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foundations of econometrics phần 10

Chúng tôi đã đề cập tại mục 13.4 là một mô hình sửa lỗi có thể được sử dụng ngay cả khi các dữ liệu bất tĩnh. Để biện minh cho khẳng định này, chúng tôi bắt đầu lại từ trường hợp đơn giản, trong đó loạt hai yt1 và yt2 được tạo ra bởi hai phương trình (14,45). Từ định nghĩa (14,41) của quá trình | 14.5 Cointegration 619 Estimation Using an ECM We mentioned in Section 13.4 that an error correction model can be used even when the data are nonstationary. In order to justify this assertion we start again from the simplest case in which the two series yt1 and yt2 are generated by the two equations 14.45 . From the definition 14.41 of the I 0 process vt2 we have Avt2 A2 1 vt-1 2 et2 14.49 We may invert equations 14.45 as follows Vti x -yti x12yt2 and Vt2 x21 yti x22yt2 14.50 where xij is the ijth element of the inverse X-1 of the matrix with typical element xij. If we use the expression for vt2 and its first difference given by equations 14.50 then equation 14.49 becomes x21Ayti X22 Ayt2 Ă2 1 x21yt-i i X22 yt-1 2 et2. Dividing by x21 and noting that the relation between the inverse matrices implies that x21x11 x22x21 0 we obtain the error-correction model Ayti 2Ayt2 A2 1 yt-i i 2 yt-1 2 e t2 14.51 where as above y2 x11 x21 is the second component of the cointegrating vector and e t2 et2 x21. Although the notation is somewhat different from that used in Section 13.3 it is easy enough to see that equation 14.51 is a special case of an ECM like 13.62 . Notice that it must be estimated by nonlinear least squares. In general equation 14.51 is an unbalanced regression because it mixes the first differences which are I 0 with the levels which are I 1 . But the linear combination yt-1 1 i 2yt-1 2 is I 0 on account of the cointegration of yt1 and yt2. The term A2 1 yt-1 1 112yt-1 2 is precisely the error-correction term of this ECM. Indeed yt-1 1 1 2yt-1 2 is the equilibrium error and it influences Ayt1 through the negative coefficient A2 1. The parameter y2 appears twice in 14.51 once in the equilibrium error and once as the coefficient of Ayt2 . The implied restriction is a consequence of the very special structure of the DGP 14.45 . It is the parameter that appears in the equilibrium error that defines the cointegrating vector not the coefficient of Ayt2 . This follows