Linear Identification of Linear Rational-Expectations Models by Exogenous Variables Reconciles Lucas and Sims - Peter A. Zadrozny (Center for Economic Studies and ifo Institute (CESifo), 2022)
Linear rational-expectations models (LREMs) are usually "forwardly" estimated. Structural coefficients are restricted in terms of deep parameters. For given deep parameters, structural equations are solved for rational-expectations solution (RES) eqs. that determine endogenous variables. For given VAR eqs. that determine exogenous variables (XVAR), RES eqs. reduce to reduced-form VAR eqs. with exogenous variables (ERF). Combined XVAR and ERF eqs. comprise reduced-form (RF) overall VAR (OVAR) eqs. of all variables. The specified, solved, and combined eqs. define a mapping from deep parameters to OVAR coefficients used to forwardly est. a LREM in terms of deep parameters. Forwardly-est. deep parameters determine forwardly-est. RES eqs. that Lucas (1976) advocated for making policy predictions. Sims (1980) called identifying restrictions on deep parameters of forwardly-est. LREMs "incredible", because he considered in-sample fits of forwardly-est. OVAR eqs. inadequate and out-of-sample policy predictions of forwardly-est. RES eqs. inaccurate. Sims (1980, 1986) instead advocated directly estimating OVAR eqs. restricted by statistical restrictions and directly using directly-est. OVAR eqs. To make policy predictions. However, if assumed or predicted out-of-sample exogenous policy variables differ significantly from predictions of their in-sample est. XVAR eqs., then, out-of-sample policy predictions of endogenous variables made with OVAR eqs. won't satisfy Lucas's critique. If directly-est. OVAR eqs. are RF eqs. of underlying RES eqs., then, identification 2 derived in the paper linearly "inversely" est. the underlying RES eqs. from the directly-est. OVAR eqs. and the inversely- est. RES eqs. can be used to make policy predictions that satisfy Lucas's critique. If Sims considered directly-est. OVAR eqs. to fit in-sample data adequately (credibly) and their inversely-est. RES eqs. To make accurate (credible) out-of-sample policy predictions, then, he should consider the inversely-est. RES eqs. and further underlying LREM- structural eqs. to be credible. Thus, inversely-est. RES eqs. by identification 2 would reconcile Lucas's advocacy for making policy predictions with RES eqs. and Sims's advocacy for directly estimating OVAR eqs.
Keywords: C32; C43; C53; C63; 330; cross-equation restrictions of rational expectations; factorization of matrix polynomials; reconciliation of Lucas's advocacy of rational-expectations modelling and policy predictions and Sims's advocacy of VAR modelling