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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 debate on monetary and fiscal policy is heavily influenced by estimates of the equilibrium real interest rate. In particular, this concerns estimates derived from a simple aggregate demand and Phillips curve model with time-varying components as proposed by Laubach and Williams (2003). For example, Summers (2014a) refers to these estimates as important evidence for a secular stagnation and the need for fiscal stimulus. Yellen (2015, 2017) has made use of such estimates in order to explain and justify why the Federal Reserve has held interest rates so low for so long. First, we re-estimate the United States equilibrium rate with the methodology of Laubach and Williams (2003). Then, we build on their approach and an alternative specification to provide new estimates for the United States, Germany, the euro area and Japan. Third, we subject these estimates to a battery of sensitivity tests. Due to the great uncertainty and sensitivity that accompany these equilibrium rate estimates, the observed decline in the estimates is not a reliable indicator of a need for expansionary monetary and fiscal policy. Yet, if these estimates are employed to determine the appropriate monetary policy stance, such estimates are better used together with the consistent estimate of the level of potential output.
The purpose of the data presented in this article is to use it in ex post estimations of interest rate decisions by the European Central Bank (ECB), as it is done by Bletzinger and Wieland (2017) [1]. The data is of quarterly frequency from 1999 Q1 until 2013 Q2 and consists of the ECB's policy rate, inflation rate, real output growth and potential output growth in the euro area. To account for forward-looking decision making in the interest rate rule, the data consists of expectations about future inflation and output dynamics. While potential output is constructed based on data from the European Commission's annual macro-economic database, inflation and real output growth are taken from two different sources both provided by the ECB: the Survey of Professional Forecasters and projections made by ECB staff. Careful attention was given to the publication date of the collected data to ensure a real-time dataset only consisting of information which was available to the decision makers at the time of the decision.