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FURTHER RESULTS ON THE ASYMPTOTICS FOR NONLINEAR TRANSFORMATIONS OF INTEGRATED TIME SERIES
Published online by Cambridge University Press: 31 March 2005
Abstract
This paper establishes various results involving functions of integrated processes. Two theorems—which improve similar results by Park and Phillips—are proved for averages of functions of an integrated process that has not been rescaled by the square root of sample size. In addition, two results are given that characterize asymptotic behavior of averages of nonintegrable functions of rescaled integrated processes; the observations close to the pole take over asymptotic behavior in that case. Throughout, we make the assumption that the innovations of the integrated process are a linear process.
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1. INTRODUCTION
This paper proves three results about functions of integrated processes. Our first result is an extension of a result in Park and Phillips (1999), where it is proved that for integrable functions T(.) and for I(1) processes xt,
where L(t,s) is a two-parameter stochastic process called (Brownian) local time. The remarkable thing about this result is that it establishes limit theory for a function of an I(1) process that has not been rescaled by n−1/2. Park and Phillips establish the preceding result under some regularity conditions on the I(1) process xt and the integrable function T(.). In this paper, we show that Park and Phillips's regularity conditions for the preceding result can be relaxed and also that their result can be extended to yield, for 0 ≤ α < ½,
A central tool for the proof of this first result is a lemma that was recently established in de Jong (2004). Also in Park and Phillips (1999), it is shown that for functions T(.) that satisfy
under conditions on R(.,.) that basically serve to ensure asymptotic negligibility of
we have
where σ2 = limn→∞ n−1Exn2. Again the interesting aspect of the preceding result is the fact that it considers integrated processes that have not been rescaled by n−1/2. Functions T(.) that satisfy the appropriate condition are called asymptotically homogeneous by Park and Phillips. The asymptotically homogeneous condition is trivially satisfied for T(x) = |x|a for a ≥ 0 but is general enough to also deal with functions such as T(x) = |x|a log|x| for all a ≥ 0. In this paper, we show the more general result that whenever for functions H(.) and ν(.) we have
in L1 sense, we have for 0 ≤ α < ½, under regularity conditions,
Therefore, we show that Park and Phillips's class of asymptotically homogeneous functions can be extended, and we consider n−αxt for 0 ≤ α < ½ instead of xt as the argument for T(.).
A third result that is proved in this paper concerns averages of the type
where m > 1. Although it has been shown in de Jong (2004) and Pötscher (2004) that under regularity conditions for locally integrable functions T(.) we have
it is yet unknown what happens to functions T(.) that are not integrable. Using a “clipping device” involving a deterministic sequence cn that converges to 0 with n, it will be proved that for m > 1,
and also that
2. ASSUMPTIONS AND RESULT FOR INTEGRABLE FUNCTIONS
Identically to Park and Phillips (1999), linear process conditions for xt are assumed and
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, we will assume that x0 is an arbitrary random variable that is independent of all wt, t ≥ 1. The main assumptions used in this paper are Assumptions 2.1 and 2.2 from Park and Phillips (1999).
Assumption 1.
.
Assumption 2.
Assumption 1 guarantees that n−1/2x[rn] ⇒ σW(r) where ⇒ denotes weak convergence in C[0,1], i.e., the space of functions that are continuous on [0,1]. Define Wn0 = n−1/2x[rn]. Then by the Skorokhod representation there exists a Wn(.) such that
. Assumption 2 in addition also guarantees a convergence rate for supr∈[0,1]|Wn(r) − W(r)|; see Park and Phillips (1999, Lemma 2.3). This result justifies that proving our results for Wn(.) suffices. Several of the manipulations in the proofs of the results in this paper require the use of local time L(.,.). Local time is a random function satisfying
See Park and Phillips (1999, pp. 271–272) and Chung and Williams (1990, Ch. 7) for more details regarding local time.
Park and Phillips (1999) establish the following result for integrable functions of integrated random variables.
THEOREM 1. Suppose that T(.) is integrable and Assumption 2 holds with p > 4. If T(.) is square integrable and satisfies the Lipschitz condition
over its support for some constants c and l > 6/(p − 2), then
For differentiable functions T(.), we need to set l = 1, implying that we need p > 8 for the theorem to work. To improve the preceding result, we needed the following useful lemma, which was established in de Jong (2004).
LEMMA 1. Under Assumption 2, for all
for some value of M,
where C and M do not depend on y, δ, or n.
Using this lemma, we were able to improve Park and Phillips's result and show the following quite general result.
