Regional President Speech

Learning and Time-Varying Macroeconomic Volatility Fabio Milani University of California, Irvine International Research Forum, ECB - June 26, 2008 Introduction Strong evidence of changes in macro volatility over time (The Great Moderation) Kim and Nelson (1999), McConnell and P´erez-Quir`os (2000), Stock and Watson (2002), Blanchard and Simon (2001) Time-Varying Volatility Conditional Standard Deviation (Inflation) 1.6 1.4 1.2 1 0.8 0.6 0.4 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Conditional Standard Deviation (Output Gap) 2 1.5 1 0.5 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Figure: Conditional Standard Deviation series for Inflation and Output Gap Introduction Need to correctly model volatility Sims and Zha (AER 2006): BVAR, Regime changes in volatilities of shocks Introduction In DSGE Models? Exogenous shocks with constant variance (Smets and Wouters JEEA 2003, AER 2007, An and Schorfheide ER 2007) DSGE with Stochastic Volatility Justiniano and Primiceri (AER forth.), Fernandez-Villaverde and Rubio-Ramirez (RES 2007) Time variation in the volatility of exogenous shocks Introduction But what explains the changing volatility? 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 2007), Lansing (2007), Bullard and Singh (2007). 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 The Model Stylized New Keynesian Model π = βE(cid:98) π +κx +u (1) t t t+1 t t x = E(cid:98) x −σ(i −E(cid:98) π )+g (2) t t t+1 t t t+1 t i = ρ i +(1−ρ )(χ π +χ x )+ε (3) t t t−1 t π,t t−1 x,t t−1 t Learning instead of RE TV Monetary Policy Expectations Formation VAR to form inflation and output expectations Perceived Law of Motion (VAR(1)): Z = a +b Z +η (4) t t t t−1 t where Z ≡ [π ,x ,i ](cid:48) t t t t ≈ Minimum State Variable solution Learning Coefficient Updating φ(cid:98) = φ(cid:98) +g R−1X (Z −X(cid:48)φ(cid:98) ) (5) t t−1 t,y t t t t t−1 R = R +g (X X(cid:48) −R ) (6) t t−1 t,y t−1 t−1 t−1 where φ(cid:98) = (a(cid:48),vec(b )(cid:48))(cid:48) and X ≡ {1,Z }t−1. t t t t t−1 0 Endogenous Time-Varying Gain Decreasing Gain if Forecast Errors are small Switch to Constant Gain if Forecast Errors become large   t−1 if J j=0 (|yt−j−Et−j−1yt−j |) < υy g = J t (7) t,y  g if P J j=0 (|yt−j−Et−j−1yt−j |) ≥ υy, y J t where y = π, x, i. (Decr. G P ain reset to 1 ) g− y 1+t Similar to Marcet-Nicolini (υ is m.a.d. of forecast errors) t Constant Gain is estimated Which situations? 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? 1. Endogenous Gain and TV Volatility 4.5 Std. Infl Std. Output Gap 4 3.5 3 2.5 2 1.5 1 0.5 0 0.05 0.1 0.15 Figure: Volatility of simulated Inflation and Output Gap as a function of the constant gain coefficient. 1. Endogenous Gain and TV Volatility Volatility typically increases in the gain Simulation (10,000 periods) Gain switches endogenously according to previous forecast errors 1. Endogenous Gain and TV Volatility Time−Varying Volatility (rolling standard deviation) 5 Std. Infl 4 Std. Gap 3 2 1 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Endogenous Time−Varying Gain 0.2 TV gain (Infl) TV gain (Gap) 0.15 0.1 0.05 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Figure: Time-Varying Volatility with Time-Varying Endogenous Gain Coefficient. 