Dynamics or diversity? An empirical appraisal of distinct means to - PowerPoint PPT Presentation

Dynamics or diversity? An empirical appraisal of distinct means to measure inflation uncertainty M. Hartmann and H. Herwartz ur Statistik und ¨ Institut f¨ Okonometrie Christian-Albrechts-Universit¨ at zu Kiel June 2nd, 2012 Bundesbank/Ifo Workshop Uncertainty and Forecasting in Macroeconomics M. Hartmann and H. Herwartz (CAU) Inflation uncertainty measurement June 2nd, 2012 1 / 20

Introduction 1 Objective 2 Measuring inflation uncertainty 3 Inflation uncertainty measures as preditors of interest rates 4 Impact of inflation uncertainty on interest rates 5 Conclusion 6 Appendix 7 M. Hartmann and H. Herwartz (CAU) Inflation uncertainty measurement June 2nd, 2012 2 / 20

Introduction Introduction (Inflation-) expectations play a key role in many economic models Examples: New Keynesian Phillips curve, consumption smoothing, firms’ investment, price setting,... ⇒ Under risk aversion, considering inflation uncertainty makes sense whenever inflation expectations are part of the model → Inflation uncertainty (IU) is unobservable → Distinct ways to measure IU have been proposed Any empirical study involving inflation risk has to motivate choice of particular uncertainty measure M. Hartmann and H. Herwartz (CAU) Inflation uncertainty measurement June 2nd, 2012 3 / 20

Objective Objective This study: Pseudo out-of-sample forecasting ’horse race’ with alternative IU measures as predictors for interest rates Objective: Empirical ranking of distinct approaches to measure inflation uncertainty (IU) Distinguish two families of IU measurement: → Dynamic approaches (e.g. (G)ARCH) → Disparity (or Dispersion) of expectations, typically based on surveys of expert forecasts, e.g. ASA-NBER Quarterly Economic Outlook Survey, ZEW survey M. Hartmann and H. Herwartz (CAU) Inflation uncertainty measurement June 2nd, 2012 4 / 20

Objective Median IU trajectories - 4 × Dynamic (above), 4 × Dispersion (below) h (0 . 1) σ τ + ℓ | τ GARCH(1,1) a τ ˆ τ +1 | τ −3 8 x 10 −3 0.015 10 x 10 0.04 9 7 0.035 8 0.01 6 0.03 7 6 0.025 5 0.005 5 0.02 4 4 0.015 3 0.01 3 0 1990 1995 2000 2005 2010 1990 1995 2000 2005 2010 1990 1995 2000 2005 2010 2 1990 1995 2000 2005 2010 ˆ s τ + ℓ | τ σ τ + ℓ | τ ¯ ξ τ + ℓ | τ ZEW-survey IU 10 0.14 0.06 0.12 9 0.12 0.05 0.1 8 0.1 7 0.04 0.08 0.08 6 0.03 0.06 5 0.06 0.02 0.04 4 0.04 3 0.01 0.02 0.02 2 0 0 0 1 1990 1995 2000 2005 2010 1990 1995 2000 2005 2010 1990 1995 2000 2005 2010 1990 1995 2000 2005 2010 The figure shows the median over 18 economies. GARCH(1,1) and ZEW-survey IU are benchmark measures from the related literature M. Hartmann and H. Herwartz (CAU) Inflation uncertainty measurement June 2nd, 2012 5 / 20

Measuring inflation uncertainty Measuring IU by means of inflation forecasting We consider forecast-based measures of IU Autoregressive (AR) scheme is among most successful models to predict inflation π t = ln( CPI t / CPI t − 4 ) iid ∼ (0 , σ 2 π t + ℓ = α 0 + α 1 t + α 2 π t + ǫ t + ℓ , t = τ − B +1 , ..., τ, ǫ t + ℓ ǫ ) (1) Predictions ˆ π τ + ℓ | τ obtained at forecast horizons ℓ ∈ { 1 , 2 , 3 , 4 } τ = T 0 − ℓ, ..., T − ℓ := rolling forecast origin, B is estimation window size time instances T 0 and T delimit period for which IU measures are obtained (1988Q1 to 2011Q1) Cross section comprises 18 developed economies (Austria, Belgium, Canada, Denmark, Finland, France, Germany, Ireland, Italy, Japan, Netherlands, Norway, Portugal, Spain, Sweden, Switzerland, UK, US) M. Hartmann and H. Herwartz (CAU) Inflation uncertainty measurement June 2nd, 2012 6 / 20

