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STATIONARITY CONDITION FOR AR INDEX PROCESS
Published online by Cambridge University Press: 12 December 2005
Abstract
The stationarity conditions for an autoregressive (AR) process in general are reduced to a remarkably simple inequality if the lag coefficients are restricted to be identical. The condition is not only analytically elegant but also applicable in checking the validity of the stationarity conditions for such a restricted AR process of any order.We are deeply indebted to Professor Paolo Paruolo, NP co-editor of Econometric Theory, and anonymous referees for constructive comments and suggestions that led to significant improvements. Errors, if any, are solely ours.
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1. PROBLEM AND MOTIVATION
The stationarity conditions for an unrestricted autoregressive (AR) process in terms of lag coefficients increase in number with the lag order. This makes it progressively more complex to identify exact bounds for lag coefficients ensuring the stationarity. To reduce the level of complexity, we consider a restricted AR process where the lag coefficients are all equal (referred to in this note as an AR index process). Then, the AR index process of order n (≥ 1) can be written as
where εt ∼ iid(0, σ2) with σ2 < ∞.
Because εt is stationary, the stationarity of yt in (1) is contingent upon the convergence of yt in the corresponding homogeneous difference equation:
In light of the Schur theorem (Chiang and Wainwright, 2005, pp. 589–590), the necessary and sufficient conditions for convergence of yt in (2) can be expressed as
where
in which Sk(α) defines a lower triangular matrix of order k whose (i,j)th element is equal to α if i > j, 1 if i = j, and 0 if i < j, whereas Tk(α) is also a lower triangular matrix of order k whose (i,j)th element is equal to α if i ≥ j and 0 if i < j. That is,
In this simplified framework, the set of convergence conditions in (3) reduces to a single inequality in terms of α and n only (established formally as a theorem in the next section):
If we let α = −φ in (3) as is usual in the econometrics literature, the convergence condition (6) is expressed in terms of φ as
In addition to its analytical elegance, either (6) or (7), depending on how the AR index process is written, is applicable in practice. The stationarity condition for an AR(1) process, yt = φyt−1 + εt with εt ∼ iid(0,σ2), known as −1 < φ < 1, is immediately obtained from (7) for n = 1. The stationarity conditions for an AR(2) process, yt = φ1 yt−1 + φ2 yt−2 + εt, are also known as φ1 + φ2 < 1; φ2 − φ1 < 1; −1 < φ2 < 1 (e.g., see Judge, Griffiths, Hill, Lütkepohl, and Lee, 1985, p. 293). The solution of these three inequalities under the restriction φ1 = φ2 = φ can be worked out as −1 < φ < ½, but the same result is immediately derived from (7) for n = 2. Similarly, the condition can be used to check the validity of the stationarity conditions for an AR index process of any higher order.
2. PROOF AND DISCUSSION
The convergence condition in (6) depends on analytical eigenvalues of Σ2k(α) that may well be useful elsewhere. Hence, we deal with them separately in a pair of lemmas in sequence prior to establishing our main result as a theorem.
DEFINITION. λi(·) denote eigenvalues of any real square matrix of order r in descending order: λ1(·) ≥ λ2(·) ≥ ··· ≥ λr(·).
LEMMA 1. For any real square matrix M of order r, λi(ξ1 M + ξ2 Ir) = ξ1λi(M) + ξ2 where ξ1 and ξ2 are nonnull constants.
Proof of Lemma 1. Because |λIr − (ξ1 M + ξ2 Ir)| = |(λ − ξ2)Ir − ξ1 M|, we can deduce λ = λi(ξ1 M + ξ2 Ir) and λ − ξ2 = λi(ξ1 M) = ξ1λi(M). Substituting out λ from either equation, λi(ξ1 M + ξ2 Ir) = ξ1λi(M) + ξ2.1
We owe conciseness of the proof to an anonymous referee.
LEMMA 2. For Σ2k(α) defined in (5), λi(Σ2k(α)) = kα for i = 1; 1 for i = 2,3,…, k − 1; 1 − α for i = k + 1, k + 2, …, 2k − 1, 2k.
Proof of Lemma 2. Define a square matrix of order 2k:
where Lk is a lower triangular matrix of order k whose (i,j)th element is equal to 1 if i ≥ j and 0 if i < j.
Then we can rewrite Σ2k(α) in terms of A2k as
Hence, in view of Lemma 1
It is clear from (10) that once analytical solutions for λi(A2k) are obtained, their counterparts for λi(Σ2k(α)) immediately follow.
The upper k rows in A2k in (8) are identical with the lower k rows, which implies rank(A2k) = k. Hence, we can reduce the lower k rows to null vectors through pre- and postmultiplications of an appropriate nonsingular matrix and its inverse:
where
Because P2k in (12) is nonsingular, A2k and B2k in (11) have the same set of eigenvalues with the same multiplicities (Magnus and Neudecker, 1990, p. 14). Hence, noting that B2k is block-diagonal,
Because Lk + Lk′ = ιι′ + Ik where ι(k × 1) = (1, 1,…,1)′, in view of Lemma 1,
in which
Recursive substitutions of (15) into (14), (14) into (13), and then (13) into (10) lead to the eigenvalues sought after:
THEOREM. For
, the necessary and sufficient condition for convergence of yt is −1/n < α < 1.
Proof of Theorem. yt in the theorem converges if and only if |Σ2k(α)| > 0 for all k (1 ≤ k ≤ n) by virtue of the Schur theorem as stated in (3). Reflecting the eigenvalues from Lemma 2 on (3),
which holds if and only if −1/n < α < 1. █
References
REFERENCES
Chiang, A.C. &
K. Wainwright
(2005)
Fundamental Methods of Mathematical Economics.
4th ed.
McGraw-Hill.
Judge, G.G.,
W.E. Griffiths,
R.C. Hill,
H. Lütkepohl, &
T. Lee
(1985)
The Theory and Practice of Econometrics.
Wiley.
Magnus, J.R. &
H. Neudecker
(1990)
Matrix Differential Calculus: With Applications in Statistics and
Econometrics.
Rev. ed.
Wiley.
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