Decomposing the Volatility Structure of Inflation

Transcription

1 Decomposing the Volatility Structure of Inflation Mengheng Li 1 Siem Jan Koopman 1,2 1 Department of Econometrics, Vrije Universiteit Amsterdam, The Netherlands 2 CREATES, Aarhus University, Denmark November 21, 2017

2 Outline Motivations and literature Model and simulated likelihood Empirical on US inflation 2 / 33

3 Motivations A generic forward-looking Taylor rule with feedback effect says r t = (1 γ)r + γr t 1 + θ[e t π t+j π ]. r t = i t E t π t+1 the short-term ex-ante interest rate. π the inflation target, such as 2% annually. r interest rate equilibrium. Apparently, inflation forecast E t π t+j is important in choosing the policy instrument parameter γ and θ. For macroeconometric study, how to de-trend and de-seasonalize the (monthly) inflation series? Hamilton (2017): You should never use HP filter. 3 / 33

4 Literature in macro DSGE model Bodenstein et al. (2008): Objective function penalizing core inflation volatility welfare maximization. Related to Mishkin (2007). Del Negro and Schorfheide (2013), Diebold et al. (2015): Inflation with SV, and technology shock with SV. Volatility response to macro shocks Goodfriend and King (1997), King and Wolman (1999), Aoki (2001): Welfare loss price change volatility in the CPI sticky components. Erceg et al. (2000), FOMC (2009), BOE (2013): Discrepancy of core and headline inflation is due to different volatility response to macro shocks, such monetary policy shock and energy crisis. Should we and how can we detrend and deseasonalize? Aadland (2005): A review. 4 / 33

5 Literature in econometrics Time-varying volatility is a key nonlinearity in macroeconomic time series Sims and Zha (2006), Justiniano and Primiceri (2008), Bloom (2009), Clark (2011), Fernandez and Rubio (2013), Curdia et al. (2014): fit of VAR, factor model, and ARIMA model are largely improved with SV. Inflation forecasting models have time-inconsistent performance without SV Stock and Watson (2007, 2008) showed that Gordon(1990) s triangle model, Harvey (1990) s local level model, Atkeson and Ohanian (2001) s random walk model, and many other univariate as well as multivariate models show episodes of good performance in forecasting. 5 / 33

6 Stock and Watson s LoL-SV Stock and Watson (2007, 2008) showed good forecasting performance of the following local level model with SV (LoL-SV) y t = µ t + exp( 1 2 hy t )ɛ t, ɛ t N (0, 1), µ t+1 = µ t + exp( 1 2 hµ t )η t, η t N (0, 1), and SV h y t, h µ t are modelled as random walks with correlated innovations, i.e. h y t+1 = hy t + σ y ζt y, h µ t+1 = hµ t + σ µ ζ t µ, [ ζ y ] [ ] [ ] t 0 1 ρ N (, ). 0 ρ 1 ζ µ t Novelty: SV in both observation and the state equation. Difficulty: They did not estimate but calibrated σ y = σ µ = 0.2 and ρ = 0. 6 / 33

7 ML estimation LoL-SV is non-linear state space model without analytical likelihood function. We propose a simulated likelihood method for estimating this model, using importance sampling to integrate out the SV ht y (transitory volatility) in the observation equation, and SV h t µ (permanent volatility) in the state transition. Conditional on ht y and h t µ, this model is a linear Gaussian state space model. Kalman filter can efficiently evaluate its likelihood. 7 / 33

8 ML estimation The conditional log-likelihood can be written as T log p(y T H y T, Hµ T ) = log p(y t Y t 1, H y t 1, Hy t 1 ), t=1 and the conditional log-likelihood contribution is log p(y t Y t 1, H y t 1, Hy t 1 ) = log Φ(v t; 0, F t ), where Φ(.; µ, σ 2 ) denotes the Gaussian density function, and v t and F t are produced by Kalman filter given h y t 1 and hµ t 1. 8 / 33

9 ML estimation We can show that the likelihood function can be written as L(Y T ; σ µ, σ y, ρ) = p(y T H y T, Hµ T )p(hy T, Hµ T )dhy T dhµ T p(yt H y T = g(y t ), Hµ T ) g(y T H y T, Hµ T )g(hy T, Hµ T Y T )dh y T dhµ T. For estimation, we maximise L (Y T ; σ µ, σ y, ρ) = g(y T ) 1 M M m=1 p(y T H y,(m) T, H µ,(m) T ) g(y T H y,(m) T, H µ,(m) T ), We propose the importance density or importance model to be a linear Gaussian state space model, so that g(y t ) is easily computed using prediction error decomposition, H µ,(m) T can be easily drawn from g(h y T, Hµ T Y T ) using simulation smoother. H y,(m) T 9 / 33

