Core Inflation and Trend Inflation. Appendix

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1 Core Inflation and Trend Inflation Appendix June 2015 (Revised November 2015) James H. Stock Department of Economics, Harvard University and the National Bureau of Economic Research and Mark W. Watson Department of Economics and the Woodrow Wilson School, Princeton University and the National Bureau of Economic Research

2 This appendix contains a detailed description of the estimation methods for the univariate and multivariate UCSVO models, and contains additional empirical results. 1. The univariate model 1.1 The model: π t = τ t + ε t τ t = τ t-1 + σ Δτ,t η τ,t ε t = σ ε,t s t η ε,t. 2 Δln( σ ) = γ ε ν ε,t Δln( ε,t 2 σ Δ τ,t ) = γ Δτ ν Δτ,t where (η ε, η τ, ν ε, ν Δτ ) are iidn(0, I 4 ), and s t ~ i.i.d and independent of (η ε, η τ, ν ε, ν Δτ ) with s t =! 1!with!probability!(1 p) U[2,10]!with!probability!p 1.2 Priors: γ ε, γ Δτ, p, τ 0, ln(σ ε,0 ), and ln(σ Δτ,0 ) are independent, with: γ ε ~ U[0,! 0.40/ and np = 12 for monthly data); np ], where np denote the number of periods per year (np = 4 for quarterly data γ Δτ ~ U[0, 0.40/ np! ]; p ~ Beta(α, β) with α = (1/(4np)) (10np) and β = [1 (1/(4np))] (10np); and (τ 0, ln(σ ε,0 ), ln(σ Δτ,0 )) ~ N(0,κI 3 ), with κ = The U[2,10] distribution for the values of s t is approximated with an equally-spaced grid of 9 points. The uniform distributions for γ are approximated by an equally spaced grid of 5 points. 1

3 1.3 Approximation of the posterior: The mean and quantiles of the posterior are approximated by MCMC draws. The details are as follows Kim-Shephard-Chib (KSC) (1998) approximate model for stochastic volatility: Background: A review of KSC approach Let x t = σ t η t and ln(σ 2 2 t ) = ln(σ t 1 ) + γν t with (η t, ν t ) ~ iidn(0,i 2 ). Then ln( x t 2 ) = ln(σ t 2 ) + ln(η t 2 ) ln(σ 2 2 t ) = ln(σ t 1 ) + γν t which is a linear state-space model with non-gaussian measurement error ln(η 2 t ) ~ ln( χ 2 1 ). KSC suggest approximating the ln( χ 2 1 ) distribution by a mixture of normals so that ln(η t 2 ) ~ n w it a it, where w it are iid (0-1) variables with w it = 1 for only one value of i at each t, i=1 and with p(w it =1)=p i. The a it variables are Gaussian with a it ~ N(µ i, σ i 2 ). KSC propose an n = 7 component Gaussian mixture to approximate the ln( χ 1 2 ) distribution and report the values of (p i, µ i, σ i ) for i = 1,, 7. Omori, Chib, Shephard, and Nakajima (2007) propose a more accurate 10-component Gaussian mixture approximation. We use the OCSN approximation Details of the MCMC Iterations: Let Y denote the observed data (π t, for t = 1,, T). Let θ denote the vector of unknown random variables for which Gibbs draws will be taken. In this model θ = [{τ t }, {w ε,i,t }, {w Δτ,i,t }, γ ε, γ Δτ, {σ ε,t }, {σ Δτ,t }, {s t }, p]. Partition θ as θ = (θ 1, θ 2, θ 3, θ 4 ) where θ 1 = {τ t }, {w ε,i,t }, {w Δτ,i,t } θ 2 = γ ε, γ Δτ,{σ ε,t }, {σ Δτ,t } θ 3 = {s t } θ 4 = p 2

