Effects of incorrect specification on the finite sample properties of full and limited information estimators in DSGE models☆
Introduction
New Keynesian DSGE models have become the workhorse monetary macroeconomic models in policy analysis and forecasting (e.g. Smets, Wouters, 2007, Schorfheide, 2013). This model class typically involves rational expectations as well as non-linearities in the structural model parameters. The econometric analysis of DSGE models has progressed in recent years and many estimation techniques – classical and Bayesian – have been proposed.
In particular, the availability of computational power strongly simplified the estimation of these models by full information (FI) methods and, thus, Bayesian estimation techniques became increasingly popular. There is now a growing interest in the econometric aspects of these models. One major concern has been potential identification difficulties (Canova, Sala, 2009, Kleibergen, Mavroeidis, 2009, Iskrev, 2010, Andrews, Mikusheva, 2015, Dufour, Khalaf, Kichian, 2013, Guerron-Quintana, Inoue, Kilian, 2013). In this paper, we concentrate on another econometric aspect of New Keynesian DSGE models: the consequences of potential misspecification.
Although FI estimation of DSGE models are currently dominant (see e.g. Ireland, 2004, An, Schorfheide, 2007 for earlier examples), LI techniques remain popular to study selective parts of a DSGE model. Notably, the literature on the New-Keynesian Phillips curve made heavily use of GMM applied to a single equation (e.g. Galí, Gertler, 1999, Eichenbaum, Fisher, 2007, Kleibergen, Mavroeidis, 2009). An important advantage of limited information techniques is that it can be applied to study certain aspects of a model and leaving apart those of little interest. But generally, limited information procedures can also be applied to a complete DSGE model to estimate its structural parameters (see e.g. Beyer et al., 2008, for an application).
In this paper we investigate the relative performance of FI and LI techniques for New Keynesian macro models. To study the potential effects of misspecification, we use a standard (and potentially misspecified) DSGE model with nominal price and wage rigidities (as in Erceg et al., 2000). Several forms of misspecification are taken into account. First, we look at the consequences of estimating a model with price rigidities but omit nominal wage rigidities. Second, we omit price indexation in the Phillips curve. Finally, we investigate the case of misspecified shocks (missing autocorrelation) in the IS curve. In a simulation study we document the properties of the different estimation techniques for point estimates and standard significance tests.
In comparing full with limited information techniques we employ FIML on the likelihood of the state-space solution of the log-linearized model as our FI method. FI methods provide the complete range of statistical properties associated with the model under investigation. Normally, this is preferable in terms of efficiency, given a correctly specified model (see e.g. Cragg, 1967, for an early contribution). LI methods do not require a fully specified model, instead it is enough to set up certain moment conditions to estimate the parameters of interest. Thus, there is the classical trade-off between efficiency and the sensitivity to model misspecification known from simultaneous equation models (see Theil, 1971, Ch. 10 for a summary). For the LI methods we consider different variants of GMM estimators. We take into account system based as well as single equation techniques. It turned out that the continuous-updating GMM (CUGMM) as advocated by Hansen et al. (1996), produces much more promising results than the standard two-step GMM estimator (which is normally employed in this context).
This paper is closely related to the work of Ruge-Murcia (2007), who compares the properties of different estimators in a stylized RBC model. Instead, we look at a New Keynesian DSGE model which forms the basis of todays macroeconometric models. Moreover, we focus on different variants of GMM (which are related to the single equation techniques). Lindé (2005) as well as Jondeau and Le Bihan (2008) also take into account model misspecification. While Jondeau and Le Bihan (2008) look at misspecification within one equation of interest, we look at the consequences of misspecification in the whole system. Lindé (2005) only takes into account quite moderate degrees of misspecification (mainly the misspecification of shocks and lagged persistence). Both, Lindé (2005) and Jondeau and Le Bihan (2008) compare ML with GMM, but use the standard two-step GMM estimator in a single equation set-up. We apply GMM to a multi-equation framework, to estimate all relevant relationships jointly and consider the CUGMM estimator as an alternative.
Our results suggest that LI techniques can be seen as a useful complement to FI methods in analyzing DSGE models (as also suggested by Fukac and Pagan, 2010). If one is interested in the estimate of structural parameters, FI procedures should be used with caution. As expected, FIML is dominant when it comes to estimating the model under the null hypothesis (without misspecification). Parameter estimates are unbiased and quite precise. Moreover, confidence intervals are very reliable in this case, i.e. they provide at least the pre-specified coverage (even in smaller samples). GMM estimates (based on CUGMM) turn out to be less efficient in that case, but remain unbiased. Estimated standard errors are also reliable in most cases and under the proposed model.
