The Conquest of U.S. Inflation: Learning and Robustness to Model Uncertainty

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1 The Conquest of U.S. Inflation: Learning and Robustness to Model Uncertainty Timothy Cogley and Thomas Sargent University of California, Davis and New York University and Hoover Institution Conquest of U.S. Inflation p. 1/46

2 The Question Conquest of U.S. Inflation p. 2/46

3 The Question After ample evidence had accumulated in the 1970s in favor the natural unemployment rate hypothesis, why did the Fed wait so long to bring inflation down? Conquest of U.S. Inflation p. 2/46

4 The Question After ample evidence had accumulated in the 1970s in favor the natural unemployment rate hypothesis, why did the Fed wait so long to bring inflation down? The picture Conquest of U.S. Inflation p. 2/46

5 Standard answers take for granted that the Fed always knew that the rational expectations version of the natural rate hypothesis was correct Conquest of U.S. Inflation p. 3/46

6 Standard answers take for granted that the Fed always knew that the rational expectations version of the natural rate hypothesis was correct Fluctuations in the natural rate (Parkin, Ireland) Conquest of U.S. Inflation p. 3/46

7 Standard answers take for granted that the Fed always knew that the rational expectations version of the natural rate hypothesis was correct Fluctuations in the natural rate (Parkin, Ireland) Expectations traps (Chari, Christiano, Eichenbaum, Albanesi) Conquest of U.S. Inflation p. 3/46

8 Standard answers take for granted that the Fed always knew that the rational expectations version of the natural rate hypothesis was correct Fluctuations in the natural rate (Parkin, Ireland) Expectations traps (Chari, Christiano, Eichenbaum, Albanesi) Imperfect estimates of the natural rate (Orphanides) Conquest of U.S. Inflation p. 3/46

9 Standard answers take for granted that the Fed always knew that the rational expectations version of the natural rate hypothesis was correct Fluctuations in the natural rate (Parkin, Ireland) Expectations traps (Chari, Christiano, Eichenbaum, Albanesi) Imperfect estimates of the natural rate (Orphanides) Evil motives of Arthur Burns (DeLong) Conquest of U.S. Inflation p. 3/46

10 Our explanation features learning Conquest of U.S. Inflation p. 4/46

11 Our explanation features learning The Fed keeps several models on the table, including ones that don t incorporate the rational expectations version of the natural rate hypothesis. Conquest of U.S. Inflation p. 4/46

12 Our explanation features learning The Fed keeps several models on the table, including ones that don t incorporate the rational expectations version of the natural rate hypothesis. The Fed updates parameters of each model and its posterior probabilities over models Conquest of U.S. Inflation p. 4/46

13 Our explanation features learning The Fed keeps several models on the table, including ones that don t incorporate the rational expectations version of the natural rate hypothesis. The Fed updates parameters of each model and its posterior probabilities over models The Fed uses dynamic programming to choose a first-period decision each period Conquest of U.S. Inflation p. 4/46

14 Anticipated utility model (Kreps) Conquest of U.S. Inflation p. 5/46

15 Anticipated utility model (Kreps) Using time t model, solve dynamic programming problem at t. Conquest of U.S. Inflation p. 5/46

16 Anticipated utility model (Kreps) Using time t model, solve dynamic programming problem at t. Take time t decision Conquest of U.S. Inflation p. 5/46

17 Anticipated utility model (Kreps) Using time t model, solve dynamic programming problem at t. Take time t decision Update model Conquest of U.S. Inflation p. 5/46

18 Anticipated utility model (Kreps) Using time t model, solve dynamic programming problem at t. Take time t decision Update model Using time t + 1 model, solve dynamic programming problem at t + 1 and... Conquest of U.S. Inflation p. 5/46

19 Three models Samuelson-Solow model (a long-run tradeoff) Conquest of U.S. Inflation p. 6/46

20 Three models Samuelson-Solow model (a long-run tradeoff) Solow-Tobin model (no long-run tradeoff) Conquest of U.S. Inflation p. 6/46

21 Three models Samuelson-Solow model (a long-run tradeoff) Solow-Tobin model (no long-run tradeoff) Lucas model (no tradeoff) Conquest of U.S. Inflation p. 6/46

