Abstract
Below Proposition 3.5 we saw that the autocorrelation sequence of any stationary ARMA process dies out at exponential rate: | ρ(h) | ≤ c g h, see (3.14). This is too restrictive for many time series of stronger persistence, which display long memory in that the autocovariance sequence vanishes at a slower rate. In some fields of economics and finance long memory is treated as an empirical stylized fact.
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Notes
- 1.
See e.g. the special issue edited by Maasoumi and McAleer (2008) in Econometric Reviews on “Realized Volatility and Long Memory”.
- 2.
For the rest of this chapter we maintain d > −1, which guarantees that {π j } converges to 0 with growing j, making the infinite expansion meaningful.
- 3.
- 4.
More complicated is the effect of changes in d if d < 0, see Hassler (2014).
- 5.
Readers not familiar with complex numbers, \(i^{2} = -1\), may skip the following equation, see also Footnote 6 in Chap. 4:
$$\displaystyle\begin{array}{rcl} T_{(1-L)^{-d}}(\lambda )& =& (1 - e^{i\lambda })^{-d}(1 - e^{-i\lambda })^{-d} {}\\ & =& (1 - e^{i\lambda } - e^{-i\lambda } + 1)^{-d} {}\\ & =& (2 - 2\cos (\lambda ))^{-d}. {}\\ \end{array}$$ - 6.
The use of ‘ ∼ ’ with a differing meaning from that one in (5.2) should not be a source for confusion.
- 7.
Use \((1 + x/n)^{n} \rightarrow e^{x}\).
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Hassler, U. (2016). Long Memory and Fractional Integration. In: Stochastic Processes and Calculus. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-23428-1_5
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