In the case of model misspecification the performance of FIML worsens substantially. When nominal wage rigidities are ignored, all parameters are heavily biased and confidence intervals are totally unreliable. More milder forms of misspecifications result in slightly better estimates, but even in the case of misspecified shocks, FIML estimates turn out to be biased and some parameter estimates are far from their true value.
GMM estimates based on CUGMM remain roughly unbiased in all considered cases, although estimated standard errors get slightly less reliable. In general, our results suggest that if one has not very strong beliefs in the usefulness of all aspects of the model, one should stick to limited information methods. In particular, CUGMM applied to multi-equations does a good job in finite samples and is much less sensitive to model misspecification than FIML based on the Kalman filter. Therefore, the classical trade-off between efficiency and potential biases due to misspecification of the different estimation strategies remain valid for standard DSGE models.
The remainder of the paper is organized as follows. The subsequent section presents the model structure. Section 3 discusses the different estimation strategies. Section 4 outlines the simulation setup. Section 5 shows our results; followed by a short discussion in Section 6. Section 7 concludes.
Section snippets
The model economy
Our starting point is a stylized New Keynesian Dynamic Stochastic General Equilibrium (DSGE) model, the workhorse structural macroeconomic model for short-term analysis. Those models are well established in the academic literature (see e.g. Christiano, Eichenbaum, Evans, 2005, Smets, Wouters, 2007) and one of the main tools for policy analysis in central banks and policy institutions.
To make things transparent and as simple as possible, we consider a small scale DSGE inspired by Erceg et al.
Maximum likelihood
The full information set up is concerned with estimating the described DSGE model by means of Maximum Likelihood (ML). Therefore, Eq. 20 is slightly modified to setting up the likelihood function and to estimate the structural parameters of the model. In particular, the Kalman filter is used for this purpose (see e.g. Ireland, 2003, Ireland, 2004, Adolfson, Lindé, 2011, for a similar strategy). For the observation equation we take output, the inflation rate, nominal wage growth and the interest
Monte Carlo simulation
To investigate the performance of the various estimation procedures, we take the DSGE model as outlined above and generate a sequence of observable variables. More precisely, we solve the model and take the matrices Π, M and G (see Eq. 20) to construct our data generating process. Let denote the data matrix comprising the endogenous variables Yt and the exogenous variables Xt, such that our data can be constructed from:
with S0 describing the initial
Results
In the following, we discuss the small-sample properties of limited and full estimation under different settings. First, we present the results under the null, i.e. when the model is correctly specified. Next, we turn to the model estimates under misspecification. In all cases we compare the point estimates with their true values to investigate the consistency properties. Therefore, we present means and medians as well as their corresponding average bias. Mean square errors (MSE) are computed
Discussion
The results presented above emphasize the usefulness of limited information estimation techniques, namely the presented CUGMM estimator. This estimator has several advantages. First, it is not limited to linear settings (like ML based on the state-space system), non-linear models can be estimated as well. Second, the robustness of the limited information technique may be used as a specification test for larger models. One can proceed in a similar way as Jondeau and Le Bihan (2008), where
Conclusion
In this paper, we systematically investigated GMM and FIML under several forms of model misspecification. We employed a standard New Keynesian model including nominal price and wage rigidities (as in Erceg, Henderson, Levin, 2000, Rabanal, Rubio-Ramirez, 2005). By estimating the correct model structure, we can show that FIML provides superior results in small samples. While GMM also provides unbiased estimates (when using a CUGMM estimator), the dispersion of this estimator is larger but
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We want to thank Gregor Bäurle, Jörg Breitung, Fabio Canova, Jean-Marie Dufour, André Kurmann, Jesper Lindé, Ludger Linnemann, Roberto Motto, Federico Ravenna, Barbara Rossi and Michael R. Wickens for valuable comments. The paper also benefited from discussions at the 12th IWH-CIREQ conference 2011, the North American Summer Meeting and the Australasian Meeting of the Econometric Society 2012 as well as at the 8th Dynare Conference 2012 and the European Meeting of the Econometric Society 2013. The views expressed in this paper are those of the authors and not necessarily those of the Swiss National Bank or the Deutsche Bundesbank.