... Econometric Methods for the Analysis of Dynamic General Equilibrium Models
|
|
- Kelly Joseph
- 6 years ago
- Views:
Transcription
1 ... Econometric Methods for the Analysis of Dynamic General Equilibrium Models 1
2 Overview Multiple Equation Methods State space-observer form Three Examples of Versatility of state space-observer form: Smoothing and filtering (estimation of output gap, real interest rate ) Handling mixed monthly/quarterly data. Connection between DSGE models and Vector Autoregressions. Limited information estimation : impulse response function matching Impulse response functions Formal connection between VARs and DSGE models Full information estimation Maximum likelihood Bayesian inference Single equation ( limited information ) methods: Introduction to Generalized Method of Moments 7
3 State Space-Observer Form Compact summary of the model, and of the data used in the analysis. Typically, data are available in log form. So, the following is useful: If x is steady state of x t : ˆx t x t x x, = x t x =1+ˆx t ³ xt = log =log(1+ˆx t ) ˆx t x Suppose we have a model solution in hand: z t = Az t 1 + Bs t s t = Ps t 1 + t,e t 0 t = D. 8
4 State Space-Observer Form... Consider example #3 in solution notes, in which z t = µ ˆKt+1,s t =ˆε t, t = e t. ˆN t Data used in analysis may include variables in z t and/or other variables. Suppose variables of interest include employment and GDP. GDP, y t : so that y t = ε t K α t N 1 α t, ŷ t =ˆε t + α ˆK t +(1 α) ˆN t = 01 α z t + α 0 z t 1 + s t Then, Y data t = µ log yt log N t = µ log y log N + µ ŷt ˆN t 9
5 State Space-Observer Form... Model prediction for data: Y data t = = µ log y + log N µ log y + log N µ ŷt ˆN t 01 α 0 1 z t + α z t s t = a + Hξ t ξ t = z t z t 1 ˆε t,a= log y log N,H= 01 αα The Observer Equation may include measurement error, w t : Y data t = a + Hξ t + w t,ew t w 0 t = R. Semantics: ξ t is the state of the system (do not confuse with the economic state (K t,ε t )!). 10
6 State Space-Observer Form... The state equation Lawofmotionofthestate,ξ t ξ t = Fξ t 1 + u t,eu t u 0 t = Q z t+1 z t s t+1 = A 0 BP I P z t z t 1 s t + B 0 I t+1, u t = B 0 I t,q= BDB0 0 BD DB 0 D,F= A 0 BP I P. 11
7 State Space-Observer Form... Summary: State-Space, Observer System - ξ t = Fξ t 1 + u t,eu t u 0 t = Q, Y data t = a + Hξ t + w t,ew t w 0 t = R. Can be constructed from model parameters θ =(β, δ,...) so F = F (θ),q= Q (θ),a= a (θ),h= H (θ),r= R (θ). 12
8 State Space-Observer Form... State space observer system very useful Estimation of θ and forecasting ξ t and Y data t Can take into account situations in which data represent a mixture of quarterly, monthly, daily observations. Software readily available on web and elsewhere. Useful for solving the following forecasting problems: Filtering: P ξ t Yt 1 data,yt 2 data,..., Y1 data,t=1, 2,..., T. Smoothing: P ξ t YT data,..., Y1 data,t=1, 2,..., T. Example: real rate of interest and output gap can be recovered from ξ t using example #5 in solution notes. 13
9 Mixed Monthly/Quarterly Observations Different data arrive at different frequencies: daily, monthly, quarterly, etc. This feature can be easily handled in state space-observer system. Example: suppose inflation and hours are monthly, t =0, 1/3, 2/3, 1, 4/3, 5/3, 2,... suppose gdp is quarterly, t =0, 1, 2, 3,... Y data t = GDP t monthly inflation t monthly inflation t 1/3 monthly inflation t 2/3 hours t hours t 1/3 hours t 2/3,t=0, 1, 2,.... that is, we can think of our data set as actually being quarterly, with quarterly observations on the first month s inflation, quarterly observations on the second month s inflation, etc. 14
10 Mixed Monthly/Quarterly Observations... Problem: find state-space observer system in which observed data are: Y data t = GDP t monthly inflation t monthly inflation t 1/3 monthly inflation t 2/3 hours t hours t 1/3 hours t 2/3,t=0, 1, 2,.... Solution: easy! 15
11 Mixed Monthly/Quarterly Observations... Model: specified at a monthly level, with solution, t =0, 1/3, 2/3,... z t = Az t 1/3 + Bs t, s t = Ps t 1/3 + t,e t 0 t = D. Monthly state-space observer system, t =0, 1/3, 2/3,... ξ t = Fξ t 1/3 + u t,eu t u 0 t = Q, u t iid t =0, 1/3, 2/3,... Y t = Hξ t,y t = y t π t. h t Note: first order vector autoregressive representation for quarterly state z } { ξ t = F 3 ξ t 1 + u t + Fu t 1/3 + F 2 u t 2/3, u t + Fu t 1/3 + F 2 u t 2/3 ~iid for t =0, 1, 2,...!! 16
12 Mixed Monthly/Quarterly Observations... Consider the following system: ξ t ξ t 1/3 = F 3 00 ξ t 1 F 2 00 ξ t 4/3 ξ t 2/3 F 00 ξ t 5/3 + I F F2 0 I F 0 0 I u t u t 1/3 u t 2/3. Define ξ t = ξ t ξ t 1/3 ξ t 2/3, F = F 3 00 F 2 00, ũ t = F 00 I F F2 0 I F 0 0 I u t u t 1/3 u t 2/3, so that ξ t = F ξ t 1 +ũ t, ũ t iid in quarterly data, t =0, 1, 2,... Eũ t ũ 0 t = Q = I F F2 0 I F D D 0 I F F2 0 0 I F 0 0 I 0 0 D 0 0 I 18
13 Mixed Monthly/Quarterly Observations... Done! State space-observer system for mixed monthly/quarterly data, for t = 0, 1, 2,... ξ t = F ξ t 1 +ũ t, ũ t iid, Eũ t ũ 0 t = Q, Y data t = H ξ t + w t,w t iid, Ew t w 0 t = R. Here, H selects elements of ξt to construct Y data t can easily handle distinction between whether quarterly data represent monthly averages (as in flow variables), or point-in-time observations on one month in the quarter (as in stock variables). Can use Kalman filter to forecast current quarter data based on first month s (day s, week s) observations. 19
