ABSTRACT

With a view to providing a tool to accurately model time series processes which may be corrupted with errors such as measurement, round-off and data aggregation, this study developed an integrated moving average (IMA) model with a transition matrix for the errors resulting in a convex combination of two ARMA errors. Datasets on interest rates in the United States and Nigeria were used to demonstrate the application of the formulated model. Basic tools such as the autocovariance function, maximum likelihood method, Newton–Raphson iterative method and Kolmogorov–Smirnov test statistic were employed to examine and fit the formulated specification to data. Test results showed that the proposed model provided a generalisation and a more flexible specification than the existing models of AR error and ARMA error in fitting time series processes in the presence of errors.

References

1. Ayodeji, I. O. (2016). A three-state Markov-modulated switching model for exchange rates. Journal of Applied Mathematics, 2016, 1–9. doi: 10.1155/2016/5061749 [Crossref], [Google Scholar]
2. Barnett, V. D. (1967). A note on linear structural relationships with both residual variances are known. Biometrika, 63, 39–50. [Google Scholar]
3. Box, G., & Jenkins, G. (1976). Time series analysis: Forecasting and control. San Francisco, CA: Holden Day. [Google Scholar]
4. Engel, C., & Hamilton, J. (1990). Long swings in the dollar: Are they in the data and do market know it? American Economic Review, 80, 687–713. [Web of Science ®], [Google Scholar]
5. Eni, D. (2013). Parameter estimation of first order IMA model in the presence of ARMA(1,1) errors. Research Journal in Engineering and Applied Sciences, 2(4), 246–50. [Google Scholar]
6. Eni, D., & Mahmud, S. A. (2008). On the parameter estimation of first order IMA model corrupted with AR (1) errors. Global Journal of Pure and Applied Sciences, 14(1), 115–20. doi: 10.4314/gjpas.v14i1.16783 [Crossref], [Google Scholar]
7. Eni, D., Ogban, G., Ekpenyong, B., & Atsu, J. (2007). On error handling for a process following AR(1) with MA(1) errors. Journal of Research in Engineering, 4, 102–4. [Google Scholar]
8. Garcia, R., & Perron, P. (1996). An analysis of the real interest rate under regime shifts. The Review of Economics and Statistics, 78, 111–125. doi: 10.2307/2109851 [Crossref], [Web of Science ®], [Google Scholar]
9. Kendall, M. G., & Stuart, A. (1961). The advanced theory of statistics. London: Griffin. [Crossref], [Google Scholar]
10. Lindley, D. V. (1947). Regression lines and the linear functional relationship. Supplement to the Journal of the Royal Statistical Society, 9, 218–44. doi: 10.2307/2984115 [Crossref], [Google Scholar]
11. Madansky, A. (1959). The fitting of straight lines when both variables are subject to error. Journal of the American Statistical Association, 54, 173–205. doi: 10.1080/01621459.1959.10501505 [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
12. Moran, P.A.P. (1971). Estimating structural and functional relationships. Journal of Multivariate Analysis, 1, 232–255. doi: 10.1016/0047-259X(71)90013-3 [Crossref], [Google Scholar]
13. Rudelson, M., & Zhou, S. (2017). Errors-in-variables models with dependent measurements. Electronic Journal of Statistics, 11, 1699–1797. doi: 10.1214/17-EJS1234 [Crossref], [Web of Science ®], [Google Scholar]
14. Schneeweiss, H., & Shalabh, H. (2007). On the estimation of the linear relation when the error variances are known. Computational Statistics & Data Analysis, 52, 1143–1148. doi: 10.1016/j.csda.2007.06.018 [Crossref], [Web of Science ®], [Google Scholar]
15. Yilmazkuday, H., & Akay, K. (2008). An analysis of Turkish economy under regime shifts. Economic Modelling, 25(5), 885–898. doi: 10.1016/j.econmod.2007.11.013 [Crossref], [Web of Science ®], [Google Scholar]