22 Variables u t unemployment Conquest of U.S. Inflation p. 7/46

23 Variables u t u t unemployment natural rate Conquest of U.S. Inflation p. 7/46

24 Variables u t u t y t unemployment natural rate actual inflation Conquest of U.S. Inflation p. 7/46

25 Variables u t u t y t x t unemployment natural rate actual inflation expected inflation Conquest of U.S. Inflation p. 7/46

26 Three models Samuelson-Solow model y t = γ 0 + γ 1 (L)y t 1 + γ 2 (L)u t + η 1t Conquest of U.S. Inflation p. 8/46

27 Three models Samuelson-Solow model Solow-Tobin model y t = γ 0 + γ 1 (L)y t 1 + γ 2 (L)u t + η 1t y t = δ 1 (L) y t 1 + δ 2 (L)(u t u t 1) + η 2t Conquest of U.S. Inflation p. 8/46

28 Three models Samuelson-Solow model Solow-Tobin model y t = γ 0 + γ 1 (L)y t 1 + γ 2 (L)u t + η 1t Lucas model y t = δ 1 (L) y t 1 + δ 2 (L)(u t u t 1) + η 2t u t u t = φ 1 (y t x t t 1 ) + φ 2 (L)(u t 1 u t) + η 3t Conquest of U.S. Inflation p. 8/46

29 Three models Samuelson-Solow model Solow-Tobin model y t = γ 0 + γ 1 (L)y t 1 + γ 2 (L)u t + η 1t Lucas model y t = δ 1 (L) y t 1 + δ 2 (L)(u t u t 1) + η 2t u t u t = φ 1 (y t x t t 1 ) + φ 2 (L)(u t 1 u t) + η 3t η it is iid N(0, σ 2 i ). Conquest of U.S. Inflation p. 8/46

30 Form of each model Y t = X t θ + ν t Conquest of U.S. Inflation p. 9/46

31 Priors p(θ, σ 2 ) = p(θ σ 2 )p(σ 2 ) The marginal prior p(σ 2 ) makes the error variance an inverse gamma variate. The conditional prior p(θ σ 2 ) makes the regression parameters a normal random vector. Conquest of U.S. Inflation p. 10/46

32 Posteriors p(θ σ 2, Z t 1 ) = N(θ t 1, σ 2 P 1 t 1 ), p(σ 2 Z t 1 ) = IG(s t 1, v t 1 ), Conquest of U.S. Inflation p. 11/46

33 Posteriors p(θ σ 2, Z t 1 ) = N(θ t 1, σ 2 P 1 t 1 ), p(σ 2 Z t 1 ) = IG(s t 1, v t 1 ), P t 1 is a precision matrix, s t 1 a scale parameter for the inverse-gamma density, and v t 1 counts degrees of freedom. Conquest of U.S. Inflation p. 11/46

34 Posteriors p(θ σ 2, Z t 1 ) = N(θ t 1, σ 2 P 1 t 1 ), p(σ 2 Z t 1 ) = IG(s t 1, v t 1 ), P t 1 is a precision matrix, s t 1 a scale parameter for the inverse-gamma density, and v t 1 counts degrees of freedom. The estimate of σ 2 is just s t 1 /v t 1. Conquest of U.S. Inflation p. 11/46

35 Posteriors After seeing outcomes at t, the central bank s updated beliefs are p(θ σ 2, Z t ) = N(θ t, σ 2 P 1 t ) p(σ 2 Z t ) = IG(s t, v t ) Conquest of U.S. Inflation p. 12/46

36 Posteriors After seeing outcomes at t, the central bank s updated beliefs are where p(θ σ 2, Z t ) = N(θ t, σ 2 P 1 t ) P t = P t 1 + X t X t, p(σ 2 Z t ) = IG(s t, v t ) θ t = P 1 t (P t 1 θ t 1 + X t Y t ). s t = s t 1 + Y t Y t + θ t 1P t 1 θ t 1 θ tp t θ t, v t = v t Conquest of U.S. Inflation p. 12/46

