The Return of the Gibson Paradox

Transcription

1 The Return of the Gibson Paradox Timothy Cogley, Thomas J. Sargent, and Paolo Surico Conference to honor Warren Weber February 212

2 Introduction Previous work suggested a decline in inflation persistence (e.g. Cogley, Sargent, and Primiceri 21) If so, Barksy s (JME 1987) analysis says we should anticipate a return of Gibson s paradox. We show that Gibson s paradox has returned, and we use a new-keynesian model to explain why.

3 Gibson s paradox Prior to WWI, nominal interest rates were highly correlated with the price level but uncorrelated with inflation. Keynes (193) called it Gibson s paradox in honor of A.H. Gibson (1923), whom Keynes said first detected the pattern. Keynes and Gibson interpreted this as contradicting the Fisher equation, as did later authors.

4 Gibson s paradox (con t.) Friedman and Schwartz (1982): The relation holds over neither World War I nor World War II. It is dubious whether it holds for the post-world War II period, particularly since the middle 196s. For the period our data cover, it holds clearly and unambiguously only for the period from 188 to 1914, and less clearly for the interwar period. Barsky (1987) corroborates Friedman and Schwartz s findings and shows that Gibson s paradox vanished by the early 197s. Barksy also demonstrates that the paradox is closely linked to inflation persistence, appearing when inflation is weakly persistent and disappearing when it is strongly persistent.

5 Characterizing Gibson s paradox, 1 We characterize the Gibson paradox in terms of low-frequency comovements between inflation and nominal interest within a finite-dimensional vector autoregression. Let {y t,z t } be a mean-zero covariance-stationary random process. For us, y t is an interest rate, R t and z t is an inflation rate, π t.

6 Characterizing Gibson s paradox, 2 We consider the infinite-order least-squares projection of y t onto past, present, and future values of z t, y t = j= where ǫ t satisfies the orthogonality conditions h j z t j +ǫ t, (1) Eǫ t z t j = j. We study the sum of distributed-lag coefficients, h() = h j. (2) j=

7 Lucas-Whiteman characterization The Gibson paradox is an apparent failure of Fisher equation. Lucas (198) characterized the Fisher equation by plotting moving averages of an interest rate against a moving average of an inflation rate: he took their lying on a 45 degree line to vindicate Fisher. Whiteman (1984) showed h() 1 means that Lucas s graphs of moving averages of interest rate plotted against moving average of inflation lie on 45 degree line. We use h() to characterize the Gibson paradox.

8 Gibson s paradox in frequency domain Let S y (ω) and S z (ω) be spectral densities of y and z, and let S yz (ω) be the cross-spectrum. The Fourier transform of {h j } can be expressed as h(ω) = j= The sum of distributed-lag coefficients is h() = j= h j e iωj = S yz(ω) S z (ω). (3) h j = S yz() S z (). (4) We use this formula because the spectral density matrix is easy to calculate for VARs and DSGE models.

9 Interpreting h() h() is the coefficient in a regression of zero-frequency component of R on zero-frequency components of π. The Gibson paradox emerges when h() is zero or negative and vanishes when h() is 1. No paradox: A coefficient of 1 means that a persistent increase in π is associated with a persistent increase in R of the same amount, in accordance with the Fisher relation. A paradox emerges when π isn t persistent. E.g., suppose π is white noise, so that expected inflation is constant. All variation in π is unexpected, hence noise for the Fisher relation. Noise in the regressor biases its coefficient toward zero.

10 Smoothness prior Vacuity of restriction for infinite dimensional VAR. Cross-frequency and cross-equation restrictions. Smoothness of h(ω) across frequencies ω [ π,π] is implied by finite dimensional VAR.

11 Inflation persistence Following Barsky (1987), we also want to keep our eye on inflation persistence. We measure inflation persistence by its first-order autocorrelation, FACF π. Similar results emerge using the normalized spectrum at frequency zero.

12 Empirical evidence on the return of Gibson s paradox We estimate a time-varying VAR a la Cogley and Sargent (25) and Primiceri (25) Data: inflation, nominal interest, output growth, and money growth for 1968.Q1-27.Q4. Earlier data are used as a training sample for eliciting a prior. To ignore effects of the financial crisis, the sample ends at 27.Q4.

