Identifjcation analysis and higher-order approximation of DSGE - PowerPoint PPT Presentation

Identifjcation analysis and higher-order approximation of DSGE models Willi Mutschler 1

Introduction

Introduction • diffjcult to maximize likelihood/posterior or minimize some (moment) • lack of vs. strength of identifjcation actually estimating the model • BUT: identifjability is a model property and can be analyzed without • Gaussian asymptotics yield poor approximations • estimators often lie on the boundary of theoretically admissible space objective function unfortunate choice of observables Identifjcation Problem • in practice, many caveats due to identifjability issues and/or an estimation and inference • lack of identifjcation leads to wrong conclusions from calibration, distribution of data (in particular moments and spectra) • distinct parameter values do not lead to a distinct probability 2 p ( Y | θ ) = p ( Y | θ 0 ) � θ = θ 0

Introduction • diffjcult to maximize likelihood/posterior or minimize some (moment) • lack of vs. strength of identifjcation actually estimating the model • BUT: identifjability is a model property and can be analyzed without • Gaussian asymptotics yield poor approximations • estimators often lie on the boundary of theoretically admissible space objective function unfortunate choice of observables Identifjcation Problem • in practice, many caveats due to identifjability issues and/or an estimation and inference • lack of identifjcation leads to wrong conclusions from calibration, distribution of data (in particular moments and spectra) • distinct parameter values do not lead to a distinct probability 2 p ( Y | θ ) = p ( Y | θ 0 ) � θ = θ 0

Introduction • diffjcult to maximize likelihood/posterior or minimize some (moment) • lack of vs. strength of identifjcation actually estimating the model • BUT: identifjability is a model property and can be analyzed without • Gaussian asymptotics yield poor approximations • estimators often lie on the boundary of theoretically admissible space objective function unfortunate choice of observables Identifjcation Problem • in practice, many caveats due to identifjability issues and/or an estimation and inference • lack of identifjcation leads to wrong conclusions from calibration, distribution of data (in particular moments and spectra) • distinct parameter values do not lead to a distinct probability 2 p ( Y | θ ) = p ( Y | θ 0 ) � θ = θ 0

Example (1): ARMA(1,1)

Example (1): ARMA(1,1) (I) • consider the following ARMA(1,1)-process 3 iid x t − φ 1 x t − 1 = ε t − φ 2 ε t − 1 , with ε t ∼ N ( 0 , σ 2 ) with parameter vector θ = ( φ 1 , φ 2 , σ ) ′ : ARMA(1,1) with θ = (0.4, 0.4, 1) 3 2 1 x 0 −1 −2 0 20 40 60 80 100 time

Example (1): ARMA(1,1) (I) • consider the following ARMA(1,1)-process 3 iid x t − φ 1 x t − 1 = ε t − φ 2 ε t − 1 , with ε t ∼ N ( 0 , σ 2 ) with parameter vector θ = ( φ 1 , φ 2 , σ ) ′ : ARMA(1,1) with θ = (0, 0, 1) 3 2 1 x 0 −1 −2 0 20 40 60 80 100 time

Example (1): ARMA(1,1) (II) • rank of Jacobian of • similar argument applies to the spectral density matrix 2 1 is not full at w.r.t 1 4 1 • autocovariance function: Γ = ( γ 0 , γ 1 , . . . , γ h ) with γ 0 = ( 1 + φ 2 2 − 2 φ 1 φ 2 ) σ 2 , 1 − φ 2 γ 1 = ( φ 1 − φ 2 )( 1 − φ 1 φ 2 ) σ 2 , 1 − φ 2 γ h = φ 1 γ h − 1

Example (1): ARMA(1,1) (II) 1 1 • similar argument applies to the spectral density matrix 4 • autocovariance function: Γ = ( γ 0 , γ 1 , . . . , γ h ) with γ 0 = ( 1 + φ 2 2 − 2 φ 1 φ 2 ) σ 2 , 1 − φ 2 γ 1 = ( φ 1 − φ 2 )( 1 − φ 1 φ 2 ) σ 2 , 1 − φ 2 γ h = φ 1 γ h − 1 • rank of Jacobian of Γ w.r.t θ is not full at φ 1 = φ 2

Example (1): ARMA(1,1) (II) 1 1 • similar argument applies to the spectral density matrix 4 • autocovariance function: Γ = ( γ 0 , γ 1 , . . . , γ h ) with γ 0 = ( 1 + φ 2 2 − 2 φ 1 φ 2 ) σ 2 , 1 − φ 2 γ 1 = ( φ 1 − φ 2 )( 1 − φ 1 φ 2 ) σ 2 , 1 − φ 2 γ h = φ 1 γ h − 1 • rank of Jacobian of Γ w.r.t θ is not full at φ 1 = φ 2