4 Gibbs draws (1) Draw θ 1 from f(θ 1 Y, θ 2, θ 3, θ 4 ) (2) Draw θ 2 from f(θ 2 Y, θ 1, θ 3, θ 4 ) (3) Draw θ 3 from f(θ 3 Y, θ 1, θ 2, θ 4 ) (4) Draw θ 4 from f(θ 3 Y, θ 1, θ 2, θ 3 ) Gibbs Draws: Details (1) Draw θ 1 from f(θ 1 Y, θ 2, θ 3, θ 4 ). That is, draw {τ t }, {w ε,i,t }, {w Δτ,i,t } from f({τ t },{w ε,i,t },{w Δτ,i,t } Y, θ 2, θ 3, θ 4 ). Write f({τ t },{w ε,i,t },{w Δτ,i,t } Y, θ 2, θ 3, θ 4 ) = f({τ t } Y, θ 2, θ 3, θ 4 ) f({w ε,i,t },{w Δτ,i,t } Y, θ 2, θ 3, θ 4, {τ t }) (1.a) Draw {τ t } from f({τ t } Y, θ 2, θ 3, θ 4 ) = f({τ t } Y, {σ ε,t }, {σ Δτ,t }, {s t }) These are draws from a linear Gaussian unobserved components model. Conditional moments can be computed from Kalman smoother, and draws can be conveniently obtained using the insights in Carter and Kohn (1994). (1.b) Draw {w ε,i,t }, {w Δτ,i,t } from f({w ε,i,t },{w Δτ,i,t } Y, θ 2, θ 3, θ 4, {τ t }). 2 (i) With {τ t } and {σ Δτ,t } known, ln(η Δτ,t ) can be calculated. Then {w Δτ,i,t } can be drawn from posterior mixture probability in a straightforward manner. (ii) With π t and τ t known, ε t can be calculated. With {ε t } and {σ ε,t } known, ln(η 2 ε,t ) can be calculated. Then {w ε,i,t } can be drawn from posterior mixture probability in a straightforward manner. (2) Draw θ 2 from f(θ 1 Y, θ 1, θ 3, θ 4 ) That is, draw γ ε, γ Δτ,{σ ε,t }, {σ Δτ,t } from f(γ ε, γ Δτ,{σ ε,t }, {σ Δτ,t } Y, θ 1, θ 3, θ 4 ). Write f(γ ε, γ Δτ,{σ ε,t }, {σ Δτ,t } Y, θ 1, θ 3, θ 4 ) = f(γ ε, γ Δτ Y, θ 1, θ 3, θ 4 ) f({σ ε,t }, {σ Δτ,t } Y, θ 1, θ 3, θ 4, γ ε, γ Δτ ). (2a) Draw γ ε, γ Δτ from f(γ ε, γ Δτ Y, θ 1, θ 3, θ 4 ) (i) with Δτ and w Δτ,i,t known, then 3

5 ln 2 (( Δτ ) t ) = ln(σ 2 Δτ,t 2 ln(σ Δτ.t n ) + w Δτ,i,t a Δτ,i,t i=1 2 ) = ln(σ Δτ.t 1 ) + γ Δτ ν Δτ,t which is a Gaussian linear state-space model indexed by γ Δτ. Using a discrete 2 prior for γ Δτ, the likelihood as a function of ln ( Δτ ) t can be calculated, which ( ) then yields the posterior probabilities for the grid of values of γ Δτ. This enables draws from the posterior. (ii) draws for γ ε are computed in a similar fashion. (2b) Draw {σ ε,t }, {σ Δτ,t } from f({σ ε,t }, {σ Δτ,t } Y, θ 1, θ 3, θ 4, γ ε, γ Δτ ). 2 Draws of ln(σ Δτ.t ) are readily computed from the linear Gaussian state-space model given in 2a. Similarly for ln(σ 2 ε.t ) (3) Draw θ 3 from f(θ 3 Y, θ 1, θ 2, θ 4 ) That is, draw {s t } from f({s t } Y, θ 1, θ 2, θ 4 ). Write ln(ε 2 t ) ln(σ 2 ε,t ) = ln(s 2 t ) + w ε,i,t a ε,i,t n i=1 With ε t, σ ε,t and w ε,i,t known, then ln(ε 2 t ) ln(σ 2 ε,t ) has a mixture of normal distribution, where the normal distributions have means that depend on the value of s t. This can be used to form the (discrete) posterior for s t, from which draws can be made. (4) Draw θ 4 from f(θ 4 Y, θ 1, θ 2, θ 3 ) That is, draw p from f(p Y, θ 1, θ 2, θ 3 ) = f(p {s t })= f(p {1(s t > 1)}), which is standard given the Beta prior for p. 4