37 Model probabilities Let α i0 = p(m i ) represent the prior probability on model i. Conquest of U.S. Inflation p. 13/46

38 Model probabilities Let α i0 = p(m i ) represent the prior probability on model i. Bayes theorem implies the posterior probability is p(m i Y t, X t ) m it p(m i ) w it, where m it is the marginalized likelihood function for model i at date t. Conquest of U.S. Inflation p. 13/46

39 Model probabilities Let α i0 = p(m i ) represent the prior probability on model i. Bayes theorem implies the posterior probability is p(m i Y t, X t ) m it p(m i ) w it, where m it is the marginalized likelihood function for model i at date t. The conditional likelihood for model i through date t is defined via a prediction error decomposition as l(y t, X t, θ, σ 2 ) = t s=1 p(y s X s, θ, σ 2 ). Conquest of U.S. Inflation p. 13/46

40 Model probabilities (2) The marginalized likelihood is m it = l(y t i, X t i, θ i, σ 2 i )p(θ i, σ 2 i )dθ i dσ 2 i Conquest of U.S. Inflation p. 14/46

41 Model probabilities (2) The marginalized likelihood is m it = l(y t i, X t i, θ i, σ 2 i )p(θ i, σ 2 i )dθ i dσ 2 i m it is the normalizing constant in Bayes theorem and can also be expressed as m it = l(y t i, Xt i, θ i, σ 2 i )p(θ i, σ 2 i ) p(θ i, σ 2 i Zt i ) Conquest of U.S. Inflation p. 14/46

42 Posterior model probabilities log w it+1 = log w it + log p(y it+1 X it+1, θ i, σ 2 i ) log p(θ i, σ 2 i Zt+1 i ) p(θ i, σ 2 i Zt i ) Conquest of U.S. Inflation p. 15/46

43 Posterior model probabilities log w it+1 = log w it + log p(y it+1 X it+1, θ i, σ 2 i ) log p(θ i, σ 2 i Zt+1 i ) p(θ i, σ 2 i Zt i ) A renormalization enforces that the model weights sum to 1 α it = w it w 1t + w 2t + w 3t Conquest of U.S. Inflation p. 15/46

44 Posterior model probabilities (2) or α it = [expr 1i (t) + expr 2i (t) + expr 3i (t)] 1 where R ji (t) = (log w jt log w it ) summarizes the weight of the evidence favoring model j relative to model i. Conquest of U.S. Inflation p. 16/46

45 Posterior model probabilities (2) or α it = [expr 1i (t) + expr 2i (t) + expr 3i (t)] 1 where R ji (t) = (log w jt log w it ) summarizes the weight of the evidence favoring model j relative to model i. Conquest of U.S. Inflation p. 17/46

46 State space models S it+j = A i (t 1)S it+j 1 + B i (t 1)x t+j t 1 + C i (t 1)η it+j (1) where (S it, A i (t 1), B i (t 1), C i (t 1)) are the state vector and system arrays for model i at time t. Conquest of U.S. Inflation p. 18/46

47 Grand model S 1t+j S 2t+j S 3t+j = + + A 1 (t 1) A 2 (t 1) A 3 (t 1) B 1 (t 1) B 2 (t 1) x t+j t 1 B 3 (t 1) C 1 (t 1) C 2 (t 1) C 3 (t 1) S 1t+j 1 S 2t+j 1 S 3t+j 1 η 1t η 2t η 3t Conquest of U.S. Inflation p. 19/46

48 Grand model S 1t+j S 2t+j S 3t+j = + + A 1 (t 1) A 2 (t 1) A 3 (t 1) B 1 (t 1) B 2 (t 1) x t+j t 1 B 3 (t 1) C 1 (t 1) C 2 (t 1) C 3 (t 1) S 1t+j 1 S 2t+j 1 S 3t+j 1 η 1t η 2t η 3t or in a more compact notation S Et+j = A E (t 1)S Et+j 1 + B E (t 1)x t+j t 1 + C E (t + j)η t Conquest of U.S. Inflation p. 19/46

49 Loss function L(M i )(t) = E t j=0 βj (S it+jm s i QM si S it+j + x t+j t 1 Rx t+j t 1) Conquest of U.S. Inflation p. 20/46

