Online Appendix for: A Bounded Model of Time Variation in Trend Inflation, NAIRU and the Phillips Curve

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1 Online Appendix for: A Bounded Model of Time Variation in Trend Inflation, NAIRU and the Phillips Curve Joshua CC Chan Australian National University Gary Koop University of Strathclyde Simon M Potter Federal Reserve Bank of New York August 6, 14 1 Summary of Contents of Online Appendix This appendix contains two sections The first is a technical appendix which describes the prior and MCMC algorithm used to estimate our bivariate model of inflation and unemployment labelled Bi-UC in the paper) It also provides additional estimation details about the other models used for comparison The second is section contains a prior predictive analysis Papers cited in this appendix are listed in the references in the paper itself Technical Appendix We remind the reader that our model is defined by: π t τ π t ) = ρ π t ) πt 1 τ π t 1 + λt u t τ u t ) + ε π t u t τ u t ) = ρ u 1 τ π t = τ π t 1 + ε τπ t τ u t = τ u t 1 + ε τu t ρ π t = ρ π t 1 + ε ρπ t λ t = λ t 1 + ε λ t ) ) ut 1 τ u t 1 + ρ u ut τ u t + ε u t, 1) 1

2 and ε π t N, e ht ) ) h t = h t 1 + ε h t, ε h t N, σ h), ε u t N, σ u) Trends are bounded through: ε τπ t ε τu t and time varying parameters through T Na π τ π t 1, b π τ π t 1;, σ τπ) T Na u τ u t 1, b u τ u t 1;, σ τu) 3) T N ρ π t 1, 1 ρ π t 1;, σ ρπ) ε λ t T N 1 λ t 1, λ t 1 ;, σ λ ) 4) ε ρπ t We also impose the stationary condition on the unemployment equation and assume ρ u 1 + ρ u < 1, ρ u ρ u 1 < 1 and ρ u < 1 We use notation where π = π 1,, π T ), u = u 1,, u T ), y = π, u ), τ π = τ π 1,, τ π T ), τ u = τ u 1,, τ u T ), ρ π = ρ π 1,, ρ π T ), λ = λ 1,, λ T ), h = h 1,, h T ), and define ε π, ε u, ε τπ, ε τu, ε h, ε ρπ, ε λ similarly In addition, let θ denote the model parameters, ie, θ = σ u, σ τπ, σ τu, σ h, σ ρπ, σ λ, a π, b π, a u, b u, ρ u 1, ρ u ) 1 The Prior We require a prior for the initial condition in every state equation and these are: τ π 1 T Na π, b π ; τ π, ω τπ), τ u 1 T Na u, b u ; τ u, ω τu), ρ π 1 T N, 1; ρ π, ω ρπ), λ 1 T N 1, ; λ, ω λ), h 1 T Nh, ω h), where τ π, ω τπ, τ u, τ u 1, ω τu, ρ π, ω ρπ, λ, ω ρu, h, and ω h are known constants In particular we choose the relatively non-informative values of τ π = 3, τ u = τ u 1 = 5, h = ρ π = λ =, ω τπ = ω τu = ω h = 5 and ω ρπ = ω ρu = ω λ = 1

3 Note that the need for τ u 1 arises from the use of an AR) specification in the unemployment equation The prior for the model parameters is specified as pθ) = pσ u)pσ h)pσ τπ)pσ τu)pσ ρπ)pσ λ)p a π ) p b π ) p a u ) p b u ) p ρ u 1) p ρ u ), where σ u IGν u, S u ), σ h IGν h, S h ), σ τπ IGν τπ, S τπ ), σ τu IGν τu, S τu ), σ ρπ IGν ρπ, S ρπ ), σ λ IGν λ, S λ ), and IG, ) denotes the inverse-gamma distribution We choose relatively small values for the degrees of freedom parameters, which imply large prior variances, ie, the priors are relatively non-informative Specifically, we set ν u = ν h = ν τπ = ν τu = ν ρπ = ν ρu = ν λ = 1 We then choose values for the scale parameters so that the parameters have the desired prior means We set S u = S h = 9, which imply prior means Eσ u) = Eσ h ) = 1 Next, we set S τπ = 18 and S τu = 9, which imply Eσ τπ) = and Eσ τu) = 1 These values are chosen to reflect the desired smoothness of the corresponding state transition For example, the prior mean for σ τπ implies that with high probability the difference between consecutive trend inflation, τ π t τ π t 1, lies within the values 3 and 3 We set S ρπ = S λ = 18, which imply prior means Eσ ρπ) = Eσ ρu) = Eσ λ ) = These values imply a relatively smooth transition for the relevant states For the bounds we use uniform priors: a π U, ), b π U3, 5), a u U3, 5), b u U6, 8) The priors for ρ u 1 and ρ u are jointly normal with mean 18, 8) and covariance matrix 5I MCMC Algorithm We extend the MCMC sampler developed in Chan, Koop and Potter 13) which in turn is an adaptation of the algorithm introduced in Chan and Strachan 1) Specifically, we sequentially draw from we suppress the dependence on π, u and u 1 ): 1 pτ π y, τ u, ρ π, λ, h, θ) pτ u y, τ π, ρ π, λ, h, θ) 3 pρ π y, τ π, τ u, λ, h, θ) 4 pλ y, τ π, τ u, ρ π, h, θ) 3