14 Matching Impulse Response Functions Set t =1for t =1, t =0otherwise Impulse response function: log deviation of data with shock from where data would have been in the absence of a shock - u t = B 0 I t, ξ t = Fξ t 1 + u t,ξ 0 =0, impulse response function = Ỹ data t = Hξ t, for t =1, 2,... Choose model parameters, θ, to match Ỹ t data VAR (more on this later). with corresponding estimate from 20
15 Connection Between DSGE s and VAR s Fernandez-Villaverde, Rubio-Ramirez, Sargent Result Vector Autoregression Y t = B 1 Y t 1 + B 2 Y t u t, where u t is iid. Matching impulse response functions strategy for building DSGE models fits VARs and assumes u t are a rotation of economic shocks (for details, see later notes). Can use the state space, observer representation to assess this assumption from the perspective of a DSGE. 21
16 Connection Between DSGE s and VAR s... System (ignoring constant terms and measurement error): ( State equation ) ξ t = Fξ t 1 + D t,d= ( Observer equation ) Y t = Hξ t. Substituting: Y t = HFξ t 1 + HD t B 0 I, Suppose HD is square and invertible. Then t =(HD) 1 Y t (HD) 1 HFξ t 1 ( ) Substitute latter into the state equation: ξ t = Fξ t 1 + D (HD) 1 Y t D (HD) 1 HFξ t 1 = h i I D (HD) 1 H Fξ t 1 + D (HD) 1 Y t. 22
17 Connection Between DSGE s and VAR s... We have: ξ t = Mξ t 1 + D (HD) 1 Y t,m= h i I D (HD) 1 H F. If eigenvalues of M are less than unity, ξ t = D (HD) 1 Y t + MD(HD) 1 Y t 1 + M 2 D (HD) 1 Y t Substituting into ( ) t =(HD) 1 Y t (HD) 1 HF h i D (HD) 1 Y t 1 + MD(HD) 1 Y t 2 + M 2 D (HD) 1 Y t or, 23
18 Connection Between DSGE s and VAR s... where Y t = B 1 Y t 1 + B 2 Y t u t, u t = HD t B j = HFM j 1 D (HD) 1,j=1, 2,... The latter is the VAR representation. Note: t is invertible because it lies in space of current and past Y t s. Note: VAR is infinite-ordered. Note: assumed system is square. Sims-Zha (Macroeconomic Dynamics) show that square-ness is not necessary. 24
19 Maximum Likelihood Estimation State space-observer system: ξ t+1 = Fξ t + u t+1,eu t u 0 t = Q, Y data t = a 0 + Hξ t + w t,ew t w 0 t = R Parameters of system: (F, Q, a 0,H,R). These are functions of model parameters, θ. Formulas for computing likelihood P Y data θ = P Y data 1,..., Y data T θ. are standard (see Hamilton s textbook). 25
20 Bayesian Maximum Likelihood Bayesians describe the mapping from prior beliefs about θ, summarized in p (θ), to new posterior beliefs in the light of observing the data, Y data. General property of probabilities: p Y data,θ = which implies Bayes rule: ½ p Y data θ p (θ) p θ Y data p Y data, p θ Y data = p Y data θ p (θ), p (Y data ) mapping from prior to posterior induced by Y data. 26
21 Bayesian Maximum Likelihood... Properties of the posterior distribution, p θ Y data. The value of θ that maximizes p θ Y data ( mode of posterior distribution). Graphs that compare the marginal posterior distribution of individual elements of θ with the corresponding prior. Probability intervals about the mode of θ ( Bayesian confidence intervals ) Other properties of p θ Y data helpful for assessing model fit. 27
22 Bayesian Maximum Likelihood... Computation of mode sometimes referred to as Basyesian maximum likelihood : ( ) θ mod e =argmax θ log p Y data θ + maximum likelihood with a penalty function. NX log [p i (θ i )] i=1 Shape of posterior distribution, p θ Y data, obtained by Metropolis-Hastings algorithm. Algorithm computes θ (1),..., θ (N), which, as N, has a density that approximates p θ Y data well. Marginal posterior distribution of any element of θ displayed as the histogram of the corresponding element {θ (i),i=1,..,n} 28
23 Metropolis-Hastings Algorithm (MCMC) We have (except for a constant): Ã f {z} θ Y N 1! = f (Y θ) f (θ). f (Y ) We want the marginal posterior distribution of θ i : h (θ i Y )= Z θ j6=i f (θ Y ) dθ j6=i,i=1,..., N. MCMC algorithm can approximate h (θ i Y ). Obtain (V produced automatically by gradient-based maximization methods): θ mod e θ =argmax θ f (Y θ) f (θ),v 1 2 f (Y θ) f (θ) θ θ 0. θ=θ 29
24 Metropolis-Hastings Algorithm (MCMC)... Compute the sequence, θ (1),θ (2),..., θ (M) (M large) whose distribution turns out to have pdf f (θ Y ). θ (1) = θ to compute θ (r), for r>1 step 1: select candidate θ (r),x, jump distribution z à }!{ draw {z} x from θ (r 1) + N 1 step 2: compute scalar, λ : λ = kn {z} 0,V N 1 f (Y x) f (x) f ³Y (r 1) θ f ³θ (r 1),kis a scalar step 3: ½ compute θ (r) : θ (r) θ = (r 1) if u>λ x if u<λ,uis a realization from uniform [0, 1] 30
25 Metropolis-Hastings Algorithm (MCMC)... Approximating marginal posterior distribution, h (θ i Y ), of θ i compute and display the histogram of θ (1) Other objects of interest: i,θ (2) i mean and variance of posterior distribution θ : Eθ ' θ 1 M MX θ (j),var(θ)' 1 M j=1 MX j=1,..., θ (M),i=1,...,N. i h θ (j) θ ih θi 0 θ (j). marginal density of Y (actually, Geweke s harmonic mean works better): f (Y ) ' 1 M MX f ³Y (j) θ f ³θ (j) j=1 31
26 Metropolis-Hastings Algorithm (MCMC)... Some intuition Algorithm is more likely to select moves into high probability regions than into low probability regions. n Set, θ (1),θ (2),..., θ (M)o, populated relatively more by elements near mode of f (θ Y ). n Set, θ (1),θ (2),...,θ (M)o, also populated (though less so) by elements far from mode of f (θ Y ). 32