Measuring inflation uncertainty Distinct ways to measure IU - 1. Dynamic measures 1.1 Predictive standard deviation � (1 + z ′ τ ( Z ′ τ Z τ ) − 1 z τ ) , ˆ σ τ + ℓ | τ = ˜ σ ǫ (2) with Z τ := design matrix of linear (AR) inflation forecasting model, z τ := most recent observations for out-of-sample forecasting. 1.2 Exponential smoothing (Zangari 1996) � h ( λ ) λ (∆ π τ ) 2 + (1 − λ )(∆ π ) 2 . τ +1 | τ = (3) t = τ − B +1 (∆ π t ) 2 , Presetting: In (3), ∆ π t = π t − π t − 1 , and (∆ π ) 2 = (1 / ( B − 1)) � τ − 1 λ ∈ { 0 . 1 , 0 . 2 } ≈ typical estimates (e.g. Bollerslev 1986) 1.3 Unanticipated volatility (Ball and Cecchetti 1990) a τ + ℓ = | ˆ ˆ π τ + ℓ | τ − π τ + ℓ | , (4) based on AR-implied inflation forecasts ˆ π τ + ℓ | τ M. Hartmann and H. Herwartz (CAU) Inflation uncertainty measurement June 2nd, 2012 7 / 20

Measuring inflation uncertainty Distinct ways to measure IU - 2. Dispersion measures 2.1 Disagreement of expectations � J � � � π j ,τ + ℓ | τ − π τ + ℓ | τ ) 2 ˆ s τ + ℓ | τ = � (1 / ( J − 1)) (ˆ (5) j =1 from j = 1 , ..., 5 linear autoregressive distributed lag (ADL) forecasting models 2.2 Average uncertainty (Zarnowitz and Lambros 1987) J � σ τ + ℓ | τ = (1 / J ) ¯ ˆ σ j ,τ + ℓ | τ (6) j =1 2.3 Augmenting the disagreement measure (cf. Lahiri and Liu 2005, Wallis 2005) ξ τ + ℓ | τ = 0 . 5(ˆ s τ + ℓ | τ + ¯ σ τ + ℓ | τ ) (7) 2.4 Alternative augmentation (cf. Lahiri and Sheng 2010) s τ + ℓ | τ + h (0 . 1) ζ τ + ℓ | τ = 0 . 5(ˆ τ +1 | τ ) (8) M. Hartmann and H. Herwartz (CAU) Inflation uncertainty measurement June 2nd, 2012 8 / 20

Inflation uncertainty measures as preditors of interest rates Forcasting by means of the ’augmented Fisher equation’ P P � � R τ + ℓ = γ 10 + γ 11 τ + γ 12 , p π τ − p +1 + γ 13 , p R τ − p +1 + p =1 p =1 P � + γ 14 , p IU τ − p + ℓ +1 | τ + e τ + ℓ , τ = T 0 − ℓ, ..., T − ℓ (9) p =1 following Levi and Makin (1979), Blejer and Eden (1979), inter alia. iid ∼ (0 , σ 2 IU τ + ℓ | τ represents a particular inflation uncertainty measure, e τ + ℓ e ) R τ + ℓ : Interest rate on 10-year government bond → Each observation R τ + ℓ from the sample period τ = T 0 − ℓ, ..., T − ℓ is predicted ℓ -steps ahead by means of a respective leave-one-out cross-validation estimate → This yields distinct forecasts of R τ + ℓ based on alternative IU measures (2) to (8) Maximum lag order P = 4 ⇒ 2 12 distinct subset models M. Hartmann and H. Herwartz (CAU) Inflation uncertainty measurement June 2nd, 2012 9 / 20