50 Grand loss function L E (t) = α 1t L(M 1 ) + α 2t L(M 2 ) + α 3t L(M 3 ) = E t 3 i=1 α it + x t+j t 1 Rx t+j t 1) j=0 βj (S it+jm s i QM si S it+j Conquest of U.S. Inflation p. 21/46

51 Grand loss function L E (t) = α 1t L(M 1 ) + α 2t L(M 2 ) + α 3t L(M 3 ) = E t 3 i=1 α it + x t+j t 1 Rx t+j t 1) j=0 βj (S it+jm s i QM si S it+j Conquest of U.S. Inflation p. 22/46

52 Grand loss function L E (t) = α 1t L(M 1 ) + α 2t L(M 2 ) + α 3t L(M 3 ) = E t 3 i=1 α it j=0 βj (S it+jm s i QM si S it+j + x t+j t 1 Rx t+j t 1) = E t j=0 βj (S E,t+jQ E S E,t+j + x t+j t 1 Rx t+j t 1) Conquest of U.S. Inflation p. 23/46

53 Grand loss function L E (t) = α 1t L(M 1 ) + α 2t L(M 2 ) + α 3t L(M 3 ) = E t 3 i=1 α it j=0 βj (S it+jm s i QM si S it+j + x t+j t 1 Rx t+j t 1) = E t j=0 βj (S E,t+jQ E S E,t+j + x t+j t 1 Rx t+j t 1) where Q E = α 1t M s 1 QM s α 2t M s 2 QM s α 3t M s 3 QM s3 Conquest of U.S. Inflation p. 23/46

54 Bellman equation subject to v t (S E ) = max x { S E Q Et S E x Rx + βev t (S E) } S E = A E (t 1)S E + B E (t 1)x + C E (t 1)η Conquest of U.S. Inflation p. 24/46

55 Bellman equation subject to v t (S E ) = max x { S E Q Et S E x Rx + βev t (S E) } S E = A E (t 1)S E + B E (t 1)x + C E (t 1)η Conquest of U.S. Inflation p. 25/46

56 Decision rule x t t 1 = f E (t 1) S Et 1 = f E (t 1) 1 S 1t 1 f 2 E(t 1)S 2t 1 f 3 E(t 1)S 3t 1. Conquest of U.S. Inflation p. 26/46

57 Decision rule x t t 1 = f E (t 1) S Et 1 = f E (t 1) 1 S 1t 1 f 2 E(t 1)S 2t 1 f 3 E(t 1)S 3t 1. If the composite model is detectable and stabilizable, then the policy rule f E can be computed using standard algorithms Conquest of U.S. Inflation p. 26/46

58 Data Inflation: log difference in chain weighted GDP deflator Unemployment: civilian unemployment rate quarterly, seasonally adjusted, 1948:Q1 2002:Q4 u t = u t (u t u t 1 ) Conquest of U.S. Inflation p. 27/46

59 Parameters β = α 10 =.98, α 20 = α 30 =.01 λ = 16 (equal weight on u, y) Training sample: first 12 years Conquest of U.S. Inflation p. 28/46

60 Lag orders Inflation Samuelson-Solow γ 1 : 4 Solow-Tobin δ 1 : 3 Lucas-Sargent φ 1 : 0 Unemployment Conquest of U.S. Inflation p. 29/46

61 Lag orders Inflation Unemployment Samuelson-Solow γ 1 : 4 γ 2 : 2 Solow-Tobin δ 1 : 3 δ 2 : 2 Lucas-Sargent φ 1 : 0 φ 2 : 2 Conquest of U.S. Inflation p. 30/46

62 Inflation and α it s 0.12 Inflation Model Weights Conquest of U.S. Inflation p. 31/46

63 Inflation and Optimal Policy Actual Inflation y t Policy Recommendation x t t Conquest of U.S. Inflation p. 32/46

64 Expected Loss from a Zero Inflation Policy Samuelson Solow Solow Tobin x 10 3 Lucas Sargent Composite Model Conquest of U.S. Inflation p. 33/46

