## E17 Forecasting and Simulation

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- DSGE (4)
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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.

This paper studies the macro-financial implications of using carbon prices to achieve ambitious greenhouse gas (GHG) emission reduction targets. My empirical evidence shows a 0.6% output loss and a rise of 0.3% in inflation in response to a 1% shock on carbon policy. Furthermore, I also observe financial instability and allocation effects between the clean and highly polluted energy sectors. To have a better prediction of medium and long-term impact, using a medium-large macro-financial DSGE model with environmental aspects, I show the recessionary effect of an ambitious carbon price implementation to achieve climate targets, a 40% reduction in GHG emission causes a 0.7% output loss while reaching a zero-emission economy in 30 years causes a 2.6% output loss. I document an amplified effect of the banking sector during the transition path. The paper also uncovers the beneficial role of pre-announcements of carbon policies in mitigating inflation volatility by 0.2% at its peak, and our results suggest well-communicated carbon policies from authorities and investing to expand the green sector. My findings also stress the use of optimal green monetary and financial policies in mitigating the effects of transition risk and assisting the transition to a zero-emission world. Utilizing a heterogeneous approach with macroprudential tools, I find that optimal macroprudential tools can mitigate the output loss by 0.1% and investment loss by 1%. Importantly, my work highlights the use of capital flow management in the green transition when a global cooperative solution is challenging.

Climate change has become one of the most prominent concerns globally. In this paper, the authors study the transition risk of greenhouse gas emission reduction in structural environmental-macroeconomic DSGE models. First, they analyze the uncertainty in model prediction on the effect of unanticipated and pre-announced carbon price increases. Second, they conduct optimal model-robust policy in different settings. They find that reducing emissions by 40% causes 0.7% to 4% output loss with 2% on average. Pre-announcement of carbon prices affects the inflation dynamics significantly. The central bank should react slightly less to inflation and output growth during the transition risk. With optimal carbon price designs, it should react even less to inflation, and more to output growth.

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.