Discussion of “The Time-Varying Volatility of Macroeconomic Fluctuations” by Justiniano and Primiceri Marco Del Negro Federal Reserve Bank of New York NYU Macroeconometrics Reading Group, March 31, 2014 Disclaimer: The views expressed are mine and do not necessarily reflect those of the Federal Reserve Bank of New York or the Federal Reserve System
Motivation: Standardized Policy shocks in Gaussian DSGE r 6 6 5 5 Exc. Kurtosis:4.3 4 4 Standard Deviations 3 3 2 2 1 1 0 0 1965 1969 1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 Marco Del Negro JP discussion 2 / 1
The Smets and Wouters DSGE Model - DSSW variant • Christiano, Eichenbaum, and Evans (2005) + several shocks. • Stochastic growth model + . . . real rigidites nominal rigidites investment adjustment costs price stickiness variable capital utilization wage stickiness partial indexation to lagged inflation + habit persistence • 7 shocks: Neutral technology, investment specific technology, labor supply, price mark-up, government spending, “discount rate” , policy. Marco Del Negro JP discussion 3 / 1
Estimating a DSGE model • Linearized DSGE = state space model • Transition equation: s t = T ( θ ) s t − 1 + R ( θ ) ǫ t • Measurement equation: y t = D ( θ ) + Z ( θ ) s t where y t and s t are the vectors of observables and states, respectively, and θ is the vector of DSGE model parameters (so-called “deep” parameters). • Likelihood p ( Y 1: T | θ ) computed using the Kalman filter. • Random-Walk Metropolis algorithm to obtain draws from the posterior p ( θ | Y 1: T ) – see Del Negro, Schorfheide, “Bayesian Macroeconometrics”, (in Handbook of Bayesian Econometrics , Koop, Geweke, van Dijk eds.) Marco Del Negro JP discussion 4 / 1
Measurement equations • y t = D ( θ ) + Z ( θ ) s t LN (( GDPC ) / LNSINDEX ) ∗ 100 Output growth = LN ((( PCEC − Durables ) / GDPDEF ) / LNSINDEX ) ∗ Consumption growth = Investment growth = LN ((( FPI + durables ) / GDPDEF ) / LNSINDEX ) ∗ 100 Real Wage growth = LN ( PRS 85006103 / GDPDEF ) ∗ 100 Hours = LN (( PRS 85006023 ∗ CE 16 OV / 100) / LNSINDEX ) ∗ 100 Inflation = LN ( GDPDEF / GDPDEF ( − 1)) ∗ 100 FFR = FEDERAL FUNDS RATE / 4 • Sample 1954:III up to 2004:IV. • Same prior p ( θ ) as DSSW. Marco Del Negro JP discussion 5 / 1
Estimating linear DSGEs with SV • Measurement: y t = D ( θ ) s + Z ( θ ) s t • Transition: s t +1 = T ( θ ) s t + R ( θ ) ε t where θ are the DSGE parameters • Shocks ε q , t = σ q σ q , t η q , t η q , t ∼ N (0 , 1), i.i.d. across q , t . log σ q , t = log σ q , t − 1 + ζ q , t , σ q , 0 = 1 , ζ q , t ∼ N (0 , ω 2 q ) • Non linear: Fernandez-Villaverde and Rubo-Ramirez (ReStud 2007,...) Marco Del Negro JP discussion 6 / 1
Inference • The joint distribution of data and observables is: p ( y 1: T | s 1: T , θ ) p ( s 1: T | ε 1: T , θ ) p ( ε 1: T | ˜ σ 1: T , θ ) σ 1: T | ω 2 q ) p ( ω 2 p (˜ q ) p ( θ ) 1:¯ 1:¯ where ˜ σ t = log σ t • Priors: • p ( θ ) ‘usual’ • IG prior for ω 2 q : � ν � νω 2 / 2 − νω 2 2 � � 2 − 1 p ( ω 2 q | ν, ω 2 ) = Γ( ν/ 2) ( ω 2 q ) − ν 2 exp 2 ω 2 q Marco Del Negro JP discussion 7 / 1