27 Metropolis-Hastings Algorithm (MCMC)... Practical issues what value should you set k to? set k so that you accept (i.e., θ (r) = x) in step 3 of MCMC algorithm are roughly 27 percent of time what value of M should you set? a value so that if M is increased further, your results do not change in practice, M =10, 000 (a small value) up to M =1, 000, 000. large M is time-consuming. Could use Laplace approximation (after checking its accuracy) in initial phases of research project. 33
28 Laplace Approximation to Posterior Distribution In practice, Metropolis-Hasting algorithm very time intensive. Do it last! In practice, Laplace approximation is quick, essentially free and very accurate. Let θ R N denote the N dimensional vector of parameters and g (θ) log f (y θ) f (θ), f (y θ) ~likelihood of data f (θ) ~prior on parameters θ ~maximum of g (θ) (i.e., mode) 34
29 Laplace Approximation to Posterior Distribution... Second order Taylor series expansion about θ = θ : where g (θ) g (θ )+g θ (θ )(θ θ ) 1 2 (θ θ ) 0 g θθ (θ )(θ θ ), Interior optimality implies: g θθ (θ )= 2 log f (y θ) f (θ) θ θ 0 θ=θ g θ (θ )=0,g θθ (θ ) positive definite Then, f (y θ) f (θ) ' f (y θ ) f (θ )exp ½ 1 ¾ 2 (θ θ ) 0 g θθ (θ )(θ θ ). 35
30 Laplace Approximation to Posterior Distribution... Note 1 (2π) N 2 g θθ (θ ) 1 2 exp ½ 1 ¾ 2 (θ θ ) 0 g θθ (θ )(θ θ ) = multinormal density for N dimensional random variable θ with mean θ and variance g θθ (θ ) 1. So, posterior of θ i (i.e., h (θ i Y ³ )) is h approximately θ i ~N θ i, g θθ (θ ) 1i. ii This formula for the posterior distribution is essentially free, because g θθ is computed as part of gradient-based numerical optimization procedures. 36
31 Laplace Approximation to Posterior Distribution... Marginal likelihood of data, y, is useful for model comparisons. Easy to compute using the Laplace approximation. Property of Normal distribution: Z 1 g (2π) N θθ (θ ) 1 2 exp ½ 1 ¾ 2 2 (θ θ ) 0 g θθ (θ )(θ θ ) dθ =1 Then, Z f (y θ) f (θ) dθ ' = f (y θ ) f (θ Z ) 1 g θθ (θ ) 1 2 (2π) N 2 Z f (y θ ) f (θ )exp 1 (2π) N 2 ½ 1 ¾ 2 (θ θ ) 0 g θθ (θ )(θ θ ) dθ g θθ (θ ) 1 2 exp ½ 1 ¾ 2 (θ θ ) 0 g θθ (θ )(θ θ ) dθ = f (y θ ) f (θ ). 1 g θθ (θ ) 1 2 (2π) N 2 37
32 Laplace Approximation to Posterior Distribution... Formula for marginal likelihood based on Laplace approximation: f (y) = Z f (y θ) f (θ) dθ ' (2π) N f (y θ ) f (θ ) 2. g θθ (θ ) 1 2 Suppose f(y Model 1) >f(y Model 2). Then, posterior odds on Model 1 higher than Model 2. Model 1 fits better than Model 2 Can use this to compare across two different models, or to evaluate contribution to fit of various model features: habit persistence, adjustment costs, etc. 38
33 Generalized Method of Moments Express your econometric estimator into Hansen s GMM framework and you get standard errors Essentially, any estimation strategy fits (see Hamilton) Works when parameters of interest, β, have the following property: E u t {z} N 1 β {z} =0,βtrue value of some parameter(s) of interest n 1 u t (β) ~ stationary stochastic process (and other conditions) n = N : exactly identified n<n: over identified 39
34 Generalized Method of Moments... Example 1: mean β = Ex t, u t (β) =β x t. Example 2: mean and variance β = μσ, Ex t = μ, E (x t μ) 2 = σ 2. then, u t (β) = μ x t (x t μ) 2 σ 2. 40
35 Generalized Method of Moments... Example 3: mean, variance, correlation, relative standard deviation where β = μ y σ y μ x σ x ρ xy λ,λ σ x /σ y, Ey t = μ y,e y t μ y 2 = σ 2 y Ex t = μ x,e(x t μ x ) 2 = σ 2 x then ρ xy = E y t μ y (xt μ x ). σ y σ x μ x x t (x t μ x ) 2 σ 2 x μ u t (β) = y y t 2 yt μ y σ 2. y σ y σ x ρ xy y t μ y (xt μ x ) σ y λ σ x 41
36 Generalized Method of Moments... Example 4: New Keynesian Phillips curve π t =0.99E t π t+1 + γs t, or, π t 0.99π t+1 γs t = η t+1 where, η t+1 =0.99 (E t π t+1 π t+1 )= E t η t+1 =0 Under Rational Expectations : η t+1 time t information, z t u t (γ) =[π t 0.99π t+1 γs t ] z t 42
37 Generalized Method of Moments... Inference about β Estimator of β in exactly identified case (n = N) Choose ˆβ to mimick population property of true β, Eu t (β) =0. Define: Solve g T (β) = 1 T TX u t (β). t=1 ˆβ : g T {z} ˆβ N 1 = 0 {z}. N 1 43
38 Generalized Method of Moments... Example 1: mean β = Ex t, Choose ˆβ so that g T ³ˆβ = 1 T u t (β) =β x t. TX t=1 u t ³ˆβ = ˆβ 1 T TX x t =0 t=1 and ˆβ is simply sample mean. 44
39 Generalized Method of Moments... Example 4 in exactly identified case Eu t (γ) =E [π t 0.99π t+1 γs t ] z t,z t ~scalar choose ˆγ so that g T ³ˆβ = 1 T TX [π t 0.99π t+1 ˆγs t ] z t =0, t=1 or. standard instrumental variables estimator: ˆγ = 1 T P T t=1 [π t 0.99π t+1 ] z t 1 T P T t=1 s tz t 45
40 Generalized Method of Moments... Key message: In exactly identified case, GMM does not deliver a new estimator you would not have thought of on your own means, correlations, regression coefficients, exactly identified IV estimation, maximum likelihood. GMM provides framework for deriving asymptotically valid formulas for estimating sampling uncertainty. 46
41 Generalized Method of Moments... Estimating β in overidentified case (N >n) Cannot exactly implement sample analog of Eu t (β) =0: g T {z} ˆβ n 1 Instead, do the best you can : = 0 {z} N 1 ˆβ =argmin β g T (β) 0 W T g T (β), where W T ~ is a positive definite weighting matrix. GMM works for any positive definite W T, but is most efficient if W T is inverse of estimator of variance-covariance matrix of g T ³ˆβ : (W T ) 1 ³ˆβ 0 = Eg T ³ˆβ g T. 47
