Bonn Summer School Advances in Empirical Macroeconomics

Transcription

1 Bonn Summer School Advances in Empirical Macroeconomics Karel Mertens Cornell, NBER, CEPR Bonn, June 2015

2 In God we trust, all others bring data. William E. Deming ( ) Angrist and Pischke are Mad About Macro (2010 JEP, The Credibility Revolution in Empirical Economics: How Better Research Design Is Taking the Con out of Econometrics) Sims, JEP 2010, Comment on Angrist and Pischke: But economics is not an experimental science

3 Overview 1. Estimating the Effects of Shocks Without Much Theory 1.1 Structural Time Series Models 1.2 Identification Strategies 2. Applications to Fiscal Shocks 2.1 Tax Policy Shocks 2.2 Government Spending Shocks 2.3 Austerity Measures 3. Two Difficulties in Interpreting SVARs 3.1 Noninvertibility 3.2 Time Aggregation 4. Systematic Tax Policy and the ZLB

4 1. Estimating the Effects of Shocks Without Much Theory General Approach: 1. Using observables z t, fit a model of expectations E[z t I t 1 ] 2. Identify meaningful shocks from innovations z t E[z t I t 1 ]. 3. Estimate dynamic causal effects of shocks/variance contributions. I t : information available to economic decision makers at time t.

5 1.1 Structural Time Series Models State Space (SS) Representation The solution of stationary linear models can generally be written as s t = Gs t 1 + Fe t z t = As t 1 + De t s t is a m 1 vector of state variables e t+1 is an l 1 vector of uncorrelated white noise, or structural shocks E [e t ] = 0, E [e t e t] = I, E [e t e s] = 0 for s t z t is an n 1 vector of variables of interest G is m m, F is m l, A is n m and D is n l

6 Stability and Stationarity The m m matrix G has all eigenvalues less than one in modulus, i.e. det(m λi ) 0 for λ 1, or equivalently det(i Mz) 0 for z 1 s t follows a stable VAR(1) process s t and z t are stationary stochastic processes, i.e. the first and second moments are time invariant.

7 Lag Operator Define the lag operator L, i.e. L k x t = x t k. s t = Gs t 1 + Fe t (I GL)s t = Fe t s t = (I GL) 1 Fe t s t = G i Fe t i i=0

8 Time Series Representations Moving Average (MA) Representation q MA(q) : z t = M(L)υ t = M i υ t i where M(L) = M 0 + M 1 L M q L q and innovations process υ t E [υ t ] = 0, E [υ t υ t] = Σ, E [υ t υ s] = 0 for s t i=0 Wold Representation Theorem Every stationary process z t can be written as an MA( ). (plus deterministic terms).

9 Stationary linear models for z t and e t can always be written as MA( ) z t = M (L)e t = M i e t i Given an SS representation {G, F, A, D}, i=0 z t = AG i 1 Fe t i + De t i=1 = A(I GL) 1 Fe t 1 + De t = ( D + A(I GL) 1 FL ) e t such that M 0 = D and M i = AG i 1 F for i 1

10 Assume n = l and Stochastic Nonsingularity: D is an n n invertible matrix. We can write a structural MA z t = M(L)υ t = M i υ t i i=0 with υ t = De t, Σ = DD, M 0 = I z t = ( I + A(I GL) 1 FD 1 L ) υ t such that M i = AG i 1 FD 1 for i 1

11 Time Series Representations Vector Autoregressive Moving Average Representation VARMA(p,q) : B(L)z t = M(L)υ t where B(L) = I B 1 L... B p L p M(L) = M 0 + M 1 L M q L q E [υ t ] = 0, E [υ t υ t] = Σ, E [υ t υ s] = 0 for s t

12 Starting from the MA representation of our linear models, z t = A(I GL) 1 Fe t 1 + De t Suppose n = m and A is invertible, A 1 z t = (I GL) 1 Fe t 1 + A 1 De t (I GL)A 1 z t = Fe t 1 + (I GL)A 1 De t Assuming D invertible we obtain a structural VARMA(1,1), z t = AGA 1 z t 1 + υ t A ( G FD 1 A ) A 1 υ t 1 Structural here means υ t = De t