Gibbs Sampler • What’s the idea? Suppose you want to draw from p ( x , y ) and you don’t know how ... • But you know how to draw from p ( x | y ) ∝ p ( x , y ) and p ( y | x ) ∝ p ( x , y ) • Gibbs sampler: you obtain draws from p ( x , y ) by drawing repeatedly from p ( x | y ) and p ( y | x ) Marco Del Negro JP discussion 8 / 1
Why does it work? • Some theory of Markov chains. • Say you want to draw from the marginal p ( x ) (note, by Bayes’ law if you have draws from the marginal you also have draws from the joint p ( x , y )). • If you find a Markov transition kernel K ( x , x ′ ) that solves the fixed point integral equation : � K ( x , x ′ ) p ( x ′ ) dx ′ p ( x ) = (and that is π ∗ -irreducible and aperiodic ) ... • Then if you generate draws x i , i = 1 , ..., m from x ′ starting from x ′ , | K ( A , x ′ ) m − p ( A ) | → 0 for any set A and any x and 1 � � h ( x i ) → h ( x ) p ( x ) dix m i Marco Del Negro JP discussion 9 / 1
Why does it work? • But wait... the Gibbs sample does provide a Markov transition kernel � K ( x , x ′ ) = p ( x | y ) p ( y | x ′ ) dy • ... that solves the fixed point integral equation : � K ( x , x ′ ) p ( x ′ ) dx ′ p ( x ) = � �� � p ( x | y ) p ( y | x ′ ) dy p ( x ′ ) dx ′ = � �� � p ( y | x ′ ) p ( x ′ ) dx ′ p ( x | y ) = dy � p ( x | y ) p ( y ) dy = p ( x ) = (and sufficient conditions for π ∗ -irreducibility and aperiodicity are usually met, see Chib and Greenberg 1996). Marco Del Negro JP discussion 10 / 1
Gibbs Sampler σ 1: T , ω 2 1) Draw from p ( θ, s 1: T , ε 1: T | ˜ 1:q , y 1: T ): 1.a) [Metropolis-Hastings] Draw from the marginal p ( θ | ˜ σ 1: T , y 1: T ) ∝ p ( y 1: T | ˜ σ 1: T , θ ) p ( θ ) where p ( y 1: T | ˜ σ 1: T , θ ) = � p ( y 1: T | s 1: T , θ ) p ( s 1: T | ε 1: T , θ ) p ( ε 1: T | ˜ σ 1: T , θ ) · d ( s 1: T , ε 1: T ) with ε t | ˜ σ 1: T ∼ N (0 , ∆ t ) ) ( 1.b) [Simulation smoother] Draw from the conditional: p ( s 1: T , ε 1: T | θ, ˜ σ 1: T , y 1: T ) Marco Del Negro JP discussion 11 / 1
σ 1: T | ε 1: T , ω 2 2) [ Kim-Sheppard-Chib] Draw from p (˜ 1:q , . . . ) by drawing from: σ 1: T | ω 2 p ( ε 1: T | ˜ σ 1: T , θ ) p (˜ q ) 1:¯ 3) Draw from p ( ω 2 σ 1: T | ω 2 q ) p ( ω 2 1:q | σ 1: T , . . . ) ∝ p (˜ q ): 1:¯ 1:¯ T ! ν + T 2 ω 2 + 1 , ν X ω 2 σ q , t − 1 ) 2 q | σ 1: T , · · · ∼ IG (˜ σ q , t − ˜ 2 2 t =1 Marco Del Negro JP discussion 12 / 1
Step 1a: Draw from p ( θ | ˜ σ 1: T , y 1: T ) • Usual MH step on p ( y 1: T | ˜ σ 1: T , θ ) p ( θ ) Marco Del Negro JP discussion 13 / 1
