This paper shows how to identify the structural shocks of a Vector Autoregression (VAR) while simultaneously estimating a dynamic stochastic general equilibrium (DSGE) model that is not assumed to replicate the data-generating process. It proposes a framework for estimating the parameters of the VAR model and the DSGE model jointly: the VAR model is identified by sign restrictions derived from the DSGE model; the DSGE model is estimated by matching the corresponding impulse response functions.

The paper above goes further. Instead of estimating one model using restrictions from the other, it estimates both the VAR and the DSGE models jointly. Is this the way to end this debate? Does this imply that we get better estimates?

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6 Responses to Matching Theory and Data: Bayesian Vector Autoregression and Dynamic Stochastic General Equilibrium Models

I value the recent trend of DSGE models towards estimation, and especially Bayesian estimation. It is a nice compromise of estimation and calibration. However, I fail to see the point of estimating jointly the DSGE model and the VAR. A VAR describes the data and should do this independently of a model, except for necessary long-run restrictions. The latter are arguably model dependent, but they are only sign restrictions.

My point really is that the argument can become circular. The (mentioned) literature has tested one class of models using results from the other. If both models are estimated jointly, what are we testing?

One thing that this paper shows is that this idea that is making rounds in some circles (and the press) that DSGE models are only calibrated is false. And the estimation here is by far not the first one to be done. People like Krugman and DeLong should really bring their macroeconomics up to level.

First, I want to commend you on creating this blog. There is much too little discussion on current research in the Economics blogging community. In particular, within the criticism of the DGE methodology, there is an obsolete idea that these models are poorly calibrated and thus of little value. As a commenter above rightfully states, the featured paper shows that one can successfully estimate such a model, and it is by far not the first one to do so. There is a strong traditional, especially in Europe, to estimate DGE models.

That said, I am still not convinced that estimating DGE models is the right thing to do in general. These models are microfounded, and some discipline is imposed on them that reduced forms models do not have. As a consequence, they are more likely to be rejected. Not because they are bad, but because of the statistically more stringent test. And remember that models are abstractions, thus we cannot take everything into account. This would both be difficult and actually make the statistical test worse, as this imposes more restrictions in a microfounded world. In a reduced form world, however, piling on variables, whether they make sense on not, will improve the fit no matter what.

Also, I do not see the point of estimating jointly the VAR and the DSGE model. In fact, I do not see the point of estimating a VAR in the first place. The most one can get out of a VAR is a description of the dynamics of the data, and a poor one at that, as it changes significantly with the slightest changes in specification.

I would like to give my opinion and respond to the Economic Logician. Although the paper could do a better job in actually showing this, my hunch is that the procedure put forward in this paper is way better than the standard way to estimate models (either max. likelihood or Bayesian methods) EXACTLY because as Ecnomic Logician makes clear that these models are misspecified and are likely to be rejected. I have seen several cases where maximizing the likelihood (or doing the Bayesian equivalent) leads to parameter estimates that generate odd predictions for variables not included as observables (e.g. an implied consumption variance that is way too high). There is no way of telling what aspects of the data make the estimation procedure choose what kind of parameter values. The procedure of Kriwoluzky, for sure, makes this mapping much more understandable. But I also strongly suspect that by looking at the IRFs of included shocks and dealing with the singularity (due to having more observables than shocks) by putting an extra error between model and empirical IRFs this paper introduces much less bias due to misspecification then other models (basically because it doesn’t require having to specify a large set of shocks just to overcome the singularity).

First, I want to thank the editor of the blog for posting my paper.

As pointed out by Wouter den Haan, the paper is not perfect and can do a better job in several dimensions. I am therefore very thankful for the comments and suggestions. In the following I would like to reply to some the comments above:

@Economic Logician

I agree with you that one should be very careful when bringing a DSGE model to the data. I suggest estimating the DSGE model by matching moments of the data (impulse response functions of the VAR model). I.e. the DSGE model is assumed to be a good description of those impulse response functions. This has the advantage that it can be misspecified in several other dimensions. In fact, as mentioned by you and Wouter den Haan it often is.

VAR model comment: As you put it, the most we can get out of VAR models are the dynamics – and this is exactly what I need in the suggested methodology. Also, as pointed out correctly, their extraction is not unique. Here, the identification and therefore the probability distribution of the dynamics depends on restrictions derived from the DSGE model. Since the restrictions depend in turn on the parameterization of the DSGE model, which is also estimated, it is necessary to estimate both distributions jointly.

@ JohnC.

The paper aims at estimating the joint posterior distribution of the DSGE model and the impulse response functions (!) of the VAR model since both depend on each other. While the VAR model describes the data, its impulse response functions cannot be estimated, but depend on additional assumptions. In the paper I suggest to derive these assumptions from a DSGE. This has the advantage that different identifying assumptions can be build into the DSGE model. Since the DSGE model is estimated it is even possible to discriminate between the different assumptions (as shown in one of my other papers “Pre-announcement and Timing: The Effects of a Government Expenditure Shock”).

