ESTIMATION OF DSGE MODELS WHEN THE DATA ARE PERSISTENT

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1 ESTIMATION OF DSGE MODELS WHEN THE DATA ARE PERSISTENT Yuriy Gorodnichenko 1 Serena Ng 2 1 U.C. Berkeley 2 Columbia University May 2008

2 Outline Introduction Stochastic Growth Model Estimates Robust Estimators The QD Conclusion

3 Two Problems Step 1 Step 2 Step 3 Specify Model Detrend Data Estimation

4 Two Problems Step 1 Step 2 Step 3 Specify Model Detrend Data Estimation MTS DD

5 Two Problems Step 1 Step 2 Step 3 Specify Model Detrend Data Estimation MTS DD Model Trend Specification (MTS) The trend assumed for the model is different from the trend in the data. - Example: Assume linear trend when series are I(1).

6 Two Problems Step 1 Step 2 Step 3 Specify Model Detrend Data Estimation MTS DD Model Trend Specification (MTS) The trend assumed for the model is different from the trend in the data. - Example: Assume linear trend when series are I(1). Data Detrending (DD) The cycle in the model and the data have different properties. - Example: Use HP filter to remove the trend in the data and technology is assumed to be a random walk.

7 Two Problems Step 1 Step 2 Step 3 Specify Model Detrend Data Estimation MTS DD Model Trend Specification (MTS) The trend assumed for the model is different from the trend in the data. - Example: Assume linear trend when series are I(1). Data Detrending (DD) The cycle in the model and the data have different properties. - Example: Use HP filter to remove the trend in the data and technology is assumed to be a random walk.

8 This paper Trend misspecification: known for univariate and linear models

9 This paper Trend misspecification: known for univariate and linear models DSGE: multivariate and non-linear analysis - Biased estimates - Spurious identification of propagation mechanisms

10 This paper Trend misspecification: known for univariate and linear models DSGE: multivariate and non-linear analysis - Biased estimates - Spurious identification of propagation mechanisms consider robust approaches to avoid these problems. T consistent estimates without prior assumption on specification of the trend.

11 Neoclassical growth model subject to max E t t=0 β t (ln C t θl t ) Y t = C t + I t = K α t 1(Z t L t ) (1 α) K t = (1 δ)k t 1 + I t Z t = exp(ḡt) exp(ut z ), ut z = ρ z ut 1 z + et z, ρ z 1 Let m t = (c t, k t, l t ) (model variables) Let m t = (c t, k t, l t ) (model trends) Let m c t = m t m t (stationary model variables)

12 Deterministic trend model (DT), ρ z < 1 c t = k t = ḡt: m t = (ḡt, ḡt, 0) m t = m t m t ( stationary)

13 Deterministic trend model (DT), ρ z < 1 c t = k t = ḡt: m t = (ḡt, ḡt, 0) m t = m t m t ( stationary) Solving E t Γ D 2 m t+1 = Γ D 0 m t + Γ D 1 m t 1 + Ψ D 1 E t u z t+1 + Ψ D 0 u z t m t = Π DT m t 1 + B DT u z t

14 Stochastic trend model (ST), ρ z = 1 ct = kt = z t ḡt + u t : mt = (ḡt + u t, ḡt + u t, 0) m t = m t mt ( stationary).

15 Stochastic trend model (ST), ρ z = 1 ct = kt = z t ḡt + u t : mt = (ḡt + u t, ḡt + u t, 0) m t = m t mt ( stationary). Solving E t Γ S 2 m t+1 = Γ S 0 m t + Γ S 1 m t 1 + Ψ S 1 E t e z t+1 + Ψ S 0 e z t m t = Π ST m t 1 + B ST e z t.

16 ST vs DT models DT: E t Γ D 2 m t+1 = Γ D 0 m t + Γ D 1 m t 1 + Ψ D 1 E t u z t+1 + Ψ D 0 u z t. ST: E t Γ S 2 m t+1 = Γ S 0 m t + Γ S 1 m t 1 + Ψ S 1 E t e z t+1 + Ψ S 0 e z t.