Step 1b (Simulation smoother) Option 1: Carter and Kohn • Since � T − 1 � � p ( s 0: T | y 1: T ) = p ( s t | s t +1 , y 1: t ) p ( s T | y 1: T ) t =0 the sequence s 1: T , conditional on y 1: T , can be drawn recursively: 1 Draw s T from p ( s T | y 1: T ) 2 For t = T − 1 , .., 0, draw s t from p ( s t | s t +1 , y 1: t ) • How do I draw from p ( s T | y 1: T )? • i) I know that s T | y 1: T is gaussian, ii) I have s T | T = E [ s T | y 1: T ] and P T | T = Var[ s T | y 1: T ] from the filtering procedure ⇒ � � s T | y 1: T ∼ N s T | T , P T | T Marco Del Negro JP discussion 14 / 1
• How do we draw from p ( s t | s t +1 , y 1: t )? We know that � s t +1 | t � P t +1 | t � �� s t +1 TP t | t � � y 1: t ∼ N P t | t T ′ � s t s t | t P t | t ( s t +1 − s t +1 | t )( s t − s t | t ) ′ � � Note: 1) easy to show that E = TP t | t , 2) we know all these matrices from the Kalman filter. • Then ... E [ s t | s t +1 , y 1: t ] = s t | t + P ′ t | t T ′ P − 1 t +1 | t ( s t +1 − s t +1 | t ) t | t T ′ P − 1 Var [ s t | s t +1 , y 1: t ] = P t | t − P ′ t +1 | t TP t | t • ... and s t | s t +1 , y 1: t ∼ N ( E [ s t | s t +1 , y 1: t ] , Var [ s t | s t +1 , y 1: t ]) Marco Del Negro JP discussion 15 / 1
Step 1b Option 2: Durbin and Koopman (Biometrika 2002) The idea: • Say you have two normally distributed random variables, x and y . You know how to (i) draw from the joint p ( x , y ) and (ii) to compute I E [ x | y ]. • You want to generate a draw from x | y 0 ∼ N ( I E [ x | y 0 ] , W ) for some y 0 . Proceed as follows: 1 Generate a draw ( x + , y + ) from p ( x , y ). By definition, x + is also a draw from p ( x | y + ) = N ( I E [ x | y + ] , W ) or, alternatively, x + − I E [ x | y + ] is a draw from N (0 , W ) . E [ x | y 0 ] + x + − I E [ x | y + ] is a draw from N ( I E [ x | y 0 ] , W ) 2 Use I Since the variables are normally distributed the scale W does not depend on the location y (draw a two dimensional normal, or review the formulas for normal updating, to convince yourself that is the case). Hence p ( x | y + ) and p ( x | y 0 ) have the same variance W , which E [ x | y 0 ] + x + − I E [ x | y + ] is a draw from N ( I E [ x | y 0 ] , W ). means that I Marco Del Negro JP discussion 16 / 1
Durbin and Koopman • Imagine you know how to compute the smoothed estimates of the shocks I E [ ε 1: T | y 1: T ] (see Koopman, Disturbance smoother for state space models, Biometrika 1993) • ... and want to obtain draws from p ( ε 1: T | y 1: T ) (again, we omit θ for notational simplicity). Proceed as follows: 1 Generate a new draw ( ε + 1: T , s + 1: T , y + 1: T ) from p ( ε 1: T , s 1: T , y 1: T ) by drawing s 0 | 0 and ε 1: T from their respective distributions, and then using the transition and measurement equations. E [ ε 1: T | y + 2 Compute I E [ ε 1: T | y 1: T ] and I 1: T ] (and I E [ s 1: T | y 1: T ] and E [ s 1: T | y + I 1: T ] if need the states); E [ ε 1: T | y 1: T ] + ε + E [ ε 1: T | y + 3 Compute I 1: T − I 1: T ] (and E [ s 1: T | y 1: T ] + s + E [ s 1: T | y + I 1: T − I 1: T ] ). Marco Del Negro JP discussion 17 / 1
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