13 VAR with drifts Y t = B,t +B 1,t Y t B p,t Y t p +ǫ t X t θ t +ǫ t. Drifting coefficients: θ t = θ t 1 +η t, where η t N(,Q) (plus reflecting barriers). Drifting volatilities: Innovations ǫ t are normally distributed with mean zero and a time-varying covariance matrix Ω t that obeys Var(ǫ t ) Ω t = A 1 t H t (A 1 t ) (5)

14 VAR, cont d Time-varying matrices H t and A t : h 1,t H t h 2,t h 3,t h 4,t, A t with h i,t being geometric random walks: lnh i,t = lnh i,t 1 +ν i,t 1 α 21,t 1 α 31,t α 32,t 1 α 41,t α 42,t α 43,t 1

15 VAR, cont d α t = α t 1 +τ t where α t [α 21,t, α 31,t,.., α 43,t ]. [u t, η t, τ t, ν t ] is distributed as u t I 4 η t τ t N (,V), V= Q S ν t Z and Z= σ 2 1 σ 2 2 σ 2 3 σ 2 4, where u t is such that ǫ t A 1 t H 1 2 t u t.

16 Empirical evidence (con t.) We describe the evolution of low-frequency comovements between inflation and nominal interest by constructing a local-to-date t approximation of the sum of projection coefficients, h Rπ,t T () = S Rπ,t T() S π,t T (), (6) using smoothed estimates of the time-varying VAR conditioned on the full sample. We also construct a local-to-date t approximation to the first-order autocorrelation for inflation FACF π implied by the VAR.

17 INFLATION AND NOMINAL INTEREST RATE: LONG RUN LINK 3 2 h ~ R,π () INFLATION PERSISTENCE.8.6 FACF π Figure: Median and 68% central posterior bands for h R,π () (top panel) and FACF π (bottom panel).

18 INFLATION AND NOMINAL INTEREST RATE: LONG RUN LINK 1 5 h ~ () in 2 R,π h ~ () in 198 R,π 1 INFLATION PERSISTENCE.8 FACF π in FACF in 198 π Figure: Joint posterior distributions for h R,π () (top panel) and FACF π (bottom panel),

19 Empirical evidence (con t.) No Gibson paradox is evident in the 197s or 198s, when the sum of lag coefficients is near or above 1. A Gibson paradox reappears after 1995, when h Rπ,t T () falls to the neighborhood of. Indeed, for the year 2, the preponderance of probability mass lies below. Consistent with Barsky (1987), the paradox emerges when inflation is weakly persistent and vanishes when it is strongly persistent.

20 A structural interpretation Game plan Estimate a new Keynesian model for two subperiods, 1968.Q Q4 and 1995.Q1-27.Q4 Verify that it approximates h R,π () and FACF π for the periods before 198 and after Conduct a number of structural counterfactuals to identify what caused the return of Gibson s paradox

21 A new Keynesian model Inspired by Rotemberg (1982) and Ireland (24) π t = β(1 α π )E t π t+1 +βα π π t 1 +κx t 1 τ e t x t = (1 α x )E t x t+1 +α x x t 1 σ(r t E t π t+1 ) +σ(1 ξ)(1 ρ a )a t m t = π t +z t + 1 σγ x t 1 γ R t + 1 γ ( χ t a t ) ỹ t = x t +ξa t ỹ t = ỹ t ỹ t 1 +z t e t = ρ e e t 1 +ε et, with ε et N(,σe) 2 a t = ρ a a t 1 +ε at, with ε at N(,σ 2 a ) χ t = ρ χ χ t 1 +ε χt, with ε χt N(,σ 2 χ) ln(z t ) z t = ε zt, with ε zt N(,σ 2 z )

22 A new Keynesian model (con t.) Variables (expressed as log deviations from steady-state values) π t = inflation R t = nominal interest rate m t = money growth x t = output gap ỹ t = detrended output ỹ t = output growth Shocks e t = markup a t = aggregate demand (preference) χ t = money demand Z t = technology

23 A new Keynesian model (con t.) Parameters β = discount factor α π = indexation of prices to past inflation α x = habit formation κ = slope of the Phillips curve σ = elasticity of intertemporal substitution τ = cost of adjusting prices ξ = inverse elasticity of labor supply γ = inverse interest elasticity of money demand

24 A new Keynesian model (con t.) Monetary policy A monetary-aggregate rule for 1968.Q Q4 m t = ρ m m t 1 +(1 ρ m )(φ π π t +φ x x t )+ε mt, ε mt N(,σm 2 ) An interest-rate rule for 1995.Q1-27.Q4 R t = ρ r R t 1 +(1 ρ r )(ψ π π t +ψ x x t )+ε Rt, ε rt N(,σR 2 )

25 Prior predictive analysis Priors for the two subsamples will be shown momentarily. Although priors for structural parameters and shocks are the same across subsamples, priors on policy coefficients differ because the policy instruments differ. This alters the implied priors for h R,π () and FACF π. We therefore want to demonstrate that The model is capable of matching h R,π () and FACF π for some parameters in the support of the respective priors. The priors do not hardwire our findings.

