1. Back to square one: Identification issues in DSGE models
Fabio Canova, Luca Sala
Marc Goñi – 19 th April

2. Informational Cascades
Introduction Generics of id Population id Sample id Diagnosting id Conclusion
31/03/2017
Motivation
In the last years DSGE models have greatly evolved with the objective of better forecasting and deriving policy implications
Development in the specification of DSGE
Comparing models with the data through the ability to match conditional dynamics in response to structural shocks
However, this inference depends crucially on identification, which has been partially ignored
This paper investigates identificability issues in DSGE models in the class of minimum distance estimators

3. Introduction Generics of id Population id Sample id Diagnosting id Conclusion
Literature Review
Choi and Phillips (1992), Stock and Wright (2000), Rosen (2006), Kleibergen and Mavroidis (2008) Beyer and Farmer (2004), Moon and Schorfheide (2007) Christiano et al (2006), Fernandez-Villaverde et al (2007), Chari et al (2008)

4. Introduction Generics of id Population id Sample id Diagnosting id Conclusion
Outline
1. Generics of Identification 2. Population Identification Christiano et al (2005), Smets and Wouters (2003) 3. Sample Identification 4. Dealing with Identification

5. Introduction Generics of identification Population id Sample id Diagnosting id Conclusion
Identification is the ability to draw inference about the parameters of the model from the data
Identification requires the objective function to have
A unique extreme at the true parameter
Sufficient curvature in all the relevant dimension
The mapping from structural parameters to the objective function is usually non-linear or doesn’t have a closed form solution

6. Problems Under Identification
Introduction Generics of identification Population id Sample id Diagnosting id Conclusion
Problems
Under Identification
If the objective function is independent of certain structural
parameters
Partial Identification
If the parameters enter the objective function only
proportionally and they cannot be separately analyzed
Weak Identification
If the objective function does not have enough curvature in
all the relevant dimensions
This problems can induce Observational Equivalence, i.e, that different models with different theoretical implications become indistinguishible

7. Source of the problems Location of the true parameters
Introduction Generics of identification Population id Sample id Diagnosting id Conclusion
Source of the problems
Location of the true parameters
Choice of the objective function
Consider the optimality conditions of a DSGE model
The unique stable RE solution is
In State Space representation

8. Introduction Generics of identification Population id Sample id Diagnosting id Conclusion
The Likelihood function provides a natural upper bound to identification of the information available in the data. Using the Kalman filter and assuming normality for the errors And, thus, an identification upper bound is Which, compared to a minimum distance objective function

9. 3. Mapping of Structural parameters and sample objective function
Introduction Generics of identification Population id Sample id Diagnosting id Conclusion
3. Mapping of Structural parameters and sample objective function
Solution mapping
Linking the parameters and the coefs. of the solution
Parameters disappear from the solution, do not have independent variability
Moment mapping
Links the coefs of the solution with the function of interest
Selection of a particular Impulse Response may poorly identify coeficients
Objective function mapping
Links the function of interest with the pop. objective function
Function may not have a unique minimum or may not display enough curvature
Data mapping
Links the pop. Objective function with the sample objective function
Estimated VAR responses may not reflect population ones

10. A simple example Solution mapping
Introduction Generics of identification Population id Sample id Diagnosting id Conclusion
A simple example
Solution mapping
Parameter a1 , a3 , a5 disappears from the solution

11. A simple example Moment mapping The impulse response takes the form of
Introduction Generics of identification Population id Sample id Diagnosting id Conclusion
A simple example
Moment mapping
The impulse response takes the form of
Even if we pick responses to all shocks, some parameters remain underidentified
a2a4 respond jointly to e3

12. A simple example Objective mapping Choosing a MD objective function
Introduction Generics of identification Population id Sample id Diagnosting id Conclusion
A simple example
Objective mapping
Choosing a MD objective function
Weak identification problems

13. Solutions Calibration
Introduction Generics of identification Population id Sample id Diagnosting id Conclusion
Solutions
Calibration
Calibrate some of the parameters based on micro-evidence…
Problem: If the calibrated parameters are partially identified small calibration differences may shift the estimates

14. Introduction Generics of identification Population id Sample id Diagnosting id Conclusion
2. Bayesian Methods Estimate structural parameters with Bayesian techniques If the parameter space is not variation free, Id. Problems can be detected by setting a more diffuse prior and checking if the posterior becomes more diffuse However, this can be driven by restrictions Bayesian methods plus tight prior produce well behaved posteriors even when the objective function behaves poorly

15. Introduction Generics of identification Population id Sample id Diagnosting id Conclusion
3. Serially Correlated Disturbances Allowing for serially correlated disturbances maintains the forward looking coefficients in the solution However, separating internal and external propagation parameters might be difficult.