I agree, from the abstract it appears as if the argument would go in circles. It is just the usual Gibbs sampling algorithm to estimate a joint distribution. In order to sample from p(x,y), you sample from p(x|y) and p(y|x).

Moreover, the restrictions derived from the DSGE model are sign restrictions. Therefore, they allow for a very wide range of possible impulse response functions of the VAR model. Also, not all variables are restricted; some variables (very often the variables of interest) are left unrestricted. Thus, the identified impulse response function of the VAR model can be quite different from the impulse response function of the DSGE model, which was used to derive the sign restrictions.

Furthermore, in order to discriminate between pairs of parameters vectors of the DSGE model and the VAR model, there is a Metropolis step between the Gibbs sampling steps, i.e. I compare the probability and keep those with higher probability. That way, the estimated parameter vectors of the DSGE model and their implied identification of the VAR model are narrower then the identification implied by the prior distribution of the DSGE model.

Concerning testing: The paper is more about answering a question like “What are the effects of …?”. It aims at providing evidence, where the DSGE model and the VAR model coincide.

I will likely always remain a sceptic of DSGE “estimation”, although it is something that needs to be done. I disagree strongly with any suggestion that estimating an economic model is not a valuable exercise, as is being suggested in the comments here. If DSGE models are ever to have meaning for policy, and in particular macro policy, they have to be estimated, full stop (or period). That they will be rejected is a reason to carry on improving these models, not for rejecting empirical validation per se.

My concerns about the estimation is that little attention appears to be paid to the data, and to the problems it may cause when modelling. There is not a single plot of any of the data series being modelled, nor any reference to their stationarity, or lack of it: By that I mean difference stationarity, or absence of structural breaks. It is well documented the adverse affect of such data ailments, hence my concern.

On top of that, there is no reporting of the actual model estimated, and no attempt to check whether the model is an accurate representation of the data: Whether the model is a good fit. This is essential if any trust is to be put in the coefficients pre-identification, let alone afterwards.

I find the recent trend towards assessing econometric models by their impulse responses deeply disturbing. Not least the identification required is untested, but importantly, as Luca Sala and Fabio Canova have shown, impulse responses are very sensitive indeed to changes in identification.

Put simply, there is no justification for departing from traditional methods of reported econometric results, other than that the author of the paper has something to hide.

There is little doubt that estimating a DSGE model will prove a difficult task given non-stationarity and given structural breaks. And the reported paper is a step forward. However, it is still a long way short of what is required if DSGE “estimation” is to establish a decent reputation for itself, a reputation that would allow the speech marks to be removed from the word estimation.

Let me the first to throw a stone.

I value the recent trend of DSGE models towards estimation, and especially Bayesian estimation. It is a nice compromise of estimation and calibration. However, I fail to see the point of estimating jointly the DSGE model and the VAR. A VAR describes the data and should do this independently of a model, except for necessary long-run restrictions. The latter are arguably model dependent, but they are only sign restrictions.

My point really is that the argument can become circular. The (mentioned) literature has tested one class of models using results from the other. If both models are estimated jointly, what are we testing?

One thing that this paper shows is that this idea that is making rounds in some circles (and the press) that DSGE models are only calibrated is false. And the estimation here is by far not the first one to be done. People like Krugman and DeLong should really bring their macroeconomics up to level.

First, I want to commend you on creating this blog. There is much too little discussion on current research in the Economics blogging community. In particular, within the criticism of the DGE methodology, there is an obsolete idea that these models are poorly calibrated and thus of little value. As a commenter above rightfully states, the featured paper shows that one can successfully estimate such a model, and it is by far not the first one to do so. There is a strong traditional, especially in Europe, to estimate DGE models.

That said, I am still not convinced that estimating DGE models is the right thing to do in general. These models are microfounded, and some discipline is imposed on them that reduced forms models do not have. As a consequence, they are more likely to be rejected. Not because they are bad, but because of the statistically more stringent test. And remember that models are abstractions, thus we cannot take everything into account. This would both be difficult and actually make the statistical test worse, as this imposes more restrictions in a microfounded world. In a reduced form world, however, piling on variables, whether they make sense on not, will improve the fit no matter what.

Also, I do not see the point of estimating jointly the VAR and the DSGE model. In fact, I do not see the point of estimating a VAR in the first place. The most one can get out of a VAR is a description of the dynamics of the data, and a poor one at that, as it changes significantly with the slightest changes in specification.

