Dynare & Bayesian Estimation
Wouter J. Den HaanLondon School of Economics
c© 2011 by Wouter J. Den Haan
August 19, 2011
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Overview of the program
• Calculate likelihood, L(YT|Ψ)• Calculate posterior, P(Ψ|YT) ∝ L(YT|Ψ)P(Ψ)• Calculate mode• Calculate preliminary info about posterior
• using quick and dirty assumption of normality
• Use MCMC to• trace the shape of P(Ψ|YT)• calculate things like confidence intervals
• Plot graphs
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Calculate Likelihood as function of psi
• Given Ψ, get first-order approximation of the model• Write the system in state-space notation• Use the Kalman filter to back out
yt = yt − E[yt|Yt−1, x1
]and Σy,t
• yt vector with ny observed values• E
[yt|Yt−1, x1
]prediction according to Kalman filter
• yt : prediction error• yt : function of all the shocks in the model
• Linearity =⇒• yt ∼ N(0, Σy,t)• likelihood of sequence can be calculated
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Calculate posterior & mode
P(Ψ|YT) ∝ L(YT|Ψ)P(Ψ)
• P(Ψ|YT) is a complex function• But its value can be calculated easily for given Ψ• =⇒ value of Ψ that attains the max can be calculated using anoptimization routine(in practice, max of P(Ψ|YT) much easier to find than max ofL(YT|Ψ) because P (Ψ) makes problem better behaved)
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Information about posterior using MCMC
• But we want to calculate objects like
E [g(Ψ)] =∫
g(Ψ)P(Ψ|YT)dΨ∫P(Ψ|YT)dΨ
• Examples, mean, standard errors, confidence intervalsthese are all integrals
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Idea behing MCMC
• Problem: We cannot draw numbers directly from P(Ψ|YT)
• Solution: Generate a sequence for ψ such that its distributionis equal to P(Ψ|YT)
Implementing MCMC is not trivial!!!
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Part I:initialization
// dynareestimate.mod
var lc, lk, lz, ly;varexo e;parameters beta, rho, alpha, nu, delta;
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Part II:set values for parameters that arenot estimated
alpha = 0.36;rho = 0.95;beta = 0.99;nu = 1;delta = 0.025;
• Values will be ignored during estimation• So only needed if you first give Dynare commands that requireparameter values
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Part III: model
model;
exp(-nu*lc)=beta*(exp(-nu*lc(+1)))*(exp(lz(+1))*alpha*exp((alpha-1)*lk)+1-delta);
exp(lc)+exp(lk)=exp(lz+alpha*lk(-1))+(1-delta)*exp(lk(-1));
lz = rho*lz(-1)+e;
end;
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Part IV (optional): analyze model solution& properties for specified parameter values
steady;Stoch_simul(order=1,nocorr,nomoments,IRF=12);
• This requires having given numerical values for all parameters
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Part V: define priors
estimated_params;alpha, inv_gamma_pdf, 0.007, inf;end;
• more alternatives given below
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Part V: initialize estimation
Tell dynare what the observables are
varobs lk;
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Part V: initialize estimation
Give initial values for steady state
initval;lc = -1.02;lk = -1.61;lz = 0;end;
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Calculate steady state
Steady state must be calculated for many different values of Ψ !!!
• Linearize the system yourself• then easy to solve for steady state
• Give the exact solution of steady state as initial values• Provide program to calculate the steady state yourself
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Calculate steady state yourself
• If your *.mod file is called xxx.mod then write a filexxx_steadystate.m
• dynare checks whether a file with this name exists and will use it• sequence of output should correspond with sequence given invar list
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Calculate steady state yourselffunction [ys,check] = modela_steadystate(ys,exe)global M_
beta = M_.params(1);rho = M_.params(2);alpha = M_.params(3);nu = M_.params(4);delta = M_.params(5);sig = M_.params(6);check = 0;
z = 1;k = ((1-beta*(1-delta))/(alpha*beta))^(1/(alpha-1));c = k^alpha-delta*k;
ys =[ c; k; z ];
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
How to create this file?