17 ST vs DT models DT: E t Γ D 2 m t+1 = Γ D 0 m t + Γ D 1 m t 1 + Ψ D 1 E t u z t+1 + Ψ D 0 u z t. ST: E t Γ S 2 m t+1 = Γ S 0 m t + Γ S 1 m t 1 + Ψ S 1 E t e z t+1 + Ψ S 0 e z t. Solutions to both models imply the same m t.

18 ST vs DT models DT: E t Γ D 2 m t+1 = Γ D 0 m t + Γ D 1 m t 1 + Ψ D 1 E t u z t+1 + Ψ D 0 u z t. ST: E t Γ S 2 m t+1 = Γ S 0 m t + Γ S 1 m t 1 + Ψ S 1 E t e z t+1 + Ψ S 0 e z t. Solutions to both models imply the same m t. But m c t = m t under DT and m c t = m t under ST. m t = m t u t

19 ST vs DT models DT: E t Γ D 2 m t+1 = Γ D 0 m t + Γ D 1 m t 1 + Ψ D 1 E t u z t+1 + Ψ D 0 u z t. ST: E t Γ S 2 m t+1 = Γ S 0 m t + Γ S 1 m t 1 + Ψ S 1 E t e z t+1 + Ψ S 0 e z t. Solutions to both models imply the same m t. But m c t = m t under DT and m c t = m t under ST. m t = m t u t - m t variables are deviations from stochastic trend - m t variables are deviation from deterministic trend.

20 ST vs DT models DT: E t Γ D 2 m t+1 = Γ D 0 m t + Γ D 1 m t 1 + Ψ D 1 E t u z t+1 + Ψ D 0 u z t. ST: E t Γ S 2 m t+1 = Γ S 0 m t + Γ S 1 m t 1 + Ψ S 1 E t e z t+1 + Ψ S 0 e z t. Solutions to both models imply the same m t. But m c t = m t under DT and m c t = m t under ST. m t = m t u t - m t variables are deviations from stochastic trend - m t variables are deviation from deterministic trend. m t (and 1 m t ) is a stationary when ρ z = 1 m t is not stationary.

21 ST vs DT models DT: E t Γ D 2 m t+1 = Γ D 0 m t + Γ D 1 m t 1 + Ψ D 1 E t u z t+1 + Ψ D 0 u z t. ST: E t Γ S 2 m t+1 = Γ S 0 m t + Γ S 1 m t 1 + Ψ S 1 E t e z t+1 + Ψ S 0 e z t. Solutions to both models imply the same m t. But m c t = m t under DT and m c t = m t under ST. m t = m t u t - m t variables are deviations from stochastic trend - m t variables are deviation from deterministic trend. m t (and 1 m t ) is a stationary when ρ z = 1 m t is not stationary.

22 Data and filtering Data actual variables d t = (c t, k t, l t ) trends d τ t = (c τ t, k τ t, l τ t ) Detrended data d c t = (c c t, k c t, l c t ) = d t d τ t

23 Data and filtering Data actual variables d t = (c t, k t, l t ) trends d τ t = (c τ t, k τ t, l τ t ) Detrended data d c t = (c c t, k c t, l c t ) = d t d τ t Popular detrending techniques: Linear Trend (LT): d c t = d t ḡt HP Trend (HP): d c t = HP(L)d t First Difference (FD): d c t = d t ḡ

24 Data and filtering Data actual variables d t = (c t, k t, l t ) trends d τ t = (c τ t, k τ t, l τ t ) Detrended data d c t = (c c t, k c t, l c t ) = d t d τ t Popular detrending techniques: Linear Trend (LT): d c t = d t ḡt HP Trend (HP): d c t = HP(L)d t First Difference (FD): d c t = d t ḡ Note that generally d τ t m t.

25 What can go wrong? 1: Solve model in terms of m t OR m t (i.e., ρ z < 1 vs ρ z = 1).

26 What can go wrong? 1: Solve model in terms of m t OR m t (i.e., ρ z < 1 vs ρ z = 1). 2: Filter/detrend data to get d c t.

