ASSESSING DSGE MODEL NONLINEARITIES S. Borağan Aruoba Luigi Bocola Frank Schorfheide December 2013
Introduction • Until recently, much of the research that estimates DSGE models used first-order approximations to the equilibrium decision rules. • This made linear models such as VARs appropriate for evaluating the restrictions of the DSGE model. • Higher-order approximations like: Fernandez-Villaverde and Rubio-Ramrez (2007 ) “Estimating Macroeconomic Models: A o Likelihood Approach” Review of Economic Studies . 2
DSGE Nonlinearities • Nonlinear features may arise endogenously or exogenously. o Curvature in utility functions, adjustment cost function and production functions can generate nonlinear decision rules of households and firms endogenously. o An example of an exogenous nonlinearity is stochastic volatility in the exogenous shocks that generate business cycle fluctuations. • The DSGE model generates cross coefficient restrictions that may or may not be correctly specified. • In principle, one could try to estimate and compare two versions of the model: o Restricted and Unrestricted • As in Linear approximations: o assessing the discrepancy between unrestricted VAR coefficient estimates and the DSGE- model-implied VAR approximation as in Smith (1993) o Or the comparison of VAR and DSGE model impulse responses as in Cogley and Nason (1994) or Christiano, Eichenbaum, and Evans (2005). 3
Nonlinear AR • In evaluating a Nonlinear DSGE models, a linear (V)AR would not be of any use. • The most popular (and empirically successful) nonlinear time-series models are those capturing time variation in the coefficients of linear time-series models o Markov switching models o Time-varying coefficient models o GARCH models o Stochastic volatility models • None of these models provides a good characterization of the nonlinearity generated endogenously by the DSGE model solution. • Quadratic AR 4
DSGE Nonlinearities • The starting point is the unique deterministic steady-state solution for a nonlinear difference equation of the form: • Which is: • Following the literature on perturbation methods, e.g., Holmes (1995) and Lombardo (2011): • Which is second-order accurate: 5
Nonlinear AR • Next step is to obtain We do this by taking a second-order Taylor expansion of the function 𝑔 around • • We also set 6
Nonlinear AR • Considering the three equations, we obtain the law of motions for • And after substitution: 7
Nonlinear AR • After reparameterization: • Which we call QAR(1,1). o The first number: the number of lags in the conditional mean function. o The second number: the number of lags that interact with the innovation. 8
Properties of QAR • Consider this alternative: • Two steady states: • The second one is the result of quadratic representation of the first difference equation . • From writing • it becomes clear that the system will become explosive if a large shock pushes above . This explosiveness can arise regardless of the value of • Kim, Kim, Schaumburg, and Sims (2008) proposed an ex-post modification of quadratic autoregressive equations to ensure that unwanted higher-order terms do not propagate forward and generate explosive behavior. • This modification is called pruning in the literature. 9
Properties of QAR • Our derivation of the QAR model automatically generates a recursively linear structure with a unique steady state and non-explosive dynamics for suitably restricted values of • The model is able to generate nonlinear dynamics that are akin to the nonlinear dynamics of DSGE models solved with perturbation methods. In particular, impulse responses are state dependent. • Moreover, the model generates conditional heteroskedasticity. 10
Empirical Analysis • Fitting the QAR(1,1) model to per capita output growth, nominal wage growth, GDP deflator inflation and federal funds rate data. • The choice of data is motivated by the DSGE model that is being evaluated subsequently. 11
Empirical Analysis • The models are estimated using five different sample periods: • high-inflation episode of the 1970s • Great Recession of 2008-09. • pre-Great-Moderation sample that ranges from 1960:Q1 to 1983:Q4 • post-Great-Moderation sample from 1984:Q1 to 2012:Q4. • Prior distributions are assumed to be normal and inverted gamma, truncated (if needed) to ensure stationarity of the model. 12
The strongest evidence for nonlinearity in GDP growth is present in the 1984-2012 sample, which includes large negative growth rates of output during the Great Recession, in the form of 13
Empirical Analysis 14
Empirical Analysis This figure highlights, that regardless of the initial state, negative shocks are more persistent than positive shocks. Moreover, both shocks are more persistent in recessions. • Median impulse responses. • One standard deviation shock. • 60% confidence interval. 15
Empirical Analysis • A DSGE Model with Asymmetric Price and Wage Adjustment Costs … 16
Empirical Analysis • Posterior Predictive Checks • Are QAR(1,1) parameter estimates obtained from data that are simulated from the estimated DSGE model similar to the estimates computed from actual data? 17
Empirical Analysis Let 𝜄 (𝑗) denote the 𝑗 -th draw from the posterior distribution of the DSGE model • parameter 𝜄 . 1. For 𝑗 = 1 to n: 1. Conditional on 𝜄 (𝑗) simulate a pre-sample of length 𝑈0 and an estimation sample of size 𝑈 from the DSGE model. 2. The second-order approximated DSGE model is simulated using the pruning algorithm described in Kim, Kim, Schaumburg, and Sims (2008). 3. A Gaussian iid measurement error is added to the simulated data. 4. Based on the simulated data, estimate the QAR(1,1) model. 2. The empirical distribution of all simulated posterior median estimated of QAR parameters (in step 1-4), approximates the posterior predictive distribution of the QAR model conditional on actual data. 3. How far are they from each other? 18
Only interest rates exhibit large discrepancies between actual and model-implied estimates of the QAR(1,1) parameters. Overall, the estimated DSGE model does not generate very strong nonlinearities. Posterior predictive distributions typically cover both positive and negative values. 19
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