Learning and Time-Varying Macroeconomic Volatility

Transcription

1 Learning and Time-Varying Macroeconomic Volatility Fabio Milani University of California, Irvine International Research Forum, ECB - June 26, 28

2 Introduction Strong evidence of changes in macro volatility over time (The Great Moderation) Kim and Nelson (1999), McConnell and Pérez-Quiròs (2), Stock and Watson (22), Blanchard and Simon (21)

3 Time-Varying Volatility Conditional Standard Deviation (Inflation) Conditional Standard Deviation (Output Gap) Figure: Conditional Standard Deviation series for Inflation and Output Gap

4 Introduction Need to correctly model volatility Sims and Zha (AER 26): BVAR, Regime changes in volatilities of shocks

5 Introduction In DSGE Models? Exogenous shocks with constant variance (Smets and Wouters JEEA 23, AER 27, An and Schorfheide ER 27) DSGE with Stochastic Volatility Justiniano and Primiceri (AER forth.), Fernandez-Villaverde and Rubio-Ramirez (RES 27) Time variation in the volatility of exogenous shocks

6 Introduction But what explains the changing volatility?

7 Scope of the paper Present a simple model with learning The learning speed (gain coefficient) of the agents is endogenous: it responds to previous forecast errors Endogenous Time-Varying Volatility Related: Branch and Evans (RED 27), Lansing (27), Bullard and Singh (27).

8 Results: 1 The changing gain induces endogenous time variation in the volatilities of the macroeconomic variables the agents try to learn 2 Evidence of time variation in endogenous gain from estimated model 3 The econometrician can spuriously find evidence of stochastic volatility if learning is not taken into account

9 The Model Stylized New Keynesian Model π t = βêtπ t+1 + κx t + u t (1) x t = Êtx t+1 σ(i t Êtπ t+1 ) + g t (2) i t = ρ t i t 1 + (1 ρ t )(χ π,t π t 1 + χ x,t x t 1 ) + ε t (3) Learning instead of RE TV Monetary Policy

10 Expectations Formation VAR to form inflation and output expectations Perceived Law of Motion (VAR(1)): Z t = a t + b t Z t 1 + η t (4) where Z t [π t, x t, i t ] Minimum State Variable solution

11 Learning Coefficient Updating φ t = φ t 1 + g t,y Rt 1 X t (Z t X t φ t 1 ) (5) R t = R t 1 + g t,y (X t 1 X t 1 R t 1 ) (6) where φ t = (a t, vec(b t ) ) and X t {1, Z t 1 } t 1.

12 Endogenous Time-Varying Gain Decreasing Gain if Forecast Errors are small Switch to Constant Gain if Forecast Errors become large P J t g t,y = 1 j= if ( y t j E t j 1 y t j ) J < υ P t y J j= g y if ( y t j E t j 1 y t j ) J υt y, where y = π, x, i. (Decr. Gain reset to 1 g 1 y +t ) Similar to Marcet-Nicolini (υ t is m.a.d. of forecast errors) Constant Gain is estimated Which situations? (7)

13 Questions: 1 Does the gain coefficient affect volatility? Can the model generate time-varying volatility in inflation and in the output gap? 2 Does the model fit U.S. data? Is there evidence of changes in the gain over time? 3 Does the omission of learning imply that researchers spuriously find stochastic volatility in the structural shocks? 4 Does the model-implied stochastic volatility resemble the SV estimated from the data? 5 What are the effects of MP on the estimated Volatility?

14 1. Endogenous Gain and TV Volatility Std. Infl Std. Output Gap Figure: Volatility of simulated Inflation and Output Gap as a function of the constant gain coefficient.

15 1. Endogenous Gain and TV Volatility Volatility typically increases in the gain Simulation (1, periods) Gain switches endogenously according to previous forecast errors

16 1. Endogenous Gain and TV Volatility 5 4 Std. Infl Std. Gap Time Varying Volatility (rolling standard deviation) TV gain (Infl) TV gain (Gap) Endogenous Time Varying Gain Figure: Time-Varying Volatility with Time-Varying Endogenous Gain Coefficient.

17 2. Bayesian Estimation Gain switches from decreasing to constant Constant Gain jointly estimated in the system Metropolis-Hastings Quarterly U.S. data, 196:I-26:I, data from 1954 to 1959 to initialize learning algorithm Uniform priors for gains

18 2. Bayesian Estimation: Priors Prior Distribution Description Param. Range Distr. Mean 95% Int. Inverse IES σ 1 R + G 1 [.12, 2.78] Slope PC κ R + G.25 [.3,.7] Discount Rate β Interest-Rate Smooth ρ pre79 [, 1] B.8 [.46,.99] Feedback to Infl. χ π,pre79 R N 1.5 [.51, 2.48] Feedback to Output χ x,pre79 R N.5 [.1,.99] Interest-Rate Smooth ρ post79 [, 1] B.8 [.46,.99] Feedback to Infl. χ π,post79 R N 1.5 [.51, 2.48] Feedback to Output χ x,post79 R N.5 [.1,.99] Std. MP shock σ ε R + IG 1 [.34, 2.81] Std. g t σ g R + IG 1 [.34, 2.81] Std. u t σ u R + IG 1 [.34, 2.81] Constant Gain infl. g π [,.3] U.15 [.7,.294] Constant Gain gap g x [,.3] U.15 [.7,.294] Constant Gain FFR g i [,.3] U.15 [.7,.294] Table 1 - Prior Distributions.