65 Dominant Eigenvalue Under Zero Inflation 1.2 Dominant Eigenvalue Under Zero Inflation Samuelson Solow Solow Tobin Lucas Sargent Stability Boundary Conquest of U.S. Inflation p. 34/46

66 Optimal Policy v. Zero Inflation, 1975.Q4 Samuelson Solow 0.2 Solow Tobin Lucas Sargent 0.08 Complete Model Forecast Horizon Forecast Horizon Conquest of U.S. Inflation p. 35/46

67 Optimal Policy v. Zero Inflation, 1975.Q4 Samuelson Solow 0.2 Solow Tobin 0.3 U (zero inflation) Lucas Sargent 0.08 Complete Model Forecast Horizon Forecast Horizon Conquest of U.S. Inflation p. 36/46

68 Optimal Policy v. Zero Inflation, 1975.Q4 Samuelson Solow 0.2 Solow Tobin U optimal Lucas Sargent 0.08 Complete Model Forecast Horizon Forecast Horizon Conquest of U.S. Inflation p. 37/46

69 Optimal Policy v. Zero Inflation, 1975.Q4 Samuelson Solow 0.2 Solow Tobin x (optimal) Lucas Sargent 0.08 Complete Model Forecast Horizon Forecast Horizon Conquest of U.S. Inflation p. 38/46

70 Optimal Policy v. Zero Inflation, 1979.Q4 0.4 Samuelson Solow 0.2 Solow Tobin Lucas Sargent 0.1 Complete Model Forecast Horizon Forecast Horizon Conquest of U.S. Inflation p. 39/46

71 Optimal Policy and Policy for Worst-Case Scenarios Optimal Policy and Policy for Worst Case Scenarios Bayesian Policy Samuelson Solow Solow Tobin Conquest of U.S. Inflation p. 40/46

72 Sensitivity of Bayesian Policy to the Discount Rate Sensitivity of the Bayesian Policy to the Discount Factor 4 percent 8 percent 12 percent 16 percent 20 percent Conquest of U.S. Inflation p. 41/46

73 Optimal Policy v. Zero Inflation, 1992.Q Samuelson Solow Solow Tobin Lucas Sargent Complete Model Forecast Horizon Forecast Horizon Conquest of U.S. Inflation p. 42/46

74 Optimal Policy v. Zero Inflation, 2002.Q Samuelson Solow Solow Tobin Lucas Sargent 0.08 Complete Model Forecast Horizon Forecast Horizon Conquest of U.S. Inflation p. 43/46

75 1978 Okun and Perry (1978) summarize things as follows: Thus, the mainline model and its empirical findings reaffirm that there is a slow-growth, high unemployment cure for inflation, but that it is an extremely expensive one....using one of Perry s successful equations as an example, an extra percentage point of unemployment would lower the inflation rate by only about 0.3 percentage point after one year and by 0.7 percentage point if maintained for three years. That extra point of unemployment would cost over a million jobs and some $60 billion of real production each year. Conquest of U.S. Inflation p. 44/46

76 1978 again Okun and Perry also summarize Perry s reasons for rejecting Feller s guesses of much lower costs in terms of unemployment that could be attained through a credible disinflationary policy: Perry believes that much of the [inflation] inertia is backward-looking rather than forward looking, and so is not susceptible to even convincing demonstrations that demand will be restrained in the future. [Perry s] own empirical evidence shows that wage developments are better explained in terms of the recent past history of wages and prices than on any assumption that people are predicting the future course of wages and prices in a way that differs from the past. (page 6) Conquest of U.S. Inflation p. 45/46

77 More 1978 beliefs Perry (1978, pp ) elaborates forcefully on his argument against an expectational interpretation of Phillips curve dynamics. Okun (1978a, p.284) says that recession will slow inflation, but only at the absurd cost in production of roughly $200 billion per point. Okun (1978b) lists the models on which this estimate is based. Many distinguished economists are represented on the list. At that time, $200 billion amounted to roughly 10 percent of GDP. Inflation averaged 7.4 percent from 1974 to 1979, and extrapolating to zero inflation implies a total cost of almost three-quarters of a year s GDP. Conquest of U.S. Inflation p. 46/46

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