## 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.

### MSC:

 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 60G50 Sums of independent random variables; random walks

Zbl 0721.62088
Full Text:

### References:

 [1] DOI: 10.1007/BF01197844 · Zbl 0695.60041 [2] Billingsey P., Probability and Measure, 2nd Edn. (1986) [3] Corradi V., Testing for stationarity-ergodicity and for comovements between nonlinear discrete time Markov processes (1994) [4] Dachuna-Castelle D., Probability and Statistics Vol. II (1986) [5] DOI: 10.1016/0304-4076(93)90050-F · Zbl 0772.62065 [6] Granger C. W. J., J. Time Ser. Anal. 12 pp 207– (1991) [7] Granger C. W. J., Oxford Bull. Econ. Stat. 53 pp 11– (1991) [8] Karatzas I., Brownian Motion and Stochastic Calculus, 2nd Edn. (1990) · Zbl 0721.90013 [9] DOI: 10.1007/BF00532611 · Zbl 0288.60034 [10] DOI: 10.2307/1427479 · Zbl 0757.60061 [11] Rosenblatt M., Markov Processes. Structure and Asymptotic Behavior. (1971) [12] DOI: 10.2307/2039821 · Zbl 0323.60077
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.