In your *.mod file include:
steady_state_model;z = 1;k = ((1-beta*(1-delta))/(alpha*beta))^(1/(alpha-1));c = k^alpha-delta*k;end;
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
When it has been created?
• Now that the file has been created you can do more things:• e.g. solve for some analytically and some numerically
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Part VI: Estimation
Actual estimation command with some of the possible options
estimation(datafile=kdata,mh_nblocks=5,mh_replic=10000,mh_jscale=3,mh_init_scale=12) lk;
• lk: (optional) name of the endogenous variables (e.g. if youwant to plot Bayesian IRFs)
• datafile: contains observables• kdata.mat or kdata.m or kdata.xls
• nobs: number of observations used (default all)• first_obs: first observation (default is first)
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
MCMC options
• mh_replic: number of observations in each MCMC sequence• mh_nblocks: number of MCMC sequences• mh_jscale: variance of the jumps in Ψ in MCMC chain
• a higher value of mh_jscale =⇒ bigger steps through thedomain of Ψ & lower acceptance ratio
• acceptance ratio should be around 0.234(according to some optimality results; see below)
• mh_init_scale: variance of initial draw• important to make sure that the different MCMC sequencesstart in different points
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Part VII: Using estimated model
Acutal estimation command with some of the possible options
shock_decomposition;
• plots graphs with the observables and the partexplained by which shock.
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Priors - format
estimated_params;parameter name, prior_shape, prior_mean,prior_standarddeviation[,prior 3rd par. value, prior 4th par. value];end
the part in [] only for some priors
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Priors - examples
alpha’s prior is Normal with mean mu and standard deviation sigma:
alpha , normal_pdf,mu,sigma;
alpha’s prior is uniform over [p3,p4]:
alpha , uniform_pdf, , , p3, p4;
Note the two spaces betweeen the commas
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Priors - innovation variances
• use "stderr e" as the parameter name for the innovationvariance.
• For example,• stderr e , uniform_pdf, , , p3, p4;
Note the two empty spaces betweeen the commas
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Priors - examples
name distribution & parameters rangenormal_pdf N(µ, σ) R
gamma_pdf G2(µ, σ, p3) [p3,+∞)beta_pdf B(µ, σ, p3, p4) [p3, p4]inv_gamma_pdf IG1(µ, σ) R+
uniform_pdf U(p3, p4) [p3, p4]
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Priors - gamma distribution
P(x− p3) = (x− p3)k−1 exp (− (x− p3) /θ)
Γ (k) θk
Γ (k) = (k− 1)! if k is an integer
Γ (k) =∫ ∞
0tk−1e−tdt o.w.
k : shape parameter
θ : scale parameter
µ = kθ
σ2 = kθ2
skewness = E (x− p3)3 /(E[(x− p3)
2])3/2
= 2/√
κ
mode : p3 + (k− 1)θ for k ≥ 1
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Priors - gamma distribution
• If k = 1 =⇒ mode at lower bound• If k = 1 =⇒ exponential• If k = degrees of freedom/2 en θ = 2 =⇒ Chi-squared• Gamma distribution is right-skewed
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Priors - gamma distribution (mean = 1)
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
k=1.2
k=3
k=4
k=1 k=5
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Priors - inverse gamma distribution-
