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ADDENDUM TO “ASYMPTOTICS FOR NONLINEAR TRANSFORMATIONS OF INTEGRATED TIME SERIES”
Published online by Cambridge University Press: 08 June 2004
Abstract
Typically in time series econometrics, for many statistics, a rescaled integrated process is replaced with Brownian motion to find the limit distribution. For averages of functions of a rescaled integrated process, Park and Phillips have shown that this remains true for functions with poles, as long as a sample-size-dependent region around the poles is excluded from consideration, the function is locally integrable, and some other regularity conditions hold. In this addendum, I show that under some regularity conditions on the function under consideration, there is no need for such a sample-size-dependent region around the pole in Park and Phillips' theorem, as long as the function under consideration is locally integrable.The author thanks four anonymous referees and Professor Phillips for their comments and suggestions and Benedikt Pötscher for suggestions and for pointing out an error in a previous version of this paper.
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1. INTRODUCTION
The continuous mapping theorem ensures that for I(1) processes xt,
for continuous functions T(.), where
denotes convergence in distribution, n−1/2x[rn] ⇒ σW(r) where W(.) denotes Brownian motion and ⇒ denotes weak convergence, and σ2 = limn→∞ n−1Exn2. The continuity of T(.) results in a mapping
from C[0,1] to
that is continuous, which implies that the continuous mapping theorem can be applied to obtain the result of equation (1). But for functions T(.) with poles such as T(x) = |x|α for −1 < α < 0, the mapping Π is not continuous. Here and everywhere in this paper, we define T(.) to equal 0 at every pole. Theorem 3.2 of Park and Phillips (1999) shows that for functions that are “regular” (as defined in that paper) and locally integrable, the result of equation (1) still holds. Among the functions that are dealt with by that theorem is, for example, T(x) = I(x > 0). Park and Phillips' Theorem 3.2, however, fails to show that
for all −1 < α < 0 and also fails to establish the result of equation (1) for T(x) = log|x|. For α ≥ 0, the preceding result follows from the continuous mapping theorem in a straightforward way, because |x|α is continuous on
for α ≥ 0. But for −1 < α < 0, the pole at 0 will make it impossible to apply the continuous mapping theorem directly, and this paper shows that under regularity conditions, the result of equation (2) still holds. Note that because the right-hand side of equation (2) is undefined for α ≤ −1, there is no hope of proving the result of equation (2) for such values of α. Instead of the result of equation (2), Park and Phillips (1999) show that
for α > −1, for a sequence δn that converges to 0 as n → ∞. In this paper, I prove the result of equation (1) for a function class that is different from that of Park and Phillips, and I show that the result of equation (2) holds under the exact same assumptions that were needed in Park and Phillips (1999) for the proof of the result of equation (3). Note that among the assumptions of this paper and Park and Phillips (1999) is one that guarantees that the increments wt of xt have an absolutely continuous distribution. In the main theorem, I show that as long as it is possible to “integrate over the poles” of the function T(.) and an additional regularity condition on T(.) holds, the result of equation (1) can still be proved. It should be noted that the result of equation (3) and a similar result for the logarithm function are established in Remark 3.5(b) in Park and Phillips (1999) and also that the result of equation (2) appears to be anticipated in Remark 3.5(c) in Park and Phillips (1999) and was suggested to me in personal communication with Professor Phillips.
2. MAIN RESULT
In this paper, as in Park and Phillips (1999), it is assumed that
where wt is generated according to
where εt is assumed to be a sequence of independent and identically distributed (i.i.d.) random variables with mean zero and where it is assumed that
. In addition, I will assume that x0 is an arbitrary random variable that is independent of all wt, t ≥ 1. The main assumption in this paper is Assumption 2.2 from Park and Phillips (1999).
Assumption 1.
(a)
for some p > 2.
(b) The distribution of εt is absolutely continuous with respect to the Lebesgue measure and has a characteristic function ψ(s) for which lims→∞ sηψ(s) = 0 for some η > 0.
Assumption 1 guarantees that n−1/2x[rn] ⇒ σW(r). To prove the main result, I needed the following key lemma.
LEMMA 1. Under Assumption 1, there exist a constant C > 0 and an index N such that for all
,
A result such as Lemma 1 is called an “integro-local limit theorem.” It was pointed out to me by a referee that the absolute continuity of the distribution of εt (which is implied by Assumption 1) and the full force of Lemma 1 may not be necessary to obtain the result of Theorem 1, which follows, and that a local limit theorem for lattice random variables (such as the theorem in Gnedenko, 1967, p. 262, paragraph 43) may also suffice for the proof of Theorem 1.
The main result of this paper is the following theorem.
THEOREM 1. Assume that
for all K > 0. In addition, assume that for some q ≥ 1 there exists a grid {a1,…,aq}, where aj < aj+1 for all j = 1,…,q − 1, such that T(.) is continuous at any
and monotone on (aj−1,aj) for j = 1,…,q + 1 (defining a0 = −∞ and aq+1 = ∞). Then under Assumption 1,
A monotone function is defined as a function that is either nondecreasing or nonincreasing. The condition
for all K > 0 is exactly the condition that T(.) is locally integrable. A consequence of Theorem 1 is the following results, which render the improvement of the result of Park and Phillips (1999) mentioned in the Introduction.
