DSGE pileups

This 2017 Journal of Economic Dynamics & Control paper by Stephen D. Morris is a methodological contribution explaining why maximum-likelihood or Bayesian estimates of DSGE structural parameters often “pile up” – concentrate at or near a boundary of the theoretically admissible parameter space, or otherwise depart from the asymptotic normal distribution – even when the model is correctly specified. Morris argues the underlying cause is weak identification of structural parameters by the reduced-form data, operating through three specific channels in DSGE models: observational equivalence (the map from structural parameters theta to reduced-form parameters pi sends multiple theta values, including economically implausible ones, to the same pi), functional boundaries created by the theta-to-pi mapping itself (rather than by the stated theoretical restrictions), and multiplicity of stable rational-expectations solutions. To study these mechanisms formally, the paper builds on the author’s own prior result (Morris 2016b) that, under stated regularity, stationarity, and left-invertibility assumptions, the ABCD state-space representation of a DSGE model has an exact finite-order VARMA(p, p-1) representation, and proposes a minimum chi-square estimator (MCSE) – asymptotically equivalent to MLE but numerically simpler and bootstrappable – together with an F-test and an overidentification chi-squared test to diagnose pileups. The paper contains no empirical application to real data; all results come from Monte Carlo experiments (T=225 quarters, N=1000 replications, mimicking a typical postwar quarterly sample) on three small illustrative models plus the Smets-Wouters (2007) medium-scale model. In the Brock-Mirman stochastic growth calibration, the discount-factor estimate beta-hat piles up at its upper boundary of 1 even when the true value is beta_0 = 0.99, with Kolmogorov-Smirnov tests rejecting Gaussianity at the 1%, 5%, and 10% levels; in the Krause-Lubik search-and-matching calibration, the match-elasticity parameter xi piles up near its lower boundary of 0.1 with a long left tail, driven by a “natural” functional boundary rather than the stated theoretical restriction; in the An-Schorfheide New Keynesian model, the CRRA coefficient tau and shock-persistence parameter rho_z show multimodal sampling distributions traceable to an observationally equivalent solution point with an economically infeasible tau of -45.4, a problem that Jeffreys priors only partly resolve and that informative conjugate priors can worsen for other parameters (Morris calls informative priors “a double-edged sword”); and in the Smets-Wouters model (41 structural parameters, VARMA(3,2) representation), all three pileup types recur simultaneously – a boundary pileup at 1 for TFP-shock persistence rho_z, a skewed distribution for the investment adjustment-cost parameter, and bimodality in the trend-growth parameter gamma – and persist under an alternative observable set that replaces hours worked with the labor income share. Morris stresses throughout that these phenomena reflect identification weakness intrinsic to DSGE models, not misspecification, and that while the MCSE framework helps diagnose pileups and enables valid bootstrap inference, it does not offer a general method to prevent them.

VARMA representation of DSGE models

This 2016 Economics Letters paper by Stephen D. Morris asks how concise the VARMA (vector autoregressive moving-average) representation of a DSGE model can be made, and whether the model’s structural parameters can be locally identified from that representation. Starting from the general “ABCD” state-space form of a DSGE model (Fernandez-Villaverde et al. 2007) – states X_t (m x 1), observables Y_t (n x 1), transition matrix A, and shock-loading matrices B, C, D – Morris shows that a prior general result (Ravenna 2007) implying every such model has a VARMA(n+m, n+m-1) representation is far larger than necessary in practice: for the Smets-Wouters (2007) model, with n=7 observables and m=12 states, that bound is a “prohibitively large” VARMA(19,18). Under Assumption 1 – observables no more numerous than states (n <= m) and an invertible shock-loading matrix D, i.e., the model is not stochastically singular – Morris derives a condition (Proposition 1) on the rank of an observability-type matrix Psi(kappa) that pins down a much lower-order VARMA(kappa+2, kappa+1) representation, where kappa is the minimum number of lags of the observables needed before the unobserved states become indirectly recoverable through them; corollaries collapse this further to VARMA(2,1), VARMA(1,1), or even VAR(1) representation under additional structure (n=m and C_X invertible; then A_Y=0 and C_Y=0; then an invertible map from X_t to Y_t). Applied to the Smets-Wouters (2007) model, checking that Psi(1) has full column rank 7 at reasonable parameter values shows the model actually has a VARMA(3,2) representation – far more concise than the VARMA(19,18) implied by Ravenna’s bound. Morris then shows that the largest subset of DSGE structural parameters that is locally identifiable from the entire model likelihood is also locally identifiable from the identifiable VARMA parameter subset alone – the nonzero AR coefficients phi together with vech of the innovation covariance matrix Omega – meaning no information outside this reduced parameter set aids local identification of the structural parameters; for Smets-Wouters, the paper reports the Jacobian of this map is 161x36 with full column rank 36 “for a range of reasonable parameterizations,” implying 36 of the model’s 41 structural parameters are locally identified this way and yielding 125 over-identifying restrictions useful for GMM-based testing. The paper is pure theory: it presents no data and no empirical estimation, its main results are stated as holding under Assumption 1 and as local (not global) identification results valid at generic rather than all parameter points, and it does not claim its VARMA(kappa+2, kappa+1) representation is the uniquely minimal one in every case.

Stephen D. Morris | Macro Paper Warehouse

DSGE pileups

VARMA representation of DSGE models