26 Prior predictive analysis (con t.) h ~ (): R, π FACF π : prior.1 pdf.5 pdf h ~ (): R, π FACF π : pdf pdf Centered 9 percent credible sets hr,π (): (.13,.96) and (-.62,1.6) FACF π : (.67,.97) and (.51,.96)

27 Prior and posterior densities Q1-1983Q4 Prior Posterior description coefficient density domain mean [5 th ; 95 th ] mean [5 th ; 95 th ] discount factor β beta [,1].99 [.981 ;.997].989 [.98 ;.997] NKPC backward-looking component απ beta [,1].5 [.171 ;.826].864 [.765 ;.972] NKPC slope κ gamma R +.15 [.14 ;.22].18 [.7 ;.141] price adjustment cost τ gamma R + 3 [1.56 ; 4.811] 2.76 [1.427 ; 3.987] IS curve backward-looking component αx beta [,1].5 [.171 ;.826].39 [.335 ;.446] elasticity of intertemporal substitution σ gamma R +.15 [.14 ;.22].1 [.69 ;.131] inverse of labour supply elasticity ξ gamma R + 3 [1.56 ; 4.811] [1.282 ; 2.923] interest elasticity of money demand γ gamma R + 3 [1.56 ; 4.811] 2.75 [2.124 ; 3.248] money growth response to inflation φπ normal R.5 [.335 ;.665].466 [.35 ;.625] money growth response to output gap φx normal R -.5 [-.665 ; -.335] [-.721 ; -.398] money growth smoothing ρm beta [,1].5 [.171 ;.826].598 [.478 ;.752] persistence of mark up shock ρe beta [,1].5 [.171 ;.826].136 [.21 ;.246] persistence of demand shock ρa beta [,1].5 [.171 ;.826].926 [.872 ;.983] persistence money demand shock ρχ beta [,1].5 [.171 ;.826].656 [.396 ;.91] standard deviation of mark up shock σe inv. gamma R +.1 [.4 ;.22].98 [.5 ;.146] standard deviation of demand shock σa inv. gamma R +.1 [.4 ;.22].45 [.3 ;.6] standard deviation of money demand shock σχ inv. gamma R +.1 [.4 ;.22].62 [.32 ;.93] standard deviation of technology shock σz inv. gamma R +.1 [.4 ;.22].63 [.47 ;.78] standard deviation of policy shock σm inv. gamma R +.1 [.4 ;.22].82 [.69 ;.95] long-run link inflation-nominal interest rate implied h R,π () R.664 [.134 ;.96].788 [.621 ;.924] inflation persistence implied FACFπ R.867 [.667 ;.973].947 [.911 ;.975] Note: Based on 1,, posterior draws using the Metropolis-Hastings algorithm.

28 The first subsample (con t.) INTEREST RATE and INFLATION: h~ () R, π φx φπ.5 1 INFLATION PERSISTENCE φx φ.5 1 π Figure: The scatter plots depict distributions of the features h R,π () and FACFπ swept out by sampling from the joint posterior distribution of the response coefficients of money growth to inflation and the output gap. All other parameters are fixed at the posterior mean.

29 Factors contributing to high inflation persistence during the first subsample A high degree of intrinsic inflation persistence, α π =.86 A positive policy response of money growth to inflation, φ π =.47 A negative policy response of money growth to output, φ x = -.57, combined with a preponderance of markup shocks. These move output and inflation in opposite directions.

30 Prior and posterior densities Q1-27Q4 Prior Posterior description coefficient density domain mean [5 th ; 95 th ] mean [5 th ; 95 th ] discount factor β beta [,1].99 [.981 ;.997].99 [.982 ;.998] NKPC backward-looking component απ beta [,1].5 [.171 ;.826].133 [.22 ;.236] NKPC slope κ gamma R +.15 [.14 ;.22].138 [.92 ;.185] price adjustment cost τ gamma R + 3 [1.56 ; 4.811] 4.9 [2.538 ; 5.472] IS curve backward-looking component αx beta [,1].5 [.171 ;.826].179 [.68 ;.286] elasticity of intertemporal substitution σ gamma R +.15 [.14 ;.22].112 [.73 ;.151] inverse of labour supply elasticity ξ gamma R + 3 [1.56 ; 4.811].972 [.526 ; 1.393] interest elasticity of money demand γ gamma R + 3 [1.56 ; 4.811] [1.658 ; 3.33] interest rate response to inflation ψπ truncated normal R 1.5 [1.1 ; 1.99] [1.276 ; 2.4] interest rate response to output gap ψx normal R.125 [. ;.25].117 [.12 ;.222] interest rate smoothing ρr beta [,1].5 [.171 ;.826].838 [.787 ;.888] persistence of mark up shock ρe beta [,1].5 [.171 ;.826].474 [.234 ;.75] persistence of demand shock ρa beta [,1].5 [.171 ;.826].921 [.868 ;.979] persistence money demand shock ρχ beta [,1].5 [.171 ;.826].638 [.342 ;.951] standard deviation of mark up shock σe inv. gamma R +.1 [.4 ;.22].5 [.31 ;.67] standard deviation of demand shock σa inv. gamma R +.1 [.4 ;.22].44 [.27 ;.61] standard deviation of money demand shock σχ inv. gamma R +.1 [.4 ;.22].52 [.3 ;.72] standard deviation of technology shock σz inv. gamma R +.1 [.4 ;.22].42 [.34 ;.5] standard deviation of policy shock σm inv. gamma R +.1 [.4 ;.22].21 [.17 ;.25] long-run link inflation-nominal interest rate implied h R,π () R.76 [-.623 ; 1.623] [ ; 1.16] inflation persistence implied FACFπ R.792 [.59 ;.96].585 [.415 ;.732] Note: Based on 1,, posterior draws using the Metropolis-Hastings algorithm.

