# zbMATH — the first resource for mathematics

Statistical analysis of cointegration vectors. (English) Zbl 0647.62102
Summary: We consider a nonstationary vector autoregressive process which is integrated of order 1, and generated by i.i.d. Gaussian errors. We then derive the maximum likelihood estimator of the space of cointegration vectors and the likelihood ratio test of the hypothesis that it has a given number of dimensions. Further we test linear hypotheses about the cointegration vectors.
The asymptotic distribution of these test statistics are found and the first is described by a natural multivariate version of the usual test for unit root in an autoregressive process, and the other is a $$\chi^2$$ test.

##### MSC:
 62P20 Applications of statistics to economics 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62E20 Asymptotic distribution theory in statistics 62H15 Hypothesis testing in multivariate analysis
Full Text:
##### References:
 [1] Ahn, S.K.; Reinsel, G.C., Estimation for partially nonstationary multivariate autoregressive models, (1987), University of Wisconsin Madison, WI [2] Anderson, T.W., An introduction to multivariate statistical analysis, (1984), Wiley New York · Zbl 0651.62041 [3] Andersson, S.A.; Brøns, H.K.; Jensen, S.T., Distribution of eigenvalues in multivariate statistical analysis, Annals of statistics, 11, 392-415, (1983) · Zbl 0517.62053 [4] Davidson, J., Cointegration in linear dynamic systems, (1986), London School of Economics London, Mimeo [5] Dickey, D.A.; Fuller, W.A., Distribution of the estimators for autoregressive time series with a unit root, Journal of the American statistical association, 74, 427-431, (1979) · Zbl 0413.62075 [6] Engle, R.F.; Granger, C.W.J., Co-integration and error correction: representation, estimation and testing, Econometrica, 55, 251-276, (1987) · Zbl 0613.62140 [7] Granger, C.J., Some properties of time series data and their use in econometric model specification, Journal of econometrics, 16, 121-130, (1981) [8] Granger, C.W.J.; Engle, R.F., Dynamic model specification with equilibrium constraints, (1985), University of California San Diego, CA, Mimeo [9] Granger, C.W.J.; Weiss, A.A., Time series analysis of error correction models, () · Zbl 0547.62060 [10] Johansen, S., Functional relations, random coefficients, and non-linear regression with application to kinetic data, () [11] Johansen, S., The mathematical structure of error correction models, Contemporary mathematics, (1988), in press · Zbl 0688.62049 [12] Phillips, P.C.B., Understanding spurious regression in econometrics, Cowles foundation discussion paper no. 757, (1985) · Zbl 0602.62098 [13] Phillips, P.C.B., Multiple regression with integrated time series, Cowles foundation discussion paper no. 852, (1987) · Zbl 0613.62109 [14] Phillips, P.C.B.; Durlauf, S.N., Multiple time series regression with integrated processes, Review of economic studies, 53, 473-495, (1986) · Zbl 0599.62103 [15] Phillips, P.C.B.; Ouliaris, S., Testing for cointegration, Cowles foundation discussion paper no. 809, (1986) · Zbl 0647.62103 [16] Phillips, P.C.B.; Ouliaris, S., Asymptotic properties of residual based tests for cointegration, Cowles foundation discussion paper no. 847, (1987) · Zbl 0733.62100 [17] Phillips, P.C.B.; Park, J.Y., Asymptotic equivalence of OLS and GLS in regression with integrated regressors, Cowles foundation discussion paper no. 802, (1986) [18] Phillips, P.C.B.; Park, J.Y., Statistical inference in regressions with integrated processes: part 1, Cowles foundation discussion paper no. 811, (1986) [19] Phillips, P.C.B.; Park, J.Y., Statistical inference in regressions with integrated processes: part 2, Cowles foundation discussion paper no. 819, (1987) [20] Rao, C.R., The theory of least squares when the parameters are stochastic and its applications to the analysis of growth curves, Biometrika, 52, 447-458, (1965) · Zbl 0203.21501 [21] Rao, C.R., Linear statistical inference and its applications, (1973), Wiley New York · Zbl 0169.21302 [22] Sims, A.; Stock, J.H.; Watson, M.W., Inference in linear time series models with some unit roots, (1986), Preprint [23] Stock, J.H., Asymptotic properties of least squares estimates of cointegration vectors, Econometrica, 55, 1035-1056, (1987) · Zbl 0651.62105 [24] Stock, J.H.; Watson, M.W., Testing for common trends, () · Zbl 0673.62099 [25] Velu, R.P.; Reinsel, G.C.; Wichern, D.W., Reduced rank models for multiple time series, Biometrika, 73, 105-118, (1986) · Zbl 0612.62121
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.