4 5 ph y, τ π, τ u, ρ π, λ, θ) 6 p θ y, τ π, τ u, ρ π, λ, h) Step 1: To derive the conditional distribution pτ π y, τ u, ρ π, λ, h, θ), we first rewrite the inflation equation as K π π = µ π + K π τ π + ε π, ε π N, Ω π ), where Ω π = diage h 1,, e h T ) and ρ π 1π τ π ) + λ 1 u 1 τ u 1 1) λ u τ u ρ ) π 1 µ π = λ 3 u 3 τ u ρ 3) π 3 1, K π = λ T u T τ u T ) ρ π T 1 Since K π = 1 for any ρ π, K π is invertible Therefore, we have ie, π u, τ u, ρ π, λ, h, θ) NKπ 1 µ π + τ π, K πω 1 π K π ) 1 ), log pπ u, τ u, ρ π, λ, h, θ) 1 ι T h 1 π K 1 π µ π τ π ) K πω 1 π K π π Kπ 1 µ π τ π ), 5) where ι T is a T 1 column of ones Similarly, rewrite the state equation for τ π as Hτ π = α π + ε τπ, where τ π α π = 1 1, H = 1 1 4

5 That is, the prior density for τ π is given by log pτ π σ τπ) 1 τ π H 1 α π ) H Ω 1 τπ Hτ π H 1 α π ) + g τπ τ π, σ τπ), 6) where a π < τ π t < b π for t = 1,, T, Ω τπ = diagω τπ, σ τπ,, σ τπ) and ) )) g τπ τ π, σ bπ aπ τπ) = log Φ Φ T log ω τπ Φ Combining 5) and 6), we obtain log pτ π y, τ u, ρ π, ρ u, λ, h, θ) 1 π K 1 π ω τπ ) bπ τ π t 1 aπ τ π t 1 Φ σ τπ σ τπ µ π τ π ) K πω 1 π K π π Kπ 1 µ π τ π ) )) 1 τ π H 1 α π ) H Ω 1 τπ Hτ π H 1 α π ) + g τπ τ π, σ τπ), 1 τ π ˆτ π ) D 1 τπ τ π ˆτ π ) + g τπ τ π, σ τπ), where a π < τ π t < b π for t = 1,, T, and D τπ = ) H Ω 1 τπ H + K πω 1 1 π K π, ˆτ π = D τπ H Ω 1 τπ α π + K πω 1 π K π π Kπ 1 µ π )) Since this conditional density is non-standard, we sample τ π via an independencechain Metropolis-Hastings MH) step Specifically, candidate draws are first obtained from the Nˆτ π, D τπ ) distribution with the precision-based algorithm discussed in Chan and Jeliazkov 9), and they are accepted or rejected via an acceptance-rejection Metropolis-Hastings ARMH) step Step : To implement Step, note that information about τ u comes from three sources: the two measurement equations for inflation and unemployment and the state equation for τ u We derive an expression for each component in turn First, write the inflation equation as: z = Λτ u + ε π, ε π N, Ω π ), 5