42 Generalized Method of Moments... This choice of weighting matrix very sensible: weight heavily those moment conditions (i.e., elements of g T ³ˆβ )that are precisely estimated pay less attention to the others. 49
43 Generalized Method of Moments... Estimator of WT 1 Note: 0 Eg T ³ˆβ g T ³ˆβ = 1 h i h i 0 T 2E u 1 ³ˆβ + u 2 ³ˆβ u T ³ˆβ u 1 ³ˆβ + u 2 ³ˆβ u T ³ˆβ ³ˆβ ³ˆβ = 1 T [T T Eu t ³ˆβ + T 1 ³ˆβ T Eu t u t 1 ³ˆβ where W 1 T 0 T 1 u t ³ˆβ + T Eu t 0 T 2 ³ˆβ + T Eu t 0 1 u t+1 ³ˆβ u t 2 ³ˆβ = 1 C (0) + T 0 C (r) =Eu t ³ˆβ u t r ³ˆβ " XT 1 r=1 0 u t+t 1 ³ˆβ T Eu t ³ˆβ ³ˆβ 0] T Eu t u t T +1 T r T is 1 T spectral density matrix at frequency zero, S 0, of u t ³ˆβ C (r)+c (r) 0 #, 50
44 Generalized Method of Moments... Conclude: W 1 T W 1 T = Eg T ³ˆβ g T ³ˆβ estimated by " [W 1 T = 1 T = 1 T Ĉ (0) + " XT 1 r=1 C (0) + T r T XT 1 r=1 T r T ³Ĉ (r)+ Ĉ (r) 0 # C (r)+c (r) 0 # = S 0 T. = 1 T Ŝ0, imposing whatever restrictions are implied by the null hypothesis, i.e., (as in ex. 4) C (r) =0,r>Rsome R. which is Newey-West estimator of spectral density at frequency zero Problem: need ˆβ to compute WT 1 and need WT 1 to compute ˆβ!! Solution - first compute ˆβ using W T = I, then iterate... 51
45 Generalized Method of Moments... Sampling Uncertainty in ˆβ. The exactly identified case By the Mean Value Theorem, g T ³ˆβ can be expressed as follows: g T ³ˆβ = g T (β 0 )+D ³ˆβ β0, where β 0 is the true value of the parameters and D = g T (β) β 0 β=β, some β between β 0 and ˆβ. Since g T ³ˆβ =0and g T (β 0 ) ã N (0,S 0 /T ), it follows: so ˆβ β 0 = D 1 g T (β 0 ), ã D ˆβ β 0 N Ã0, 0 S0 1D! 1 T 52
46 Generalized Method of Moments... The overidentified case. An extension of the ideas we have already discussed. Can derive the results for yourself, using the delta function method for deriving the sampling distribution of statistics. Hamilton s text book has a great review of GMM. 53
Bayesian Inference for DSGE Models. Lawrence J. Christiano
Bayesian Inference for DSGE Models Lawrence J. Christiano Outline State space-observer form. convenient for model estimation and many other things. Bayesian inference Bayes rule. Monte Carlo integation.
More informationBayesian Inference for DSGE Models. Lawrence J. Christiano
Bayesian Inference for DSGE Models Lawrence J. Christiano Outline State space-observer form. convenient for model estimation and many other things. Bayesian inference Bayes rule. Monte Carlo integation.
More informationBayesian Inference for DSGE Models. Lawrence J. Christiano
Bayesian Inference for DSGE Models Lawrence J. Christiano Outline State space-observer form. convenient for model estimation and many other things. Preliminaries. Probabilities. Maximum Likelihood. Bayesian
More informationAdvanced Econometrics III, Lecture 5, Dynamic General Equilibrium Models Example 1: A simple RBC model... 2
Advanced Econometrics III, Lecture 5, 2017 1 Contents 1 Dynamic General Equilibrium Models 2 1.1 Example 1: A simple RBC model.................... 2 1.2 Approximation Method Based on Linearization.............
More informationDSGE Methods. Estimation of DSGE models: Maximum Likelihood & Bayesian. Willi Mutschler, M.Sc.
DSGE Methods Estimation of DSGE models: Maximum Likelihood & Bayesian Willi Mutschler, M.Sc. Institute of Econometrics and Economic Statistics University of Münster willi.mutschler@uni-muenster.de Summer
More informationAmbiguous Business Cycles: Online Appendix
Ambiguous Business Cycles: Online Appendix By Cosmin Ilut and Martin Schneider This paper studies a New Keynesian business cycle model with agents who are averse to ambiguity (Knightian uncertainty). Shocks
More informationLabor-Supply Shifts and Economic Fluctuations. Technical Appendix
Labor-Supply Shifts and Economic Fluctuations Technical Appendix Yongsung Chang Department of Economics University of Pennsylvania Frank Schorfheide Department of Economics University of Pennsylvania January
More informationDynamic Factor Models and Factor Augmented Vector Autoregressions. Lawrence J. Christiano
Dynamic Factor Models and Factor Augmented Vector Autoregressions Lawrence J Christiano Dynamic Factor Models and Factor Augmented Vector Autoregressions Problem: the time series dimension of data is relatively
More informationB y t = γ 0 + Γ 1 y t + ε t B(L) y t = γ 0 + ε t ε t iid (0, D) D is diagonal
Structural VAR Modeling for I(1) Data that is Not Cointegrated Assume y t =(y 1t,y 2t ) 0 be I(1) and not cointegrated. That is, y 1t and y 2t are both I(1) and there is no linear combination of y 1t and
More informationMassachusetts Institute of Technology Department of Economics Time Series Lecture 6: Additional Results for VAR s
Massachusetts Institute of Technology Department of Economics Time Series 14.384 Guido Kuersteiner Lecture 6: Additional Results for VAR s 6.1. Confidence Intervals for Impulse Response Functions There
More informationMonetary and Exchange Rate Policy Under Remittance Fluctuations. Technical Appendix and Additional Results
Monetary and Exchange Rate Policy Under Remittance Fluctuations Technical Appendix and Additional Results Federico Mandelman February In this appendix, I provide technical details on the Bayesian estimation.