16. Population Idenification
Introduction Generics of id Population identification Sample id Diagnosting id Conclusion
Population Idenification
Even when the true model is known, identification problems may make inference not feasible
Consider a standard DSGE (Christiano et al, Dedola and Neri, Smets and Wouters). The analytical mapping between structural parameters and the objective function is no longer available
Instead examine the slope of the distance function in a neighborhood of the true parameter

17. Introduction Generics of id Population identification Sample id Diagnosting id Conclusion
Model

18. Preliminary identification evidence
Introduction Generics of id Population identification Sample id Diagnosting id Conclusion
Preliminary identification evidence
Let
and
For each parameter in compute the elasticity of the distance function to it by varying it while holding fixed the rest of the parameters to the true value.
Preliminary evidence for weak identification: Although distance functions have a unique minimum at the true parameter, variation within the neighborhood is small

20. Size of Identification Problems
Introduction Generics of id Population identification Sample id Diagnosting id Conclusion
Size of Identification Problems
Check how severe weak identification actually is and to test if it can yield observational equivalence.
1. Construct the distribution of the distance function
Pick randomly vector parameters from before and
compute the distance between its Impulse responses and 5
benchmark IR
True models: benchmark model with monetary shocks,
either p, w stickiness or indexation out and benchmark with
monetary and technology shocks
2. Pick draws in the 0.1 percentile of the distribution

22. Case 5: Same results when more shocks are added
Introduction Generics of id Population identification Sample id Diagnosting id Conclusion
Case 1: Large intervals make difficult to infer how important p, w stickiness and indexation are
Case 1-4: As intervals are similar, it is difficult to infer which friction matters (observational equivalence)
Case 5: Same results when more shocks are added
So, weak and partial identification problems are severe and may induce observational equivalence

23. Introduction Generics of id Population identification Sample id Diagnosting id Conclusion
Remark
These poorly identified DSGE model may be good for forecasting but not optimal for policy inference
In the presence of such a weak and partial identification problems, one needs to bring information external to the dynamics to be able to interpret the estimates

24. Introduction Generics of id Population id Sample identification Diagnosting id Conclusion
Sample problems
What are the effects of these population identification issues when the analysis has to be conducted with sample data, rather than populational.
Simulate 500 time series from the true model
Estimate a 6-variable VAR with 6 lags, identify monetary shocks and construct the data based Impulse Responses
Avoid non-invertibility and correctly identify shocks
Estimate theoretical parameters minimizing the distance with the VAR based Impulse Responses

25. Results Mean estimates do not depend on sample size
Introduction Generics of id Population id Sample identification Diagnosting id Conclusion
Results
Mean estimates do not depend on sample size
Standard errors and biases decrease with sample size, but
are large
Bimodal distributions of parameter estimates with peaks at
the boundaries
That is, standard asymptotic approximations seem not reliable

27. Introduction Generics of id Population id Sample identification Diagnosting id Conclusion
Results
Model based responses fall in the range of VAR-based responses confidence bands
Mean estimates are statistically and economically different from the true ones (observational equivalence)
That is, the technique of showing model responses in VAR-based confidence bands may lead to wrong inference in the presence of population identification problems

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Alternatives
Under certain conditions, asymptotic methods to compute estimates are robust to Identification problem (Stock and Wright, Kleibergen and Mavroeidis)
Rosen (2006) methodology yields similar results here

29. Introduction Generics of id Population id Sample id Diagnosting identification Conclusion
Theory
Consider the following mapping of structural parameters to sample objective function
First order Taylor expansion
To translate information from the function to the parameter we need to be invertible
Thus, if the rank is not full: under identification
If the eigenvalues are small: weak and partial identification

30. 2. Split the problem in two
Introduction Generics of id Population id Sample id Diagnosting identification Conclusion
2. Split the problem in two
First find the θ that minimizes the distance between data and model VAR parameters (solution mapping) and then find the reduced form parameters parameters that make the IR close
Check the rank and the eigenvalues of and

31. 3. Split the problem in three
Introduction Generics of id Population id Sample id Diagnosting identification Conclusion
3. Split the problem in three
When only one estimate is available, the problem can be splitted into a solution mapping, moment mapping and data mapping
Compute F and G by calibrating θ and then performing sensitivity analysis
Compare fixed and estimated parameter for identificability issues (as before)

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Practical issues
Methods to test for the rank and the size of the eigenvalues
Anderson (1984)
Estimates of eigenvalues have asymptotic normal distribution.
Therefore, test if the smallest eigenvalue is different from 0
We can normalize the test by using the ratio of the sum of the
smallest eigenvalues over the sum of all eigenvalues
Concentration Statistics
Measures the curvature of the objective function around θo
For large values of accept that the objective function
has an optimum at 0

33. Standard errors not useful
Introduction Generics of id Population id Sample id Diagnosting identification Conclusion
Standard errors not useful
Relatively small std. errors can coexist with identification
issues
Identification analysis should preceed estimation
Standard errors do not infere if the problem is model or
data based

34. An application Applying this methods to our example
Introduction Generics of id Population id Sample id Diagnosting identification Conclusion
An application
Applying this methods to our example
G’G matrix has one eigenvalue representing 99.9% of the trace
G’F’FG smallest 13 eigenvalues account for 0.001% of the trace
Thus, solution mapping (GG) is the source of the problem
Gs’Gs (mapping structural coefs with LOM coefs) has one
eigenvalue representing 99.9% of the trace
Thus, the source of the identification problem is the insensibility of the Law of Motion coeficients to the structural parameters

35. Solution Model reespecification Reparametrize commonly used functions
Introduction Generics of id Population id Sample id Diagnosting identification Conclusion
Solution
Model reespecification
Reparametrize commonly used functions
Choice of different pivotal points around which log linearize
Use Higher order approximations

36. Introduction Generics of id Population id Sample id Diagnosting identification Conclusion
Identification problems ignored for a long time, with consequences in policy recommendations
Tools given here to be applied before structural estimation
Montecarlo Methods may help
When choosing the objective function, choose the most informative
Bringing more data only helps if the problem is data based
Robust methods exists but only give you intervals