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A (pedagogical) note on the log-linearization of functions of several variables

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  • Solis-Garcia, Mario

Abstract

Dynamic stochastic general equilibrium (DSGE) models are analytically intractable and numerical methods must be used to approximate a solution. A key input shared by many solution methods is log-linearization. While the basics of the procedure have been extensively documented, applying the methodology to complicated functions of model variables remains uncharted territory, often resulting in cumbersome and error-prone calculations. This paper offers a procedure – the log-linear product approach – that automates and simplifies this task, as I show with a full working example. The procedure relies on the basic fact that the product of second order terms is zero when dealing with a linear expansion.

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  • Solis-Garcia, Mario, 2021. "A (pedagogical) note on the log-linearization of functions of several variables," International Review of Economics Education, Elsevier, vol. 37(C).
    • Handle: RePEc:eee:ireced:v:37:y:2021:i:c:s1477388021000025
    DOI: 10.1016/j.iree.2021.100210
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    References listed on IDEAS

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    1. Lucas, Robert E, Jr & Prescott, Edward C, 1971. "Investment Under Uncertainty," Econometrica, Econometric Society, vol. 39(5), pages 659-681, September.
    2. Harald Uhlig, 1995. "A toolkit for analyzing nonlinear dynamic stochastic models easily," Discussion Paper / Institute for Empirical Macroeconomics 101, Federal Reserve Bank of Minneapolis.
    3. Ellen R. McGrattan, 2012. "Capital Taxation During the U.S. Great Depression," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 127(3), pages 1515-1550.
    4. Lawrence J. Christiano & Martin Eichenbaum & Charles L. Evans, 2005. "Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy," Journal of Political Economy, University of Chicago Press, vol. 113(1), pages 1-45, February.
    5. Blanchard, Olivier Jean & Kahn, Charles M, 1980. "The Solution of Linear Difference Models under Rational Expectations," Econometrica, Econometric Society, vol. 48(5), pages 1305-1311, July.
    6. Fernández-Villaverde, J. & Rubio-Ramírez, J.F. & Schorfheide, F., 2016. "Solution and Estimation Methods for DSGE Models," Handbook of Macroeconomics, in: J. B. Taylor & Harald Uhlig (ed.), Handbook of Macroeconomics, edition 1, volume 2, chapter 0, pages 527-724, Elsevier.
    7. Uhlig, H.F.H.V.S., 1995. "A toolkit for analyzing nonlinear dynamic stochastic models easily," Other publications TiSEM cc1b2469-9d2f-445a-a2b3-1, Tilburg University, School of Economics and Management.
    8. J. B. Taylor & Harald Uhlig (ed.), 2016. "Handbook of Macroeconomics," Handbook of Macroeconomics, Elsevier, edition 1, volume 2, number 2.
    9. Marimon, Ramon & Scott, Andrew (ed.), 1999. "Computational Methods for the Study of Dynamic Economies," OUP Catalogue, Oxford University Press, number 9780198294979.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    DSGE models; Log-linear approximation;

    JEL classification:

    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools
    • E00 - Macroeconomics and Monetary Economics - - General - - - General

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