27 What can go wrong? 1: Solve model in terms of m t OR m t (i.e., ρ z < 1 vs ρ z = 1). 2: Filter/detrend data to get d c t. 3: Estimate parameters using d c t

28 What can go wrong? 1: Solve model in terms of m t OR m t (i.e., ρ z < 1 vs ρ z = 1). 2: Filter/detrend data to get d c t. 3: Estimate parameters using d c t

29 True Assumed Model Model Variables Filter Problems 1 DT DT m t LT -

30 True Assumed Model Model Variables Filter Problems 1 DT DT m t LT - 2 ST ST 1 m t FD -

31 True Assumed Model Model Variables Filter Problems 1 DT DT m t LT - 2 ST ST 1 m t FD - 3 DT DT m t HP (DD)

32 True Assumed Model Model Variables Filter Problems 1 DT DT m t LT - 2 ST ST 1 m t FD - 3 DT DT m t HP (DD) 4 ST ST m t HP (DD)

33 True Assumed Model Model Variables Filter Problems 1 DT DT m t LT - 2 ST ST 1 m t FD - 3 DT DT m t HP (DD) 4 ST ST m t HP (DD) 5 ST DT m t LT (MTS)

34 True Assumed Model Model Variables Filter Problems 1 DT DT m t LT - 2 ST ST 1 m t FD - 3 DT DT m t HP (DD) 4 ST ST m t HP (DD) 5 ST DT m t LT (MTS) 6 DT ST, m t HP (DD),(MTS)

35 Paper Technology Filter Estimator Ireland (01), Kim (00) AR Detrend MLE Del Negro et al (07) Unit Root Diff+Norm Bayesian Smets and Wouters (03) AR HP Bayesian Mcgratten et al (97) AR Detrend+HP MLE Christiano-Eichenbaum Unit root HP GMM Burnside et al (93) AR AR GMM Altig et al (04) AR Diff+Norm GMM GMM: covariance structure euler equations impulse response

36 Covariance Structure Estimation 1: Compute Ω d (j), the sample autocovariance at lag j = 0, 1. ω d = (vech(ω d (0)) vec(ω d (1)) )

37 Covariance Structure Estimation 1: Compute Ω d (j), the sample autocovariance at lag j = 0, 1. ω d = (vech(ω d (0)) vec(ω d (1)) ) 2: Solve the chosen rational expectations model for a guess of Θ. Use the solution to compute Ω m j, the model implied autocovariances, ( m c t = m t or m t ). Let ω m (Θ) = (vech(ω m 0 ) vec(ω m 1 ) ).

38 Covariance Structure Estimation 1: Compute Ω d (j), the sample autocovariance at lag j = 0, 1. ω d = (vech(ω d (0)) vec(ω d (1)) ) 2: Solve the chosen rational expectations model for a guess of Θ. Use the solution to compute Ω m j, the model implied autocovariances, ( m c t = m t or m t ). Let ω m (Θ) = (vech(ω m 0 ) vec(ω m 1 ) ). 3: Θ = argminθ ω d ω m (Θ).

39 Estimates of α (True=.33) MTS LT HP HP FD DD m t m t m t 1 m t ρ z =

40 Estimates of α (True=.33) MTS LT HP HP FD DD m t m t m t 1 m t ρ z =

41 Estimates of α (True=.33) MTS LT HP HP FD DD m t m t m t 1 m t ρ z =

42 Estimates of α (True=.33) MTS LT HP HP FD DD m t m t m t 1 m t ρ z =

43 Estimates of ρ MTS LT HP HP FD DD m t m t m t 1 m t ρ z =

44 Estimates of σ (true = 1.0) MTS LT HP HP FD DD m t m t m t 1 m t ρ z =

45 Incredible estimates may motivate the researcher to introduce - spurious propagation mechanisms - spurious dynamics of forcing variables

46 Incredible estimates may motivate the researcher to introduce - spurious propagation mechanisms - spurious dynamics of forcing variables Examples: - capital adjustment costs, variable capital utilization

