Post on 20-Dec-2015
transcript
Marco Del Negro, Frank Schorfheide, Frank Smets, and Raf Wouters (DSSW)
On the Fit of New-Keynesian Models
Discussion by:
Lawrence Christiano
Objective:
• Provide a scalar measure of the fit of a Dynamic, Stochastic, General Equilibrium Model (DSGE).
• Apply measure of fit to an empirically important example.
• Consider a vector autoregression (VAR):
• Least squares estimation:
1m yt
1 k xt
km
1m ut , Eutut
XX 1XY
1TY X
Y X
• DSGE model implication for VAR– DSGE model parameters – θ
• Hybrid model
XX 1TEXX, XY 1
TEXY, YY 1
TEYY
XX 1 XY , YY YX XX 1 XY
, XX T XX 1XY T XY
, ~analogously defined.
• Models:
• Marginal likelihood:
• Best fit:
• Finding:
• Conclusion: ‘Evidence of model misspecification’
DSGE model
small VAR model
small Hybrid Model
L Y, parameters
LY|parameters, Pparametersdparameters
maxsmall
L Y,
0.75,1.5
Questions• Although the marginal likelihood is a sensible
way to assess fit in principle…
– Compromises are required for tractability
– How compelling are the assumptions about likelihood function, priors…
• How severe is the evidence against the model when
– Even if DSGE model were true, unrestricted VAR might fit better in a small sample
• Is the Hybrid model useful?
DSSW Assume the Likelihood of the Data is Gaussian
• Fit a four-lag, 7 variable VAR using US data, 1955Q4-2006Q1.
• Compute skewness and kurtosis statistics for each of 7 VAR disturbances
• There is strong evidence against normality assumption
Disturbance kurtosis skewness
statistic, s probs s| normal statistic, s probs s| normal
logCt/Yt 1.40 0.18% -0.17 84.64%
logI t/Yt 1.59 0.12% 0.07 33.04%
PCE inflationt 0.79 2.00% 0.41 0.68%
logYt/lt logWt/Pt 1.04 0.66% 0.39 1.22%
R t 11.01 0.00% 1.62 0.00%
logl t 1.27 0.40% 0.11 26.46%
GDP growtht 1.85 0.02% -0.02 55.00%
Prior on VAR Parameters• Gaussian Likelihood is a function only of
VAR parameters:
• How do DSGE model parameters enter?– They control the priors on VAR parameters:
LY| ,
P , | ,
Prior on VAR Parameters…
DensityIn case DSGE modelIs true
Density in caseDSGE model isfalse
Prior on VAR Parameters…• Do DSSW priors fairly capture notion that DSGE model might be
false?
• Another possibility:
– If preferred DSGE model is false, some other DSGE model is true.
– Must specify a prior over alternative DSGE models. Induced priors over VAR parameters likely to be different from Normal/Wishart assumption of DSSW
– Problem: Most likely, could not even describe alternative DSGE models, much less assign priors to them! Presumably, this would lead us even further away from DSSW.
• These concerns about the DSSW priors would be mere quibbles if their approach were the only one to assessing model fit.
– But, there are other approaches– More on this later…
And DSGE Model Fit• Priors for DSGE:
• Marginal likelihood:
P
L Y,
, LY| , P , | , P d , d
L Y,
Huge integration problem, made trivial by:
Normal assumption on LY| , Normal/Wishart assumptions P , | ,
,
LY| , P , | , d , P d
Questions
• How severe is the evidence against the model when
• To answer this, studied multiple artificial data samples generated from a simple DSGE model
Simple (Long-Plosser) Model• Setup:
• Experiment:
E0 t 0
0.99t logCt expxt 1 l t
1 ,
Ct Kt 1 Kt1/3expzt l t 2/3 Yt,
xt, zt ~iid mean zero, variance x2, z2
, x2, z2 , true 1,0.022, 0. 022
Uniform priors: ~0,2, x2, z2~0.0001,0.0007
200 observations on: log Ytl t
, logKt 1 .
Results• Doing DSSW calculations on artificial data
• Implications
– DSSW evidence of misspecification occurs 1/3 of the time, even though DSGE model is true.
– Misspecification of likelihood seems not to matter.
. 33, 0.5, 0.75, 1, 1.25, 1.5, 2, 5
Prob 5 E | 5
Normal Disturbances 33.4 1.43
Kurtotic Disturbances 34.9 1.33
Note:Monte Carlo standard error on prob, 1.5
Interpretation of Results
• Why do DSSW find evidence against DSGE model, even when the model is true?
• One answer: In finite samples, unrestricted VAR often fits substantially better than true VAR implied by DSGE.
Interpretation of Results…• Interior typically occur in samples
where VAR fits substantially better than true model
E LR test of DSGE model|
0.33 28.8
0.50 17.2
0.75 15.3
1.00 13.23
1.25 12.2
2.00 10.6
5.00 7.5
Conclusion• DSSW rule:
– ‘We have evidence of misspecification whenever the peak of the marginal likelihood function is attained at a finite value.’
– with high probability, this rule leads to overly pessimistic assesment of models.
• What can we learn from about fit of DSGE models?
– Requires doing simulation experiments in more elaborate models.
– Poses significant computational challenges.
Conclusion….• Marginal likelihood provides a sensible measure
of fit in principle, however
– Assumptions required for tractability render marginal likelihood hard to interpret.
– The hybrid model is selected by marginal likelihood criterion – why should it be taken seriously?
• A less sophisticated, but more transparent and easy to interpret measure of fit:
– Out of Sample Root Mean Square Errors.
Most likelyModel,
P |M1 Other model
P|M2
Prior on model 2: P(M2 )
Prior on model 1: P(M1 )
P , P ,
marginal prior over VAR parameters: P ,
Prior on VAR Parameters…• The alternative priors would presumably be very different (e.g.,
multimodal).
• In practice, we don’t know what other model might be true (this is a basic fact about research!)
– How would we even think of priors in this case?– Robust control?
• Placing priors on VAR parameters conditional on model being false seems very difficult.
– Is the DSSW approach the right one?
• If DSSW approach were the only way to assess model fit, concerns about plausibility of prior would have less force
– But, there are other approaches– More on this later…