2. Bayesian Estimation Gain switches from decreasing to constant Constant Gain jointly estimated in the system Metropolis-Hastings Quarterly U.S. data, 1960:I-2006:I, data from 1954 to 1959 to initialize learning algorithm Uniform priors for gains 2. Bayesian Estimation: Priors PriorDistribution Description Param. Range Distr. Mean 95%Int. InverseIES σ−1 R+ G 1 [.12,2.78] SlopePC κ R+ G .25 [.03,.7] DiscountRate β .99 − .99 − Interest-RateSmooth ρpre79 [0,1] B .8 [.46,.99] FeedbacktoInfl. χπ,pre79 R N 1.5 [.51,2.48] FeedbacktoOutput χx,pre79 R N .5 [.01,.99] Interest-RateSmooth ρpost79 [0,1] B .8 [.46,.99] FeedbacktoInfl. χπ,post79 R N 1.5 [.51,2.48] FeedbacktoOutput χx,post79 R N .5 [.01,.99] Std.MPshock σε R+ IG 1 [.34,2.81] Std.gt σg R+ IG 1 [.34,2.81] Std.ut σu R+ IG 1 [.34,2.81] ConstantGaininfl. g π [0,0.3] U .15 [.007,.294] ConstantGaingap g x [0,0.3] U .15 [.007,.294] ConstantGainFFR g i [0,0.3] U .15 [.007,.294] Table 1 - Prior Distributions. 2. Bayesian Estimation: Results PosteriorDistribution Description Parameter Mean 95%Post.Prob.Int. InverseIES σ−1 6.04 [4.17-9.14] SlopePC κ 0.021 [0.0026-0.054] DiscountFactor β 0.99 - IRSpre-79 ρpre79 0.937 [0.85-0.99] FeedbackInfl.pre79 χπ,pre−79 1.30 [0.83-1.81] FeedbackGappre79 χx,pre−79 0.66 [0.29-1.13] IRSpost-79 ρpost79 0.93 [0.88-0.97] FeedbackInfl.post79 χπ,post−79 1.66 [1.19-2.11] FeedbackGappost79 χx,post−79 0.48 [0.07-0.85] Autoregr.Cost-pushshock ρu 0.39 [0.27-0.49] Autoregr.Demandshock ρg 0.85 [0.78-0.92] Std.Cost-pushshock σu 0.89 [0.81-0.98] Std.Demandshock σg 0.65 [0.59-0.72] Std.MPshock σε 0.97 [0.88-1.07] Constantgain(Infl.) gπ 0.082 [0.078-0.09] Decreasinggain(Infl.) t−1 - - Constantgain(Gap) gx 0.073 [0.06-0.082] Decreasinggain(Gap) t−1 - - Constantgain(FFR) gi 0.003 [0,0.023] Decreasinggain(FFR) t−1 - - Table 2 - Posterior Distributions: baseline case with J =4. 2. Bayesian Estimation: Time-Varying Gain Endogenous Time−Varying Gain − Inflation 0.08 0.06 0.04 0.02 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Endogenous Time−Varying Gain − Output Gap 0.08 0.06 0.04 0.02 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Figure: Endogenous Time-Varying Gain Coefficients (estimated constant gain). Baseline Case Is it a good idea to use this learning rule? Is it dominated by alternatives? EndogenousTVGain DecreasingGain ConstantGain Inflation 0.94 0.97 0.98 OutputGap 0.88 1.00 0.91 Table 6 - RMSEs. Optimality Tests. I ≡ 1(Y < Y(cid:98) ) = α+βY(cid:98) +u (8) t+1,t t+1,t t+1,t t+1,t t+1 Back out Loss Function 2. Bayesian Estimation: Time-Varying Gain Endogenous Time−Varying Gain − Inflation 0.08 0.06 0.04 0.02 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Endogenous Time−Varying Gain − Output Gap 0.08 0.06 0.04 0.02 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Figure: Endogenous Time-Varying Gain Coefficients (estimated constant gain). Case with J =20 2. Bayesian Estimation: Time-Varying Gain 0.035 0.03 P P r o io st r e D ri i o s r t r D ib i u st t r io ib n ution gπ 0.025 0.02 0.015 0.01 0.005 0 0.05 0.06 0.07 0.08 0.09 0.1 0.02 Posterior Distribution gx Prior Distribution 0.015 0.01 0.005 0 0.05 0.06 0.07 0.08 0.09 0.1 Figure: Constant Gain Coefficients: Prior and Posterior Distributions. 2. Bayesian Estimation: Time-Varying Gain Endogenous Time−Varying Gain − Inflation 0.1 0.08 0.06 0.04 0.02 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Endogenous Time−Varying Gain − Output Gap 0.06 0.05 0.04 0.03 0.02 0.01 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Figure: Endogenous Time-Varying Gain Coefficients (Case with low and high constant gain coefficients only). 