6 where z t = π t τ π t ) ρ π t π t 1 τ π t 1) λ t u t, z = z 1,, z T ), and Λ = diag λ 1,, λ T ) Hence, ignoring any terms not involving τ u, we have log pπ u, τ u, ρ π, λ, h, θ) 1 z Λτ u ) Ω 1 π z Λτ u ) 7) The second component comes from the unemployment equation, which can be written as: K u u = µ u + K u τ u + ε u, ε u N, Ω u ), where Ω u = I T σ u and 1 ρ u 1u τ u ) + ρ u u 1 τ u 1) ρ u ρ u u τ u 1 1 ) ρ u ρ u 1 1 µ u =, K u = ρ u ρ u 1 1 ρ u ρ u 1 1 Thus, ignoring any terms not involving τ u, we have log pu τ u, θ) 1 u K 1 u µ u τ u ) K uω 1 u K u u Ku 1 µ u τ u ) 8) The third component is contributed by the state equation for τ u : log pτ u σ τu) 1 τ u H 1 α u ) H Ω 1 τu Hτ u H 1 α u ) + g τu τ u, σ τu), 9) where α u = τ u,,, ), a u < τ u t < b u for t = 1,, T, Ω τu = diagω τu, σ τu,, σ τu) and ) )) g τu τ u, σ bu au τu) = log Φ Φ T log ω τu Φ ω τu ) bu τ u t 1 au τ u t 1 Φ σ τu σ τu )) 6

7 Combining 7), 8) and 9), we obtain log pτ u y, τ π, ρ u 1, ρ u, λ, h, θ) 1 z Λτ u ) Ω 1 π z Λτ u ) 1 u K 1 u µ u τ u ) K uω 1 u K u u Ku 1 µ u τ u ) 1 τ u H 1 α u ) H Ω 1 τu Hτ u H 1 α u ) + g τu τ u, σ τu), 1 τ u ˆτ u ) D 1 τu τ u ˆτ u ) + g τu τ u, σ τu), where a u < τ u t < b u for t = 1,, T, and D τu = H Ω 1 τu H + K uω 1 u K u + Λ Ω 1 π Λ ) 1, ˆτ u = D τu H Ω 1 τu α u + K uω 1 u K u u Ku 1 µ u ) + Λ Ω 1 π z) Again, we sample τ u via an ARMH step with candidate draws obtained from Nˆτ u, D τu ) Step 3: Next, we derive an expression for pρ π y, τ π, τ u, λ, h, θ) First, let π t = π t τ π t, u t = u t τ u t, π = π 1,, π T ), and u = u 1,, u T ) Then the measurement equation for inflation can be rewritten as π + Λu = X π ρ π + ε π, ε π N, Ω π ), where X π = diagπ,, π T 1 ) and Λ = diag λ 1,, λ T ) From the state equation for ρ π we also have Hρ π = ε ρπ Therefore, using a similar argument as before, we have pρ π y, τ π, τ u, ρ u, λ, h, θ) 1 ρπ ˆρ π ) D 1 ρπ ρ π ˆρ π ) + g ρπ ρ π, σ ρπ), where < ρ π t < 1 for t = 1,, T, g ρπ ρ π, σ ρπ) = T ) )) 1 ρ π log Φ t 1 ρ π Φ t 1, D ρπ = H Ω 1 ρπ H + X πω 1 π X π ) 1, ˆρ π = D ρπ X πω 1 π π + Λu ), 7 σ ρπ σ ρπ

8 and Ω ρπ = diagω ρπ, σ ρπ,, σ ρπ) As before, we implement an ARMH step with approximating density Nˆρ π, D ρπ ) Step 4: Using the same argument as before, we have pλ y, τ π, τ u, ρ π, h, θ) 1 λ ˆλ) D 1 λ λ ˆλ) + g λ λ, σ λ), where 1 < λ t < for t = 1,, T, g λ λ, σ λ) = T log Φ λt 1 σ λ ) )) 1 λt 1 Φ, D λ = H Ω 1 λ H + ) X uω 1 1 π X u, ˆλ = Dλ X uω 1 π w, X u = diagu,, u T 1 ), w = π 1 ρ π 1π,, π T ρπ T π T 1 ), and Ω λ = diagω λ, σ λ,, σ λ ) As before, we implement an ARMH step with approximating density Nˆρ u, D ρu ) Step 5: For Step 5, we use the algorithm in Chan and Strachan 1) to sample from ph y, τ π, τ u, ρ π, λ, θ) Step 6: We draw from θ in separate blocks, mainly using standard results for the regression model We use notation where θ x for all parameters in θ except for x Using standard linear regression results, it can be shown that ρ u = ρ u 1, ρ u ) is a bivariate truncated normal: pρ u y, τ π, τ u, ρ π, λ, h, θ ρ u) 1 ρu ˆρ u ) D 1 ρu ρ u ˆρ u ) + g ρu ρ u ), with the stationarity constraints ρ u 1 +ρ u < 1, ρ u ρ u 1 < 1 and ρ u < 1, where D ρu = V 1 ρu σ λ u u 1 ) + X ux u /σ 1 u, ˆρ u = D ρu X u /σ u 1 u u, X u = u T 1 u T A draw from this truncated normal distribution can be obtained via acceptancerejection sampling with proposal from Nˆρ u, D ρu ) To sample from the error variances, first note that they are conditionally independent given the data and the states Hence, we can sample each element one by one 8

