×

Polynomial cointegration. Estimation and test. (English) Zbl 0806.62096

Summary: This paper develops statistical tools to analyze the multivariate time series which can be represented by a polynomial error correction model as introduced by the authors in Econ. Theory 9, 329-342 (1993). We propose an identification criterion for the error correction terms, which fits with the estimation procedure. The estimation proceeds in a number of steps, through repeated applications of principal component analysis: test for the overall cointegration dimension and its decomposition into the dimensions of the error correction terms of various degrees, estimation of the error correction terms themselves, and finally estimation of the full model, given the previous results, by ordinary least squares and overall specification test. The asymptotic distribution of the test statistics at each of these various steps is nonstandard, and we provide statistical tables of its main percentiles, for the two cases where there is (or not) a constant term on the right-hand side of the model and the degree of the polynomial error correction terms is at most two.

MSC:

62P20 Applications of statistics to economics
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62H25 Factor analysis and principal components; correspondence analysis
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Dickey, D. A.; Fuller, W. A.: Distribution of the estimators for auto regressive time series with a unit root. Journal of the American statistical association 74, 427-431 (1979) · Zbl 0413.62075
[2] Engle, R. F.; Granger, C. W. J.: Co-integration and error correction: representation, estimation and testing. Econometrica 55, 251-276 (1987) · Zbl 0613.62140
[3] Granger, C. W. J.: Some properties of time series data and their use in econometric model specification. Journal of econometrics, 121-130 (1981)
[4] Granger, C. W. J.; Lee, T. H.: Investigation of production, sales and inventory relationships using multi cointegration and non-symmetric error correction models. Journal of applied econometrics 4, 145-159 (1989)
[5] Granger, C. W. J.; Lee, T. H.: Multi cointegration. Advances in econometrics: cointegration, spurious regressions and unit roots (1989)
[6] Granger, C. W. J.; Weiss, A. A.: Time series analysis of error correcting models. Studies in econometrics, time series and multivariate statistics, 225-278 (1983) · Zbl 0547.62060
[7] Gregoir, S.; Laroque, G.: Multivariate time series: A general error correction representation theorem. Econometric theory 9, 329-342 (1993)
[8] Johansen, S.: Statistical analysis of cointegration vectors. Journal of economic dynamics and control 12, 231-254 (1988) · Zbl 0647.62102
[9] Johansen, S.: Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59, 1551-1580 (1991) · Zbl 0755.62087
[10] Johansen, S.: A statistical analysis of cointegration for \(I(2)\) variables. Econometric theory (1991) · Zbl 1274.62597
[11] Johansen, S.: Identifying restrictions of linear equations. Mimeo (1992)
[12] Johansen, S.; Juselius, K.: Identification of the long-run and the short-run structure: an application to the ISLM model. Journal of econometrics (1994) · Zbl 0806.62097
[13] Magnus, J. R.; Neudecker, H.: Matrix differential calculus. (1988) · Zbl 0651.15001
[14] Park, J. Y.: Canonical cointegrating regressions. Econometrica 60, 119-143 (1992) · Zbl 0749.62047
[15] Phillips, P. C. B.; Hansen, B. E.: Statistical inference in instrumental variables regression with \(I(1)\) processes. Review of economic studies 57, 99-125 (1990) · Zbl 0703.62098
[16] Phillips, P. C. B.; Loretan, M.: Estimating long-run economic equilibria. Review of economic studies 58, 407-436 (1991) · Zbl 0734.62107
[17] Phillips, P. C. B.; Ouliaris, S.: Asymptotic properties of residual based tests for cointegration. Econometrica 58, 165-193 (1990) · Zbl 0733.62100
[18] Saikkonen, P.: Asymptotically efficient estimation of cointegration regressions. Econometric theory 7, 1-21 (1991)
[19] Sims, C. A.; Stock, J. H.; Watson, M. W.: Inference in linear time series with some unit roots. Econometrica 58, 113-144 (1990) · Zbl 0724.62087
[20] Stock, J. H.; Watson, M. W.: A simple MLE of cointegrating vectors in higher order integrated systems. Econometrica 61, 783-820 (1993) · Zbl 0801.62097
[21] Yoo, B. S.: Multi-cointegrated time series and generalized error-correction models. (1986)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.