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Time series analysis of error-correction models. (English) Zbl 0547.62060
Studies in econometrics, time series, and multivariate statistics, Commem. T. W. Anderson’s 65th Birthday, 255-278 (1983).
[For the entire collection see Zbl 0528.00008.]
The error-correction models considered here take the form \[ (1)\quad (1- B)^ da_ 1(B)y_ t=m_ 1+\beta (y_{t-1}-Ax_{t-1})+(1-B)^ db_ 1(B)x_ t+c_ 1(B)\epsilon_{1t} \]
\[ (2)\quad (1-B)^ da_ 2(B)x_ t=m_ 2+c_ 2(B)\epsilon_{2t} \] where \(\epsilon_{1t}\), \(\epsilon_{2t}\) are a pair of independent, zero-mean white noise series with finite variances, \(m_ 1\), \(m_ 2\) are constants, B is the lag operator so that \(B^ kz_ t=z_{t-k}\), \(a_ 1(B)\), \(b_ 1(B)\), \(c_ 1(B)\) are finite polynomials in B with \(a_ 1(1)\neq 0\), \(b_ 1(1)\neq 0\), etc. and \(a_ 1(0)=a_ 2(0)=c_ 1(0)=c_ 2(0)=1\), d is either 0 or 1.
A series \(x_ t\) is called integrated of order d, denote \(x_ t\sim I(d)\), if it has a univariate ARIMA(p,d,q) model of the form (3) \((1- B)^ dg_ p(B) x_ t=h_ q(B)a_ t\), where \(g_ p(B)\), \(h_ q(B)\) are finite polynomials in B of orders p, q, respectively, and \(a_ t\) is white noise.
Several types of models derived from (1), (2) and (3) are illustrated, e.g., (a) one-way causal model, (b) multi-component co-integrated series, and (c) bivariate feedback models. Some tests for co-integration are proposed and applied to illustrative economic data.
Reviewer: J.K.Sengupta

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62P20 Applications of statistics to economics