47 Incredible estimates may motivate the researcher to introduce - spurious propagation mechanisms - spurious dynamics of forcing variables Examples: - capital adjustment costs, variable capital utilization - taste shocks

48 Incredible estimates may motivate the researcher to introduce - spurious propagation mechanisms - spurious dynamics of forcing variables Examples: - capital adjustment costs, variable capital utilization - taste shocks - internal habit: U(C t ) = ln(c t φc t 1 ) L t φ = 0 but φ 0

49 Incredible estimates may motivate the researcher to introduce - spurious propagation mechanisms - spurious dynamics of forcing variables Examples: - capital adjustment costs, variable capital utilization - taste shocks - internal habit: U(C t ) = ln(c t φc t 1 ) L t φ = 0 but φ 0 - correlated growth shocks: u t = φ u t 1 + e t φ = 0 but φ 0

50 Two Observations Problem (DD): ω d and ω m (Θ) are based on incompatible filters Implication: E( ω d ω m (Θ 0 )) 0

51 Two Observations Problem (DD): ω d and ω m (Θ) are based on incompatible filters Implication: E( ω d ω m (Θ 0 )) 0 Solution: apply the same filter to the model variables and the data series.

52 Two Observations Problem (DD): ω d and ω m (Θ) are based on incompatible filters Implication: E( ω d ω m (Θ 0 )) 0 Solution: apply the same filter to the model variables and the data series. Problem (MTS): ω d, ω m (Θ) are based on the same but incorrect trend specification Implication: ω d possibly not well defined when data are I(1)

53 Two Observations Problem (DD): ω d and ω m (Θ) are based on incompatible filters Implication: E( ω d ω m (Θ 0 )) 0 Solution: apply the same filter to the model variables and the data series. Problem (MTS): ω d, ω m (Θ) are based on the same but incorrect trend specification Implication: ω d possibly not well defined when data are I(1) Solution: Let the data decide whether DT or ST is the correct model.

54 Estimation Strategy solve a model that nests DT and ST.

55 Estimation Strategy solve a model that nests DT and ST. Start with DT solution: m t = Π DT m t 1 + B DT u z t u z t = ρ z u z t 1 + e z t find an estimator that has standard properties without knowing a priori if the data are non-stationary.

56 The QD Estimator Quasi-differencing operator: ρz = 1 ρ z L ρz m t = Π DT ρz m t 1 + B DT e z t

57 The QD Estimator Quasi-differencing operator: ρz = 1 ρ z L ρz m t = Π DT ρz m t 1 + B DT e z t ρz m t is stationary at all ρ 0 z 1.

58 The QD Estimator Quasi-differencing operator: ρz = 1 ρ z L ρz m t = Π DT ρz m t 1 + B DT e z t ρz m t is stationary at all ρ 0 z 1. Compute Ω m ρ(j), the autocovariance at lag j of m t

59 The QD Estimator Quasi-differencing operator: ρz = 1 ρ z L ρz m t = Π DT ρz m t 1 + B DT et z ρz m t is stationary at all ρ 0 z 1. Compute Ω m ρ(j), the autocovariance at lag j of m t Let Ω m ρ(j) = Ωm ρ(j) Ωm ρ(0)

60 The QD Estimator Quasi-differencing operator: ρz = 1 ρ z L ρz m t = Π DT ρz m t 1 + B DT et z ρz m t is stationary at all ρ 0 z 1. Compute Ω m ρ(j), the autocovariance at lag j of m t Let Ω m ρ(j) = Ωm ρ(j) Ωm ρ(0) Compute Ω m ρ(j), the autocovariance at lag j of m t

61 The QD Estimator Quasi-differencing operator: ρz = 1 ρ z L ρz m t = Π DT ρz m t 1 + B DT et z ρz m t is stationary at all ρ 0 z 1. Compute Ω m ρ(j), the autocovariance at lag j of m t Let Ω m ρ(j) = Ωm ρ(j) Ωm ρ(0) Compute Ω m ρ(j), the autocovariance at lag j of m t Let Ω d ρ(j) = Ωd ρ(j) Ωd ρ(0)