10 ML estimation The (conditional) importance density g(h t Y t ) takes the Gaussian canonical form g(h t Y t ; ψ) = exp (r t + b th t 12 ) h tc t h t, where h t collects all SV series at time t. r t is a normalizing constant, so importance parameters are b t and C t. It can be shown that this results from with yt = h t + ɛ t, ɛ N (0, Ct 1 ), h t+1 = δ + φh t + η t, y t = C 1 t b t, C t and b t are functions of Y T. The efficiency of the simulated likelihood method crucially depends on the choice of C t and b t. If determined, one can evaluate g(y T ), g(y T H T ), and easily draw from g(h t Y T ). 10 / 33

11 ML estimation To determine C t and b t : Our method is numerically accelerated importance sampling, built on the efficient importance sampling method of Richard (2007). We minimize the variance of distance between log-densities. min λ 2 t (y t, h t ; ψ)ω t g(h t YT ; ψ)dh t, b t,c t where λ t = logp(v t h t ; ψ) logg(y t h t ; ψ) constant, ω t = p(v t h t ; ψ) g(y t h t ; ψ). This is a weighted least square problem. Straightfoward iteration leads to optimal importance density. 11 / 33

12 E g (h t Y T ) Figure: Iteration 0 12 / 33

13 E g (h t Y T ) Figure: Iteration 1 13 / 33

14 E g (h t Y T ) Figure: Iteration 2 14 / 33

15 E g (h t Y T ) Figure: Iteration 3 15 / 33

16 E g (h t Y T ) Figure: Iteration 4 16 / 33

17 E g (h t Y T ) Figure: Iteration 5 17 / 33

18 ML estimation The parameters of the importance model are determined via a modified Numerically Accelerated Importance Sampling algorithm of Koopman et al. (2015). Figure: Log-likelihood as a function of (hyper)parameters. 18 / 33

19 Estimation and signal extraction of LoL-SV 4 (i) (ii) trend quaterly inflation memory index (iii) transitory volatility (iv) permanent volatility Figure: Main fit from the U.S. quarterly inflation series. (i) Inflation series and its trend component µ t; (ii) Memory index m t as defined in Section 2.3; (iii) Transitory volatility exp(h y t /2); (iv) Permanent volatility exp(h µ t /2). Green dashed lines indicate the 95% confidence bands. 19 / 33

20 Estimation and signal extraction of ARUC-SV Working paper of Bank of England last month (Cecchetti et al. 2017): ARUC-SV (autoregressive unobserved components model with SV) model (y t µ t ) = φ(y t 1 µ t 1 ) + σt y ɛ t, ɛ t N(0, 1), µ t+1 = µ t + σ t µ η t, η t N(0, 1), log σ y t+1 = log σy t + σ y ζt y, ζt y N(0, 1), log σ µ t+1 = log σµ t + σ µ ζ t µ, ζ t µ N(0, 1). y t : U.K. CPI inflation; µ t : stochastic trend, core inflation component; y t µ t : AR(1) inflation gap. 20 / 33

21 Estimation and signal extraction of ARUC-SV 7.5 (i) (ii) trend inflation 5.0 inflation cycle (iii) transitory volatility (iv) permanent volatility Figure: Main fit from the U.S. quarterly inflation series. (i) Inflation series and its trend component µ t; (ii) Inflation gap y t µ t; (iii) Transitory volatility exp(h y t /2); (iv) Permanent volatility exp(h µ t /2). Green dashed lines indicate the 95% confidence bands. 21 / 33

22 Example: LLS-OTSSV Monthly core inflation from 1957:1-2015:1, not seasonally adjusted (i) (ii) Figure: (i) The monthly U.S. core CPI and (ii) the first difference of log CPI (inflation). 22 / 33

23 Example: LLS-OTSSV The model is local level plus seasonal model with SV (LLS-OTSSV), y t = µ t + γ t + exp( 1 2 hy t )ɛ t, h y t+1 = α y + φ y h y t + σ y ζ y t µ t+1 = µ t + exp( 1 2 hµ t )η µ t, h µ t+1 = hµ t + σ µ ζ µ t, γ t+1 = (γ t + γ t γ t 10 ) + exp( 1 2 hγ t )η γ t, h γ t+1 = hγ t + σ γ ζ γ t, with (ɛ y t, η µ t, η γ t ) being uncorrelated standard Gaussian random variables and independent on (ζ y t, ζ µ t, ζ γ t ), for t = 1,..., n and n = 695. Furthermore, we also model correlations among SV series by E(ζ y t ζ µ t ) = ρ yµ. There are eight parameters in the model, namely ψ = (α y, φ y, σ y, σ µ, σ γ, ρ yµ ). 23 / 33