Example (2): Simple DSGE Model

0 A 1 lie within unit 0 A 1 E t y t 0 A 1 0 E t Example (2): Simple DSGE Model(I) A y t circle, implies: 1 • stationary solution of the model, i.e. Eigenvalues of A t t S t D t M 1 1 A 1 1 t 0 1 A j t j A 1 1 A 0 j t 0 1 E t y t t • consider a simple purely forward looking log-linearized model x t t t t or 1 0 1 1 0 0 1 E t r t A 0 t 0 1 E t x t 1 y t A 1 E t r t 0 1 1 0 0 0 0 5 r t = ψπ t + ε M ( TR ) x t = E t x t + 1 − 1 τ ( r t − E t π t + 1 ) + ε D ( IS ) π t = β E t π t + 1 + κ x t + ε S ( PC )

0 A 1 lie within unit 0 A 1 E t y t 0 A 1 0 E t circle, implies: 1 • stationary solution of the model, i.e. Eigenvalues of A t t t A 1 • consider a simple purely forward looking log-linearized model A 0 0 1 1 0 0 0 y t Example (2): Simple DSGE Model(I) 1 j A t 0 1 A j t 1 1 y t A 0 j t 0 1 A 1 0 5 or r t t A 0 0 1 1 t x t 1 0 0 1 t r t = ψπ t + ε M ( TR ) x t = E t x t + 1 − 1 τ ( r t − E t π t + 1 ) + ε D ( IS ) π t = β E t π t + 1 + κ x t + ε S ( PC )           − ψ ε M E t r t + 1           = + ε D           E t x t + 1 τ τ − κ π t β E t π t + 1 ε S � �� � � �� � � �� � � �� � � �� � ε t E t y t + 1

Example (2): Simple DSGE Model(I) 0 1 A 0 r t x t y t 0 0 0 1 0 • consider a simple purely forward looking log-linearized model 0 0 A 1 t t t circle, implies: 0 1 1 0 t t t or 1 5 1 r t = ψπ t + ε M ( TR ) x t = E t x t + 1 − 1 τ ( r t − E t π t + 1 ) + ε D ( IS ) π t = β E t π t + 1 + κ x t + ε S ( PC )           − ψ ε M E t r t + 1           = + ε D           E t x t + 1 τ τ − κ π t β E t π t + 1 ε S � �� � � �� � � �� � � �� � � �� � ε t E t y t + 1 • stationary solution of the model, i.e. Eigenvalues of A − 1 0 A 1 lie within unit � ∞ y t = A − 1 0 A 1 E t y t + 1 + A − 1 ( A − 1 0 A 1 ) j A − 1 0 E t ε t + j = A − 1 0 ε t = 0 ε t j = 0

Example (2): Simple DSGE Model(II) , all parameters are identifjable 0 t t t • some insights • some parameters ( ) do not enter solution, thus do not enter the likelihood (or any other objective) • if we fjx (or calibrate) • some parameters enter as products, e.g. 1 , identifying them separately may be impossible (depending on choice of observables) • is already the product of several other structural parameters (Calvo or Rotemberg) • restrictions necessary to ensure regularity (Eigenvalues inside unit circle) imply bounds involving all structural parameters, i.e. parameter space is not variation free • solution/data-generating-process/reduced-form 6 1 1 r t x t 1       κψ ψ ε M        = − 1 − 1 ε D      κψ τ τ τ + 1 − κ π t κ ε S τ � �� � A − 1

Example (2): Simple DSGE Model(II) , all parameters are identifjable 0 t t t • some insights • some parameters ( ) do not enter solution, thus do not enter the likelihood (or any other objective) • if we fjx (or calibrate) • some parameters enter as products, e.g. 1 , identifying them separately may be impossible (depending on choice of observables) • is already the product of several other structural parameters (Calvo or Rotemberg) • restrictions necessary to ensure regularity (Eigenvalues inside unit circle) imply bounds involving all structural parameters, i.e. parameter space is not variation free • solution/data-generating-process/reduced-form 6 1 1 r t x t 1       κψ ψ ε M        = − 1 − 1 ε D      κψ τ τ τ + 1 − κ π t κ ε S τ � �� � A − 1

Example (2): Simple DSGE Model(II) , all parameters are identifjable • solution/data-generating-process/reduced-form 0 t t t • some insights likelihood (or any other objective) • if we fjx (or calibrate) • some parameters enter as products, e.g. 1 , identifying them separately may be impossible (depending on choice of observables) • is already the product of several other structural parameters (Calvo or Rotemberg) • restrictions necessary to ensure regularity (Eigenvalues inside unit circle) imply bounds involving all structural parameters, i.e. parameter space is not variation free 1 6 1 x t r t 1       κψ ψ ε M        = − 1 − 1 ε D      κψ τ τ τ + 1 − κ π t κ ε S τ � �� � A − 1 • some parameters ( β ) do not enter solution, thus do not enter the