6 1.3.3 Number of draws and accuracy The MCMC iterations described above were initialized with 10,000 iterations. We then carried out 50,000 iterations, saving every 10 draws, yielding 5,000 draws from which the various posterior statistics were estimated. We carried out this process twice using independent starting values for PCE-all; this yielded two estimates of the posterior mean of {τ t }. The mean absolute difference between the two sets of estimates over all t was 0.01 and the largest absolute difference was less that Results The posterior means for τ t, σ ε,t, σ Δτ, and s t are plotted in the paper. The posterior distributions for γ ε, γ Δτ, and p are summarized in the following tables: Value Prior Prob Table A.1: Posterior distribution of γ ε and γ Δτ Posterior Probability γ ε PCE-all PCExE PCExFE PCE-all PCExE PCExFE γ Δτ Table A.2: Posterior distribution of p (selected quantiles) Quantile PCE-all PCExE PCExFE

7 2. The 17-component multivariate model 2.1 The model: π i,t = α i, τ,t τ c,t + α i, ε,t ε c,t + τ i,t + ε i,t, τ c,t = τ c,t-1 + σ Δτ,c,t η τ,c,t ε c,t = σ ε,c,t s c,t η ε,c,t τ i,t = τ i,t-1 + σ Δτ,i,t η τ,i,t ε i,t = σ ε,i,t s i,t η ε,i,t α i, τ,t = α i,τ, t-1 + λ i, τζ i,τ,t and α i, ε,t = α i,ε,t-1 + λ i, εζ i,ε,t Δln( σ Δ ) = γ Δτ,c ν Δτ, c,t, Δln( σ ) = γ ε,c ν ε, c,t, Δln( σ Δ ) = γ Δτ,i ν Δτ,i,t, and Δln( σ ) = γ ε,i ν ε,i,t, τ, ct, ε, ct, τ,, it ε,, it 2.2 Priors The priors for γ ε,c, γ ε,i, γ Δτ,c, γ Δτ,i, p c, p i, τ i,0, ln(σ ε,i,0 ), and ln(σ Δτ,i,0 ) are the same priors used in the univariate model and described above. The values of τ c,0, ln(σ ε,c,0 ), and ln(σ Δτ,c,0 ) are set to zero. These are normalizations as described in the text. Let α Δτ,0 denote the n 1 vector of factor loadings at t = 0. The prior is α Δτ,0 ~ N(0, κ 2 ll'+κ 2! 1 2 I n ), where l is and n 1 vector of 1s, κ 1 = 10 and κ 2 = 0.4. The same prior is used for α ε,0 and the two vectors of factor loadings are independent. Independent inverse Gamma priors are used for the λ parameters. The scale and shape parameters so that the prior corresponds to T prior prior observations with s 2 Prior = ω 2 /T Prior. We use ω =0.25 and T prior = T/ Approximation of the posterior: The MCMC iterations proceed much as for the univariate model, with the following exception. θ 1 now contains θ 1 ={τ c,t }, {τ i,t },{w ε,i,t }, {w Δτ,i,t }, {ε c,t }, {α i, τ,t}, {α i, ε,t}, λ τ, and λ ε Partition θ 1 as (θ 1a, θ 1b ) with θ 1a = {τ c,t }, {τ i,t }, {ε c,t }, {α i, τ,t}, {α i, ε,t}, λ τ, λ ε, and θ 1b = {w ε,i,t }, {w Δτ,i,t }. As in the univariate model, θ 1 is drawn from f(θ 1 Y, θ 2, θ 3, θ 4 ) by drawing 6

