Abstract
In this chapter three further topics are considered in some detail: models where the orders of integration of the series are not the same, estimation of models with I(2) variables, and models where the order of integration is fractional.
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Notes
- 1.
Many series are often limited to lie in the range [0,1] as is the case for market interest rates. The question of non-stationarity in this context is further complicated by the notion of what an extreme value might be. Maybe one should consider the performance of bond prices upon which the rate of return of the safe asset is derived. Then again the non-stationarity may be a function of the process of aggregation or the pricing formula. In practice all models are not identified, the models estimated are always approximations and the modellers task is to limit the degree of non-identification (Sargan 1983).
- 2.
Here our analysis is restricted to the case where trends are possible via unrestricted intercepts in the conventional cointegration analysis (μ 1 ≠0), but there are no quadratic trends. Otherwise, the second step of the I(2) estimator has a restricted intercept (μ 2 = 0). This is the case considered by Johansen (1995) and unlike Paruolo (1996) it restricts our discussion to a single table. In the empirical example considered by Paruolo (1996), he concludes that the selection of results associated with 1 Q r, s is quite consistent when a pre-analysis of the data suggests that there are trends in the differences (μ 1 ≠0), but not the second differences (μ 2 = 0). 
- 3.
For the example considered by Paruolo, inference progressed in a straightforward manner, by sequentially moving past each test statistic a table at a time. However, here the progress is more complicated, even when one only considers the table of tests associated with μ 1 ≠0. 
- 4.
Q r ∗ is significant for r = 1 (119. 69 > 93. 92 ) and r = 2 (68. 861 > 68. 68).
- 5.
The critical values for this test can be found in Johansen (1995).
- 6.
This follows from the coherence of this direction with a similar test procedure that arises by the conditional application of the tests by Johansen. Furthermore, Johansen (1995) has shown that the specific to general ordering is optimal in relation to the Johansen CI(1, 1) test; and the same applies in relation to the I(2) test. However, Nielsen (2009) suggests that care is taken when the Johansen procedure is considered as there are further unit roots or explosive series.
- 7.
This is an extension of the I(1) case where α ζ ζ −1 β ′ = α ∗ β ∗′, meaning that the estimated loadings and cointegrating vectors are not distinguished from any non-singular matrix product. That is [α, β ′] and [α ∗, β ∗′] are observationally equivalent. Now this problem is further complicated in the I(2) case.
- 8.
The values of d estimated are found to be sensitive to the bandwidth m. A common assumption made in the literature on evaluating standard errors in cointegrating regressions is to set the bandwidth to a third of the sample, \(m = \frac{T} {3}\). Alternatively, Henry and Robinson (1996) provide some methods for the selection of m. 
- 9.
Assuming the impact of cointegration resides in the MA component of the model, the notation from Chap. 2 is applied; this result depends on \(C(L) = \Gamma (L)^{-1}\Theta (L)\) with the roots to \(\Gamma (L)\) lying outside the unit circle and the roots of \(\Theta (L)\) outside or in the unit circle. Otherwise the results related to spectral division (Gohberg et al. 1983) or to unimodular matrices in Chap. 3 apply.
- 10.
For the purposes of this chapter, where appropriate, the notation of Robinson (2008) is followed.
- 11.
It is of interest to note that both estimators are asymptotically normal.
- 12.
The Geweke and Porter-Hudak (1983) method computes − d by least squares as the slope parameter from an equation that regresses the log of the periodogram at frequency 2π j∕T against the log{4sin 2(π j∕T)} over the sample j = 1, …, m where in this case convention is followed and m=T . 5. 
- 13.
Commonly used functions in this case are the Haar and Daubechies wavelet transform (Percival and Walden 2000). Akansu et al. (1990) have shown that finite impulse response quadrature mirror filters ( FIR-QMF) give rise to a set of basis functions that are able to perfectly reconstruct a series, while the binomial QMFs developed in this article are shown to be the same filters as those that derive from the discrete orthonormal Daubechies wavelet transform.
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Hunter, J., Burke, S.P., Canepa, A. (2017). Models with Alternative Orders of Integration. 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_8
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