THEOREM 2. Suppose Assumption 2 holds. Also assume that |T(x)| ≤ R(x) and assume that R(.) is integrable, continuous on
, and monotone on (0,∞) and (−∞,0). If T(.) is continuous, then for 0 ≤ α < ½,
Comparing this theorem to Park and Phillips's theorem, it can be seen that we have completely removed their Lipschitz-continuity condition and weakened it to continuity and, in addition, their requirement on p has been removed. Also, weights n−α for 0 ≤ α < ½ are allowed for. Although no R(.) function such as is present in Theorem 2 is explicitly used in their Theorem 1, from Park and Phillips's proof it is clear that existence of such a function is implied. Therefore, Theorem 2 is a “clean” improvement to Park and Phillips's Theorem 1. A simple example of a function T(.) that satisfies our conditions but violates those of Park and Phillips (1999) is T(x) = (1 − x2)1/2I(−1 ≤ x ≤ 1).
3. ASYMPTOTICALLY HOMOGENEOUS FUNCTIONS
In this section, we improve Park and Phillips's (1999) result for asymptotically homogeneous functions. Park and Phillips assume that
and they show that
if either
a. |R(x,λ)| ≤ a(λ)P(x), where lim supλ→∞ a(λ)/ν(λ) = 0 and P is locally integrable, or
b. |R(x,λ)| ≤ b(λ)Q(λx), where lim supλ→∞ b(λ)/ν(λ) < ∞ and Q is locally integrable and vanishes at infinity, i.e., Q(x) → 0 as |x| → ∞.
In this paper, we redefine their notion of an asymptotically homogeneous function, as follows.
DEFINITION 1. A function T(.) is called asymptotically homogeneous if for all K > 0 and some function H(.),
Obviously from the dominated convergence theorem it follows that if for some ν(.) and H(.), pointwise in x,
and |ν(λ)−1T(λx)| ≤ G(x) for a locally integrable function G(.), then T(.) is asymptotically homogeneous. In the discussion that follows, we will call a function monotone regular if for some {a1,…,aq}, T(.) is monotone on (aj,aj+1) for j = 0,…,q (setting a0 = −∞ and aq+1 = ∞).
The main result of this section is the following theorem.
THEOREM 3. Suppose Assumption 1 holds. Also assume that T(.) is asymptotically homogeneous. In addition, assume that H(.) is continuous and T(.) is monotone regular. Then, for 0 ≤ α < ½,
It is also possible to show that our definition of an asymptotically homogeneous function is more general than Park and Phillips's. Under assumption a preceding Definition 1,
as λ → ∞ if P(.) is locally integrable. Under assumption b preceding Definition 1,
as λ → ∞, because
by boundedness of Q(.) (which is also assumed in Park and Phillips, 1999). Therefore, obviously the set of functions that is “asymptotically homogeneous” in this paper is wider than in Park and Phillips (1999). But clearly, most functions that one may expect to be useful for applications should be expected to already be in Park and Phillips's class of asymptotically homogeneous functions, and the main achievement of our analysis is the redefinition of the class of asymptotically homogeneous functions to as large as possible a collection of functions. It appears to us that the preceding result should be close to the limits of what should be possible in this setting, and for the authors of this paper, it is hard to see how the preceding definition of the class of asymptotically homogeneous functions can be relaxed further to yield an even larger function class that generates similar behavior.
4. NONINTEGRABLE FUNCTIONS
In de Jong (2004) and Pötscher (2004) it is proved that under regularity conditions, in spite of possible poles in T(.), as long as
for all K > 0, we have
These results raise the question as to what will happen if a nonintegrable function of an integrated process is used for T(.) in statistics of the form
This issue appears to have never been tackled before in either the statistics or the econometrics literature. This section explores this issue for functions
for m > 1. As it turns out and is perhaps to be expected, the observations “close to zero” take over the limit behavior of the statistic in this case. We will need a “clipping device,” and we construct statistics similar to those constructed in Park and Phillips (1999) for integrable functions. Our first result is the following theorem.
THEOREM 4. Let cn = n−(2p+1)/3p+η for some η > 0 such that −(2p + 1)/3p + η < 0. In addition, assume that
for some m > 1. Let
. Then under Assumption 2,
Clearly, in the preceding theorem dn = (m − 1)−1(cn1−m − 1), but we choose the preceding formulation to bring out better the origin of our rescaling factor dn. The proof of the following “two-sided” version of the preceding theorem is analogous and therefore omitted.
THEOREM 5. Let cn = n−(2p+1)/3p+η for some η > 0 such that −(2p + 1)/3p + η < 0. Assume that
for some m > 1. Let
. Then under Assumption 2,
The preceding theorems leave the issue wide open as to what function class the preceding theorem can be extended. The line of proof employed in the Appendix may allow for some generalization, but it is not clear to the authors what the outer limits are for which a result such as the preceding theorem might hold.
As a referee pointed out, it may also be of interest to attempt to show convergence toward
. Note that this expression is well defined, whereas the limit as ε approaches zero from above of
does not exist; the former result can be seen to follow from the occupation time formula and a modulus of continuity type result for the Brownian local time. The techniques used in this paper, however, seem not suited toward proving a result of this type.