62 The QD Estimator Quasi-differencing operator: ρz = 1 ρ z L ρz m t = Π DT ρz m t 1 + B DT et z ρz m t is stationary at all ρ 0 z 1. Compute Ω m ρ(j), the autocovariance at lag j of m t Let Ω m ρ(j) = Ωm ρ(j) Ωm ρ(0) Compute Ω m ρ(j), the autocovariance at lag j of m t Let Ω d ρ(j) = Ωd ρ(j) Ωd ρ(0) Θ = argminθ vec( Ωd ρ( ρ )) vec( Ω m ρ)

63 The QD Estimator Quasi-differencing operator: ρz = 1 ρ z L ρz m t = Π DT ρz m t 1 + B DT e z t ρz m t is stationary at all ρ 0 z 1. Compute Ω m ρ(j), the autocovariance at lag j of m t Let Ω m ρ(j) = Ωm ρ(j) Ωm ρ(0) Compute Ω m ρ(j), the autocovariance at lag j of m t Let Ω d ρ(j) = Ωd ρ(j) Ωd ρ(0) Θ = argminθ vec( Ωd ρ( ρ )) vec( Ω m ρ) Stationary moments give normal results. QD 0 : do not normalize by Ω 0.

64 The DT Estimator Start with DT solution: m t = Π DT m t 1 + B DT u z t. m t = Π DT m t 1 + B DT u z t (1) ρ z not constrained to 1. Possible overdifferncing. m t is stationary for ρ z 1. Let ω m ( ) = (vech(ω m 1 ( )) vec(ω m 1 ( )) ) Let ω d ( ) = (vech(ω d 1( )) vec(ω d 1( )) ) Θ = argminθ ω d ( ) ω m ( )

65 A HD (Hybrid Estimator) Start with DT solution: m t = Π DT m t 1 + B DT u z t. m t is stationary for ρ z 1.

66 A HD (Hybrid Estimator) Start with DT solution: m t = Π DT m t 1 + B DT ut z. m t is stationary for ρ z 1. ρz m t is stationary for ρ z 1.

67 A HD (Hybrid Estimator) Start with DT solution: m t = Π DT m t 1 + B DT u z t. m t is stationary for ρ z 1. ρz m t is stationary for ρ z 1. Match ω m (Θ) the covariance of ( ρz m t, m t j ) with ω d the covariance ( ρz d t, d t j ).

68 The HP-HP Estimator m t = Π DT m t 1 + B DT u z t. HP-Filter/Band Pass:H(L) m t = H(L) m t d t = H(L) d t

69 The HP-HP Estimator m t = Π DT m t 1 + B DT u z t. HP-Filter/Band Pass:H(L) m t = H(L) m t d t = H(L) d t Match autocovvariances of m t with autocovariaces d t

70 The HP-HP Estimator m t = Π DT m t 1 + B DT u z t. HP-Filter/Band Pass:H(L) m t = H(L) m t d t = H(L) d t Match autocovvariances of m t with autocovariaces d t autocovariances of the filtered data (easy)

71 The HP-HP Estimator m t = Π DT m t 1 + B DT u z t. HP-Filter/Band Pass:H(L) m t = H(L) m t d t = H(L) d t Match autocovvariances of m t with autocovariaces d t autocovariances of the filtered data (easy) autocovariances of the filtered model variables (ifft)

72 Estimates of α (capital share=.33) ρ z DT QD 0 QD HD HP Filter ρ ρ HD HP

73 Estimates of ρ z ρ z DT QD 0 QD HD HP Filter ρ ρ HD HP

74 Estimates of σ ρ z DT QD 0 QD HD HP Filter ρ ρ HD HP

75 Figure: ρ z = 0.95, T = α QD 0 HD LT HP QD LT N(0,1) σ ρ

76 Figure: ρ z = α QD 0 HD LT HP QD LT N(0,1) σ ρ

77 Properties of the Robust Estimators: T ( θ θ) = GT ( θ) T ḡ(θ 0 ) + o p (1). i T ḡ(θ0 ) d N(0, S)

78 Properties of the Robust Estimators: T ( θ θ) = GT ( θ) T ḡ(θ 0 ) + o p (1). i T ḡ(θ0 ) d N(0, S) ii G T ( θ) p G(θ 0 ) θ is asymptotically normal distributed T ( θ) d (G 0G 0 ) 1 G 0N(0, S). FD and HP-HP: use sample moments of stationary variables. Result is standard.

