Advertisement

Inference Based on the Vector Autoregressive and Moving Average Model

  • Yuzo HosoyaEmail author
  • Kosuke Oya
  • Taro Takimoto
  • Ryo Kinoshita
Chapter
  • 657 Downloads
Part of the SpringerBriefs in Statistics book series (BRIEFSSTATIST)

Abstract

Based on the stationary vector ARMA process, this chapter shows how the partial measures of interdependence introduced in Sect.  3.3 are numerically evaluated and applied to practical situations. Section 4.1 discusses the statistical inference on those measures using the standard asymptotic theory of the Whittle likelihood inference for stationary multivariate ARMA processes. The point is the use of simulation-based estimations of the covariance matrix of each measure-related statistic. In Sect. 4.2, we investigate the small sample performance of partial one-way effect measure estimates using Monte Carlo data generated by a pair of trivariate data generating processes, the VAR(2) and VARMA(1,1) models. All model parameter estimates are produced using an improved version of the Takimoto and Hosoya (2004, 2006) procedure. The partial frequency-wise measures of the one-way effect are evaluated using spectral factorization, and the parameters are substituted with a modified Whittle estimate. To illustrate the analysis of interdependence in the frequency domain, Sect. 4.3 provides an empirical analysis of US interest rates and economic growth data.

Keywords

Inference procedure Monte Carlo Wald test Partial measures of interdependence Small sample performance US macroeconomic data Vector ARMA process 

References

  1. Assenmacher-Wesche, K., Gerlach, S., & Sekine, T. (2008). Monetary factors and inflation in Japan. Journal of the Japanese and International Economies, 22, 343–363.CrossRefGoogle Scholar
  2. Breitung, J., & Candelon, B. (2006). Testing for short- and long-run causality: A frequency-domain approach. Journal of Econometrics, 132, 363–378.CrossRefzbMATHMathSciNetGoogle Scholar
  3. Dufour, J.M., & Pelletier, D. (2011). Practical methods for modelling weak VARMA prcoesses: Identification, estimation and specialization with macroeconomic application, working paper.Google Scholar
  4. Durbin, J. (1960). The fitting of time-series models. International Statistical Review, 33, 233–244.CrossRefzbMATHGoogle Scholar
  5. Geweke, J. (1984). Measures of conditional linear dependence and feedback between time series. Journal of the American Statistical Association, 79, 907–915.CrossRefzbMATHMathSciNetGoogle Scholar
  6. Granger, C. W. J. (1997). The ET interview: Professor Clive Granger. Econometric Theory, 13, 253–303.CrossRefGoogle Scholar
  7. Granger, C. W. J. (1999). Empirical Modeling in Economics: Specification and Evaluation. Cambridge: Cambridge University Press.Google Scholar
  8. Gronwald, M. (2009). Reconsidering the macroeconomics of the oil price in Germany: Testing for causality in the frequency domain. Empirical Economics, 36, 441–453.CrossRefGoogle Scholar
  9. Hamilton, J. D., & Kim, D. H. (2002). A reexamination of the predictability of economic activity using the yield spread. Journal of Money, Credit and Banking, 34, 340–360.CrossRefGoogle Scholar
  10. Hannan, E. J., & Rissanen, J. (1982). Recursive estimation of mixed autoregressive-moving average order. Biometrika, 69, 81–94.CrossRefzbMATHMathSciNetGoogle Scholar
  11. Hannan, E. J., & Kavalieris, L. (1984a). A method for autoregressive-moving average estimation. Biometrika, 71, 273–280.CrossRefzbMATHMathSciNetGoogle Scholar
  12. Hannan, E. J., & Kavalieris, L. (1984b). Multivarate linear time series models. Advances of Applied Probability, 16, 492–561.Google Scholar
  13. Hosoya, Y. (1997). A limit theory for long-range dependence and statistical inference on related models. The Annals of Statistics, 25, 105–137.Google Scholar
  14. Hosoya, T., & Takimoto, T. (2010). A numerical method for factorizing the rational spectral density matrix. Journal of Time Series Analysis, 31, 229–240.zbMATHMathSciNetGoogle Scholar
  15. Johansen, S. (1995). Likelihood-based Inference in Cointegrated Autoregressive Models. Oxford: Oxford University Press.Google Scholar
  16. Judd, K. L. (1999). Numerical Methods in Economics. Cambridge: The MIT Press.zbMATHGoogle Scholar
  17. Kim, K. A., & Limpaphayom, P. (1997). The effect of economic regimes on the relation between term structures and real activity in Japan. Journal of Economic and Business, 49, 379–392.CrossRefGoogle Scholar
  18. Sims, C. A. (1980). Comparison of interwar and postwar business cycles: Monetarism reconsidered. The American Economic Review, 70, 250–257.Google Scholar
  19. Staiger, D., Stock, J. H., & Watson, M. W. (1997). The NAIRU, unemployment and monetary policy. Journal of Economic Perspective, 11, 33–49.CrossRefGoogle Scholar
  20. Stock, J. H., & Watson, M. W. (2003). Forecasting output and inflation: The role of asset prices, Journal of Economic Literature, XLI, 788–829.Google Scholar
  21. Takimoto, T., & Hosoya, Y. (2004). A three-step procedure for estimating and testing cointegrated ARMAX models. The Japanese Economic Review, 55, 418–450.CrossRefMathSciNetGoogle Scholar
  22. Takimoto, T., & Hosoya, Y. (2006). Inference on the cointegration rank and a procedure for VARMA root-modification. Journal of Japan Statistical Society, 36, 149–171.CrossRefzbMATHMathSciNetGoogle Scholar
  23. Wheelock, D. C., & Wohar, M. E. (2009). Can the term spread predict output growth and recession? Federal Reserve Bank of St. Louis Review, September/October, Part 1, 419–440.Google Scholar
  24. Whittle, P. (1952). Some results in time series analysis. Skandinavisk Aktuarietidskrift, 1–2, 48–60.zbMATHMathSciNetGoogle Scholar
  25. Whittle, P. (1953). The analysis of multiple stationary time series. Journal of the Royal Statistical Society B, 15, 125–139.zbMATHMathSciNetGoogle Scholar
  26. Yao, F., & Hosoya, Y. (2000). Inference on one-way effect and evidence in Japanese macroeconomic data. Journal of Econometrics, 98, 225–255.CrossRefzbMATHMathSciNetGoogle Scholar
  27. Yap, S. F., & Reinsel, G. C. (1995). Estimation and testing for unit roots in a partially nonstationary vector autoregressive moving average model. Journal of the American Statistical Association, 90, 253–267.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Yuzo Hosoya
    • 1
    Email author
  • Kosuke Oya
    • 2
  • Taro Takimoto
    • 3
  • Ryo Kinoshita
    • 4
  1. 1.Tohoku UniversitySendaiJapan
  2. 2.Osaka UniversityToyonakaJapan
  3. 3.Kyushu UniversityFukuokaJapan
  4. 4.Tokyo Keizai UniversityKokubunjiJapan

Personalised recommendations