Inflation uncertainty measures as preditors of interest rates Subset modelling by Bayesian model averaging (BMA) Averaging forecasts improves predictive accuracy (Bates and Granger 1969, Timmermann 2005, Wright 2009) Combine forecasts from m = 1 , ..., M = 2 12 reformulations of augmented Fisher equation: M R ( m ) ˆ � w ∗ m ˆ R τ + ℓ | τ = τ + ℓ | τ , (10) m =1 w m � w ∗ L m ( γ ( m ) ) p m ( γ ( m ) ) d γ ( m ) . m = and w m = (11) � m w m L m ( γ ( m ) ) := likelihood function, p m ( γ ( m ) ) := a-priori distribution of γ ( m ) Based on log-likelihood l ( γ ( m ) ) = ln L ( γ ( m ) ), posterior probabilities w m in (11) can be approximated as γ ( m ) ) − n m ln ˆ w m = l (ˆ 2 ln( T − T 0 ) , (12) γ ( m ) := (Q)ML estimator of γ ( m ) and n m stands for the number of right hand side ˆ variables in model m . � � γ ( m ) ) − n m Forecast combination weights obtain as w m in (11) by exp l (ˆ 2 ln( T − T 0 ) M. Hartmann and H. Herwartz (CAU) Inflation uncertainty measurement June 2nd, 2012 10 / 20

Inflation uncertainty measures as preditors of interest rates Performance criterion Forecast ranking based on absolute forecast error (AE) | e • τ + ℓ | τ | = | ˆ R • τ + ℓ | τ − R τ + ℓ | (13) σ τ + ℓ | τ , h ( λ ) ’ • ’ represents IU measures ˆ τ +1 | τ , ˆ a τ , ˆ s τ + ℓ | τ , ¯ σ τ + ℓ | τ , ξ τ + ℓ | τ , ζ τ + ℓ | τ , max(IU) , min(IU) , median(IU) , mean( TS ) , mean( DS ). → Frequency by which IU measure • produces forecasts among the 3 best (Stock and Watson 1999): T − ℓ 18 TOP3 • = (1 / (( T − T 0 + 1) × 18)) i ,τ + ℓ | ≤ | e (3) � � I( | e • i ,τ + ℓ | ) , (14) τ = T 0 − ℓ i =1 where | e (3) i ,τ + ℓ | is the 3rd smallest AE and I( · ) is the indicator function M. Hartmann and H. Herwartz (CAU) Inflation uncertainty measurement June 2nd, 2012 11 / 20

Inflation uncertainty measures as preditors of interest rates TOP3 • frequencies Dynamic measures Dispersion measures ℓ = 1 ℓ = 2 ℓ = 3 ℓ = 4 ℓ = 1 ℓ = 2 ℓ = 3 ℓ = 4 ˆ σ τ + ℓ | τ 21.45 24.16 25.32 25.19 ˆ 21.51 20.09 20.54 20.74 s τ + ℓ | τ h (0 . 1) 23.32 22.03 21.77 21.77 σ τ + ℓ | τ ¯ 22.35 27.65 28.62 27.97 τ +1 | τ h (0 . 2) 23.26 21.77 17.57 18.09 ς τ + ℓ | τ 15.96 18.15 20.80 21.90 τ +1 | τ a τ ˆ 26.94 23.06 22.93 23.13 ζ τ + ℓ | τ 19.44 20.22 22.22 20.93 TS 28.81 24.22 21.38 20.99 DS 19.06 18.28 18.99 21.12 Further IU statistics max(IU) 15.70 16.86 20.80 20.09 median 20.74 23.00 22.48 23.39 min(IU) 19.51 21.83 20.16 17.12 ◦ 22.16 19.51 18.22 19.32 Cell entries represent the frequencies in which distinct IU measures lead to forecasts which are among the 3 most accurate ones. The row labelled as ’ ◦ ’ reports respective ranking frequencies for a forecasting model without an IU term. M. Hartmann and H. Herwartz (CAU) Inflation uncertainty measurement June 2nd, 2012 12 / 20