Remark 1. Under Assumption 1,
for all α > −1. This result follows immediately from Theorem 1 by choosing q = 1 and a1 = 0 and noting that |x|α is monotone on (−∞,0) and (0,∞) and that
for all K > 0 for α > −1.
Remark 2. Under Assumption 1,
. This follows by again choosing a1 = 0 and q = 1 and noting that
for all K > 0.
As far as this author is aware, the result of Remark 2 has not been established elsewhere. Remark 2 may be useful for the analysis of inference procedures when a logarithm transformation has been incorrectly applied to a time series process.
Theorem 1 suggests the theoretically interesting question as to what will be “minimal” conditions, in addition to the integrability condition on T(.), for the result of equation (1) to hold. After learning about the research of this note, Pötscher (2003) has worked on finding minimal conditions on T(.) and minimal weak dependence conditions that will allow the result of equation (1) to go through. The proof as presented in the Appendix seems to rely on the assumed monotonicity to obtain Theorem 1. In addition, another problem of interest is to establish what happens to the statistic of equation (1) if T(.) should not be locally integrable (e.g., T(x) = |x|α for α < −1).
This note generated some comments and suggestions for extensions and future research by referees and readers. Referees have suggested ways to consider relaxation of the monotonicity condition on T(.) and the relaxation of my weak dependence conditions (even such as to include fractionally integrated processes). These issues are dealt with in Pötscher (2003). In addition, the referees suggested a line of proof that might be capable of dealing with innovation distributions that are not necessarily continuous, and extensions to multivariate functionals or functionals with more complex singularities, using a generalized function approach to additive functionals. Also, they brought to my attention an important paper by Borodin and Ibragimov (1995). This paper contains a result of the type of equation (1), but as Pötscher (2003) points out, their results do not allow for functions with poles as considered in this paper and in Pötscher (2003).
APPENDIX: PROOFS
Proof of Lemma 1. Note that by Theorem 6.2.1 of Chung (1974)
where fn(.) is the characteristic function of n−1/2xn. Now, note that
and therefore
This in turn implies that
Therefore, noting that sηψ(s) → 0 as |s| → ∞ for some η > 0, for λ such that
we have
say, where in the last inequality I set ξ = sn−1/2 and used the inequality |exp(ia) − exp(ib)| ≤ |a − b|. To deal with T2, note that because
we have
for k ≥ M for some M. Therefore for n ≥ N = M + [2/η + 1] , under Assumption 1,
say, where cn → 0 as n → ∞ because
C′ denotes some positive constant not depending on y, δ, or n, and the second inequality follows from the assumptions made on ψ(.) in Assumption 1 and the fact that for j in the given index range, by the inequality of equation (A.7),
The result of equation (A.9) follows because by Theorem 6.4.7 of Chung (1974), characteristic functions of absolutely continuous random variables will not equal 1 at any point other than ξ = 0 if the underlying distribution is absolutely continuous.
We now apply Theorem 11.6 of Davidson (1994) to deal with T1. Recalling that
, it follows that for |s|n−1/2 ≤ λ,
The first inequality here was Theorem 11.6 of Davidson (1994), and the third uses E|X|2I(|X| > K) ≤ E|X|pK2−p. Therefore,
for |s|n−1/2 ≤ λ. Next, again assume that n ≥ N, where N is defined after equation (A.7). Therefore, for
,
where the last inequality uses that log(1 + x) ≤ x for x > −1 and it is easily verified that for
,
This implies that
and because by the definition of N,
it now follows that for n large enough,
In conclusion, we have now shown that for large enough n,
for some C > 0 not depending on y, δ, or n, which establishes the result. █
Proof of Theorem 1. First note that
if for all δ > 0 and some
, and
This is because
Note that the second and third terms in the last equation converge to 0 by assumption and the first term can be bounded by
which equals 0 by assumption. For simplicity, I will assume that σ = 1 throughout this proof.
Next, I will argue that for any j = 0,…,q,
The joint convergence in distribution of the q + 1 terms can be easily proved using the reasoning as given subsequently for a single term.
Now define, for N as in Lemma 1, small enough δ > 0, and for j = 1,…,q − 1,
and
Note that the argument for j = 0 and j = q is analogous (although for the proof, one needs to use that max1≤t≤n n−1/2|xt| = OP(1)). Then by Theorem 3.2 in Park and Phillips (1999), it follows that for all δ > 0, as n → ∞,
because Tδ(x) is regular and locally integrable; the oP(1) assertion follows by noting that by the monotonicity of T(.) on (aj,aj+1), for j = 1,…,q − 1,
and again the argument for j = 0 and j = q is analogous. Furthermore, as δ → 0, by the occupation times formula,
(where L(t,s) denotes the Brownian local time; see Park and Phillips, 1999) because ∫ajaj+1|T(s)|ds < ∞ by assumption. Finally, note that because of absolute continuity of the distributions of xt,
I will only deal with the second term, because the third term can be dealt with analogously. Defining δk = δ2−k, the expectation of that term can be bounded by
for some constant C > 0, where the first inequality follows from monotonicity of T(.) on (aj−1,aj), the second uses Lemma 1, and the third uses the definition of δk. The last expression obviously converges to 0 as δ → 0 under the assumption of the theorem, which completes the proof of this theorem. █
References
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