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Structural Analysis of Vector Autoregressive Models

Author

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  • Christis Katsouris
Abstract
This set of lecture notes discuss key concepts for the Structural Analysis of Vector Autoregressive models for the teaching of a course on Applied Macroeconometrics with Advanced Topics.

Suggested Citation

  • Christis Katsouris, 2023. "Structural Analysis of Vector Autoregressive Models," Papers 2312.06402, arXiv.org, revised Feb 2024.
  • Handle: RePEc:arx:papers:2312.06402
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    1. Pesaran, M. Hashem & Shin, Yongcheol & Smith, Richard J., 2000. "Structural analysis of vector error correction models with exogenous I(1) variables," Journal of Econometrics, Elsevier, vol. 97(2), pages 293-343, August.
    2. Clément de Chaisemartin & Xavier D'Haultfœuille, 2020. "Two-Way Fixed Effects Estimators with Heterogeneous Treatment Effects," American Economic Review, American Economic Association, vol. 110(9), pages 2964-2996, September.
    3. Dominik Bertsche & Robin Braun, 2022. "Identification of Structural Vector Autoregressions by Stochastic Volatility," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 40(1), pages 328-341, January.
    4. Jean-Marie Dufour, 2003. "Identification, weak instruments, and statistical inference in econometrics," Canadian Journal of Economics, Canadian Economics Association, vol. 36(4), pages 767-808, November.
    5. Eli Ben-Michael & Avi Feller & Jesse Rothstein, 2021. "The Augmented Synthetic Control Method," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 116(536), pages 1789-1803, October.
    6. Iavor Bojinov & Neil Shephard, 2019. "Time Series Experiments and Causal Estimands: Exact Randomization Tests and Trading," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(528), pages 1665-1682, October.
    7. Brüggemann, Ralf & Jentsch, Carsten & Trenkler, Carsten, 2016. "Inference in VARs with conditional heteroskedasticity of unknown form," Journal of Econometrics, Elsevier, vol. 191(1), pages 69-85.
    8. Chang, Jinyuan & Chen, Song Xi & Chen, Xiaohong, 2015. "High dimensional generalized empirical likelihood for moment restrictions with dependent data," Journal of Econometrics, Elsevier, vol. 185(1), pages 283-304.
    9. Ivana Komunjer & Serena Ng, 2011. "Dynamic Identification of Dynamic Stochastic General Equilibrium Models," Econometrica, Econometric Society, vol. 79(6), pages 1995-2032, November.
    10. Gergely Ganics & Atsushi Inoue & Barbara Rossi, 2021. "Confidence Intervals for Bias and Size Distortion in IV and Local Projections-IV Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 39(1), pages 307-324, January.
    11. Jonas E. Arias & Jesús Fernández- Villaverde & Juan F. Rubio-Ramírez & Minchul Shin, 2023. "The Causal Effects of Lockdown Policies on Health and Macroeconomic Outcomes," American Economic Journal: Macroeconomics, American Economic Association, vol. 15(3), pages 287-319, July.
    12. Pagan, A.R. & Pesaran, M. Hashem, 2008. "Econometric analysis of structural systems with permanent and transitory shocks," Journal of Economic Dynamics and Control, Elsevier, vol. 32(10), pages 3376-3395, October.
    13. Markku Lanne & Keyan Liu & Jani Luoto, 2023. "Identifying Structural Vector Autoregression via Leptokurtic Economic Shocks," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 41(4), pages 1341-1351, October.
    14. Melard, Guy & Roy, Roch & Saidi, Abdessamad, 2006. "Exact maximum likelihood estimation of structured or unit root multivariate time series models," Computational Statistics & Data Analysis, Elsevier, vol. 50(11), pages 2958-2986, July.
    15. Chatfield, Chris, 1993. "Calculating Interval Forecasts: Reply," Journal of Business & Economic Statistics, American Statistical Association, vol. 11(2), pages 143-144, April.
    16. Helmut Lütkepohl & Mika Meitz & Aleksei Netšunajev & Pentti Saikkonen, 2021. "Testing identification via heteroskedasticity in structural vector autoregressive models," The Econometrics Journal, Royal Economic Society, vol. 24(1), pages 1-22.
    17. Patrick Fève & Alain Guay, 2010. "Identification of Technology Shocks in Structural Vars," Economic Journal, Royal Economic Society, vol. 120(549), pages 1284-1318, December.
    18. Mark Bognanni, 2018. "A Class of Time-Varying Parameter Structural VARs for Inference under Exact or Set Identification," Working Papers (Old Series) 1811, Federal Reserve Bank of Cleveland.
    19. Mikkel Plagborg-Møller & Christian K. Wolf, 2022. "Instrumental Variable Identification of Dynamic Variance Decompositions," Journal of Political Economy, University of Chicago Press, vol. 130(8), pages 2164-2202.
