Abstract
For policy decisions, capturing seasonal effects in impulse responses are important for the correct specification of dynamic models that measure interaction effects for policy-relevant macroeconomic variables. In this paper, a new multivariate method is suggested, which uses the score-driven quasi-vector autoregressive (QVAR) model, to capture seasonal effects in impulse response functions (IRFs). The nonlinear QVAR-based method is compared with the existing linear VAR-based method. The following technical aspects of the new method are presented: (i) mathematical formulation of QVAR; (ii) first-order representation and infinite vector moving average, VMA (∞), representation of QVAR; (iii) IRF of QVAR; (iv) statistical inference of QVAR and conditions of consistency and asymptotic normality of the estimates. Control data are used for the period of 1987:Q1 to 2013:Q2, from the following policy-relevant macroeconomic variables: crude oil real price, United States (US) inflation rate, and US real gross domestic product (GDP). A graphical representation of seasonal effects among variables is provided, by using the IRF. According to the estimation results, annual seasonal effects are almost undetected by using the existing linear VAR tool, but those effects are detected by using the new QVAR tool.
Funding source: Comunidad de Madrid
Award Identifier / Grant number: MadEco-CM S2015/HUM-3444
Funding source: Ministerio de Economía, Industria y Competitividad
Award Identifier / Grant number: ECO2016-00105-001
Award Identifier / Grant number: MDM 2014-0431
Funding source: Universidad Francisco Marroquín
Acknowledgment
Previous versions of this paper were presented in “Recent Advances in Econometrics: International Conference in Honor of Luc Bauwens” (Brussels, 19–20 October 2017), GESG Research Seminar (Guatemala City, 9 November 2017), “Workshop in Time Series Econometrics” (Zaragoza, 12–13 April 2018), and “International Conference on Statistical Methods for Big Data” (Madrid, 7–8 July 2018). The authors are thankful to the reviewer and the editor of the journal, Luc Bauwens, Matthew Copley, Antoni Espasa, Eric Ghysels, Joachim Grammig, Andrew Harvey, Søren Johansen, Òscar Jordà, Bent Nielsen, Eric Renault, Genaro Sucarrat, and Ruey Tsay. All remaining errors are our own. Blazsek and Licht acknowledge funding from Universidad Francisco Marroquín. Escribano acknowledges funding from Ministerio de Economía, Industria y Competitividad (ECO2016-00105-001 and MDM 2014-0431), and Comunidad de Madrid (MadEco-CM S2015/HUM-3444).
References
Alemany, N., V. Aragó, and E. Salvador. 2019. “The Influence of Intraday Seasonality on Volatility Transmission Pattern.” Quantitative Finance 19 (7): 1179–97, https://doi.org/10.1080/14697688.2018.1563304.Search in Google Scholar
Alsmeyer, G. 2003. “On the Harris Recurrence of Iterated Random Lipschitz Functions and Related Convergence Rate Results.” Journal of Theoretical Probability 16 (1): 217–47, https://doi.org/10.1023/a:1022290807360.10.1023/A:1022290807360Search in Google Scholar
Barsky, R. B., and L. Kilian. 2004. “Oil and the Macroeconomy since the 1970s’.” Journal of Economic Perspectives 18 (4): 115–34, https://doi.org/10.1257/0895330042632708.Search in Google Scholar
Blanchard, O. J. 2002. “Comments on “Do We Really Know that Oil Caused the Great Stagnation? A Monetary Alternative” by Robert Barsky and Lutz Kilian.” In NBER Macroeconomics Annual, 183–92, edited by B. S. Bernanke, and K. Rogoff. Cambridge, MA: MIT Press.10.1086/654440Search in Google Scholar
Blazsek, S., and A. Escribano. 2017. “Score-Driven Nonlinear Multivariate Dynamic Location Models.” Department of Economics, University Carlos III of Madrid, Working Paper 17-08.Search in Google Scholar
Box, G. E. P., and G. M. Jenkins. 1970. Time Series Analysis, Forecasting and Control. San Francisco, CA: Holden-Day.Search in Google Scholar
