Abstract
Now we consider some applications of the propositions from the previous chapter. In particular, {e t } and {x t } are integrated of order 0 and integrated of order 1, respectively, cf. the definitions above Proposition 14.2. It turns out that the regression of a time series on a linear trend leads to asymptotically Gaussian estimators.
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Notes
- 1.
Many more procedures have been developed over the last decades, notably the test by Elliott et al. (1996) with certain optimality properties.
- 2.
Through numerous works by Peter Phillips the functional central limit theory has found its way into econometrics. This kind of limiting distributions was then celebrated as “non-standard asymptotics”; meanwhile it has of course become standard.
- 3.
Equivalently, one might feed detrended data into the ADF regression above.
- 4.
For the following calculation of s 2 we divide by n without correcting for degrees of freedom, which does not matter asymptotically (\(n \rightarrow \infty )\).
- 5.
“Uncentered”, as the regression is calculated without intercept.
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Hassler, U. (2016). Trends, Integration Tests and Nonsense Regressions. In: Stochastic Processes and Calculus. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-23428-1_15
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