Abstract
As already mentioned in Chap. 7, many raw economic time series are nonstationary and become stationary only after some transformation. The most common of these transformations is the formation of differences, perhaps after having taken logs. In most cases first differences are sufficient to achieve stationarity. The stationarized series can then be analyzed in the context of VAR models as explained in the previous chapters. However, many economic theories are formalized in terms of the original series so that we may want to use the VAR methodology to infer also the behavior of the untransformed series. Yet, by taking first differences we loose probably important information on the levels. Thus, it seems worthwhile to develop an approach which allows us to take the information on the levels into account and at the same time take care of the nonstationary character of the variables. The concept of cointegration tries to achieve this double requirement.
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Notes
- 1.
A more recent interesting application of this model is given by the work of Beaudry and Portier (2006).
- 2.
This condition could be relaxed and replaced by the condition \(\sum _{j=0}^{\infty }j^{2}\|\Psi _{j}\|^{2} < \infty \). In addition, this condition is an important assumption for the application of the law of large numbers and for the derivation of the asymptotic distribution (Phillips and Solo 1992).
- 3.
The distribution of X 0 is thereby chosen such that \(\beta 'X_{0} =\beta '\widetilde{\Psi }(\mathrm{L})Z_{0}\).
- 4.
The seasonal unit roots are the roots of \(z^{s} - 1 = 0\) where s denotes the number of seasons. These roots can be expressed as \(\cos (2k\pi /s) +\iota \sin (2k\pi /s)\), \(k = 0, 1,\ldots,s - 1\).
- 5.
- 6.
If the VAR model (16.9) contains further deterministic components besides the constant, these components have to be accounted for in these regressions.
- 7.
This two-stage least-squares procedure is also known as partial regression and is part of the Frisch-Waugh-Lowell Theorem (Davidson and MacKinnon 1993, 19–24).
- 8.
Thereby we make use of the following equality for partitioned matrices:
$$\displaystyle{ \det \left (\begin{array}{*{10}c} A_{11} & A_{12}\\ A_{ 21} & A_{22}\\ \end{array} \right ) =\det A_{11}\det (A_{22}-A_{21}A_{11}^{-1}A_{ 12}) =\det A_{22}\det (A_{11}-A_{12}A_{22}^{-1}A_{ 21}) }$$where A 11 and A 22 are invertible matrices (see for example Meyer 2000, p. 475).
- 9.
- 10.
It is instructive to compare theses cases to those of the unit root test (see Sect. 7.3.1).
- 11.
The tables by MacKinnon et al. (1999) allow for the possibility of exogenous integrated variables.
- 12.
The choice of the variables used for normalization turns out to be important in practice. See the application in Sect. 16.5.
- 13.
The degrees of freedom are computed according to the formula: \(s(n - r) = 1(4 - 3) = 1\).
- 14.
The degrees of freedom are computed according to the formula: \(r(n - s) = 3(4 - 3) = 3\).
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Neusser, K. (2016). Cointegration. In: Time Series Econometrics. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-32862-1_16
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