Brandt, A. 1986. “The Stochastic Equation Yn+1 = AnYn + Bn with Stationary Coefficients.” Advances in Applied Probability 18 (1): 211–20.10.2307/1427243Search in Google Scholar
Creal, D., S. J. Koopman, and A. Lucas. 2011. “A Dynamic Multivariate Heavy-Tailed Model for Time-Varying Volatility and Correlations.” Journal of Business & Economic Statistics 29 (4): 552–63, https://doi.org/10.1198/jbes.2011.10070.Search in Google Scholar
Creal, D., S. J. Koopman, and A. Lucas. 2013. “Generalized Autoregressive Score Models with Applications.” Journal of Applied Econometrics 28 (5): 777–95, https://doi.org/10.1002/jae.1279.Search in Google Scholar
Creal, D., B. Schwaab, S. J. Koopman, and A. Lucas. 2014. “Observation-Driven Mixed-Measurement Dynamic Factor Models with An Application to Credit Risk.” The Review of Economics and Statistics 96 (5): 898–915, https://doi.org/10.1162/rest_a_00393.Search in Google Scholar
Elton, J. H. 1990. “A Multiplicative Ergodic Theorem for Lipschitz maps.” Stochastic Processes and their Applications 34 (5): 39–47.10.1016/0304-4149(90)90055-WSearch in Google Scholar
Dickey, D. A., and W. A. Fuller. 1979. “Distribution of the Estimators for Autoregressive Time Series with a Unit Root.” Journal of the American Statistical Association 74 (36): 427–31, https://doi.org/10.2307/2286348.Search in Google Scholar
Franses, P. H., S. Hylleberg, and H. S. Lee. 1995. “Spurious Deterministic Seasonality.” Economics Letters 48 (3–4): 249–56, https://doi.org/10.1016/0165-1765(94)00638-i.Search in Google Scholar
Gouriéroux, C., A. Monfort, and A. Trognon. 1984. “Pseudo Maximum Likelihood Methods: Theory.” Econometrica 52 (3): 661–700, https://doi.org/10.2307/1913471.Search in Google Scholar
Hamilton, J. D. 1994. Time Series Analysis. Princeton, NJ: Princeton University Press.10.1515/9780691218632Search in Google Scholar
Harvey, A. C. 2013. Dynamic Models for Volatility and Heavy Tails. Cambridge, UK: Cambridge University Press.10.1017/CBO9781139540933Search in Google Scholar
Harvey, A., and A. Luati. 2014. “Filtering with Heavy Tails.” Journal of the American Statistical Association 109 (507): 1112–22, https://doi.org/10.1080/01621459.2014.887011.Search in Google Scholar
Harvey, A., and A. Scott. 1994. “Seasonality in Dynamic Regression Models.” The Economic Journal 104 (427): 1324–45, https://doi.org/10.2307/2235451.Search in Google Scholar
Herwartz, H., and H. Lütkepohl. 2000. “Multivariate Volatility Analysis of VW Stock Prices.” International Journal of Intelligent Systems in Accounting, Finance & Management 9 (1): 35–54, https://doi.org/10.1002/(sici)1099-1174(200003)9:1<35::aid-isaf176>3.0.co;2-v.10.1002/(SICI)1099-1174(200003)9:1<35::AID-ISAF176>3.0.CO;2-VSearch in Google Scholar
Jordà, Ò. 2005. “Estimation and Inference of Impulse Responses by Local Projections.” The American Economic Review 95 (1): 161–82, https://doi.org/10.1257/0002828053828518.Search in Google Scholar
Kilian, L. 2008. “A Comparison of the Effects of Exogenous Oil Supply Shocks on Output and Inflation in the G7 Countries.” Journal of the European Economic Association 6 (1): 78–121, https://doi.org/10.1162/jeea.2008.6.1.78.Search in Google Scholar
Kilian, L., and H. Lütkepohl. 2017. Structural Vector Autoregressive Analysis. Cambridge, UK: Cambridge University Press.10.1017/9781108164818Search in Google Scholar
Lütkepohl, H. 2005. New Introduction to Multivariate Time Series Analysis. Berlin, Germany: Springer-Verlag.10.1007/978-3-540-27752-1Search in Google Scholar
Straumann, D., and T. Mikosch. 2006. “Quasi-Maximum-Likelihood Estimation in Conditionally Heteroscedastic Time Series: A Stochastic Recurrence Equations Approach.” The Annals of Statistics 34: 2449–95.10.1214/009053606000000803Search in Google Scholar
White, H. 1984. Asymptotic Theory for Econometricians. San Diego, CA: Academic Press.Search in Google Scholar
Supplementary Material
The online version of this article offers supplementary material (https://doi.org/10.1515/jem-2020-0003).
© 2020 Walter de Gruyter GmbH, Berlin/Boston