8 (a) θ 1a from f(θ 1a Y, θ 2, θ 3, θ 4 ) and then (b) θ 1b from f(θ 1a Y, θ 2, θ 3,, θ 4, θ 1a ). The draws from (b) are just as in the univariate model. The difference in is (a). Here are the details: Decompose θ 1a as θ 1a,1 = {τ c,t }, {τ i,t }, {ε c,t } θ 1a,2 = {α i, τ,t}, {α i, ε,t} θ 1a,3 = λ τ, λ ε. These are drawn in step (a) using Gibbs sampling (so this is Gibbs-within-Gibbs ). Steps: (1a.1) Draw {τ c,t }, {τ i,t }, {ε c,t } from f({τ c,t }, {τ i,t }, {ε c,t } Y, θ 2, θ 3, θ 4, θ 1a,2, θ 1a,3 ) Which are draws from factors from a linear Gaussian SS model. (1a.2) Draw {α i, τ,t}, {α i, ε,t} from f({α i, τ,t}, {α i, ε,t} Y, θ 2, θ 3, θ 4, θ 1a,1, θ 1a,3 ) Which are draws from factors from a linear Gaussian SS model. (1a.3) Draw λ τ, λ ε from f(λ τ, λ ε Y, θ 2, θ 3, θ 4, θ 1a,1, θ 1a,2 ) These are posterior draws from f(λ τ, λ ε {α i, τ,t}, {α i, ε,t}). When there is no TVP, λ τ = λ ε = 0 and step (1a.3) is skipped. 2.4 Results Parameters for common factors: Figure 3 in the text shows posterior estimates for τ t, σ Δτ,c,t, σ ε,c,t, and s c,t. The remaining parameter "common" parameters are γ Δτ,c, γ ε,c, and p c, and their posteriors are summarized in the tables below. 7

9 Value Table A.3: Posterior distribution for γ Δτ,c and γ ε,c Prior Posterior Probability Prob γ Δτ,c γ ε,c Table A.4: Posterior distribution of p c (selected quantiles) 16% 50% 67%

10 Results for sector-specific parameters: Table A.5: Posterior distribution for γ Δτ,i Value of γ Prior probability Sector Posterior probability Motor vehicles and parts Furn. & dur. household equip Rec. goods & vehicles Other durable goods Food & bev. for off-premises consumption Clothing & footwear Gasoline & other energy goods Other nondurables goods Housing excl. gas & elec. util Gas & electric untilies Health care Transportation services Recreation services Food serv. & accom Fin. services & insurance Other services NPISH

11 Table A.6: Posterior distribution for γ ε,i Value of γ Prior probability Sector Posterior probability Motor vehicles and parts Furn. & dur. household equip Rec. goods & vehicles Other durable goods Food & bev. for off-premises consumption Clothing & footwear Gasoline & other energy goods Other nondurables goods Housing excl. gas & elec. util Gas & electric untilies Health care Transportation services Recreation services Food serv. & accom Fin. services & insurance Other services NPISH

12 Table A.7: Posterior distribution of p i (selected quantiles) Sector 16% 50% 67% Motor vehicles and parts Furn. & dur. household equip Rec. goods & vehicles Other durable goods Food & bev. for off-premises consumption Clothing & footwear Gasoline & other energy goods Other nondurables goods Housing excl. gas & elec. util Gas & electric untilies Health care Transportation services Recreation services Food serv. & accom Fin. services & insurance Other services NPISH

13 Figure A.1 12

14 Figure A.2 13

15 Figure A.3 14

16 Figure A.4 15

17 Figure A.5 16

18 Figure A.6 17

19 Figure A.7 18

20 Figure A.8 19

21 Figure A.9 20

22 Figure A.10 21

23 Figure A.11 22

24 Figure A.12 23

25 Figure A.13 24

26 Figure A.14 25

27 Figure A.15 26

28 Figure A.16 27

29 Figure A.17 28

30 3. The 3-component multivariate model This model has the same structure and uses the same priors as the 17-component model but uses only 3 components. The results for the model are given below. Value Table A.8: Posterior distribution for γ Δτ,c and γ ε,c Prior Posterior Probability Prob γ Δτ,c γ ε,c Table A.9: Posterior distribution of p c (selected quantiles) 16% 50% 67%

