Back to square one: Identification issues in DSGE models

Fabio Canova, Luca Sala

Marc Goñi – 19 th April

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

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)

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Outline

1. Generics of Identification

2. Population Identification

Christiano et al (2005), Smets and Wouters (2003)

3. Sample Identification

4. Dealing with Identification

Introduction Generics of id Population id Sample id Diagnosting id Conclusion

Identification

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

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

Introduction Generics of identification Population id Sample id Diagnosting id Conclusion

Source of the problems

1. Location of the true parameters

2. Choice of the objective function

Consider the optimality conditions of a DSGE model

The unique stable RE solution is

In State Space representation

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

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

Introduction Generics of identification Population id Sample id Diagnosting id Conclusion

A simple example

• Solution mapping

Parameter a1 , a3 , a5 disappears from the solution

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

Introduction Generics of identification Population id Sample id Diagnosting id Conclusion

A simple example

• Objective mapping

Choosing a MD objective function

Weak identification problems

Introduction Generics of identification Population id Sample id Diagnosting id Conclusion

Solutions

1. 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

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

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.

Introduction Generics of identification Population id Sample id Diagnosting id Conclusion

Introduction Generics of id Population identification Sample id Diagnosting id Conclusion

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

Population Idenification

Model

Introduction Generics of id Population identification Sample id Diagnosting id Conclusion

Introduction Generics of id Population identification Sample id Diagnosting id Conclusion

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

Preliminary identification evidence

Introduction Generics of id Population identification Sample id Diagnosting id Conclusion

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 100000 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

Size of Identification Problems

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

Introduction Generics of id Population identification Sample id Diagnosting id Conclusion

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

Remark

Introduction Generics of id Population id Sample identification Diagnosting id Conclusion

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

Sample problems

Introduction Generics of id Population id Sample identification Diagnosting id Conclusion

-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

Results

Introduction Generics of id Population id Sample identification Diagnosting id Conclusion

-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

Results

Introduction Generics of id Population id Sample identification Diagnosting id Conclusion

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

Alternatives

Introduction Generics of id Population id Sample id Diagnosting identification Conclusion

Consider the following mapping of structural parameters to sample objective function

1.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

Theory

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

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)

Introduction Generics of id Population id Sample id Diagnosting identification Conclusion

Methods to test for the rank and the size of the eigenvalues

1.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

2.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

Practical issues

Introduction Generics of id Population id Sample id Diagnosting identification Conclusion

3.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

Introduction Generics of id Population id Sample id Diagnosting identification Conclusion

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

An application

Introduction Generics of id Population id Sample id Diagnosting identification Conclusion

Model reespecification

-Reparametrize commonly used functions

-Choice of different pivotal points around which log linearize

-Use Higher order approximations

Solution

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

Conclusion