    20. Lanne, Markku & Saikkonen, Pentti, 2002. "Threshold Autoregressions for Strongly Autocorrelated Time Series," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(2), pages 282-289, April.
    21. James H. Stock & Mark W. Watson, 2018. "Identification and Estimation of Dynamic Causal Effects in Macroeconomics Using External Instruments," Economic Journal, Royal Economic Society, vol. 128(610), pages 917-948, May.
    22. Markku Lanne & Henri Nyberg, 2016. "Generalized Forecast Error Variance Decomposition for Linear and Nonlinear Multivariate Models," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 78(4), pages 595-603, August.
    23. Diebold, Francis X. & Inoue, Atsushi, 2001. "Long memory and regime switching," Journal of Econometrics, Elsevier, vol. 105(1), pages 131-159, November.
    24. Chatfield, Chris, 1993. "Calculating Interval Forecasts," Journal of Business & Economic Statistics, American Statistical Association, vol. 11(2), pages 121-135, April.
    25. Jean‐Marie Dufour, 2003. "Identification, weak instruments, and statistical inference in econometrics," Canadian Journal of Economics/Revue canadienne d'économique, John Wiley & Sons, vol. 36(4), pages 767-808, November.
    26. Lanne, Markku & Lütkepohl, Helmut, 2010. "Structural Vector Autoregressions With Nonnormal Residuals," Journal of Business & Economic Statistics, American Statistical Association, vol. 28(1), pages 159-168.
    27. Cheng, Tingting & Gao, Jiti & Yan, Yayi, 2019. "Regime switching panel data models with interactive fixed effects," Economics Letters, Elsevier, vol. 177(C), pages 47-51.
    28. Sascha Alexander Keweloh, 2021. "A Generalized Method of Moments Estimator for Structural Vector Autoregressions Based on Higher Moments," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 39(3), pages 772-782, July.
    29. Fries, Sébastien & Zakoian, Jean-Michel, 2019. "Mixed Causal-Noncausal Ar Processes And The Modelling Of Explosive Bubbles," Econometric Theory, Cambridge University Press, vol. 35(6), pages 1234-1270, December.
    30. Moon, Hyungsik Roger & Schorfheide, Frank, 2002. "Minimum Distance Estimation Of Nonstationary Time Series Models," Econometric Theory, Cambridge University Press, vol. 18(6), pages 1385-1407, December.
    31. Yu-Chin Hsu & Tsung-Chih Lai & Robert P. Lieli, 2022. "Counterfactual Treatment Effects: Estimation and Inference," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 40(1), pages 240-255, January.
    32. Katarzyna Budnik & Gerhard Rünstler, 2023. "Identifying structural VARs from sparse narrative instruments: Dynamic effects of US macroprudential policies," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 38(2), pages 186-201, March.
    33. Aigner, Dennis J & Balestra, Pietro, 1988. "Optimal Experimental Design for Error Components Models," Econometrica, Econometric Society, vol. 56(4), pages 955-971, July.
    34. Lanne, Markku & Meitz, Mika & Saikkonen, Pentti, 2017. "Identification and estimation of non-Gaussian structural vector autoregressions," Journal of Econometrics, Elsevier, vol. 196(2), pages 288-304.
    35. Leena Kalliovirta & Mika Meitz & Pentti Saikkonen, 2015. "A Gaussian Mixture Autoregressive Model for Univariate Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 36(2), pages 247-266, March.
    36. Dolado, Juan J., 1992. "A note on weak exogeneity in VAR cointegrated models," Economics Letters, Elsevier, vol. 38(2), pages 139-143, February.
    37. Davis, Richard A. & Song, Li, 2020. "Noncausal vector AR processes with application to economic time series," Journal of Econometrics, Elsevier, vol. 216(1), pages 246-267.
    38. Christiane Baumeister & James D. Hamilton, 2019. "Structural Interpretation of Vector Autoregressions with Incomplete Identification: Revisiting the Role of Oil Supply and Demand Shocks," American Economic Review, American Economic Association, vol. 109(5), pages 1873-1910, May.
    39. Christiane Baumeister & James D. Hamilton, 2015. "Sign Restrictions, Structural Vector Autoregressions, and Useful Prior Information," Econometrica, Econometric Society, vol. 83(5), pages 1963-1999, September.
    40. Yuriy Gorodnichenko & Byoungchan Lee, 2020. "Forecast Error Variance Decompositions with Local Projections," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 38(4), pages 921-933, October.
    41. Regis Barnichon & Christian Brownlees, 2019. "Impulse Response Estimation by Smooth Local Projections," The Review of Economics and Statistics, MIT Press, vol. 101(3), pages 522-530, July.