31 Results for sector-specific parameters: Table A.10: Posterior distribution for γ Δτ,i Value of γ Prior probability Sector Posterior probability Core Food Energy Table A.11: Posterior distribution for γ ε,i Value of γ Prior probability Sector Posterior probability Core Food Energy Table A.12: Posterior distribution of p i (selected quantiles) Sector 16% 50% 67% Core Food Energy

32 Figure A.18 31

33 Figure A.19 32

34 Figure A.20 33

35 3. Monthly Models We also estimated the same univariate and multivariate models using monthly inflation rates. We do not report detailed results for these models, but do report the forecasting performance of these models using recursively computed posterior mean trend estimates. Table A13: Monthly and quarterly MSFE (1990 end-of-sample), excluding 2008:Q End of Sample, excluding 2008Q4 4 quarter/12 monthahead forecasts 8 quarter/24 month - ahead forecasts 12 quarter/36 month - ahead forecasts Quarterly Model Monthly Model Quarterly Model Monthly Model Quarterly Model Monthly Model Multivariate UCSVO Forecasts 17comp 0.62 (0.10) 0.78 (0.14) 0.49 (0.07) 0.68 (0.13) 0.42 (0.08) 0.62 (0.14) 3comp 0.59 (0.09) 0.75 (0.12) 0.49 (0.08) 0.67 (0.13) 0.43 (0.10) 0.62 (0.14) Univariate UCSVO Forecasts PCE-all 0.66 (0.10) 0.70 (0.12) 0.63 (0.13) 0.64 (0.13) 0.57 (0.14) 0.60 (0.14) PCExE 0.59 (0.10) 0.67 (0.11) 0.50 (0.08) 0.55 (0.09) 0.45 (0.10) 0.52 (0.10) PCExFE 0.61 (0.10) 0.66 (0.10) 0.49 (0.08) 0.51 (0.08) 0.45 (0.11) 0.49 (0.10) 4. Calculating the approximated weights plotted in Figure 5 of the paper Ignoring outliers, and conditional on the parameters {α i, τ,t}, {α i, ε,t}, {σ Δτ,c,t }, {σ ε,c,t }, {σ Δτ,i,t }, and{σ ε,i,t }, the model is π i,t = α i, τ,t τ c,t + α i, ε,t ε c,t + τ i,t + ε i,t, τ c,t = τ c,t-1 + σ Δτ,c,t η τ,c,t ε c,t = σ ε,c,t η ε,c,t τ i,t = τ i,t-1 + σ Δτ,i,t η τ,i,t ε i,t = σ ε,i,t η ε,i,t which is a linear Gaussian state-space model. The Kalman filter provides estimates τ c,t t and τ i,t t, 17 so that the filtered estimate of the aggregate trend is τ t t = w it ( α i,τ,t τ c,t t + τ i=1 i,t t ). Because the model is linear and Gaussian, the filtered estimates are linear functions of current and lagged values of π it, where the time varying weights depend on {α i, τ,t}, {α i, ε,t}, {σ Δτ,c,t }, {σ ε,c,t }, 17 t 1 i=1 j=0 {σ Δτ,i,t },{σ ε,i,t }, and {w it }. That is τ t t = ω ij,t π it j, where ω ij,t are the weights. The values plotted in Figure 5 are ω = 3 ω 3! i,t / ω ij,t ij,t, where the weights ω ij,t are j=0 17 k=1 computed using the full-sample posterior means of {α i, τ,t}, {α i, ε,t}, {σ Δτ,c,t }, {σ ε,c,t }, {σ Δτ,i,t }, and {σ ε,i,t }. j=0 34

36 Additonal References Carter, C. K., and R. Kohn On Gibbs Sampling for State Space Models.Biometrika 81 (3):