More informationCombining Macroeconomic Models for Prediction
Combining Macroeconomic Models for Prediction John Geweke University of Technology Sydney 15th Australasian Macro Workshop April 8, 2010 Outline 1 Optimal prediction pools 2 Models and data 3 Optimal pools
More informationTime Series Analysis. James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY
Time Series Analysis James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY PREFACE xiii 1 Difference Equations 1.1. First-Order Difference Equations 1 1.2. pth-order Difference Equations 7
More informationTAKEHOME FINAL EXAM e iω e 2iω e iω e 2iω
ECO 513 Spring 2015 TAKEHOME FINAL EXAM (1) Suppose the univariate stochastic process y is ARMA(2,2) of the following form: y t = 1.6974y t 1.9604y t 2 + ε t 1.6628ε t 1 +.9216ε t 2, (1) where ε is i.i.d.
More informationVAR Models and Applications
VAR Models and Applications Laurent Ferrara 1 1 University of Paris West M2 EIPMC Oct. 2016 Overview of the presentation 1. Vector Auto-Regressions Definition Estimation Testing 2. Impulse responses functions
More informationInference when identifying assumptions are doubted. A. Theory B. Applications
Inference when identifying assumptions are doubted A. Theory B. Applications 1 A. Theory Structural model of interest: A y t B 1 y t1 B m y tm u t nn n1 u t i.i.d. N0, D D diagonal 2 Bayesian approach:
More informationTime Series Analysis. James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY
Time Series Analysis James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY & Contents PREFACE xiii 1 1.1. 1.2. Difference Equations First-Order Difference Equations 1 /?th-order Difference
More informationDynamic Factor Models, Factor Augmented VARs, and SVARs in Macroeconomics. -- Part 2: SVARs and SDFMs --
Dynamic Factor Models, Factor Augmented VARs, and SVARs in Macroeconomics -- Part 2: SVARs and SDFMs -- Mark Watson Princeton University Central Bank of Chile October 22-24, 208 Reference: Stock, James
More informationVector Auto-Regressive Models
Vector Auto-Regressive Models Laurent Ferrara 1 1 University of Paris Nanterre M2 Oct. 2018 Overview of the presentation 1. Vector Auto-Regressions Definition Estimation Testing 2. Impulse responses functions
More informationEstimation and Inference on Dynamic Panel Data Models with Stochastic Volatility
Estimation and Inference on Dynamic Panel Data Models with Stochastic Volatility Wen Xu Department of Economics & Oxford-Man Institute University of Oxford (Preliminary, Comments Welcome) Theme y it =
More information... Solving and Estimating Dynamic General Equilibrium Models Using Log Linear Approximation
... Solving and Estimating Dynamic General Equilibrium Models Using Log Linear Approximation 1 Overview To Estimate and Analyze Dynamic General Equilibrium Models, Need Solution. Will Review a Linearization
More informationEstimation of moment-based models with latent variables
Estimation of moment-based models with latent variables work in progress Ra aella Giacomini and Giuseppe Ragusa UCL/Cemmap and UCI/Luiss UPenn, 5/4/2010 Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)Moments
More informationξ t = Fξ t 1 + v t. then λ is less than unity in absolute value. In the above expression, A denotes the determinant of the matrix, A. 1 y t 1.
Christiano FINC 520, Spring 2009 Homework 3, due Thursday, April 23. 1. In class, we discussed the p th order VAR: y t = c + φ 1 y t 1 + φ 2 y t 2 +... + φ p y t p + ε t, where ε t is a white noise with
More informationInference when identifying assumptions are doubted. A. Theory. Structural model of interest: B 1 y t1. u t. B m y tm. u t i.i.d.
Inference when identifying assumptions are doubted A. Theory B. Applications Structural model of interest: A y t B y t B m y tm nn n i.i.d. N, D D diagonal A. Theory Bayesian approach: Summarize whatever
More informationChapter 6. Maximum Likelihood Analysis of Dynamic Stochastic General Equilibrium (DSGE) Models
Chapter 6. Maximum Likelihood Analysis of Dynamic Stochastic General Equilibrium (DSGE) Models Fall 22 Contents Introduction 2. An illustrative example........................... 2.2 Discussion...................................
More informationOpen Economy Macroeconomics: Theory, methods and applications
Open Economy Macroeconomics: Theory, methods and applications Lecture 4: The state space representation and the Kalman Filter Hernán D. Seoane UC3M January, 2016 Today s lecture State space representation
More informationMarkov Chain Monte Carlo Methods
Markov Chain Monte Carlo Methods John Geweke University of Iowa, USA 2005 Institute on Computational Economics University of Chicago - Argonne National Laboaratories July 22, 2005 The problem p (θ, ω I)
More information(I AL BL 2 )z t = (I CL)ζ t, where
ECO 513 Fall 2011 MIDTERM EXAM The exam lasts 90 minutes. Answer all three questions. (1 Consider this model: x t = 1.2x t 1.62x t 2 +.2y t 1.23y t 2 + ε t.7ε t 1.9ν t 1 (1 [ εt y t = 1.4y t 1.62y t 2
More informationEconometrics of Panel Data
Econometrics of Panel Data Jakub Mućk Meeting # 6 Jakub Mućk Econometrics of Panel Data Meeting # 6 1 / 36 Outline 1 The First-Difference (FD) estimator 2 Dynamic panel data models 3 The Anderson and Hsiao
More informationThe Kalman filter, Nonlinear filtering, and Markov Chain Monte Carlo
NBER Summer Institute Minicourse What s New in Econometrics: Time Series Lecture 5 July 5, 2008 The Kalman filter, Nonlinear filtering, and Markov Chain Monte Carlo Lecture 5, July 2, 2008 Outline. Models
More informationAssessing Structural VAR s
