Partial autocorrelation functions of the fractional ARIMA processes with negative degree of differencing

Abstract

Let {Xn : ∈Z} be a fractional ARIMA(p,d,q) process with partial autocorrelation function α(·). In this paper, we prove that if d∈(−1/2,0) then |α(n)|~|d|/n as n→∞. This extends the previous result for the case 0<d<1/2

AR and MA representation of partial autocorrelation functions, with applications

“…This last theorem as well as Theorem 2.3 is an improvement of earlier work [24][25][26] for long-memory or FARIMA processes, asserting that…”

“…In a number of the specific examples of processes with long memory we treat, here and in [24][25][26], we observe behaviour of the form α n ∼ d/n (n → ∞) (d/n) (by the representation of α n , we are able to improve the estimate of this type in [24][25][26], where only |α n | was considered-we were unable to determine its sign). In (d/n), and throughout the paper, a n ∼ b n as n → ∞ means lim n→∞ a n /b n = 1.…”

“…We can write the property (d/n) as d = lim n→∞ nα n , suggesting how to estimate the parameter d, which is important in a process with long memory. See [26,Sect. 5] for numerical calculation.…”

“…Assuming (PND), we define Notice that b m j here is equal to that in [26] but it corresponds to b m+1 j−1 in [24,25]. Recall U n and V n from Sect.…”

Financial Markets with Memory II: Innovation Processes and Expected Utility Maximization

“…The above type of method was used in [8] in a simpler framework, i.e., that of discrete-time stationary processes, to obtain a representation of mean-squared prediction error. See [9] and [10] for subsequent results in the same framework. Now, unlike in these references, we develop a similar method to prove the prediction formula itself, rather than a representation of prediction error, for continuous-time processes with stationary increments.…”