19 2. Bayesian Estimation: Results Posterior Distribution Description Parameter Mean 95% Post. Prob. Int. Inverse IES σ [ ] Slope PC κ.21 [ ] Discount Factor β.99 - IRS pre-79 ρ pre [ ] Feedback Infl. pre79 χ π,pre [ ] Feedback Gap pre79 χ x,pre [ ] IRS post-79 ρ post79.93 [ ] Feedback Infl. post79 χ π,post [ ] Feedback Gap post79 χ x,post [.7-.85] Autoregr. Cost-push shock ρ u.39 [ ] Autoregr. Demand shock ρ g.85 [ ] Std. Cost-push shock σ u.89 [ ] Std. Demand shock σ g.65 [ ] Std. MP shock σ ε.97 [ ] Constant gain (Infl.) g π.82 [.78-.9] Decreasing gain (Infl.) t Constant gain (Gap) g x.73 [.6-.82] Decreasing gain (Gap) t Constant gain (FFR) g i.3 [,.23] Decreasing gain (FFR) t Table 2 - Posterior Distributions: baseline case with J = 4.

20 2. Bayesian Estimation: Time-Varying Gain Endogenous Time Varying Gain Inflation Endogenous Time Varying Gain Output Gap Figure: Endogenous Time-Varying Gain Coefficients (estimated constant gain). Baseline Case

21 Is it a good idea to use this learning rule? Is it dominated by alternatives? Endogenous TV Gain Decreasing Gain Constant Gain Inflation Output Gap Table 6 - RMSEs. Optimality Tests. I t+1,t 1(Y t+1,t < Ŷt+1,t) = α + βŷt+1,t + u t+1 (8) Back out Loss Function

22 2. Bayesian Estimation: Time-Varying Gain.8 Endogenous Time Varying Gain Inflation Endogenous Time Varying Gain Output Gap Figure: Endogenous Time-Varying Gain Coefficients (estimated constant gain). Case with J = 2

23 2. Bayesian Estimation: Time-Varying Gain Posterior Distribution g π Prior Distribution Posterior Distribution g x Prior Distribution Figure: Constant Gain Coefficients: Prior and Posterior Distributions.

24 2. Bayesian Estimation: Time-Varying Gain.1 Endogenous Time Varying Gain Inflation Endogenous Time Varying Gain Output Gap Figure: Endogenous Time-Varying Gain Coefficients (Case with low and high constant gain coefficients only).

25 2. Bayesian Estimation: Forecast Errors 4 Forecast Errors Inflation Forecast Errors Output Gap Forecast Errors FFR Figure: Forecast errors for inflation, output gap, and federal funds rate (absolute values).

26 2. Bayesian Estimation: Forecast Errors 3 2 Inflation Mean Absolute Forecast Error ν t π Output Gap 3 2 Mean Absolute Forecast Error ν t x FFR 6 4 Mean Absolute Forecast Error ν t i Figure: Rolling Mean Absolute Forecast errors vs. Updated ν t for inflation, output gap, and federal funds rate series.

27 3. If learning is neglected: The volatility of shocks may be overestimated Possible to spuriously find Stochastic Volatility

28 3. Test for ARCH/GARCH Effects Endogenous TV Gain No Learning J = 4 J = 2 ARCH(1) GARCH(1,1) ARCH(1) GARCH(1,1) ARCH(1) GARCH(1,1) Inflation Output Gap Table 7 - Test for the existence of ARCH/GARCH effects (5% significance): proportion of rejections of the null hypothesis of no ARCH/GARCH effects.

29 4. Volatility.12.1 Max. Std. Inflation eq. Residuals Max. Std. Output Gap eq. Residuals Figure: Maximum rolling Standard Deviation of residuals across simulations: Kernel Density Estimation.

30 4. The Great Moderation Ratio Ratio Endogenous TV Gain No Learning Data Baseline J = 2 CG Std. Infl Std. Infl (Std. OutputGap ) (Std. Output Gap ) Table 8 - The Great Moderation: ratio of standard deviations for inflation and output gap in the second versus the first part of the simulated samples (median across simulations).

31 5. Monetary Policy, Learning, and Volatility Simulation for χ π = [,..., 5]: Related: Benati-Surico (27) 1.8 Fraction of Switches to a Constant Gain χ π.8.7 Average Gain in Sample χ π.9.8 % Rejections no ARCH Effects χ π Figure: Effects of Monetary Policy on Volatility.

32 5. Bernanke - Great Moderation Speech I am not convinced that the decline in macroeconomic volatility of the past two decades was primarily the result of good luck. changes in monetary policy could conceivably affect the size and frequency of shocks hitting the economy, at least as an econometrician would measure those shocks changes in inflation expectations, which are ultimately the product of the monetary policy regime, can also be confused with truly exogenous shocks in conventional econometric analyses. some of the effects of improved monetary policies may have been misidentified as exogenous changes in economic structure or in the distribution of economic shocks.

33 6. TV Volatility: Learning or Exogenous Shocks? Test ARCH/GARCH in DSGE Model Innovations now Output Gap Inflation DSGE-RE ARCH ARCH DSGE-TV Gain ARCH No ARCH

34 6. TV Volatility: Learning or Exogenous Shocks? Innovation in Inflation Equation: Rolling Std. Under Learning/TV Gain Under RE Innovation in Output Gap Equation: Rolling Std. Under Learning/TV Gain Under RE Figure: Rolling Std. estimated innovations under RE and Learning

35 Conclusions Strong Evidence of Stochastic Volatility in the economy Usually Exogenous Learning with endogenous TV gain (depends on previous forecast errors) Endogenous Stochastic Volatility Gain often larger in pre-1984 sample Overestimation of TV in volatility of exogenous shocks.

36 Future Directions How much volatility can learning explain? (estimate DSGE model with learning and TV volatility). More serious attempt to match volatility series in the data. Different ways to model endogenous gain/ Optimality Interactions Policy/Learning/Volatility