... Assessing Structural VAR s by Lawrence J. Christiano, Martin Eichenbaum and Robert Vigfusson Zurich, September 2005 1 Background Structural Vector Autoregressions Address the Following Type of Question:
More informationMID-TERM EXAM ANSWERS. p t + δ t = Rp t 1 + η t (1.1)
ECO 513 Fall 2005 C.Sims MID-TERM EXAM ANSWERS (1) Suppose a stock price p t and the stock dividend δ t satisfy these equations: p t + δ t = Rp t 1 + η t (1.1) δ t = γδ t 1 + φp t 1 + ε t, (1.2) where
More informationBayesian Estimation of DSGE Models: Lessons from Second-order Approximations
Bayesian Estimation of DSGE Models: Lessons from Second-order Approximations Sungbae An Singapore Management University Bank Indonesia/BIS Workshop: STRUCTURAL DYNAMIC MACROECONOMIC MODELS IN ASIA-PACIFIC
More informationGraduate Macro Theory II: Notes on Quantitative Analysis in DSGE Models
Graduate Macro Theory II: Notes on Quantitative Analysis in DSGE Models Eric Sims University of Notre Dame Spring 2011 This note describes very briefly how to conduct quantitative analysis on a linearized
More informationESTIMATION of a DSGE MODEL
ESTIMATION of a DSGE MODEL Paris, October 17 2005 STÉPHANE ADJEMIAN stephane.adjemian@ens.fr UNIVERSITÉ DU MAINE & CEPREMAP Slides.tex ESTIMATION of a DSGE MODEL STÉPHANE ADJEMIAN 16/10/2005 21:37 p. 1/3
More informationAssessing Structural VAR s
... Assessing Structural VAR s by Lawrence J. Christiano, Martin Eichenbaum and Robert Vigfusson Columbia, October 2005 1 Background Structural Vector Autoregressions Can be Used to Address the Following
More informationGeneralized Method of Moments (GMM) Estimation
Econometrics 2 Fall 2004 Generalized Method of Moments (GMM) Estimation Heino Bohn Nielsen of29 Outline of the Lecture () Introduction. (2) Moment conditions and methods of moments (MM) estimation. Ordinary
More informationAssessing Structural VAR s
... Assessing Structural VAR s by Lawrence J. Christiano, Martin Eichenbaum and Robert Vigfusson Yale, October 2005 1 Background Structural Vector Autoregressions Can be Used to Address the Following Type
More informationEcon 583 Final Exam Fall 2008
Econ 583 Final Exam Fall 2008 Eric Zivot December 11, 2008 Exam is due at 9:00 am in my office on Friday, December 12. 1 Maximum Likelihood Estimation and Asymptotic Theory Let X 1,...,X n be iid random
More informationSpring 2017 Econ 574 Roger Koenker. Lecture 14 GEE-GMM
University of Illinois Department of Economics Spring 2017 Econ 574 Roger Koenker Lecture 14 GEE-GMM Throughout the course we have emphasized methods of estimation and inference based on the principle
More informationA Bayesian perspective on GMM and IV
A Bayesian perspective on GMM and IV Christopher A. Sims Princeton University sims@princeton.edu November 26, 2013 What is a Bayesian perspective? A Bayesian perspective on scientific reporting views all
More informationA Very Brief Summary of Statistical Inference, and Examples
A Very Brief Summary of Statistical Inference, and Examples Trinity Term 2008 Prof. Gesine Reinert 1 Data x = x 1, x 2,..., x n, realisations of random variables X 1, X 2,..., X n with distribution (model)
More informationAssessing Structural VAR s
... Assessing Structural VAR s by Lawrence J. Christiano, Martin Eichenbaum and Robert Vigfusson Minneapolis, August 2005 1 Background In Principle, Impulse Response Functions from SVARs are useful as
More informationDSGE Methods. Estimation of DSGE models: GMM and Indirect Inference. Willi Mutschler, M.Sc.
DSGE Methods Estimation of DSGE models: GMM and Indirect Inference Willi Mutschler, M.Sc. Institute of Econometrics and Economic Statistics University of Münster willi.mutschler@wiwi.uni-muenster.de Summer
More informationEstimating Macroeconomic Models: A Likelihood Approach
Estimating Macroeconomic Models: A Likelihood Approach Jesús Fernández-Villaverde University of Pennsylvania, NBER, and CEPR Juan Rubio-Ramírez Federal Reserve Bank of Atlanta Estimating Dynamic Macroeconomic
More informationSUPPLEMENT TO MARKET ENTRY COSTS, PRODUCER HETEROGENEITY, AND EXPORT DYNAMICS (Econometrica, Vol. 75, No. 3, May 2007, )
Econometrica Supplementary Material SUPPLEMENT TO MARKET ENTRY COSTS, PRODUCER HETEROGENEITY, AND EXPORT DYNAMICS (Econometrica, Vol. 75, No. 3, May 2007, 653 710) BY SANGHAMITRA DAS, MARK ROBERTS, AND
More informationSolving a Dynamic (Stochastic) General Equilibrium Model under the Discrete Time Framework
Solving a Dynamic (Stochastic) General Equilibrium Model under the Discrete Time Framework Dongpeng Liu Nanjing University Sept 2016 D. Liu (NJU) Solving D(S)GE 09/16 1 / 63 Introduction Targets of the
More informationDavid Giles Bayesian Econometrics
David Giles Bayesian Econometrics 1. General Background 2. Constructing Prior Distributions 3. Properties of Bayes Estimators and Tests 4. Bayesian Analysis of the Multiple Regression Model 5. Bayesian
More informationBayesian Estimation of DSGE Models
Bayesian Estimation of DSGE Models Stéphane Adjemian Université du Maine, GAINS & CEPREMAP stephane.adjemian@univ-lemans.fr http://www.dynare.org/stepan June 28, 2011 June 28, 2011 Université du Maine,
More informationNonlinear and/or Non-normal Filtering. Jesús Fernández-Villaverde University of Pennsylvania
Nonlinear and/or Non-normal Filtering Jesús Fernández-Villaverde University of Pennsylvania 1 Motivation Nonlinear and/or non-gaussian filtering, smoothing, and forecasting (NLGF) problems are pervasive
More informationProblem Set 2 Solution Sketches Time Series Analysis Spring 2010
Problem Set 2 Solution Sketches Time Series Analysis Spring 2010 Forecasting 1. Let X and Y be two random variables such that E(X 2 ) < and E(Y 2 )
More informationNews Shocks: Different Effects in Boom and Recession?
