This paper considers inference in log-linearized dynamic stochastic general equilibrium (DSGE) models with weakly (including un-) identified parameters. The framework allows for analysis using only part of the spectrum, say at the business cycle frequencies. First, we characterize weak identification from a frequency domain perspective and propose a score test for the structural parameter vector based on the frequency domain approximation to the Gaussian likelihood. The construction heavily exploits the structures of the DSGE solution, the score function, and the information matrix. In particular, we show that the test statistic can be represented as the explained sum of squares from a complex-valued Gaussu2013Newton regression, where weak identification surfaces as (imperfect) multicollinearity. Second, we prove that asymptotically valid confidence sets can be obtained by inverting this test statistic and using chi-squared critical values. Third, we provide procedures to construct uniform confidence bands for the impulse response function, the time path of the variance decomposition, the individual spectrum, and the absolute coherency. Finally, a simulation experiment suggests that the test has adequate size even with relatively small sample sizes. It also suggests it is possible to have informative confidence sets in DSGE models with unidentified parameters, particularly regarding the impulse response functions. Although the paper focuses on DSGE models, the methods are applicable to other dynamic models with well defined spectra, such as stationary (factor-augmented) vector autoregressions.
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