Abstract
In this chapter, we consider the question of long-run exogeneity and the related issue of identification. In our opinion, detection of the exogenous variables in the long run or the short run is a precursor to any attempt to structurally identify economic or financial phenomena.
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Notes
- 1.
We would like to thank Graham Mizon for his discussion of this issue.
- 2.
For the cointegrating exogenous case in Hunter (1992) r 1 = 1 and n 1 = 3.
- 3.
- 4.
Hence, α and β are 6 × 2 dimensioned matrices.
- 5.
The restrictions for strict exogeneity (SE) of the oil price are α ij  = 0 and β ij  = 0 for i, j = 1, 2.
- 6.
In fact the same test can be used by rerunning the Johansen procedure with the variable to be tested for WE being placed first.
- 7.
The restrictions for CE are α 51 = 0, α 61 = 0 and β j2 = 0 for j = 1, …, 4.
- 8.
The WE restrictions for the model in Hunter (1992) are α i1 = 0 for i = 4, 5, 6 and α j2 = ω j1 α 42 +ω j2 α 52 +ω j3 α 62 for j = 1, 2, 3.
- 9.
Strong exogeneity augments the sub-block WE restriction above by β i2 = 0 for i = 1, 2, 3.
- 10.
If an equation has exactly j i  = n 1 − 1 restrictions, then it has enough restrictions to be exactly identified. When j i  > n 1 − 1, it has enough restrictions to be over-identified, but without the appropriate number it will be under or not identified.
- 11.
The Hessian is the second derivative of the likelihood, which provides an estimate of the variance-covariance matrix of the parameters. If some parameters are ill-defined, then the likelihood is flat and the Hessian matrix singular. Then some parameters in the model are not identified. Perfect multi-collinearity is a special case of this and it occurs when two or more variables are related and their parameters cannot be independently estimated and as a result are not identifiable.
- 12.
- 13.
Usually, two equations or blocks of equations with different parameterizations have the same value for their likelihoods. In general, there exists at least one model with r exactly identifying restrictions for each equation with a likelihood value, the same as the unrestricted likelihood.
- 14.
The approach described here was first outlined for the I(1) case in Hunter and Simpson (1995).
- 15.
- 16.
The additional restriction is required to solve for all of the parameters and of the eight restrictions implied by \(\beta _{r}^{{\prime}} = \left [\begin{array}{cccccc} \beta _{11} & \beta _{21} & -\beta _{21} & -\beta _{21} & 0 & \beta _{61}\\ 0 & 0 & 0 & 0 & \beta _{52 } & -\beta _{52} \end{array} \right ]\) only six are binding. The test associated with this structure for β is χ 6 2 = 6. 8291, which is accepted at the 5% level based on a p-value=[0.3369].
- 17.
For every row and column of \(\Pi\) selected, there is an equivalent r dimensioned sub-matrix of α and β. To determine an appropriate orientation of the system the sub-matrices selected need to be of full rank.
- 18.
In the case where more complex restrictions apply, then the general restriction condition and procedure in Doornik and Hendry (2001) apply.
- 19.
Here β is identifiable for the restrictions in (I) when the selected columns of \(\Pi\) yield a matrix A of rank r.
- 20.
For n = 4, a more complex example, the approach discussed above can be shown to identify. Let:
$$\displaystyle{ \beta ^{{\prime}} = \left [\begin{array}{cccc} a&0 & b & c \\ d&e&f &0 \end{array} \right ]\text{ and }B = H_{2} = \left [\begin{array}{cc} a&0\\ d & e \end{array} \right ]. }$$Following Boswijk (1996), identifiability is lost when a normalization is invalid (i.e. a = 0 ⇒ rank(H 2) < r), but with this new restriction [α: β] is over-identified as j = 3 > r 2 − r. Selecting a new orientation, ensuring the generic result associated with Theorem 9 holds, then:
$$\displaystyle{ \beta _{(1)}^{{\prime}} = \left [\begin{array}{cccc} 0 & 0 & b & c \\ d&e&f &0 \end{array} \right ]\text{ and }B_{(1)} = \left [\begin{array}{cc} b & c\\ f &0 \end{array} \right ]. }$$This orientation is rejected when x t  ∼ I(1), f = 0 and α is not identifiable. But the following orientation for x t  ∼ I(1), implies:
$$\displaystyle{ \beta _{(2)}^{{\prime}} = \left [\begin{array}{cccc} 0 & 0 & b & c \\ d&e&0 & 0 \end{array} \right ],\text{ }B_{(2)} = \left [\begin{array}{cc} 0 & b\\ e &0 \end{array} \right ] }$$and rank(B) = r. Now, [α: β (2)] is always empirically identified and identifiable.