9 Now, both pσ u y, τ π, τ u, ρ π, λ, h, θ σ u ) and pσ h y, τ π, τ u, ρ π, λ, h, θ σ h ) are standard inverse-gamma densities, respectively: ) σ u y, τ π, τ u, ρ π, λ, h, θ σ u ) IG ν u + T, S u + 1 T u t ρ u 1u t 1 ρ u u t ) t=1 ) σ h y, τ π, τ u, ρ π, ρ u, λ, h, θ σ h ) IG ν h + T 1, S h + 1 T h t h t 1 ) Next, the log conditional density for σ τπ is given by logpσ τπ y, τ π, τ u, ρ π, λ, h, θ σ τπ ) ν τπ + 1) log σ τπ S τπ σ τπ T 1 log σ τπ 1 σ τπ T τ π t τ π t 1) + g τπ τ π, σ τπ), which is a non-standard density To proceed, we implement an MH step with the proposal density ) IG ν τπ + T 1, S τπ + 1 T τ π t τ π t 1) Similarly, the log conditional density for σ τu is given by logpσ τu y, τ π, τ u, ρ π, λ, h, θ σ τu ) ν τu + 1) log σ τu S τu σ τu T 1 log σ τu 1 σ τu T τ u t τ u t 1) + g τu τ u, σ τu) Again, a draw from pσ τu y, τ π, τ u, ρ π, λ, h, θ σ τu ) is obtained via an MH step with the proposal density ) IG ν τu + T 1, S τu + 1 T τ u t τ u t 1) The remaining error variances, σ ρπ and σ λ, are sampled analogously To draw from the bounds a π, b π, a u and b u, we use a Griddy-Gibbs sampler which is the same as the one used in Chan, Koop and Potter 13) Specifically, since each of the bounds has finite support, we can evaluate its conditional density on a fine grid, which can then be used to construct the associated cumulative distribution function Finally, a draw from the conditional density can be obtained via the inverse-transform method The reader is referred to our earlier paper for details 9

10 3 Specification and Estimation Details for Other Models The other models used in our forecast comparisons are mostly restricted special cases of Bi-UC and all specification and prior details are identical to Bi-UC except that the relevant restriction is imposed The exceptions to this are discussed in this sub-section The VAR) is specified as: y t = µ + B 1 y t 1 + B y t + ε t, where y t = π t, u t ) and ε t = ε π t, ε u t ) N, Σ) We use a relatively noninformative prior For µ the prior is N, 1) For the VAR coeffi cients, we assume each is N,1) and[ all are, a priori, ]) uncorrelated with one another 14 The prior for Σ is IW 1, so that the prior mean of the error 7 variances in the two equations are and 1, respectively We also use a VAR) with stochastic volatility: y t = µ + B 1 y t 1 + B y t + A 1 ε t, where ε π t N, e ht ), ε u t N, σ u) and ) 1 A = a 1 The VAR coeffi cients have the same prior as the VAR) without stochastic volatility All details including prior) relating to h t and σ u are exactly as in Bi-UC and a has a N, 1) prior The VARs are estimated using MCMC methods as outlined, eg, in Koop and Korobilis 9) For Bi-UC-TVP-ρ u, all details are identical to Bi-UC except that we need to specify the initial conditions for ρ u 1t and ρ u t which are now timevarying These are both assumed to be N, 5) The priors for the two error variances in the two state equations are both IG1, 9), a relatively noninformative choice implying prior means of 1 The VAR) with Minnesota Prior model is specified as follows Let β = vecµ, B 1, B ) ) and X t = I 1, y t 1, y t ), and let b denote the least 1

