Weak identification is likely to be prevalent in multi-equation macroeconomic models such as in dynamic stochastic general equilibrium setups. Identification difficulties cause the breakdown of standard asymptotic procedures, making inference unreliable. While the extensive econometric literature now includes a number of identification-robust methods that are valid regardless of the identification status of models, these are mostly limited-information-based approaches, and applications have accordingly been made on single-equation models such as the New Keynesian Phillips Curve.

In this paper, we develop a set of identification-robust econometric tools that, regardless of the model's identification status, are useful for estimating and assessing the fit of a system of structural equations. In particular, we propose a vector auto-regression (VAR) based estimation and testing procedure that relies on inverting identification-robust multivariate statistics. The procedure is valid in the presence of endogeneity, structural constraints, identification difficulties, or any combination of these, and also provides summary measures of fit. Furthermore, it has the additional desirable features that it is robust to missing instruments, errors-in-variables, the specification of the data generating process, and the presence of contemporaneous correlation in the disturbances.

We apply our methodology, using U.S. data, to the standard New Keynesian model such as the one studied in Clarida, Gali, and Gertler (1999). We find that, despite the presence of identification difficulties, our proposed method is able to shed some light on the fit of the considered model and, particularly, on the nature of the NKPC. Notably our results show that (i) confidence intervals obtained using our system-based approach are generally tighter than their single-equation counterparts, and thus are more informative, (ii) most model coefficients are significant at conventional levels, and (iii) the NKPC is preponderantly forward-looking, though not purely so.