79 Understanding the QD How is QD different from previous approaches? a one-step procedure (c.f. Fukac and Pagan 2006)

80 Understanding the QD How is QD different from previous approaches? a one-step procedure (c.f. Fukac and Pagan 2006) uses long- and short-run variation (c.f. Cogley 2001)

81 Understanding the QD How is QD different from previous approaches? a one-step procedure (c.f. Fukac and Pagan 2006) uses long- and short-run variation (c.f. Cogley 2001) can handle multiple I(1) shocks J shocks processes (1 ρ j L)u jt = e jt

82 Understanding the QD How is QD different from previous approaches? a one-step procedure (c.f. Fukac and Pagan 2006) uses long- and short-run variation (c.f. Cogley 2001) can handle multiple I(1) shocks J shocks processes (1 ρ j L)u jt = e jt Quasi-differencing operator: ρ (L) = J j=1 (1 ρ jl). key feature: both sample and model moments depend on Θ = (ρ z, Θ )

83 A Stripped Down Model Inelastic Labor Supply: y t (1 v 0 kkl)(1 ρ 0 L) = (1 b y L)e t vkk 0 does not depend on ρ0 vkz 0 depends on ρ0. b y (ρ 0, vkk 0 ) = v kk 0 α0 v kz depends on ρ 0.

84 A Stripped Down Model Inelastic Labor Supply: y t (1 v 0 kkl)(1 ρ 0 L) = (1 b y L)e t vkk 0 does not depend on ρ0 vkz 0 depends on ρ0. b y (ρ 0, vkk 0 ) = v kk 0 α0 v kz depends on ρ 0. Let ỹ 0 t = (1 ρ 0 L)y t = u t, ỹ 0 t is ARMA(1,1). (1 v 0 kkl)u t = e t + b 0 ye t 1. autocovariances ω m easily computed.

85 For arbitrary ρ, Sample Autocovariances: ỹ t = (1 ρl)y t. ω d j ( ρ ) = 1 T = 1 T T ỹ t ỹ t j t=1 T ỹt 0 ỹ t j 0 (ρ ρ 0 )y t 1 ỹt j 0 t=1 (ρ ρ 0 )y t j 1 ỹ 0 t + (ρ ρ 0 ) 2 y t 1 y t j 1

86 Objective Function Let ḡ = (ḡ1,... ḡl ). For j = 1,... L: ḡj (ρ) = ωj d ( ρ ) Ω m j ( r ho) = (ωj d ( ρ) ω0 d ( ρ )) (ωj m ( ρ ) ω0 m ( ρ)), where = ḡ j (ρ 0 ) + G Tj(ρ 0 )(ρ ρ 0 ) + H Tj( ρ)(ρ ρ 0 ) 2. G Tj(ρ) = ωd j ( ρ ) ρ ωm 0 ( ρ ) ρ 2 HTj(ρ) = 2 ωj d( ρ ) 2 ω0 m ( ρ ) ρ 2 ρ 2

87 Objective Function Let ḡ = (ḡ1,... ḡl ). For j = 1,... L: ḡj (ρ) = ωj d ( ρ ) Ω m j ( r ho) = (ωj d ( ρ) ω0 d ( ρ )) (ωj m ( ρ ) ω0 m ( ρ)), where = ḡ j (ρ 0 ) + G Tj(ρ 0 )(ρ ρ 0 ) + H Tj( ρ)(ρ ρ 0 ) 2. G Tj(ρ) = ωd j ( ρ ) ρ ωm 0 ( ρ ) ρ 2 HTj(ρ) = 2 ωj d( ρ ) 2 ω0 m ( ρ ) ρ 2 ρ 2 Need to show that H (ρ) = O (1) ρ 1.