    42. Karel Mertens & Morten O. Ravn, 2013. "The Dynamic Effects of Personal and Corporate Income Tax Changes in the United States," American Economic Review, American Economic Association, vol. 103(4), pages 1212-1247, June.
    43. Peter Boswijk, H., 1994. "Testing for an unstable root in conditional and structural error correction models," Journal of Econometrics, Elsevier, vol. 63(1), pages 37-60, July.
    44. Lanne, Markku & Saikkonen, Pentti, 2013. "Noncausal Vector Autoregression," Econometric Theory, Cambridge University Press, vol. 29(3), pages 447-481, June.
    45. Leandro M. Magnusson & Sophocles Mavroeidis, 2014. "Identification Using Stability Restrictions," Econometrica, Econometric Society, vol. 82(5), pages 1799-1851, September.
    46. Eric M. Leeper & Todd B. Walker & Shu‐Chun Susan Yang, 2013. "Fiscal Foresight and Information Flows," Econometrica, Econometric Society, vol. 81(3), pages 1115-1145, May.
    47. Petrova, Katerina, 2022. "Asymptotically valid Bayesian inference in the presence of distributional misspecification in VAR models," Journal of Econometrics, Elsevier, vol. 230(1), pages 154-182.
    48. Paruolo, Paolo, 2002. "Asymptotic Inference On The Moving Average Impact Matrix In Cointegrated I (2) Var Systems," Econometric Theory, Cambridge University Press, vol. 18(3), pages 673-690, June.
    49. Carlos A. Vegh & Guillermo Vuletin, 2015. "How Is Tax Policy Conducted over the Business Cycle?," American Economic Journal: Economic Policy, American Economic Association, vol. 7(3), pages 327-370, August.
    50. Hausman, Jerry A & Taylor, William E, 1983. "Identification in Linear Simultaneous Equations Models with Covariance Restrictions: An Instrumental Variables Interpretation," Econometrica, Econometric Society, vol. 51(5), pages 1527-1549, September.
    51. Paruolo, Paolo, 2000. "Asymptotic Efficiency Of The Two Stage Estimator In I (2) Systems," Econometric Theory, Cambridge University Press, vol. 16(4), pages 524-550, August.
    52. Tom Beernaert & Nicolas Soenen & Rudi Vander Vennet, 2023. "ECB Monetary Policy and the Term Structure of Bank Default Risk," JRFM, MDPI, vol. 16(12), pages 1-22, December.
    53. Phillips, Peter C. B., 1998. "Impulse response and forecast error variance asymptotics in nonstationary VARs," Journal of Econometrics, Elsevier, vol. 83(1-2), pages 21-56.
    54. Krampe, J. & Paparoditis, E. & Trenkler, C., 2023. "Structural inference in sparse high-dimensional vector autoregressions," Journal of Econometrics, Elsevier, vol. 234(1), pages 276-300.
    55. Lanne, Markku & Lütkepohl, Helmut & Maciejowska, Katarzyna, 2010. "Structural vector autoregressions with Markov switching," Journal of Economic Dynamics and Control, Elsevier, vol. 34(2), pages 121-131, February.
    56. Mikkel Plagborg‐Møller & Christian K. Wolf, 2021. "Local Projections and VARs Estimate the Same Impulse Responses," Econometrica, Econometric Society, vol. 89(2), pages 955-980, March.
    57. Miao, Ke & Phillips, Peter C.B. & Su, Liangjun, 2023. "High-dimensional VARs with common factors," Journal of Econometrics, Elsevier, vol. 233(1), pages 155-183.
    58. Cheng Hsiao, 1997. "Statistical Properties of the Two-Stage Least Squares Estimator Under Cointegration," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 64(3), pages 385-398.
    59. Medeiros, Marcelo C. & Mendes, Eduardo F., 2016. "ℓ1-regularization of high-dimensional time-series models with non-Gaussian and heteroskedastic errors," Journal of Econometrics, Elsevier, vol. 191(1), pages 255-271.
    60. Karel Mertens & MortenO. Ravn, 2010. "Measuring the Impact of Fiscal Policy in the Face of Anticipation: A Structural VAR Approach," Economic Journal, Royal Economic Society, vol. 120(544), pages 393-413, May.
    61. Peter C. B. Phillips & Bruce E. Hansen, 1990. "Statistical Inference in Instrumental Variables Regression with I(1) Processes," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 57(1), pages 99-125.
    62. Pretis, Felix, 2021. "Exogeneity in climate econometrics," Energy Economics, Elsevier, vol. 96(C).
    63. Montiel Olea, José L. & Stock, James H. & Watson, Mark W., 2021. "Inference in Structural Vector Autoregressions identified with an external instrument," Journal of Econometrics, Elsevier, vol. 225(1), pages 74-87.