11 squares estimator of β We fix Σ = Σ, where Σ = T 1 T t=1 y t X t b) y t X t b) For β, we consider the prior β N, V mn ), where ) V mn = diag a 3 s 1, a 1, a s 1, a 1 s 4, a s 1 4s, a 3 s, a s s 1, a 1, a s, a 1 4s 1 4 and s i is the i-th diagonal element of Σ, i = 1, We set a 1 = 1, a = 5, and a 3 = 1 The bivariate random walk model is: π t = π t 1 + ɛ π t, u t = u t 1 + ɛ u t, where ɛ π t N, σ π) and ɛ u t N, σ u) are independent This model is a special case of Bi-UC, where ρ u 1 = 1, ρ u =, τ π t = τ u t = λ t =, ρ π t = 1 for t = 1,, T, and the errors are independent and homoskedastic The UCSV-AR) model is: π t = τ π t + ɛ π t, u t = ρ u 1u t 1 + ρ u u t + ɛ u t, where ɛ π t N, e ht ) and ɛ u t N, σ u) are independent The inflation trend τ π t and log-volatility h t follow independent random walks This model is a special case of Bi-UC, where τ u t = λ t = ρ π t =, = 1 for t = 1,, T 3 Prior Predictive Analysis To convince the reader of the sensibility of our model and prior, this subsection presents results from a prior predictive analysis We begin by computing the predictive densities for future trend inflation and future NAIRU, τ π T +k and τ u T +k respectively, with k = under the Bi- UC model The results are reported in Figure 3 and 3 These figures should that the predictive densities produced by our model are sensible and show the role of the bounds tightening these densities in a sensible manner 11

12 15 1 π = π = no bounds a π =, b π =5 a π =1, b π = π = 3 π = Figure 1: Predictive densities for τ π T +k under the Bi-UC model where k = 1

13 15 1 u = u = 5 no bounds a u =3, b u =8 a u =4, b u = u = 6 u = Figure : Predictive densities for τ u T +k where k = under the Bi-UC model Next, we preform a prior predictive analysis This involves taking a draw from the prior using the prior described in the Technical Appendix) and then simulating from the state equations Given the drawn parameters, states and an initial value for inflation and the unemployment rate we set π = 3 and u = u 1 = 55), an artificial dataset of inflation and unemployment can be generated This is repeated 1 4 times and, for each generated data set, we compute various features of interest such as quantiles, variance, autocorrelations, etc in order to build up the prior cumulative distribution function cdf) for each of these features Tables 1 and present these cdf s evaluated at the feature of interest calculated for the observed data set It can be seen that all of the features of the data can be well explained by our model Our model does worst at explaining the autocorrelation patterns in the unemployment rate series However, even for this case, it does as well as an unbounded version of our model To formally compare Bi-UC and Bi-UC-NoBound, we compute the log 13

14 Table 1: Prior cdfs of features for π t Feature Bi-UC Bi-UC-NoBound 16%-tilde median %-tilde variance fraction of π t < fraction of π t > lag 1 autocorrelation lag 4 autocorrelation MA coeffi cient Table : Prior cdfs of features for u t Feature Bi-UC Bi-UC-NoBound 16%-tilde median %-tilde variance fraction of u t < fraction of u t > lag 1 autocorrelation lag 4 autocorrelation 9 93 MA coeffi cient

15 Bayes factors using the prior predictive densities for various combinations of the features of interest considered in Tables 1 and In particular, we divide the features into three groups: Quantile" includes the first three features of interest 16%-tilde, median and 84%-tilde), Spread and Drift" includes the next three variance, fraction of y t <, and fraction of y t > 1 for y t = π t or u t ), and Dynamics" include the last three features of interest lag 1 autocorrelation, lag 4 autocorrelation and MA coeffi cient) The results are reported in Table 3 The fact that the inclusion of bounds leads to a more parsimonious model without causing the fit of the model to deteriorate), leads to log Bayes factors strongly in favor of our bounded model for inflation For the unemployment rate, the bounds play less of a role and our model is performing roughly as well as its unbounded version Table 3: Log Bayes factors in favor of Bi-UC against Bi-UC-NoBound Quantile Spread and drift Dynamics All π t u t