88 Recall ωj d ( ρ ) = ωj d( ρ ) ω0 d ( ρ ) and 2 ω d j ( ρ ) ρ 2 = 2 1 T T y t 1 y t j 1. t=1 When ρ 0 = 1 and y t = y t 1 + u t y 2 t 1 = y t 1 y t j 1 + y t 1 (u t 1 + u t u t j+1 ). Implication 2 ωj d( ρ ) 2 ω0 d ( ρ ) 1 ρ 2 ρ 2 T H Tj(ρ) = 2 T T y t 1 (u t 1 + u t u t j t=1 T y t 1 (u t u t j+1 ) = O p (1). t=1

89 Consistency From Wu (1981): Lemma Let Θ be the parameter of interest and θ 0 denote the true value of θ. Suppose that for any δ > 0 lim inf T inf Θ Θ 0 δ(q T (Θ) Q T (Θ 0 )) > 0 a.s. or in probability. Then Θ T a.s. Θ 0 (or in probability) as T.

90 Asymptotic Distribution: ρ 1 From first order condition G T ( ρ)(ḡ ( ρ)) = 0 T ḡ ( ρ) = T ḡ (ρ 0 )+G T (ρ 0 ) T ( ρ ρ 0 )+ 1 T H 2 T ( ρ)( ρ ρ 0 ) 2 where ρ [ ρ, ρ 0 ]. Since T ( ρ ρ) = O p (1), T ( ρ ρ 0 ) = (GT ( ρ)g T (ρ 0 )) 1 GT (ρ 0 ) T ḡ (ρ 0 ) + o p (1) = K0 N(0, S ) = N(0, K0 S K0 ).

91 rho = N(0,1) 0.45 T=200 T= rho = 0.9 T=200 T=

92 rho = N(0,1) 0.4 T=200 T= rho = 0.98 T=200 T=

93 rho = N(0,1) 0.4 T=200 T= rho = 1 T=200 T=

94 ρ = 1, T = N(0,1) modified QD 0.6 QD ρ = 1, T = 1000 modified QD QD

95 Figure: Base Case: T MSE, ρ = α QD^0 HD DeltaDT HP QD 8 ρ 10

96 Figure: Base Case: T MSE, ρ = α QD^0 HD DeltaDT HP QD 0.25 ρ 10

97 Figure: Base Case: T MSE, ρ = α QD^0 HD DeltaDT HP QD 0.01 ρ 10

98 Figure: Other Models : T MSE, ρ = Habit formation: φ QD^0 HD DeltaDT HP QD 6 Serial correlation in technology growth rate: κ 10

99 Figure: Other Models: T MSE, ρ = Habit formation: φ QD^0 HD DeltaDT HP QD 6 Serial correlation in technology growth rate: κ 10

100 Concluding remarks Problems (DD) & (MTS) can lead to strongly biased estimates.

101 Concluding remarks Problems (DD) & (MTS) can lead to strongly biased estimates. We propose robust estimation techniques - Quasi-differencing - First differencing - Hybrid Quasi and First differencing - HP-HP

102 Concluding remarks Problems (DD) & (MTS) can lead to strongly biased estimates. We propose robust estimation techniques - Quasi-differencing - First differencing - Hybrid Quasi and First differencing - HP-HP T consistent and normal for all ρ 1. - transform both model variables and data series - transformed variables are stationary under the null for any ρ 1

103 Concluding remarks Problems (DD) & (MTS) can lead to strongly biased estimates. We propose robust estimation techniques - Quasi-differencing - First differencing - Hybrid Quasi and First differencing - HP-HP T consistent and normal for all ρ 1. - transform both model variables and data series - transformed variables are stationary under the null for any ρ 1 Proposed estimators work well in simulations.

ESTIMATION OF DSGE MODELS WHEN THE DATA ARE PERSISTENT. May 7, 2008

ESTIMATION OF DSGE MODELS WHEN THE DATA ARE PERSISTENT Yuriy Gorodnichenko Serena Ng May 7, 28 Abstract An active area of research in macroeconomics is to take DSGE models to the data. These models are