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On the accuracy of linear DSGE solution methods and the consequences for log-normal asset pricing
(2021)
This paper demonstrates a failure of standard, generalized Schur (or QZ) decomposition based solutions methods for linear dynamic stochastic general equilibrium (DSGE) models when there is insufficient eigenvalue separation about the unit circle. The significance of this is demonstrated in a simple production-based asset pricing model with external habit formation. While the exact solution afforded by the simplicity of the model matches post-war US consumption growth and the equity premium, QZ-based numerical solutions miss the later by many annualized percentage points.
This paper presents and compares Bernoulli iterative approaches for solving linear DSGE models. The methods are compared using nearly 100 different models from the Macroeconomic Model Data Base (MMB) and different parameterizations of the monetary policy rule in the medium-scale New Keynesian model of Smets and Wouters (2007) iteratively. I find that Bernoulli methods compare favorably in solving DSGE models to the QZ, providing similar accuracy as measured by the forward error of the solution at a comparable computation burden. The method can guarantee convergence to a particular, e.g., unique stable, solution and can be combined with other iterative methods, such as the Newton method, lending themselves especially to refining solutions.
The authors relax the standard assumption in the dynamic stochastic general equilibrium (DSGE) literature that exogenous processes are governed by AR(1) processes and estimate ARMA (p,q) orders and parameters of exogenous processes. Methodologically, they contribute to the Bayesian DSGE literature by using Reversible Jump Markov Chain Monte Carlo (RJMCMC) to sample from the unknown ARMA orders and their associated parameter spaces of varying dimensions.
In estimating the technology process in the neoclassical growth model using post war US GDP data, they cast considerable doubt on the standard AR(1) assumption in favor of higher order processes. They find that the posterior concentrates density on hump-shaped impulse responses for all endogenous variables, consistent with alternative empirical estimates and the rigidities behind many richer structural models. Sampling from noninvertible MA representations, a negative response of hours to a positive technology shock is contained within the posterior credible set. While the posterior contains significant uncertainty regarding the exact order, the results are insensitive to the choice of data filter; this contrasts with the authors’ ARMA estimates of GDP itself, which vary significantly depending on the choice of HP or first difference filter.
The authors present and compare Newton-based methods from the applied mathematics literature for solving the matrix quadratic that underlies the recursive solution of linear DSGE models. The methods are compared using nearly 100 different models from the Macroeconomic Model Data Base (MMB) and different parameterizations of the monetary policy rule in the medium-scale New Keynesian model of Smets and Wouters (2007) iteratively. They find that Newton-based methods compare favorably in solving DSGE models, providing higher accuracy as measured by the forward error of the solution at a comparable computation burden. The methods, however, suffer from their inability to guarantee convergence to a particular, e.g. unique stable, solution, but their iterative procedures lend themselves to refining solutions either from different methods or parameterizations.
Highlights
• Six Newton methods for solving matrix quadratic equations in linear DSGE models.
• Compared to QZ using 99 different DSGE models including Smets and Wouters (2007).
• Newton methods more accurate than QZ with comparable computation burden.
• Apt for refining solutions from alternative methods or nearby parameterizations.
Abstract
This paper presents and compares Newton-based methods from the applied mathematics literature for solving the matrix quadratic that underlies the recursive solution of linear DSGE models. The methods are compared using nearly 100 different models from the Macroeconomic Model Data Base (MMB) and different parameterizations of the monetary policy rule in the medium-scale New Keynesian model of Smets and Wouters (2007) iteratively. We find that Newton-based methods compare favorably in solving DSGE models, providing higher accuracy as measured by the forward error of the solution at a comparable computation burden. The methods, however, suffer from their inability to guarantee convergence to a particular, e.g. unique stable, solution, but their iterative procedures lend themselves to refining solutions either from different methods or parameterizations.
The authors propose a new method to forecast macroeconomic variables that combines two existing approaches to mixed-frequency data in DSGE models. The first existing approach estimates the DSGE model in a quarterly frequency and uses higher frequency auxiliary data only for forecasting. The second method transforms a quarterly state space into a monthly frequency. Their algorithm combines the advantages of these two existing approaches.They compare the new method with the existing methods using simulated data and real-world data. With simulated data, the new method outperforms all other methods, including forecasts from the standard quarterly model. With real world data, incorporating auxiliary variables as in their method substantially decreases forecasting errors for recessions, but casting the model in a monthly frequency delivers better forecasts in normal times.
We present determinacy bounds on monetary policy in the sticky information model. We find that these bounds are more conservative here when the long run Phillips curve is vertical than in the standard Calvo sticky price New Keynesian model. Specifically, the Taylor principle is now necessary directly - no amount of output targeting can substitute for the monetary authority’s concern for inflation. These determinacy bounds are obtained by appealing to frequency domain techniques that themselves provide novel interpretations of the Phillips curve.
The authors examine the effectiveness of labor cost reductions as a means to stimulate economic activity and assesses the differences which may occur with the prevailing exchange rate regime. They develop a medium-scale three-region DSGE model and show that the impact of a cut in the employers’ social security contributions rate does not vary significantly under different exchange rate regimes. They find that both the interest rate and the exchange rate channel matters. Furthermore, the measure appears to be effective even if it comes along with a consumption tax increase to preserve long-term fiscal sustainability.
Finally, they assess whether obtained theoretical results hold up empirically by applying the local projection method. Regression results suggest that changes in employers’ social security contributions rates have statistically significant real effects – a one percentage point reduction leads to an average cumulative rise in output of around 1.3 percent in the medium term. Moreover, the outcome does not differ significantly across the different exchange rate regimes.
Die Abhandlung ist eine überarbeitete und erweiterte Fassung der vom Institute for Monetary and Financial Stability am 19. Juni 2006 veranstalteten Guest Lecture des Autors zum Thema "Demystifying Hedge Funds"