• If X has a gamma(k, θ) =⇒1/X has an inverse gamma distribution(κ, 1/θ)
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Priors - inverse gamma distribution
P(x) = x−k−1 θk exp (−θ/x)Γ(k)
k : shape parameter
θ : scale parameter
µ = θ/ (k− 1) for k > 1
σ2 = θ2/((k− 1)2 (k− 2)
)for k > 2
skewness = 4√(k− 2)/ (k− 3) for k > 3
mode : θ/ (k+ 1)
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Priors - beta distribution on [0,1]
P(x) =xφ−1 (1− x)θ−1
B (φ, θ)
B(k, θ) =Γ (φ+ θ)
Γ (φ) Γ (θ)φ : shape parameter
θ : shape parameter
µ = φ/ (φ+ θ)
σ2 = φθ/((φ+ θ)2 (φ+ θ + 1)
)skewness =
2 (θ − φ)√
φ+ θ + 1(φ+ θ + 2)
√φθ
mode :φ− 1
φ+ θ − 2for φ > 1, θ > 1
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Priors - beta distribution
• If φ < 1, θ < 1 =⇒ U-shaped• If φ > 1, θ > 1 =⇒ unimodal• If φ < 1, θ ≥ 1 or φ = 1, θ > 1 =⇒ strictly decreasing
• If φ = 1, θ > 2 =⇒ strictly convex• If φ = 1, 1 < θ < 2 =⇒ strictly concave
• If φ > 1, θ ≤ 1 or φ = 1, θ < 1 =⇒ strictly increasing
• If θ = 1, φ > 2 =⇒ strictly convex• If θ = 1, 1 < φ < 2 =⇒ strictly concave
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Priors - beta distribution (mean = 2/5)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
k=0.35
k=1.5
k=0.8
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Brooks & Gelman 1989 statistics
• MCMC: should generate sequence as if drawn from P(Ψ|YT)
• Tough to check• Minimum requirement is that distribution is same
• for different parts of the same sequence• across sequences (if you have more than one)
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Brooks & Gelman 1989 statistics
• Ψij the ith draw (out of I) in the jth sequence (out of J)
• Ψ·j mean of jth sequence• Ψ·· mean across all available data
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Between variance
B =1
J− 1
J
∑j=1
(Ψ·j − Ψ··
)2
• B is an estimate of the variance of the mean (σ2/I)=⇒ B = BI is an estimate of the variance
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Within variance
W =1J
J
∑j=1
1I
I
∑t=1
(Ψtj − Ψ·j
)2
W =1J
J
∑j=1
1I− 1
I
∑t=1
(Ψtj − Ψ·j
)2
• W and W are estimates of the variance (averaged acrossstreams)
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
What do we need?
1 Between variance should go to zerolimI−→∞B −→ 0
2 Within variance should settle downlimI−→∞W −→ constant
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
What do we need
reported by Dynare
red line : Wblue line : VAR = W+ B
We need
1 red and blue line to get close
2 red line to settle down
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Univariate extensions
• The above can be done for any moment, not just the variance• Dynare reports three alternatives
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Multivariate extension
• For each moment of interest you can calculate the multivariateversion
• e.g. covariance matrix for the variance• these higher-dimensional objects have to be transformed intoscalar objects that can be plotted
• See: Brooks and Gilman 1989
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Acceptance rate
• The MCMC chain has to cover the whole state space• Roberts, Gelman, & Gilks (1997): optimal acceptance rate =0.234
• Great but · · ·• optimality is an asymptotic result (if dimension of Ψ −→ ∞)• optimality relies on assumption that elements of Ψ areindependent (or another assumption replacing this one).