    64. Richard T. Baillie & George Kapetanios, 2013. "Estimation and inference for impulse response functions from univariate strongly persistent processes," Econometrics Journal, Royal Economic Society, vol. 16(3), pages 373-399, October.
    65. Guay, Alain, 2021. "Identification of structural vector autoregressions through higher unconditional moments," Journal of Econometrics, Elsevier, vol. 225(1), pages 27-46.
    66. Paruolo, Paolo & Rahbek, Anders, 1999. "Weak exogeneity in I(2) VAR systems," Journal of Econometrics, Elsevier, vol. 93(2), pages 281-308, December.
    67. Iavor Bojinov & Ashesh Rambachan & Neil Shephard, 2021. "Panel experiments and dynamic causal effects: A finite population perspective," Quantitative Economics, Econometric Society, vol. 12(4), pages 1171-1196, November.
    68. Meitz, Mika & Saikkonen, Pentti, 2021. "Testing for observation-dependent regime switching in mixture autoregressive models," Journal of Econometrics, Elsevier, vol. 222(1), pages 601-624.
    69. Christian Gouriéroux & Alain Monfort & Jean-Paul Renne, 2020. "Identification and Estimation in Non-Fundamental Structural VARMA Models," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 87(4), pages 1915-1953.
    70. Karel Vermeulen & Stijn Vansteelandt, 2015. "Bias-Reduced Doubly Robust Estimation," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(511), pages 1024-1036, September.
    71. Hecq, Alain & Voisin, Elisa, 2021. "Forecasting bubbles with mixed causal-noncausal autoregressive models," Econometrics and Statistics, Elsevier, vol. 20(C), pages 29-45.
    72. Andrews, Beth & Davis, Richard A., 2013. "Model identification for infinite variance autoregressive processes," Journal of Econometrics, Elsevier, vol. 172(2), pages 222-234.
    73. Markku Lanne & Jani Luoto, 2021. "GMM Estimation of Non-Gaussian Structural Vector Autoregression," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 39(1), pages 69-81, January.
    74. Kuersteiner, Guido M., 2005. "Automatic Inference For Infinite Order Vector Autoregressions," Econometric Theory, Cambridge University Press, vol. 21(1), pages 85-115, February.
    75. Òscar Jordà, 2005. "Estimation and Inference of Impulse Responses by Local Projections," American Economic Review, American Economic Association, vol. 95(1), pages 161-182, March.
    76. Serena Ng & Pierre Perron, 2001. "LAG Length Selection and the Construction of Unit Root Tests with Good Size and Power," Econometrica, Econometric Society, vol. 69(6), pages 1519-1554, November.
    77. Phillips, P C B, 1991. "Optimal Inference in Cointegrated Systems," Econometrica, Econometric Society, vol. 59(2), pages 283-306, March.
    78. Kilian,Lutz & Lütkepohl,Helmut, 2018. "Structural Vector Autoregressive Analysis," Cambridge Books, Cambridge University Press, number 9781107196575, September.
    79. Kalliovirta, Leena & Meitz, Mika & Saikkonen, Pentti, 2016. "Gaussian mixture vector autoregression," Journal of Econometrics, Elsevier, vol. 192(2), pages 485-498.
    80. Gafarov, Bulat & Meier, Matthias & Montiel Olea, José Luis, 2018. "Delta-method inference for a class of set-identified SVARs," Journal of Econometrics, Elsevier, vol. 203(2), pages 316-327.
    81. Lutz Kilian & Yun Jung Kim, 2011. "How Reliable Are Local Projection Estimators of Impulse Responses?," The Review of Economics and Statistics, MIT Press, vol. 93(4), pages 1460-1466, November.
    82. Thórarinn G. Pétursson & Torsten Sløk, 2001. "Wage formation and employment in a cointegrated VAR model," Econometrics Journal, Royal Economic Society, vol. 4(2), pages 1-2.
    83. Saikkonen, Pentti, 1995. "Problems with the Asymptotic Theory of Maximum Likelihood Estimation in Integrated and Cointegrated Systems," Econometric Theory, Cambridge University Press, vol. 11(5), pages 888-911, October.
    84. Christis Katsouris, 2023. "Limit Theory under Network Dependence and Nonstationarity," Papers 2308.01418, arXiv.org, revised Aug 2023.
    85. Pretis, Felix, 2020. "Econometric modelling of climate systems: The equivalence of energy balance models and cointegrated vector autoregressions," Journal of Econometrics, Elsevier, vol. 214(1), pages 256-273.
    86. José Luis Montiel Olea & Mikkel Plagborg‐Møller, 2021. "Local Projection Inference Is Simpler and More Robust Than You Think," Econometrica, Econometric Society, vol. 89(4), pages 1789-1823, July.
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