I would like to give my opinion and respond to the Economic Logician. Although the paper could do a better job in actually showing this, my hunch is that the procedure put forward in this paper is way better than the standard way to estimate models (either max. likelihood or Bayesian methods) EXACTLY because as Ecnomic Logician makes clear that these models are misspecified and are likely to be rejected. I have seen several cases where maximizing the likelihood (or doing the Bayesian equivalent) leads to parameter estimates that generate odd predictions for variables not included as observables (e.g. an implied consumption variance that is way too high). There is no way of telling what aspects of the data make the estimation procedure choose what kind of parameter values. The procedure of Kriwoluzky, for sure, makes this mapping much more understandable. But I also strongly suspect that by looking at the IRFs of included shocks and dealing with the singularity (due to having more observables than shocks) by putting an extra error between model and empirical IRFs this paper introduces much less bias due to misspecification then other models (basically because it doesn’t require having to specify a large set of shocks just to overcome the singularity).

First, I want to thank the editor of the blog for posting my paper.

As pointed out by Wouter den Haan, the paper is not perfect and can do a better job in several dimensions. I am therefore very thankful for the comments and suggestions. In the following I would like to reply to some the comments above:

@Economic Logician

I agree with you that one should be very careful when bringing a DSGE model to the data. I suggest estimating the DSGE model by matching moments of the data (impulse response functions of the VAR model). I.e. the DSGE model is assumed to be a good description of those impulse response functions. This has the advantage that it can be misspecified in several other dimensions. In fact, as mentioned by you and Wouter den Haan it often is.

VAR model comment: As you put it, the most we can get out of VAR models are the dynamics – and this is exactly what I need in the suggested methodology. Also, as pointed out correctly, their extraction is not unique. Here, the identification and therefore the probability distribution of the dynamics depends on restrictions derived from the DSGE model. Since the restrictions depend in turn on the parameterization of the DSGE model, which is also estimated, it is necessary to estimate both distributions jointly.

@ JohnC.

The paper aims at estimating the joint posterior distribution of the DSGE model and the impulse response functions (!) of the VAR model since both depend on each other. While the VAR model describes the data, its impulse response functions cannot be estimated, but depend on additional assumptions. In the paper I suggest to derive these assumptions from a DSGE. This has the advantage that different identifying assumptions can be build into the DSGE model. Since the DSGE model is estimated it is even possible to discriminate between the different assumptions (as shown in one of my other papers “Pre-announcement and Timing: The Effects of a Government Expenditure Shock”).

I agree, from the abstract it appears as if the argument would go in circles. It is just the usual Gibbs sampling algorithm to estimate a joint distribution. In order to sample from p(x,y), you sample from p(x|y) and p(y|x).

Moreover, the restrictions derived from the DSGE model are sign restrictions. Therefore, they allow for a very wide range of possible impulse response functions of the VAR model. Also, not all variables are restricted; some variables (very often the variables of interest) are left unrestricted. Thus, the identified impulse response function of the VAR model can be quite different from the impulse response function of the DSGE model, which was used to derive the sign restrictions.

Furthermore, in order to discriminate between pairs of parameters vectors of the DSGE model and the VAR model, there is a Metropolis step between the Gibbs sampling steps, i.e. I compare the probability and keep those with higher probability. That way, the estimated parameter vectors of the DSGE model and their implied identification of the VAR model are narrower then the identification implied by the prior distribution of the DSGE model.

Concerning testing: The paper is more about answering a question like “What are the effects of …?”. It aims at providing evidence, where the DSGE model and the VAR model coincide.

I will likely always remain a sceptic of DSGE “estimation”, although it is something that needs to be done. I disagree strongly with any suggestion that estimating an economic model is not a valuable exercise, as is being suggested in the comments here. If DSGE models are ever to have meaning for policy, and in particular macro policy, they have to be estimated, full stop (or period). That they will be rejected is a reason to carry on improving these models, not for rejecting empirical validation per se.

My concerns about the estimation is that little attention appears to be paid to the data, and to the problems it may cause when modelling. There is not a single plot of any of the data series being modelled, nor any reference to their stationarity, or lack of it: By that I mean difference stationarity, or absence of structural breaks. It is well documented the adverse affect of such data ailments, hence my concern.

On top of that, there is no reporting of the actual model estimated, and no attempt to check whether the model is an accurate representation of the data: Whether the model is a good fit. This is essential if any trust is to be put in the coefficients pre-identification, let alone afterwards.

I find the recent trend towards assessing econometric models by their impulse responses deeply disturbing. Not least the identification required is untested, but importantly, as Luca Sala and Fabio Canova have shown, impulse responses are very sensitive indeed to changes in identification.

Put simply, there is no justification for departing from traditional methods of reported econometric results, other than that the author of the paper has something to hide.

There is little doubt that estimating a DSGE model will prove a difficult task given non-stationarity and given structural breaks. And the reported paper is a step forward. However, it is still a long way short of what is required if DSGE “estimation” is to establish a decent reputation for itself, a reputation that would allow the speech marks to be removed from the word estimation.