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
What to watch while Dynare runs
Plots here from two examples
1 As good as it gets: estimate only 1 parameter
2 Estimate all parameters
• Both cases 5,000 observations• No misspecification of the model
• i.e., artificial data
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
What to watch while Dynare runs
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
50
100
150
200
250SE_e
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
What to watch while Dynare runs
• When Dynare gets to the MCMC part a window opens tellingyou
• in which MCMC run you are• which fraction has been completed• most importantly the acceptance rate
• The acceptance rate should be "around" 0.234• a relatively low acceptance rate makes it more likely that theMCMC travels through the whole domain of Ψ
• acceptance rate too high =⇒ increase mh_jscale
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Tables
• RESULTS FROM POSTERIOR MAXIMIZATION
• generated before MCMC part• most important is the mode, the other stuff is based onnormality assumptions which are typically not valid
• ESTIMATION RESULTS (based on MCMC)
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Graphs
• Prior• MCMC diagnostics (see below)• Prior & posterior densities• Shocks implied at the mode• Observable and corresponding implied value
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Output for first example
• Estimate only 1 parameter, standard deviation innovation• correctly specified (neoclassical growth) model• 5,000 observations
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Univariate MCMC diagnostics I
1000 2000 3000 4000 5000 6000 7000 8000 9000 100001.6
1.8
2x 10 4 SE_e (Interval)
1000 2000 3000 4000 5000 6000 7000 8000 9000 100004
6
8x 10 9 SE_e (m2)
1000 2000 3000 4000 5000 6000 7000 8000 9000 100004
6
8x 10 13 SE_e (m3)
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Multivariate MCMC diagnostics II
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
1.4
1.6Interval
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000.4
0.6
0.8m2
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
1
2m3
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Posterior densities
0.005 0.01 0.015 0.02 0.025 0.03 0.0350
1000
2000
3000
4000
5000
6000
SE_e
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Shocks
1 0 5 0 1 1 0 0 1 1 5 0 1 2 0 0 1 2 5 0 1 3 0 0 1 3 5 0 1 4 0 0 1 4 5 0 1 5 0 0 0 .0 2 5
0 .0 2
0 .0 1 5
0 .0 1
0 .0 0 5
0
0 .0 0 5
0 .0 1
0 .0 1 5
0 .0 2
0 .0 2 5e
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Observables and implied values
500 1000 1500 2000 2500 3000 3500 4000 4500 50003.56
3.58
3.6
3.62
3.64
3.66
3.68
3.7
3.72
3.74
3.76lk
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Output for second example
• Estimate all parameters• correctly specified (neoclassical growth) model• 5,000 observations
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Second example - MCMC
5000 100000
0.005
0.01delta (Interval)
5000 100000
1
2x 10 5delta (m2)
5000 100000
5x 10 8delta (m3)
5000 100000
0.005
0.01rho (Interval)
5000 100000
0.5
1x 10 5rho (m2)
5000 100000
2
4x 10 8rho (m3)
5000 100000
0.2
0.4nu (Interval)
5000 100000
0.005
0.01nu (m2)
5000 100000
0.5
1x 10 3nu (m3)
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Second example - Posteriors
0.01 0.02 0.030
500
SE_e
0.3 0.4 0.50
10
20
alpha
0.98 0.99 10
50
100
beta
0.02 0.04 0.060
100
delta
0.7 0.8 0.90
100
200rho
1 2 30
5
nu
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
How to make the MCMC pics look better?
Problem:
• Parameters not well identified, possibly because the dynamicsof the model are too simple; capital is not much more than ascaled up version of productivity
• More data doesn’t seem to help
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Are dynamics caused by model or shocks?
• What explains the data, the shocks or the model?• How much propagation does the model really have?• Two examples:
• Standard RBC• Christiano, Motto, Rostagno
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Propagation in standard RBC
Policy rule in DSGE model:
xt+1 = a0 +A1xt +A2shockst
• Propagation, i.e., economic theory is all in A1
• Exogenous stuff is in A2shockst
• Economics important =⇒
xt+1 = a0 + A2shockst
should give a bad fit
• no matter what values of a0 and a1 used
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Check importance of economics in yourmodel
• Let
arg mina0,a1
T
∑t=2
(xt+1 − a0 − A2shockst
)2
• plot {xt+1, xt+1}
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Example
• Propagation in standard growth model• Why would the endogenous variables not follow driving process
1 for 1?
