Least Absolute Deviations pp 77-108 | Cite as

# LAD in Autoregression

Chapter

## Abstract

This chapter is devoted to stationary, k where...,U for a certain unique choice of b

^{th}order autoregressions$$ {X_n} = {a_0} + {a_1}{X_{n - 1}} + \ldots + {a_k}{X_{n - k}} + {U_n} $$

(1)

_{-1},U_{0},U_{1},... is an i.i.d. sequence of random variables. The first question that arises is: what conditions on a and the U_{i}will assure that there actually exists a stationary sequence {X_{i}} satisfying (1)? Using (1) recursively for X_{n-1}in (1), then again for X_{n-2}, etc., one obtains after N substitutions,$$ {X_n} = d(N) + \sum\limits_{j = 0}^N {{b_j}{U_{n - 1}}} + \sum\limits_{j = N + 1}^{N + k} {{c_j}(N){X_{n - j}}} $$

(2)

_{j},c_{j}(N); i.e., b_{0}= 1, b_{1}= a_{1}, etc.## Keywords

Error Distribution Stationary Sequence Ergodic Theorem Stationarity Condition Double Exponential## Preview

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## Notes

- 1.in the classical case of (1.1), E(U
_{i}^{2}) < ∞ is assumed [this certainly implies (1.5)]. With stationarity, (3), Σ b_{j}U_{n-j}converges in L_{2}, therefore in probability. Kanter and Steigert 1974) gave the first generalization to the case where U_{i}is attracted to a non-normal stable law of index α. The condition (1.5) and Lemma 1.1 is from Yohai and Maronna (1977). Being in dom(a) is much more restrictive than the Yohai-Maronna condition, (1.5). However they impose, though don’t use, symmetry of U_{i}.Google Scholar - 2.The theorem in (1.9) began in Kanter and Steiger (1974) with ĉ → a in probability. At the 1974 Brasov Conference [Kanter and Steiger (1977)] they noticed that their consistency proof actually implied the rate N
^{δ}(ĉ_{N}-a) → 0 in probability, δ < min(1/α,(2-α)/α). Kanter and Hannan (1977) cleaned up the bound on 6 and established convergence with probability 1. Under a weaker condition than U_{i}∈ Dom(*α*), namely (1.5), Yohai and Maronna established a weaker result, namely that N^{δ}(ĉ_{N}-a) → 0 in probability, δ < 1/2. Thus a wider class of processes is embraced at the expense of weakening the convergence rate. Again, we cite Feller (1971) as a convenient source for details about stable laws.Google Scholar - 3.Theorem 2.1 is from Gross and Steiger (1979). The statement of Theorem 2.2 was one of two conjectures they made.Google Scholar
- 4.Theorems 2.2 and 2.3 are from An and Chen (1982). The proof that N
^{-t}S_{2}→ ∞ in (1.12) is similar to theirs. Our proof that N^{-t}S_{1}→ 0 in (1.11) is much simpler. Curiously, they give an example, which, if correct, would seem to invalidate their method of proof of Theorem 2.2.Google Scholar - 5.All that is needed for Theorem 2.3 is that N
_{p}times the expression in (2.29) converge to zero. The original proof of this is more complicated than ours.Google Scholar

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© Birkhäuser Boston, Inc. 1983