central bank Other model-based nowcast (factor model, bridging equation, etc.) Various approaches may be used for such combination Treat nowcast as data (Faust and Wright (2009, 2013)) Treat nowcast as noisy indicator of current quarter (Del Negro and Schorfheide (2013)) Impose nowcast as condition in VAR forecast (Aastveit, et al. (2014)) Todd Clark (FRBC) Entropic Tilting June / 18

3 This paper We examine the use of entropic tilting (ET) to combine BVAR forecasts with external nowcasts Variables: GDP growth, unemployment, GDP inflation, T-bill rate Nowcasts from SPF and from nowcasting models Tilting based on not just means but also variances Assessment of accuracy of point and density forecasts Analysis of effects on longer-horizon forecast uncertainty Precedents Robertson, Tallman, and Whiteman (2005) introduce ET to economic forecasting Cogley, Morozov, and Sargent (2005) apply ET with BVAR and Bank of England forecasts Altavilla, Giacomini, and Ragusa (2013) use ET to combine term structure and survey forecasts of short rates Todd Clark (FRBC) Entropic Tilting June / 18

4 Outline 1 Model and Data 2 Entropic Tilting Implementation 3 Results Forecast accuracy Forecast uncertainty

5 Model and Data Quarterly forecasting model: BVAR(4) with stochastic volatility pÿ y t B 0 ` B i y t i ` v t i 1 v t A 1 Λt 0.5 ɛ t, ɛ t Np0, I k q, Λ t diagpλ 1,t,..., λ k,t q logpλ i,t q logpλ i,t 1 q ` ν i,t, i 1, k ν t pν 1,t, ν 2,t,..., ν k,t q 1 Np0, Φq Todd Clark (FRBC) Entropic Tilting June / 18

6 Model and Data Forecasts are generated and evaluated with real-time data Forecast sample: 1-5 quarters ahead, 1988:Q3 through 2013:Q2 Model estimates use a sample of 1955:Q1 through t Forecast actuals: 2nd available in vintage data set Point forecasts (RMSE) and density forecasts (CRPS) CRPS t py o t`h q ż 8 8 `F pzq 1ty o t`h ď zu 2 dz, Todd Clark (FRBC) Entropic Tilting June / 18

7 Basic ET Methodology Start with a sample of MCMC forecast draws, f : ty i u I i 1 Moment condition Egpyq ḡ ET is a functional optimization problem: min f PF KLICp f, f q subject to E f gpyq ḡ That is, look for distribution f which is close to f in terms of Kullback-Leibler divergence (KLIC) satisfies the moment condition ḡ The tilting solution is given by exp γ 1gpy i q π i ř I i 1 exp pγ 1gpy i qq γ arg min γ Iÿ i 1 exp γ 1 pgpy i q ḡq Todd Clark (FRBC) Entropic Tilting June / 18

8 Specifics of Our ET Implementation small m: Define y t:t`4 y t, y t`1, y t`2, y t`3, y t`4 1 Tilt to make mean of y t equal nowcast (point forecast) small m/v: big m: Tilt to make mean of y t equal nowcast (point forecast) and make variance of y t equal nowcast variance big m/v: Define y t:t`4 to contain all 20 forecasts (4 variables and 5 horizons) Tilt to make mean of y i,t, i 1,..., 4, equal nowcast Tilt to make mean of y i,t, i 1,..., 4 equal nowcast and make variance of y i,t, i 1,..., 4 equal nowcast variance Todd Clark (FRBC) Entropic Tilting June / 18

9 ET Example: GDP Growth, 2008:Q4 small m: gpy t:t`4 q y t, ḡ 2.9 small m/v: gpy t:t`4 q y t, py t ` 2.9q 2 1, ḡ 2.9, Raw MCMC SPF m SPF m/v count 400 count count value value value Todd Clark (FRBC) Entropic Tilting June / 18