News Shocks: Different Effects in Boom and Recession? Maria Bolboaca, Sarah Fischer University of Bern Study Center Gerzensee June 7, 5 / Introduction News are defined in the literature as exogenous changes
More informationThis note introduces some key concepts in time series econometrics. First, we
INTRODUCTION TO TIME SERIES Econometrics 2 Heino Bohn Nielsen September, 2005 This note introduces some key concepts in time series econometrics. First, we present by means of examples some characteristic
More informationPrediction Using Several Macroeconomic Models
Gianni Amisano European Central Bank 1 John Geweke University of Technology Sydney, Erasmus University, and University of Colorado European Central Bank May 5, 2012 1 The views expressed do not represent
More informationChapter 1. Introduction. 1.1 Background
Chapter 1 Introduction Science is facts; just as houses are made of stones, so is science made of facts; but a pile of stones is not a house and a collection of facts is not necessarily science. Henri
More informationSignaling Effects of Monetary Policy
Signaling Effects of Monetary Policy Leonardo Melosi London Business School 24 May 2012 Motivation Disperse information about aggregate fundamentals Morris and Shin (2003), Sims (2003), and Woodford (2002)
More informationTime-Varying Parameters
Kalman Filter and state-space models: time-varying parameter models; models with unobservable variables; basic tool: Kalman filter; implementation is task-specific. y t = x t β t + e t (1) β t = µ + Fβ
More informationLong-Run Covariability
Long-Run Covariability Ulrich K. Müller and Mark W. Watson Princeton University October 2016 Motivation Study the long-run covariability/relationship between economic variables great ratios, long-run Phillips
More informationOnline appendix to On the stability of the excess sensitivity of aggregate consumption growth in the US
Online appendix to On the stability of the excess sensitivity of aggregate consumption growth in the US Gerdie Everaert 1, Lorenzo Pozzi 2, and Ruben Schoonackers 3 1 Ghent University & SHERPPA 2 Erasmus
More informationGMM, HAC estimators, & Standard Errors for Business Cycle Statistics
GMM, HAC estimators, & Standard Errors for Business Cycle Statistics Wouter J. Den Haan London School of Economics c Wouter J. Den Haan Overview Generic GMM problem Estimation Heteroskedastic and Autocorrelation
More informationSingle Equation Linear GMM with Serially Correlated Moment Conditions
Single Equation Linear GMM with Serially Correlated Moment Conditions Eric Zivot November 2, 2011 Univariate Time Series Let {y t } be an ergodic-stationary time series with E[y t ]=μ and var(y t )
More informationVector Autoregressive Model. Vector Autoregressions II. Estimation of Vector Autoregressions II. Estimation of Vector Autoregressions I.
Vector Autoregressive Model Vector Autoregressions II Empirical Macroeconomics - Lect 2 Dr. Ana Beatriz Galvao Queen Mary University of London January 2012 A VAR(p) model of the m 1 vector of time series
More informationComprehensive Examination Quantitative Methods Spring, 2018
Comprehensive Examination Quantitative Methods Spring, 2018 Instruction: This exam consists of three parts. You are required to answer all the questions in all the parts. 1 Grading policy: 1. Each part
More informationSwitching Regime Estimation
Switching Regime Estimation Series de Tiempo BIrkbeck March 2013 Martin Sola (FE) Markov Switching models 01/13 1 / 52 The economy (the time series) often behaves very different in periods such as booms
More informationEmpirical Market Microstructure Analysis (EMMA)
Empirical Market Microstructure Analysis (EMMA) Lecture 3: Statistical Building Blocks and Econometric Basics Prof. Dr. Michael Stein michael.stein@vwl.uni-freiburg.de Albert-Ludwigs-University of Freiburg
More informationWhither News Shocks?
Discussion of Whither News Shocks? Barsky, Basu and Lee Christiano Outline Identification assumptions for news shocks Empirical Findings Using NK model used to think about BBL identification. Why should
More informationNon-Markovian Regime Switching with Endogenous States and Time-Varying State Strengths
Non-Markovian Regime Switching with Endogenous States and Time-Varying State Strengths January 2004 Siddhartha Chib Olin School of Business Washington University chib@olin.wustl.edu Michael Dueker Federal
More information2.5 Forecasting and Impulse Response Functions
2.5 Forecasting and Impulse Response Functions Principles of forecasting Forecast based on conditional expectations Suppose we are interested in forecasting the value of y t+1 based on a set of variables
More informationBayesian Regression Linear and Logistic Regression
When we want more than point estimates Bayesian Regression Linear and Logistic Regression Nicole Beckage Ordinary Least Squares Regression and Lasso Regression return only point estimates But what if we
More informationDSGE-Models. Limited Information Estimation General Method of Moments and Indirect Inference
DSGE-Models General Method of Moments and Indirect Inference Dr. Andrea Beccarini Willi Mutschler, M.Sc. Institute of Econometrics and Economic Statistics University of Münster willi.mutschler@uni-muenster.de
More informationBonn Summer School Advances in Empirical Macroeconomics
Bonn Summer School Advances in Empirical Macroeconomics Karel Mertens Cornell, NBER, CEPR Bonn, June 2015 In God we trust, all others bring data. William E. Deming (1900-1993) Angrist and Pischke are Mad
More informationPANEL DISCUSSION: THE ROLE OF POTENTIAL OUTPUT IN POLICYMAKING
PANEL DISCUSSION: THE ROLE OF POTENTIAL OUTPUT IN POLICYMAKING James Bullard* Federal Reserve Bank of St. Louis 33rd Annual Economic Policy Conference St. Louis, MO October 17, 2008 Views expressed are
More informationBayesian Estimation of DSGE Models 1 Chapter 3: A Crash Course in Bayesian Inference
1 The views expressed in this paper are those of the authors and do not necessarily reflect the views of the Federal Reserve Board of Governors or the Federal Reserve System. Bayesian Estimation of DSGE
More informationFinancial Econometrics and Volatility Models Estimation of Stochastic Volatility Models
Financial Econometrics and Volatility Models Estimation of Stochastic Volatility Models Eric Zivot April 26, 2010 Outline Likehood of SV Models Survey of Estimation Techniques for SV Models GMM Estimation
More informationSequential Monte Carlo Methods (for DSGE Models)
Sequential Monte Carlo Methods (for DSGE Models) Frank Schorfheide University of Pennsylvania, PIER, CEPR, and NBER October 23, 2017 Some References These lectures use material from our joint work: Tempered
More informationARIMA Modelling and Forecasting
