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**Nonlinear transformations of integrated time series: A reconsideration.**
*(English)*
Zbl 0837.62065

Summary: In this paper I reconsider two of the questions raised by C. W. J. Granger and J. Hallman [ibid. 12, No. 3, 207-224 (1991; Zbl 0721.62088)]: (i) If \(X_t\) is \(I(1)\) and \(Z_t = h(X_t)\), is \(Z_t\) also \(I(1)\)? (ii) Can \(X_t\) and \(h(X_t)\) be cointegrated? The distinction between \(I(1)\) and \(I(0)\) processes is replaced by the distinction between long memory and short memory processes, where for short memory \(I\) mean strong mixing. By exploiting the fact that random walks (with positive trend component) are martingales (submartingales) and are also first-order Markov, I show that

(a) unbounded convex (concave) and strictly monotonic transformations of random walks are always long memory processes, (b) polynomial, strictly convex (concave) transformations of random walks display a unit root component, but the first differences of such transformations need not be short memory, and (c) \(X_t\) and \(h(X_t)\), with \(h\) an unbounded convex (concave) or strictly monotonic function, can never be cointegrated.

(a) unbounded convex (concave) and strictly monotonic transformations of random walks are always long memory processes, (b) polynomial, strictly convex (concave) transformations of random walks display a unit root component, but the first differences of such transformations need not be short memory, and (c) \(X_t\) and \(h(X_t)\), with \(h\) an unbounded convex (concave) or strictly monotonic function, can never be cointegrated.

### MSC:

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

60G50 | Sums of independent random variables; random walks |

### Keywords:

cointegration; Doob decomposition; Markov property; submartingales; strong mixing; short memory processes; martingales; random walks; long memory processes; unit root component### Citations:

Zbl 0721.62088
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\textit{V. Corradi}, J. Time Ser. Anal. 16, No. 6, 539--549 (1995; Zbl 0837.62065)

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### References:

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[6] | Granger C. W. J., J. Time Ser. Anal. 12 pp 207– (1991) |

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