DSGE Model

Definition

A Dynamic Stochastic General Equilibrium (DSGE) model is a micro-founded macroeconomic model in which optimizing households and firms interact in general equilibrium under rational expectations, subject to a small number of exogenous structural shocks (technology, preferences, monetary and fiscal policy, markups). Unlike the atheoretical Vector Autoregression or the exclusion-restricted Macroeconometric Model, every equation of a DSGE model derives from an explicit intertemporal decision problem, so its reduced-form coefficients are non-linear functions of a small set of "deep" (taste and technology) parameters. The dominant modern class — the New Keynesian medium-scale model (Christiano-Eichenbaum-Evans 2005, Smets-Wouters 2003) — grafts nominal rigidities (sticky prices and wages, habit formation, adjustment costs) onto a Real Business Cycle core.

Key Ideas

Micro-foundations and deep parameters. Behavioral equations follow from utility and profit maximization; policy invariance is claimed at the level of taste/technology parameters, not reduced-form coefficients (the Lucas critique).
General equilibrium. All markets clear simultaneously; there are no "incredible" a priori exclusion restrictions of the Cowles-Commission type criticized in Structural Identification.
Rational expectations. Agents' subjective forecasts equal the model's own conditional expectations, generating the cross-equation restrictions that discipline forward-looking dynamics.
Few structural shocks, many observables. A handful of orthogonal economic shocks drive a larger vector of observables, which raises the Invertibility Problem when one tries to recover the shocks from data.
A restricted VAR. A log-linearized DSGE is observationally equivalent to a tightly cross-equation-restricted (generically infinite-order) VAR — the basis for both its estimation and its comparison against unrestricted VARs.

How It Works

Solution: from optimality conditions to state space

The equilibrium is a system of non-linear expectational difference equations (Euler equations, resource constraints, a policy rule such as a Taylor rule). Log-linearizing around the deterministic steady state yields a linear rational-expectations system whose solution is a state-space model — the $(A,B,C,D)$ transition/measurement form of State-Space Representation. Fernández-Villaverde, Rubio-Ramírez and Sargent (2005) give the exact map from these matrices to the VAR the model implies via the Kalman Filter innovations representation; the mapping and its invertibility eigenvalue condition ( $\rho(A-BD^{-1}C)<1$ ) are worked through in the DSGE-to-VAR bridge sections of Vector Autoregression and Impulse Response Function and are not repeated here.

Estimation: prior × likelihood

Because the log-linear solution is a linear-Gaussian state-space model, the Kalman filter delivers the exact likelihood of the deep parameters $\theta$ ; Bayesian estimation then combines it with a prior $p(\theta)$ elicited from micro evidence and steady-state knowledge. The posterior is non-standard (the mapping from $\theta$ to the state-space matrices is highly non-linear), so it is explored by MCMC — typically a single-block random-walk Metropolis-Hastings Algorithm over all parameters at once, since the Kalman filter marginalizes the latent states and closed-form full conditionals are unavailable (Strid 2010). Non-linearly solved DSGE models replace the Kalman filter with a particle filter and require particle-MCMC. Adolfson et al. (2005) estimate a 15-variable small-open-economy DSGE this way and show its sharp, significant impulse responses make it usable for policy conditioning where a Bayesian VAR's bands are too wide.

Identification of structural shocks

Identification is achieved by the theory itself: the cross-equation restrictions of the rational-expectations solution fix the mapping from shocks to observables, so no Cholesky ordering or sign restriction is imposed. When invertibility holds, the DSGE's own impulse responses coincide exactly with a structurally identified VAR's — the model supplies the identification matrix $G=D$ (see Impulse Response Function). This is the disciplined counterpart to the "incredible identification" problem of Structural Identification.

The DSGE–VAR bridge

Two complementary uses of the equivalence "DSGE ≈ restricted VAR" recur:

VAR responses as targets. In the Lucas program articulated by Christiano-Eichenbaum-Evans (1999), identified-VAR impulse responses are stylized facts a DSGE must reproduce. Boivin-Giannoni (2006) estimate a sticky-price DSGE by minimum-distance matching of its responses to VAR responses, then run counterfactual policy-rule swaps.
DSGE prior for a VAR (Del Negro–Schorfheide). Rather than impose the model's cross-equation restrictions exactly, one centers a VAR prior on the restrictions the DSGE implies, with a single hyperparameter $\lambda$ governing how tightly the VAR is pulled toward the model ( $\lambda\to\infty$ recovers the exact DSGE; $\lambda\to 0$ an unrestricted VAR). This "DSGE-VAR" is the concrete realization of what Zha (2008) flags as the outstanding open challenge — estimating VARs with DSGE-implied cross-equation restrictions. It is a theory-driven counterpart to the Minnesota Prior and to the DSGE-implied steady-state prior of Steady State VAR.

Why It Matters

Policy analysis under the Lucas critique. Because deep parameters are (assumed) invariant to policy, DSGE models can evaluate counterfactual policy rules — e.g., the systematic-policy experiments of Boivin-Giannoni (2006) on the Monetary Policy Shocks transmission mechanism — where reduced-form models cannot.
Central-bank forecasting and storytelling. Estimated DSGEs are used alongside BVARs for conditional forecasts and historical decompositions, giving an economically interpretable narrative that atheoretical VARs lack (Adolfson et al. 2005).
Discipline on theory. Bayesian likelihood-based estimation lets DSGE models be compared to data and to each other by Marginal Data Density — the quantitative standard Sims (1996) argued RBC calibration ("computations, not experiments") lacked.

Open Questions

Invertibility / fundamentalness. When shocks outnumber observables or news/anticipated shocks are present, VAR innovations need not span the economic shocks; DSGE-VAR comparisons based on response-matching are then compromised (Invertibility Problem).
Is DSGE identification also "incredible"? Rational-expectations cross-equation restrictions are themselves untestable in small samples, so DSGE models may inherit a version of the identification critique they were meant to escape (see Macroeconometric Model).
Non-linearity and computation. Non-linear solution (occasionally-binding constraints, the zero lower bound, stochastic volatility) forces particle-filter likelihoods whose cost and discontinuity strain standard MCMC (Strid 2010).
Misspecification vs. fit. Adding frictions to improve fit risks parameters that proxy for misspecification rather than genuine structure.