ARIMA Modelling and Forecasting Economic time series often appear nonstationary, because of trends, seasonal patterns, cycles, etc. However, the differences may appear stationary. Δx t x t x t 1 (first
More informationTesting Restrictions and Comparing Models
Econ. 513, Time Series Econometrics Fall 00 Chris Sims Testing Restrictions and Comparing Models 1. THE PROBLEM We consider here the problem of comparing two parametric models for the data X, defined by
More informationMacroeconomics Theory II
Macroeconomics Theory II Francesco Franco FEUNL February 2016 Francesco Franco Macroeconomics Theory II 1/23 Housekeeping. Class organization. Website with notes and papers as no "Mas-Collel" in macro
More informationThe Metropolis-Hastings Algorithm. June 8, 2012
The Metropolis-Hastings Algorithm June 8, 22 The Plan. Understand what a simulated distribution is 2. Understand why the Metropolis-Hastings algorithm works 3. Learn how to apply the Metropolis-Hastings
More informationKalman filtering and friends: Inference in time series models. Herke van Hoof slides mostly by Michael Rubinstein
Kalman filtering and friends: Inference in time series models Herke van Hoof slides mostly by Michael Rubinstein Problem overview Goal Estimate most probable state at time k using measurement up to time
More information10. Time series regression and forecasting
10. Time series regression and forecasting Key feature of this section: Analysis of data on a single entity observed at multiple points in time (time series data) Typical research questions: What is the
More informationNowcasting Norwegian GDP
Nowcasting Norwegian GDP Knut Are Aastveit and Tørres Trovik May 13, 2007 Introduction Motivation The last decades of advances in information technology has made it possible to access a huge amount of
More informationECON 4160, Spring term Lecture 12
ECON 4160, Spring term 2013. Lecture 12 Non-stationarity and co-integration 2/2 Ragnar Nymoen Department of Economics 13 Nov 2013 1 / 53 Introduction I So far we have considered: Stationary VAR, with deterministic
More informationChapter 1. GMM: Basic Concepts
Chapter 1. GMM: Basic Concepts Contents 1 Motivating Examples 1 1.1 Instrumental variable estimator....................... 1 1.2 Estimating parameters in monetary policy rules.............. 2 1.3 Estimating
More informationSession 2B: Some basic simulation methods
Session 2B: Some basic simulation methods John Geweke Bayesian Econometrics and its Applications August 14, 2012 ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation
More informationTitle. Description. Remarks and examples. stata.com. stata.com. Introduction to DSGE models. intro 1 Introduction to DSGEs and dsge
Title stata.com intro 1 Introduction to DSGEs and dsge Description Remarks and examples References Also see Description In this entry, we introduce DSGE models and the dsge command. We begin with an overview
More informationEco517 Fall 2014 C. Sims MIDTERM EXAM
Eco57 Fall 204 C. Sims MIDTERM EXAM You have 90 minutes for this exam and there are a total of 90 points. The points for each question are listed at the beginning of the question. Answer all questions.
More informationSentiment Shocks as Drivers of Business Cycles
Sentiment Shocks as Drivers of Business Cycles Agustín H. Arias October 30th, 2014 Abstract This paper studies the role of sentiment shocks as a source of business cycle fluctuations. Considering a standard
More informationECON 4160: Econometrics-Modelling and Systems Estimation Lecture 7: Single equation models
ECON 4160: Econometrics-Modelling and Systems Estimation Lecture 7: Single equation models Ragnar Nymoen Department of Economics University of Oslo 25 September 2018 The reference to this lecture is: Chapter
More informationMacroeconometric modelling
Macroeconometric modelling 2 Background Gunnar Bårdsen CREATES 16-17. November 2009 Models with steady state We are interested in models with a steady state They need not be long-run growth models, but
More informationUnderstanding Regressions with Observations Collected at High Frequency over Long Span
Understanding Regressions with Observations Collected at High Frequency over Long Span Yoosoon Chang Department of Economics, Indiana University Joon Y. Park Department of Economics, Indiana University
More informationY t = log (employment t )
Advanced Macroeconomics, Christiano Econ 416 Homework #7 Due: November 21 1. Consider the linearized equilibrium conditions of the New Keynesian model, on the slide, The Equilibrium Conditions in the handout,
More informationChapter 2. Some basic tools. 2.1 Time series: Theory Stochastic processes
Chapter 2 Some basic tools 2.1 Time series: Theory 2.1.1 Stochastic processes A stochastic process is a sequence of random variables..., x 0, x 1, x 2,.... In this class, the subscript always means time.
More informationBayesian Computations for DSGE Models
Bayesian Computations for DSGE Models Frank Schorfheide University of Pennsylvania, PIER, CEPR, and NBER October 23, 2017 This Lecture is Based on Bayesian Estimation of DSGE Models Edward P. Herbst &
More informationGMM and SMM. 1. Hansen, L Large Sample Properties of Generalized Method of Moments Estimators, Econometrica, 50, p
GMM and SMM Some useful references: 1. Hansen, L. 1982. Large Sample Properties of Generalized Method of Moments Estimators, Econometrica, 50, p. 1029-54. 2. Lee, B.S. and B. Ingram. 1991 Simulation estimation
More informationCPSC 540: Machine Learning
CPSC 540: Machine Learning MCMC and Non-Parametric Bayes Mark Schmidt University of British Columbia Winter 2016 Admin I went through project proposals: Some of you got a message on Piazza. No news is
More informationParameter estimation and forecasting. Cristiano Porciani AIfA, Uni-Bonn
Parameter estimation and forecasting Cristiano Porciani AIfA, Uni-Bonn Questions? C. Porciani Estimation & forecasting 2 Temperature fluctuations Variance at multipole l (angle ~180o/l) C. Porciani Estimation
More informationTaylor Rules and Technology Shocks
Taylor Rules and Technology Shocks Eric R. Sims University of Notre Dame and NBER January 17, 2012 Abstract In a standard New Keynesian model, a Taylor-type interest rate rule moves the equilibrium real
More informationSequential Monte Carlo Methods (for DSGE Models)
Sequential Monte Carlo Methods (for DSGE Models) Frank Schorfheide University of Pennsylvania, PIER, CEPR, and NBER October 23, 2017 Some References Solution and Estimation of DSGE Models, with J. Fernandez-Villaverde
More informationReview. DS GA 1002 Statistical and Mathematical Models. Carlos Fernandez-Granda
Review DS GA 1002 Statistical and Mathematical Models http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall16 Carlos Fernandez-Granda Probability and statistics Probability: Framework for dealing with
More information