13 Time Series Representations Vector Autoregressive Representation VAR(p) : B(L)z t = υ t z t = B 1 z t B p z t p + υ t where B(L) = I B 1 L... B p L p E [υ t ] = 0, E [υ t υ t] = Σ, E [υ t υ s] = 0 for s t

14 Fernandez-Villaverde, Rubio-Ramirez, Sargent and Watson (2007): Start from the SS representation of our models, s t = Gs t 1 + Fe t z t = As t 1 + De t Assume n = l and Stochastic Nonsingularity: D is an n n invertible matrix. When D is nonsingular, e t = D 1 (z t As t 1 )

15 Substituting s t = ( G FD 1 A ) s t 1 + FD 1 z t Invertibility (in the Past): The eigenvalues of G FD 1 A are strictly less then one in modulus. Under this condition we can write s t = ( G FD 1 A ) i FD 1 z t i i=0

16 Substituting z t = A ( G FD 1 A ) i 1 FD 1 z t i + De t i=1 such that B i = A ( G FD 1 A ) i 1 FD 1 υ t = De t In practice, lag truncation: z t = p i=1 B iz t i + υ t Stochastic Nonsingularity: No big deal (measurement errors) Invertibility (in the Past): Choice of z t is important! If the invertibility condition does not hold and we estimate a VAR: υ t De t

17 Alternatively, start from the MA representation z t = M(L)υ t or VARMA representation B(L)z t = M(L)υ t and invert M(L) to obtain a VAR representation. (Note: it is assumed that M(L) is a square matrix of rational functions.)

18 S(L)B(L)z t = S(L)M(L)υ t M(L) is invertible in the past, i.e. there is an S(L) such that S(L)M(L) = I and S(L) only has nonnegative powers of L, if det(m(l)) 0 for L 1. See Hansen and Sargent (1980, 1991), Lippi and Reichlin (1993), Forni and Gambetti (2014)

19 MA representation of our models: z t = ( I + A(I GL) 1 FD 1 L ) υ t Using the matrix determinant lemma, det ( I + A(I GL) 1 FD 1 L ) = det ( I (G FD 1 A)L ) /det(i GL) The condition that is equivalent to det ( I (G FD 1 A)L ) 0 for L 1 det ( (G FD 1 A) Iz ) 0 for z 1 or G FD 1 A must have all eigenvalues strictly less then one in modulus (same as Fernandez-Villaverde et al. 2007).

20 VARMA representation of our models: (I AGA 1 L)z t = ( I A ( G FD 1 A ) A 1 L ) υ t 1 The condition that is equivalent to det ( I A ( G FD 1 A ) A 1 L ) 0 for L 1 det (( G FD 1 A ) Iz ) 0 for z 1 or G FD 1 A must have all eigenvalues strictly less then one in modulus (same as Fernandez-Villaverde et al. 2007).

21 Example: New Keynesian Model See Clarida, Gali and Gertler (1999) and the Woodford (2003) and Gali (2008) books E t ŷ gap t+1 = φ ππ t E t π t+1 u t (Eq. Euler) π t = κŷt gap + βe t π t+1 v t (Phillips curve) where κ > 0, φ π > 1, 0 β < 1 y gap t : output gap, π t : inflation v t : cost push shocks u t : other shocks (technology, govt spending, taxes, monetary policy,...) u t and v t are stationary exogenous processes

22 Iterating forward, [ ŷ gap ] t π t = E t j=0 C (j+1) [ ut+j v t+j ] where C 1 = φ π κ [ 1 1 βφπ κ β + κ ] Note C 1 has eigenvalues strictly less than one in modulus. Why?

23 Suppose the shocks follow a VAR(1) process. [ ] [ ] ut ut 1 = v t }{{} Λ + v t 1 }{{} Ω }{{} G F s t e t ] [ ŷ gap t } π t {{ } z t = j=0 C (j+1) Λ j [ ut v t ] [ ut 1 ] = (C Λ) 1 Λ + (C Λ) 1 Ω }{{} v t 1 }{{} A }{{} D s t 1 e t Note that G FD 1 A = 0.