31 The second subsample (con t.) ~ INTEREST RATE and INFLATION: hr, π() ψ x ψπ INFLATION PERSISTENCE 1.8 ψ x ψ π Figure: The scatter plots depict distributions of the features h R,π () and FACFπ swept out by sampling from the joint posterior distribution of the response coefficients of the interest rate to inflation and the output gap. All other parameters are fixed at the posterior mean.

32 Posteriors for h R,π () and FACF π Densities for h R,π () and FACF π both move to the left in the later subsample. h ~ (): R, π FACF π : prior posterior pdf h ~ (): R, π.5 FACF π : pdf pdf pdf

33 Differences between the two subsamples In the second subsample, Most of the shocks are less volatile, and hard-to-manage markup shocks account for a smaller proportion of output variation. There is less intrinsic inflation persistence, with α π falling from.86 to.13. The monetary-policy rule satisfies the Taylor principle and engages in a high degree of interest smoothing.

34 Counterfactuals Outline Good-luck hypothesis: Swap the shock processes in the two subsamples. Good-policy hypothesis: Swap the monetary-policy rules in the two subsamples. Examine changes in private-sector parameters other than those governing the shocks.

35 The good-luck hypothesis Great Inflation Post-1995 h R,π () FACF π h R,π () FACF π Baseline Model Swap shock variances Replacing the shock variances in the baseline model with those from the other subsample is unsuccessful. A Gibson paradox remains absent in the first subsample and still reappears in the second. It follows that the Gibson paradox can t be explained by altered shock variances.

36 The good-policy hypothesis Great Inflation Post-1995 h R,π () FACF π h R,π () FACF π Baseline Model Swap policies Replacing the policy rule in the baseline model with that of the other subsample is partially but not entirely successful. Under the post-1995 monetary policy, a Gibson paradox would have emerged in the first subsample, but inflation would have been too persistent. Under the 197s monetary policy, the Gibson regression coefficient h() and inflation autocorrelation would both have been higher in the second subsample. It follows that there is more to the story than just a change in monetary policy, at least if we interpret things narrowly.

37 Changes in NKPC parameters Great Inflation Post-1995 h R,π () FACF π hr,π () FACF π Baseline Model Swap NKPCs Replacing NKPC parameters in the baseline model with those from the other subsample is also partially successful. The Gibson regression coefficient h() would have been lower in the 197s and inflation less persistent with the post-1995 NKPC. A Gibson paradox would not have reappeared after 1995 and inflation would have been more persistent with the NKPC of the 197s. The key change is the decline in α π from.86 to.13, which reduces the degree of intrinsic inflation persistence.

38 A failure of invariance? Why did NKPC parameters change? Hard to say, because the model treats them as primitives. One respectable interpretation, however, is that they changed because of the change in policy. In particular, the significant decline in estimates of α π after the Volcker disinflation might be taken as prima facie evidence that it is not structural in the sense of Lucas (1976). If that is so, then both NKPC and policy coefficients must be swapped in order properly to assess the effects of a change in monetary policy.

39 NKPC plus Policy Great Inflation Post-1995 h R,π () FACF π hr,π () FACF π Baseline Model Swap NKPC + Policies Swapping both NKPC and policy parameters completes our accounting. A Gibson paradox would have emerged in the 197s and inflation would have been less persistent under the post-1995 monetary policy and NKPC. A Gibson paradox would not have reappeared after 1995 and inflation would have been more persistent under the 197s monetary policy and NKPC.

40 Conclusion Our counterfactuals point to a change in monetary policy broadly interpreted as being the main reason for the return of Gibson s paradox. To make this work, we must invoke an unmodeled nonlinearity linking NKPC parameters to parameters of the monetary policy rule. Although plausible, we are uncomfortable about manipulating the model in this way, for those parameters are usually presumed to be structural. As such, they are critical for determining the properties of inflation, both directly and through their influence on monetary policy. To the extent that these key parameters fail to be invariant, we worry about the model s reliability for predicting the consequences of policies unseen in the samples used for estimation.