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Output & current productivity shock
0 50 100 150 200 250 300 350 400 450 5000.24
0.22
0.2
0.18
0.16
0.14
0.12
0.1
0.08Actual & fitted output
actualfitted: no lags
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Hours & current productivity shock
0 50 100 150 200 250 300 350 400 450 5001.485
1.48
1.475
1.47
1.465
1.46
1.455
1.45
1.445
1.44Actual & fitted hours
actualfitted: no lags
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Consumption & current productivity shock
0 50 100 150 200 250 300 350 400 450 5000.3
0.28
0.26
0.24
0.22
0.2
0.18
0.16
0.14Actual & fitted consumption
actualfitted: no lags
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Investment & current productivity shock
0 50 100 150 200 250 300 350 400 450 5000.048
0.05
0.052
0.054
0.056
0.058
0.06Actual & fitted investment
actualfitted: no lags
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Capital & current productivity shock
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 01 . 9
1 . 9 2
1 . 9 4
1 . 9 6
1 . 9 8
2
2 . 0 2
2 . 0 4
2 . 0 6
2 . 0 8
2 . 1
2 n d o rd e r p e r t u rb a t i o n
5 t h o rd e r p ro j e c t i o n s
m e t h o d
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Adding 20 lagged values of the shock
0 50 100 150 200 250 300 350 400 450 5002.1
2.15
2.2
2.25Actual & fitted capital
actualfitted: no lagsfitted: 20 lags
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Adding 40 lagged values of shock
0 50 100 150 200 250 300 350 400 450 5002.1
2.15
2.2
2.25Actual & fitted capital
actualfitted: no lagsfitted: 20 lagsfitted: 40 lags
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
R2
Only current +20 lags +40 lagsOutput 0.958 0.993 0.999Hours 0.825 0.971 0.994Consumption 0.947 0.991 0.998Investment 0.966 0.994 0.999Capital 0.288 0.880 0.976
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Second example
• Christiano, Motto, Rostagno:• "Financial Factors in Economic Fluctuations"
• Quite complex model to model interaction between financialintermediation and real activity
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Using current shocks & 1 & 2 lags
0 10 20 30 40 50 60 70 80 90 100200
150
100
50
0
50
100
150
200CONSUMPTION
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Using current shocks & 1 & 2 lags
0 10 20 30 40 50 60 70 80 90 1004000
3000
2000
1000
0
1000
2000
3000
4000CREDIT
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Using current shocks & 1 & 2 lags
0 10 20 30 40 50 60 70 80 90 1000.02
0.015
0.01
0.005
0
0.005
0.01
0.015DP
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Using current shocks & 1 & 2 lags
0 10 20 30 40 50 60 70 80 90 1000.015
0.01
0.005
0
0.005
0.01
Dpip
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Using current shocks & 1 & 2 lags
0 10 20 30 40 50 60 70 80 90 1000.6
0.5
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4OIL PRICE
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Using current shocks & 1 & 2 lags
0 10 20 30 40 50 60 70 80 90 100400
300
200
100
0
100
200
300
400INVESTMENT
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Using current shocks & 1 & 2 lags
0 10 20 30 40 50 60 70 80 90 1001000
500
0
500
1000
1500M1
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Using current shocks & 1 & 2 lags
0 10 20 30 40 50 60 70 80 90 1002500
2000
1500
1000
500
0
500
1000
1500
2000M3
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Using current shocks & 1 & 2 lags
0 10 20 30 40 50 60 70 80 90 1000.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
1.2RISK PREMIUM
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Using current shocks & 1 & 2 lags
0 10 20 30 40 50 60 70 80 90 100100
50
0
50
100
150RESERVES
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Using current shocks & 1 & 2 lags
0 10 20 30 40 50 60 70 80 90 1001.5
1
0.5
0
0.5
1
1.5
2
2.5x 104 STOCKS
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Using current shocks & 1 & 2 lags
0 10 20 30 40 50 60 70 80 90 1000.15
0.1
0.05
0
0.05
0.1WAGE RATE
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Using current shocks & 1 & 2 lags
0 10 20 30 40 50 60 70 80 90 100500
400
300
200
100
0
100
200
300
400GDP
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Using current shocks & 1 & 2 lags
0 10 20 30 40 50 60 70 80 90 10050
40
30
20
10
0
10
20
30
40HOURS
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Using current shocks & 1 & 2 lags
0 10 20 30 40 50 60 70 80 90 1003
2
1
0
1
2
3TERM PREMIUM
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
R2
Only current +1 lags +2 lagsConsumption 0.87 0.95 0.97Credit 0.95 0.97 0.98Inflation 0.51 0.81 0.90Inflation_inv 0.63 1.0 1.0Oil price 0.21 1.0 1.0Investment 0.89 0.96 0.99risk premium 0.86 0.93 0.98
hours in RBC 0.825 0.971 0.994
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
R2
Only current +1 lags +2 lagsM1 0.98 1.00 1.00M3 0.96 0.99 0.99reserves 0.99 1.0 1.0stock market 0.93 0.97 0.98wage rate 0.94 0.98 0.99GDP 0.83 0.92 0.97hours 0.94 0.98 0.99term premium 0.80 0.94 0.98
hours in RBC 0.825 0.971 0.994
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Shocks versus the model
• Is this bad?• Maybe not, but perceived wisdom– and the language in thepaper– suggests that propagation is very important
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Shocks versus the modelExplaining US vs EA
DSGE model:
zust+1 − zea
t+1 = a0 + A2(shocksust − shocksea
t )
• !!! Use the same model for US and EA• Only differences are the shocks?