24 State Space representation: [ ] [ ] ut ut 1 = Λ + Ωe v t v t t 1 [ ŷ gap ] [ ] t = (C Λ) 1 ut 1 Λ + (C Λ) 1 Ωe π t v t t 1 Moving Average Representation [ ŷ gap ] t = π t (C Λ) 1 Λ i Ωe t i i=0 VAR/VARMA Representation [ ŷ gap ] [ t = (C Λ) 1 ŷ gap ] Λ (C Λ) t 1 + (C Λ) 1 Ωe π t π t t 1

25 Reduced Form Parameters SS : s t = Gs t 1 + Hυ t G, A, H, Σ z t = As t 1 + υ t MA(q) : z t = M(L)υ t M i, Σ VARMA(p,q) : B(L)z t = M(L)υ t B i, M i, Σ VAR(p) : B(L)z t = υ t B i, Σ where E [υ t ] = 0, E [υ t υ t] = Σ, E [υ t υ s] = 0 for s t Note: SS, MA and VARMA require additional normalizations. When well-specified, these are all models that allow us to separate expectations E[z t I t 1 ] and innovations υ t = z t E[z t I t 1 ].

26 Estimation of Reduced Form Parameters VAR estimation Some references: Hamilton, 1994, Time Series Analysis Luetkepohl, 2005, A New Introduction to Time Series Analysis Brockwell and Davis, 2006, Time Series: Theory and Methods Aoki, 1990, State Space Modeling of Time Series Durbin and Koopman, 2012, Time Series Analysis by State Space Methods

27 Local Uniquess In matrix form [ ŷ gap ] [ E t+1 ŷ gap ] [ ] t ut t = C π t+1 π t v t, C 1 β [ β + κ βφπ 1 κ 1 ] The companion matrix C has the characteristic polynomial P(ϕ) = ϕ 2 tr(c)ϕ + det(c) tr(c) = 1 + 1/β + κ/β > 1 det(c) = (1 + κφ π )/β > 1 which has roots outside the unit circle if tr(c) < 1 + det(c) or φ π > 1 (Taylor Principle) Back

28 Estimating a VAR Sample of T + p observations of an n 1 vector z t : {z t p+1, z t p+2,..., z T 1, z T } Define the n T matrix z such that: z = [ ] z 1 z 2 z T Define a np 1 vector Z t : z t 1 z t 2 Z t =. z t p Let Z be a np T matrix collecting T observations of Z t : Z = [ Z 1 Z 2 Z p ]

29 Let υ be a n T matrix of n 1 residuals υ t : υ = [ υ 1 υ 2 υ T ] Let B be a n np matrix of coefficients: Introduce the vectorization operator: B = [ B 1 B 2 B p ] z = vec(z) u = vec(υ) Where z is a nt 1 vector of the stacked columns of z. The variance-covariance matrix of u is: Var(υ) Σ = I T Σ

30 Generalized Least Squares Re-write the reduced form VAR(p) as: Or as: z = BZ + u z = (Z I n )β + u Where is the Kronecker product. We can estimate β with Generalized Least Squares (GLS): u Σ 1 u = (z (Z I n )β) Σ 1 (z (Z I n )β) = z Σ 1 z + β (Z I n )Σ 1 (Z I n )β 2β (Z I n )Σ 1 z = z (I T Σ 1 )z + β (ZZ Σ 1 )β 2β (Z Σ 1 )z

31 First order condition: 2(ZZ Σ 1 )β 2(Z Σ 1 )z = 0 The GLS estimator is therefore: ˆβ = ((ZZ ) 1 Z I n )z This is the same as OLS or ML. Asymptotic normality: T ( ˆβ β) d N (0, Γ 1 Σ) where Γ = plimzz /T and the estimators are ˆΓ = ZZ T 1 ˆΣ = T np 1 z(i T Z (ZZ ) 1 Z)z Back