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I provide a solution method in the frequency domain for multivariate linear rational expectations models. The method works with the generalized Schur decomposition, providing a numerical implementation of the underlying analytic function solution methods suitable for standard DSGE estimation and analysis procedures. This approach generalizes the time-domain restriction of autoregressive-moving average exogenous driving forces to arbitrary covariance stationary processes. Applied to the standard New Keynesian model, I find that a Bayesian analysis favors a single parameter log harmonic function of the lag operator over the usual AR(1) assumption as it generates humped shaped autocorrelation patterns more consistent with the data.
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.
This paper applies structure preserving doubling methods to solve the matrix quadratic underlying the recursive solution of linear DSGE models. We present and compare two Structure-Preserving Doubling Algorithms ( SDAs) to other competing methods – the QZ method, a Newton algorithm, and an iterative Bernoulli approach – as well as the related cyclic and logarithmic reduction algorithms. Our comparison is completed 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 both SDAs perform very favorably relative to QZ, with generally more accurate solutions computed in less time. While we collect theoretical convergence results that promise quadratic convergence rates to a unique stable solution, the algorithms may fail to converge when there is a breakdown due to singularity of the coefficient matrices in the recursion. One of the proposed algorithms can overcome this problem by an appropriate (re)initialization. This SDA also performs particular well in refining solutions of different methods or from nearby parameterizations.
This paper develops and implements a backward and forward error analysis of and condition numbers for the numerical stability of the solutions of linear dynamic stochastic general equilibrium (DSGE) models. Comparing seven different solution methods from the literature, I demonstrate an economically significant loss of accuracy specifically in standard, generalized Schur (or QZ) decomposition based solutions methods resulting from large backward errors in solving the associated matrix quadratic problem. This is illustrated in the monetary macro model of Smets and Wouters (2007) and two production-based asset pricing models, a simple model of external habits with a readily available symbolic solution and the model of Jermann (1998) that lacks such a symbolic solution - QZ-based numerical solutions miss the equity premium by up to several annualized percentage points for parameterizations that either match the chosen calibration targets or are nearby to the parameterization in the literature. While the numerical solution methods from the literature failed to give any indication of these potential errors, easily implementable backward-error metrics and condition numbers are shown to successfully warn of such potential inaccuracies. The analysis is then performed for a database of roughly 100 DSGE models from the literature and a large set of draws from the model of Smets and Wouters (2007). While economically relevant errors do not appear pervasive from these latter applications, accuracies that differ by several orders of magnitude persist.
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.
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 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.
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.
The term structure of interest rates is crucial for the transmission of monetary policy to financial markets and the macroeconomy. Disentangling the impact of monetary policy on the components of interest rates, expected short rates, and term premia is essential to understanding this channel. To accomplish this, we provide a quantitative structural model with endogenous, time-varying term premia that are consistent with empirical findings. News about future policy, in contrast to unexpected policy shocks, has quantitatively significant effects on term premia along the entire term structure. This provides a plausible explanation for partly contradictory estimates in the empirical literature.
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.