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
R2
Only current +1 lags +2 lagsConsumption 0.72 0.88 0.94Credit 0.83 0.90 0.94Inflation 0.53 0.66 0.76Inflation_inv 0.31 1.0 1.0Oil price 0.23 1.0 1.0Investment 0.88 0.94 0.96risk premium 0.86 0.92 0.95
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
R2
Only current +1 lags +2 lagsM1 0.96 0.98 0.99M3 0.75 0.87 0.92stock market 0.75 0.87 0.92wage rate 0.80 0.87 0.92GDP 0.83 0.93 0.96term premium 0.93 0.96 0.98
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Very tricky issue
• Data have trends• Methodology works with stationary data
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Simple solution
• Put stochastic trend in model• Use first differences of model variables
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Disadvantages of the simple solution
• Info about level not used in estimation• e.g., consumption ≈ 2/3 output• =⇒ less parameters are identified
• ∆-filter emphasizes high frequency• measurement error shock could absorb this
• Not obvious how to put the right trend in model• Not obvious data are consistent with balanced growth
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Alternative I
Detrend data using
yt = a0 + a1t+ a2t2 + uy
Advantage
• Each observable can have its own trend
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Alternative II
Explicitly model trend as part of estimation problem
yobst = ytrendt + ymodelt
ytrendt = µ+ ytrendt−1 + ey,t
ymodelt is cyclical component determined by usual linearized equations
Advantage
• You can be very flexible in writing process for trend• Different variables can have different trends
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Alternative II
Same but written differently
y∗obst = ∆ytrendt + ∆ymodelt
∆ytrendt = µ+ ey,t
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Alternative III ??
Detrend data using HP or Band-Pass filter
yobs-filteredt = B(L)yobst
Problem
• B(L) is a two-sided filter =⇒ econometrically suspicious• yobs-filteredt has different properties than model data =⇒• apply same filter to model data
ymodel-filteredt = ymodelt
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Estimating misspecified models
• shocks versus observables• would wedges work?
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
Alternatives to Bayesian estimation
• Maximum likelihood• Calibration• GMM• SMM & indirect inference
Overview Basics MCMC Generated output Shocks versus model Trends The big issues
References
• of course: www.dynare.org• Brooks, S.P. and A. Gelman, 1998, General methods for monitoringconvergence of iterative simulations, Journal of Computational andGraphical Statistics.
• Griffoli, T.M., Dynare user guide• Roberts, G.O., and A. Gelman, and W.R. Gilks, Weak convergenceand optimal scaling of random walk metropolis algorithms, TheAnnals of Applied Probability
• the "0.234" paper
• Roberts, G.O., and J.S. Rosenthal, 2004, General state spaceMarkov chains and MCMC algorithms, Probability Surveys.
• more advanced articles describing formal properties
Top Related