24 Example: LLS-OTSSV Parameter LLS LLS-D LLS-OTSSV LLS-OTSSV-D α [-5.150, ] [-5.057, ] φ y [0.975, 0.993] [0.967, 0.983] σ y [0.133, 0.156] [0.120, 0.142] [0.126, 0.218] [0.135, 0.228] σ µ [0.031, 0.054] [0.032, 0.053] [0.082, 0.217] [0.082, 0.216] σ γ [0.020, 0.033] [0.022, 0.037] [0.092, 0.152] [0.079, 0.139] ρ yµ [0.496, 0.780] [0.426, 0.761] D 1 (1974:2) [0.107, 0.731] [0.123, ] D 2 (1974:11) [-0.576, 0.042] [-0.587, ] D 3 (1980:7) [-1.521, ] [-1.469, ] D 4 (1981:9) [-0.732, ] [-0.723, ] D 5 (1982:8) [-0.730, ] [-0.745, ] Normality Box-Ljung H(n/3) LL / 33

25 Example: LLS-OTSSV 5.0 (i) (ii) 1.00 (iii) (iv) (v) (vi) Figure: Graphic diagnostics for LLS and LLS-OTSSV based on standardized residuals: (i): LLS standardized residuals; (ii) LLS autocorrelogram of residuals (solid lines) and squared residuals; (iii) LLS scaled cumulative sum of squared residuals; (iv)-(vi) LLS-OTSSV counterparts. 25 / 33

26 Example: LLS-OTSSV 0.5 (i) irregular 0.4 (ii) transitory volatility (iii) trend (iv) trend volatility (v) 0.25 seasonal (vi) 0.04 seasonal volatility Figure: Main fit from the U.S. monthly inflation series. 26 / 33

27 LLS-OTSSV Model Variants Parameter LLS-TSSV LLS-OSV LLS-OSSV LLS-OTSV α [-5.374, ] [-5.437, ] [-5.157, ] φ y [0.978, 0.996] [0.983, 0.993] [0.968, 0.999] σ y [0.095, 0.112] [0.127, 0.246] [0.135, 0.224] [0.103, 0.234] σ µ [0.216, 0.411] [0.016, 0.028] [0.018, 0.031] [0.089, 0.295] σ γ [0.074, 0.213] [0.021, 0.035] [0.047, 0.130] [0.022, 0.033] ρ yµ [0.426, 0.761] ρ µγ [0.176, 0.490] ρ yγ [-0.365, 0.057] Normality Box-Ljung H(n/3) LL / 33

28 Figure: Main fit from the U.S. monthly inflation series under LLS-OTSSV variants. 28 / 33

29 Forecasting evaluation Four model specifications No SV at all, LLS; Only transitory volatility: LLS-OSV; Only permanent volatility: LLS-TSSV; Both: LLS-OTSSV; Four forecasting horizons h Monthly, h = 1; Quarterly: h = 3; Semiannually: h = 6; Annually: h = 12. Three types of forecast Point forecast; Density forecast. 29 / 33

30 Point forecast MFE MAFE RMSE h = 1 h = 3 h = 6 h = 12 h = 1 h = 3 h = 6 h = 12 h = 1 h = 3 h = 6 h = 12 LLS LLS-OSV LLS-TSSV LLS-OTSSV (i) (ii) 0.50 Data LLS-OSV LLS-OTSSV LLS LLS-TSSV LLS LLS-OSV LLS-TSSV LLS-OTSSV (iii) (iv) 1.00 LLS LLS-OSV LLS LLS-OSV LLS-TSSV LLS-OTSSV LLS-TSSV Figure: (i) One-step ahead forecast of four models. (ii) Forecast errors. (iii) Cumulative difference of absolute forecast errors. (iv) Recursive standard deviation plot of forecast errors. 30 / 33

31 Density forecast We firstly consider overall calibration based on PIT, similar to Diebold et al. (1998). Let d h t (.) denote an h-step ahead forecasting density with distribution function D h t (.). The PIT of h-step ahead forecast y t+h t is D h t (y t+h ) = yt+h M m=1 d h t (s)ds p(y t Y t 1, H y,(m) t 1, Hµ,(m) t 1 ) g(y t Y t 1, H y,(m) t 1, Hµ,(m) t 1 (m) )Φ(v t+h ; 0, F (m) t+h ). If the forecasting density d h t (.) is correctly calibrated, then D h t (.) s are uniformly distributed random variables in the unit interval. 31 / 33

32 Density forecast Figure: Histograms of one-step and three-step ahead forecasting density PIT D 1 t (y t+1 ) and D 3 t (y t+3 ). Top to bottom row are one- and three-step ahead PIT. We group PIT s into five equal-sized bins each of which should contain exactly 20% of PIT s under uniformity. 32 / 33

33 Conclusion One should take into account of SV when forecasting inflation. Static models usually show episodes of satisfactory performance. We propose a structural state space model which explicitly decomposes a time series into unobserved components with SV. An efficient simulated likelihood estimation procedure is developed. Besides good forecasting performance, this model provides a natural way for de-trending and de-seasonalisation. 33 / 33