10 Gaussian Analytics of Multi-Horizon/Multi-Variable Effects y t:t`4 y t, y t`1, y t`2, y t`3, y t`4 1 Npθ, Σq Consider the tilted density f which imposes that θ 1 µ 1 and Σ 1,1 Ω 1,1 f N pµ, Ωq, with parameters: µ 2:5 θ 2:5 ` Σ 1 1,1 Σ 1,2:5 pµ 1 θ 1 q Ω 2:5,2:5 Σ 2:5,2:5 Σ 2:5,1 Σ 1 1,1 Σ 1,2:5 ˆ Ω 2:5,1 Σ 2:5,1 Σ 1 1,1 Ω 1,1, ˆ 1 Ω1,1 Σ 1,1 Ω 1,1 0: Standard conditional forecast solution If Ω 1,1 ă Σ 1,1 and Σ 2:5,1 0, tilting reduces the variance of the forecasts at other horizons The impact on µ 2:5 and Ω 2:5,2:5 mainly depends on Σ 2:5,1 Todd Clark (FRBC) Entropic Tilting June / 18

11 Results Using SPF Nowcasts: Forecast Accuracy RMSEs, forecast h 1Q h 2Q h 3Q h 4Q h 5Q GDP growth BVAR ET, small m ET, big m Inflation BVAR ET, small m ET, big m Tilting toward SPF nowcasts consistently improves BVAR accuracy at horizons 2Q-3Q Todd Clark (FRBC) Entropic Tilting June / 18

12 Results Using SPF Nowcasts: Forecast Accuracy RMSEs, forecast h 1Q h 2Q h 3Q h 4Q h 5Q Unemployment rate BVAR ET, small m ET, big m T-bill rate BVAR ET, small m ET, big m Persistent variables (UR, T-bill): Nowcast tilting improves forecasts up to 5Q ahead Tilting based on the system of variables ( big ) yields results very similar to tilting variable by variable ( small ) Todd Clark (FRBC) Entropic Tilting June / 18

13 Results Using SPF Nowcasts: Forecast Accuracy CRPSs, forecast h 1Q h 2Q h 3Q h 4Q h 5Q GDP growth BVAR ET, small m ET, small m/v Inflation BVAR ET, small m ET, small m/v Tilting toward SPF nowcasts consistently improves BVAR accuracy, more so at shorter horizons Tilting toward both means and variances of nowcasts ( small m/v ) rather than just means ( small m ) yields small additional gains in density accuracy Todd Clark (FRBC) Entropic Tilting June / 18

14 Results Using SPF Nowcasts: Forecast Accuracy CRPSs, forecast h 1Q h 2Q h 3Q h 4Q h 5Q Unemployment rate BVAR ET, small m ET, small m/v T-bill rate BVAR ET, small m ET, small m/v Todd Clark (FRBC) Entropic Tilting June / 18

15 Assessing Multi-Step Uncertainty Pushing a BVAR forecast toward a nowcast can affect uncertainty of longer-horizon forecasts Some common approaches ignore the uncertainty around the nowcast and effects on subsequent horizons Treating nowcast as data Conventional Gaussian conditional forecasting An advantage of tilting is that it does not distort multi-step uncertainty. We examine ability of tilting toward nowcasts to improve estimates of forecast uncertainty. Todd Clark (FRBC) Entropic Tilting June / 18

16 Results Using SPF Nowcasts: Forecast Uncertainty Length (70%) and coverage (70%), metric forecast h 1Q h 2Q h 3Q h 4Q h 5Q GDP growth Length small m Length small m/v Coverage small m Coverage small m/v Todd Clark (FRBC) Entropic Tilting June / 18

17 Results Using SPF Nowcasts: Forecast Uncertainty Length (70%) and coverage (70%), metric forecast h 1Q h 2Q h 3Q h 4Q h 5Q Unemployment rate Length small m Length small m/v Coverage small m Coverage small m/v Tilting toward nowcast variance shortens prediction intervals It also improves coverage rates and CRPSs Effects are greater at shorter horizons and for more persistent variables Todd Clark (FRBC) Entropic Tilting June / 18

18 Summary ET systematically improves the accuracy of point and density forecasts Payoff depends in part on variable persistence ET using means and variances is a little better than using just means Tilting based on the system of variables is similar to variable-by-variable tilting Tilting can help for accurately estimating forecast uncertainty Shortens prediction intervals, more so for shorter horizons and persistent variables Todd Clark (FRBC) Entropic Tilting June / 18