DSGE pileups

A pileup occurs when the sampling distribution of a DSGE structural-parameter estimator concentrates at or near a boundary of the admissible parameter space, is multimodal, or is otherwise non-Gaussian – even when the model is correctly specified – which calls into question standard asymptotic-normal confidence intervals and tests. Morris frames the moving-average unit-root problem (|theta-hat| = 1 when theta and 1/theta are observationally equivalent under an invertibility restriction) as the canonical example of this phenomenon, then argues DSGE models face an analogous but potentially worse problem: DSGE pileup regions can be entire subsets of the parameter space rather than isolated points, and can appear as multimodal rather than purely boundary distributions.

Morris depicts three overlapping sets in parameter space – Theory, Determinacy, and Identification – and locates two distinct pileup mechanisms in their overlap: (i) the true parameter is theoretically admissible but its identification-equivalent counterpart lies outside the theoretical restrictions, pushing the MLE toward the boundary; and (ii) the true parameter and an observationally equivalent point both lie inside the admissible region, leaving the MLE indeterminate between them. These two configurations correspond, respectively, to the boundary-estimate and multimodal pileup types documented in the paper’s later Monte Carlo examples.

Building on the author’s companion paper (Morris 2016b), Morris shows that under a set of regularity, stationarity, and left-invertibility assumptions on a DSGE model’s ABCD state-space representation, the model’s observables have an exact finite-order VARMA(p, p-1) representation whose reduced-form parameters pi are identifiable and can be consistently estimated. This VARMA representation is the vehicle for everything that follows: it lets Morris write the DSGE structural-to-reduced-form mapping g: Theta -> pi explicitly and study where g creates observational equivalence or boundary behavior.

The MCSE minimizes a quadratic form T(pi-hat_MLE - g(theta))‘I-hat(pi-hat_MLE - g(theta)) over theta, where pi-hat_MLE is the maximum-likelihood estimate of the VARMA reduced-form parameters and I-hat is their estimated asymptotic information matrix; it is asymptotically equivalent to direct MLE of the structural model but has two practical advantages Morris exploits throughout the paper. First, it is numerically simpler because it separates estimation into a first-stage (reduced-form VARMA) MLE and a second-stage minimum-distance step; second, its bootstrap distribution remains valid even when a parameter is subject to pileup, whereas the asymptotic theory underlying conventional standard errors is unreliable in that case – a point Morris qualifies by noting that global identification (assumed throughout the paper’s examples) is what keeps the bootstrap valid, since bootstrap inference can otherwise fail under weak instruments.

Calibrating the three-parameter Brock-Mirman model at alpha_0 = 1/3, beta_0 = 0.99, sigma_0 = 0.1 and simulating 1,000 replications of T=225 quarterly observations, the discount-factor estimate beta-hat piles up at its theoretical upper boundary of 1 even though the true value is comfortably interior at 0.99, while the underlying ARMA(1,1) reduced-form estimates remain Gaussian; Kolmogorov-Smirnov tests reject normality of beta-hat at the 1%, 5%, and 10% levels. Because the model is just-identified here (the MCSE coincides with the MLE), Morris uses this example to establish the “boundary estimate” pileup type in its cleanest form and to show the associated small-sample bias of alpha and beta toward 1.

In the six-parameter Krause-Lubik (2010) search-and-matching calibration, the match-elasticity parameter xi develops a long left tail and piles up near 0.1 – not the theoretical boundary of Θ (0 < xi < 1), but a “natural” boundary created by the model’s own functional form – a “skew” pileup distinct from the pure boundary case. Morris traces this to a “natural boundary” created by the functional correspondence between two other parameters (A_x and C_y, jointly identifiable but not individually identifiable) rather than by the stated theoretical restriction on xi itself – so xi piles up even in replications where A_x and C_y are estimated well, meaning the pileup originates in the shape of the structural-to-reduced-form mapping, not just in the stated parameter bounds.

In the 14-parameter An-Schorfheide (2007) New Keynesian model, Morris finds an observationally equivalent solution point theta_0 implying an economically infeasible CRRA risk-aversion coefficient of tau = -45.4; without bounds on the parameter space this produces a genuinely bimodal sampling distribution in tau and the shock-persistence parameter rho_z (with rho_r and psi_pi also affected), and even after imposing the theoretical bounds that exclude theta_0, the proximity of the true parameter to the boundary keeps multimodality alive in rho_z and rho_gz.** On the Bayesian side, Jeffreys priors reduce multimodality in tau but leave it “very much apparent” in rho_z; informative conjugate priors can eliminate the rho_z multimodality when only mildly informative (prior concentration c=1000) but, when made more informative (c=100), instead push the posterior for rho_rz to accumulate on the wrong side of zero – leading Morris to describe informative priors as “a double-edged sword” rather than a general fix.

Yes: applying the same Monte Carlo design (T=225, N=1000) to the Smets-Wouters (2007) model – 41 structural parameters mapped to a VARMA(3,2) representation with 252 identifiable reduced-form parameters, after fixing five parameters to preserve identifiability of the rest – reproduces all three pileup types simultaneously: a boundary pileup at 1 for TFP-shock persistence rho_z, a long right-tailed skew for the investment adjustment-cost parameter, and clear bimodality for the average trend-growth parameter gamma. Replacing hours worked with a labor-income-share proxy among the observables changes some pileups but does not eliminate those in beta, the price-indexation parameter, or gamma, leading Morris to conclude that the choice of observables can matter but is not, by itself, an easy fix.

No – Morris is explicit that the boundary estimates, skew, and multimodality found in the Smets-Wouters exercise are “intimately related to the identifiability properties of the model” rather than evidence of misspecification, and the paper does not claim to have found a way to prevent pileups in general. What the MCSE/VARMA framework does provide is diagnosis: an F-test on exclusion restrictions in the VARMA autoregressive polynomial (for example, testing the rho_z boundary pileup) and a chi-squared overidentification test from the MCSE criterion, both of which can help distinguish a genuine pileup from misspecification; Morris describes the approach as “useful in the very common situation that such pileups should arise,” not as a cure.