Optimization-based macroeconomic models, and, in particular, dynamic stochastic general equilibrium (DSGE) setups, are popular nowadays for analyzing a multitude of macroeconomic questions such as the effects of monetary policy. But as models of this sort become increasingly complex, featuring many types of markets, various rigidities, and different non-linearities, the decision of whether to use a limited or full information (LI or FI) approach for estimation becomes a central question for model developers. Indeed, there appears to be a conflict in the conclusions of available published studies based on one or the other method. A striking example is the ongoing debate in the sizeable empirical literature with regard to the importance of the forward-looking component of the New Keynesian Phillips Curve (NKPC) equation. Recent contributions to this discussion include Gal?, Gertler, and Lopez-Salido (2005) that uses LI methods, and Linde (2005) that uses FI methods, and which report opposite outcomes with respect to the forward-looking nature of the curve.
The LI/FI trade-off is an enduring econometric problem, often presented as one of weighing specification bias versus efficiency, but there are also other concerns. In particular, advances in econometrics regarding weak-instruments and weak-identification have revealed that the latter plague LI and FI methods equally, thus presenting a set of new challenges for applied researchers.
The macroeconomic literature acknowledges the LI/FI trade-off to some extent, often presenting it as one of deciding between Instrumental Variable (IV) or maximum likelihood estimation (MLE); see, for example, Jondeau and LeBihan (2008). Furthermore, published studies in the field are also familiar with the fact that weak instruments effects are critical to IV-based model performance. However, the implications of weak-identification on MLE seem to be less understood, and indeed often confused with issues related to very large estimated standard errors or poorly-approximated test cut-off points. While it may be argued that likelihood-ratio (LR) criteria have more attractive finite sample properties than, for example, IV-based Wald-type ones, and in particular, size correction techniques have a much better chance of success with LR statistics (see Dufour 1997), it should be emphasized that standard MLE and full-information maximum likelihood (FIML) inference are not immune to weak-identification problems.
The complications arise largely because nonlinearities can impose discontinuous param-eter restrictions that cause the breakdown of standard asymptotic procedures. Given the connection between the parameters of the underlying theoretical model and those of the estimated econometric model, and given the identifying constraints imposed on the model, econometric versions of macroeconomic models are often highly nonlinear. The more rich and complex the macroeconomic model, the more likely it is that standard regularity conditions will not fully hold. In this case, even when MLE is used for the estimation, resorting to usual t-type significance tests or Wald-type confidence intervals will lead to the same problems that plague GMM and linear or nonlinear IV; see the surveys of Stock, Wright, and Yogo (2002) and Dufour (2003). As may be checked from these studies, identification difficulties will not always lead to huge regular standard errors that would alert the researcher to the problem. Instead, spuriously tight confidence intervals could occur, often concentrated on wrong parameter values, thus leading to wrong inference.
Weak-instruments and weak-identification concerns have led to the development of socalled identification-robust procedures, i.e. procedures that achieve significance or confidence-level control (at least asymptotically) whether the statistical model is weakly or strongly identified, or whether instruments are weak or strong.
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Structural Multi-Equation Macroeconomic Models: Identification-Robust Estimation and Fit