2. Bayesian Estimation: Forecast Errors Forecast Errors Inflation 4 3 2 1 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Forecast Errors Output Gap 4 3 2 1 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Forecast Errors FFR 10 5 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Figure: Forecast errors for inflation, output gap, and federal funds rate (absolute values). 2. Bayesian Estimation: Forecast Errors Inflation 3 M νπean Absolute Forecast Error 2 t 1 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Output Gap 3 M νxean Absolute Forecast Error 2 t 1 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 FFR 6 M νiean Absolute Forecast Error 4 t 2 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Figure: Rolling Mean Absolute Forecast errors vs. Updated ν for t inflation, output gap, and federal funds rate series. 3. If learning is neglected: The volatility of shocks may be overestimated Possible to spuriously find Stochastic Volatility 3. Test for ARCH/GARCH Effects EndogenousTVGain NoLearning J=4 J=20 ARCH(1) GARCH(1,1) ARCH(1) GARCH(1,1) ARCH(1) GARCH(1,1) Inflation 0.517 0.61 0.48 0.56 0.05 0.06 OutputGap 0.785 0.89 0.85 0.90 0.045 0.05 Table 7 - Test for the existence of ARCH/GARCH effects (5% significance): proportion of rejections of the null hypothesis of no ARCH/GARCH effects. 4. Volatility 0.012 Max. Std. Inflation eq. Residuals 0.01 0.008 0.006 0.004 0.002 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 0.012 Max. Std. Output Gap eq. Residuals 0.01 0.008 0.006 0.004 0.002 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Figure: Maximum rolling Standard Deviation of residuals across simulations: Kernel Density Estimation. 4. The Great Moderation EndogenousTVGain NoLearning Data Baseline J=20 CG Ratio Std.Infl.1985−2006 0.39 0.42 0.43 1.00 0.35 Std.Infl.1960−1984 Ratio (Std.OutputGap1985−2006) 0.42 0.52 0.54 1.00 0.50 (Std.Output Gap1960−1984) 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). 5. Monetary Policy, Learning, and Volatility Simulation for χ = [0,...,5]: π Related: Benati-Surico (2007) 1 Fraction of Switches to a Constant Gain 0.8 0.6 0.4 0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 χ π 0.08 Average Gain in Sample 0.07 0.06 0.05 0.04 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 χ π 0.9 % Rejections no ARCH Effects 0.8 0.7 0.6 χ 0.5 π 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure: Effects of Monetary Policy on Volatility. 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. 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 6. TV Volatility: Learning or Exogenous Shocks? Innovation in Inflation Equation: Rolling Std. 2 Under Learning/TV Gain Under RE 1.5 1 0.5 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Innovation in Output Gap Equation: Rolling Std. 1.4 Under Learning/TV Gain 1.2 Under RE 1 0.8 0.6 0.4 0.2 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Figure: Rolling Std. estimated innovations under RE and Learning 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. 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

James Bullard (2008, June 26). Regional President Speech. Speeches, Federal Reserve. https://whenthefedspeaks.com/doc/regional_speeche_20080627_james_bullard

@misc{wtfs_regional_speeche_20080627_james_bullard,
  author = {James Bullard},
  title = {Regional President Speech},
  year = {2008},
  month = {Jun},
  howpublished = {Speeches, Federal Reserve},
  url = {https://whenthefedspeaks.com/doc/regional_speeche_20080627_james_bullard},
  note = {Retrieved via When the Fed Speaks corpus}
}