Abstract
In this paper we estimate a small structural model, in order to forecast the key macroeconomic variables of output growth and underlying inflation. In contrast to models with purely statistical foundations, the Bayesian Vector Autoregressive Dynamic Stochastic General Equilibrium (BVAR-DSGE) model, uses the theoretical information of a DSGE model to offset insample overfitting. We compare the forecast performance of BVAR-DSGE model with Minesota VAR and independently estimates DSGE model. The open economy DSGE model of Lubik and Schorfheide (2007) is implemented to provide prior information for the VAR.
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Notes
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The rate at which the autoregressive coefficients converge to zero is determined by the largest eigenvalue of \(A - BD^{-1}C\). If this eigenvalue is close to unity, a low order VAR is likely to be a poor approximation to the infinite-order VAR implied by the DSGE model. If one or more of the eigenvalues of \(A - BD^{-1}C\) are exactly equal to one in modulus, y t does not have a VAR representation, i.e, the autoregressive coefficients do not converge to zero as the number of lags tend to infinity. Often, roots on the unit circle indicate that the observables have been overdifferenced.
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A VAR approximation of the DSGE model can be obtained from restriction functions that relate the DSGE model parameters to the VAR parameters: \(\varPhi ^{{\ast}}\left (\theta \right ) = \varGamma _{xx}^{{\ast}-1}\left (\theta \right )\varGamma _{xy}^{{\ast}}\left (\theta \right )\), \(\sum _{u}^{{\ast}}\left (\theta \right ) = \varGamma _{yy}^{{\ast}}\left (\theta \right ) -\varGamma _{yx}^{{\ast}}\left (\theta \right )\varGamma _{xx}^{{\ast}-1}\left (\theta \right )\varGamma _{xy}^{{\ast}}\left (\theta \right )\).
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Our prior has hierarchical structure. We conduct a posterior predictive analysis in the spirit of Gelman et al. [10].
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This λ represents the weight of the restrictions from the model imposed by the econometrician and it tells how much the economic model DSGE, is able to explain the real data.
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A consequence of this is that some of the real variables (such as output) are normalized by technology before the log-linearisation.
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Implying that output depends on the expectations of future both home and abroad, the real interest rate, the expected changes in terms of trade and technology growth.
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Movements in the output gap (\(y_{t} -\bar{y}_{t})\), affect inflation as they are associated with changes in real marginal costs. Changes in the terms of trade enter the Phillip curve reflecting the fact that some consumer goods are imported.
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We assume Spain is too small to have a significant influence on the ECB’s Taylor rule. Thus, changes in Spanish conditions do not affect R t , which is determined by the Taylor rule above, evaluated at the observed values of euro area variables. Justiniano and Preston [14] include output growth as an additional argument in their policy rule.
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By this specification, we pin down the small open economy as a system affected by foreign data generating processes but which has no perceptible influence on the rest of the world.
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Technology is assumed to grow ate the rate z t .
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To interpret this table, note that if the entry in a particular cell is less than one, then the BVAR-DSGE outperforms the corresponding benchmark model. Diebold and Mariano [6] provide a general framework for such tests.
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The DSGE and DSGE-VAR forecasts are based on 100,000 Metropolis Hastings draws starting from the posterior mode. More detail about algorithm can be found in Koop [15].
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Notice that all the parameters in the DSGE model and the DSGE-VAR including the hyperparameter λ, that is re-estimated in each recursion.
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Based on univariate root mean squared forecast error (RMSE).
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This is the case for the one-quarter and eight quarters-ahead forecasts UVAR model. Compared to the DSGE and the Minnesota VAR models, the BVAR-DSGE forecasting outperforms inflation and output growth respectively at any horizon. Adolfson [1] examine out-of-sample forecast performance for DSGE models of the euro area.
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Sánchez, M.S. (2015). Modern Forecasting of NOEM Models. In: Bourguignon, JP., Jeltsch, R., Pinto, A., Viana, M. (eds) Dynamics, Games and Science. CIM Series in Mathematical Sciences, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-16118-1_34
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