- 21.
Notice, that price homogeneity is a long-run property of LEPT. This means that in the short-run agents may mistake relative and absolute price movements. However, long-run pricing that does not satisfy this property would not appear to be consistent with competitive behaviour.
- 22.
The matrices α ij and β ij have the dimensions n i × r j , for i = 1, 2 and j = 1, 2. For example, the matrix β is partitioned into two blocks of columns, β . 1 of dimensions n × r 1, and β . 2 of dimensions n × r 2, then each block is itself cut into two blocks of rows.
- 23.
In the limit there are r such sub-blocks, which leads to the identification case considered by Boswijk (1992) where \(\alpha = \left [\begin{array}{c} I\\ 0 \end{array} \right ].\)
- 24.
The original source of the data is the National Institute of Economic Research, kindly passed on to us by Paul Fisher and Ken Wallis.
- 25.
The model in Hunter (1992) is massively over-identified. It is possible to identify subject to restrictions on both α and β. Here we will concentrate on identification from α alone.
- 26.
The discovery of four valid solutions implies that the model has four over-identifying restrictions.
- 27.
If the determinant is tested for any sub-matrix of α then it is found that no such combination with a non-zero determinant appears to exist.
- 28.
It is also interesting to speculate that the plot of the IRF appears to converge to values that look like the solution to a difference equation. In the cointegration case there is at least one unit root or a single stochastic trend. If there are roots that are complex, and this may arise when there is a second order difference equation, then to preclude a solution that is not real the roots must consist of a complex conjugate pair which may be of unit modulus. These ought to relate back to the roots of the VAR, but this is associated with differences and not simply inverses, because the VAR has real roots, one of which is a unit root.
- 29.
Roots in the paper are the reciprocals of those normally reported, thus a root less than one in modulus is a stationary root. On this basis the roots of the process are, respectively: {0.5, 0.5, 0.5, 0.5}, {0.5, 0.5, 0.95, 0.95}, {0.5, 0.5, 0.99, 0.99}, {0.5, 0.5, 1.0, 1.0}, {1.0, 1.0. 1.0, 1.0}.
- 30.
An alternative study would be one based on perturbations of the cointegrated model, model 4, that retained the common feature, but moved it from being at the unit root to being further outside the unit circle. This would mean that the processes became stationary, and more solidly so, but retained the reduced rank property key to cointegration. In this way, it is possible to isolate two aspects of the problem with potentially different impacts: stationarity and common features (reduced rank).
- 31.
- 32.
Figure 5.2d also shows clearly that, in this case, under-specification of the cointegrating rank is not harmful to forecast performance (including imposing unit roots but non-stationarity), whereas over-specification leads to a deterioration in forecasting performance.
- 33.
Though they do not establish whether it is the imposition of any false restriction that matters, or that of unit roots in particular. This is the point made by Clements and Hendry. They also do not consider if the near unit root is a common feature, or if restricting it to being so would be advantageous.
- 34.
The information criterion can also be written in terms of the eigenvalues of the underlying problem, and hence in terms of the test statistics.
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Hunter, J., Burke, S.P., Canepa, A. (2017). Structure and Evaluation. In: Multivariate Modelling of Non-Stationary Economic Time Series. Palgrave Texts in Econometrics. Palgrave Macmillan, London. https://doi.org/10.1057/978-1-137-31303-4_5
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