APPENDIX: PROOFS
Throughout this section, to improve readability, we will assume for every proof that σ2 = 1. We use the following definitions, which are identical to those in Park and Phillips (1999):
In the proofs that follow, M and C are the constants from Lemma 1. The following lemma from Park and Phillips (1999) was needed to prove our results.
LEMMA 2. Under Assumption 2, as n → ∞,
for πn ≥ νn n−2(p+1)/3p and any ε > 0.
Proof. See Park and Phillips (1999).
We are now in a position to prove the main theorems of this paper.
Proof of Theorem 2. Define TK(x) = T(x)I(|x| ≤ K), TK′(x) = T(x)I(x > K), and TK′′(x) = T(x)I(x < −K). We will show that
and the same argument, mutatis mutandis, will hold for
. Then, we will show that for all K > 0,
and the result then follows (for a formal proof that this is sufficient, see, e.g., the start of the proof of Theorem 1 of de Jong, 2004). Let M be as defined in Lemma 1. To show the result of equation (A.1), note that for all K > 0,
as n → ∞, and
as K → ∞, where
under the assumptions of the theorem because
as K → ∞. The first inequality follows from the assumed boundedness of |T(.)| by R(.) and the assumed monotonicity of R(.), and the second is an application of Lemma 1. This completes the proof of the result of equation (A.1). The remainder of the proof follows the line of proof of Park and Phillips (1999, proof of Theorem 5.1), but some modifications will be made. To show the result of equation (A.2) and thereby make the proof of Theorem 2 complete, define for δ > 0
and note that for all K > 0,
and therefore
as δ → 0 by continuity of T(.), where the second inequality is Lemma 1. Therefore, we can consider
instead of
. Now
because by the Cauchy–Schwartz inequality,
where the second inequality is Lemma 2. Therefore, it suffices to consider
Now note that
by the second part of Lemma 2. Therefore,
implying that it suffices to analyze
As n → ∞,
as explained in the text following Lemma 2.5 of Park and Phillips (1999). In addition, as δ → 0, by continuity of T(.),
Therefore,
implying that the condition of equation (A.2) is now verified. This completes the proof. █
For the proof of Theorem 3, we need the following lemma.
LEMMA 3. Under Assumption 1,
Proof of Lemma 3. For any δ > 0, for n large enough, supr∈[0,1]|Wn(r) − W(r)| ≤ δ almost surely, implying that for n large enough
where the equality follows from the occupation times formula (see Park and Phillips, 1999, Lemma 2.4) and because
is a well-defined random variable. Because δ can be chosen arbitrarily small, the result now follows. █
Proof of Theorem 3. Because sup1≤t≤n|Wn(t/n)| = Op(1), it now suffices to show that for any K > 0,
Define
and note that
by Lemma 3. Now for all δ > 0, let
We will show that limδ→0 lim supn→∞|Sjnδ − Sj+1,nδ| = 0 almost surely for j = 1,…,4. By the monotone regular condition, we can act as if T(.) is monotone without loss of generality. For |S1 − S2nδ| we then have
and as δ → 0, the last term disappears because of continuity of H(.), the second inequality follows from monotonicity of T(.), and the third follows by our definition of an asymptotically homogeneous function. To show that limδ→0 lim supn→∞|S2nδ − S3nδ| = 0 almost surely, note that
almost surely under our assumptions and by the definition of cn. For |S3nδ − S4nδ| we have
By the earlier argument,
and therefore it suffices to show that as δ → 0,
By the occupation times formula, the preceding expression satisfies
by uniform continuity of L(1,.) on [−K,K]. Finally, for |S4nδ − S5|, we have
by the definition of an asymptotically homogeneous function, which completes the proof. █
The following lemma is needed for the proof of Theorem 4.
LEMMA 4. For any sequence bn such that cn = o(bn), under the assumptions of Theorem 4,
Proof of Lemma 4. This result follows because
Now because T(x) = |x|−mI(x > 0), the last expression equals
and because m > 1 and cn = o(bn), the result now follows. A similar argument will hold for a lower bound, which then completes the proof of the lemma. █
Proof of Theorem 4. Note that, for bn = cn1−1/m−α for some α > 0 small enough that bn → 0 and dn−1T(bn) → 0 as n → ∞,
and the second term is op(1) because
by assumption. Now note that trivially, for all δ > 0,
An upper bound for the last term is
Similarly, a lower bound is
We will only consider the upper bound and determine its limit, but the argument for the lower bound is identical and renders the same limit. By Lemma 2,
Now, by Lemma 4,
and therefore the expression of equation (A.4) converges to zero in probability if
First, note that by assumption cn−1n−1 → 0 and that the second part of the preceding expression is
by assumption. Therefore, it suffices to consider
Now by the comment following Lemma 2.5 in Park and Phillips (1999),
if δcn ≥ n−(2p−1)/3p+η for some η > 0, which is the case by assumption for n large enough. Therefore, we only need consider
Now by Lemma 4, it follows that by choosing δ arbitrarily small, the limit distribution will be arbitrarily close to L(1,0); and noting that the same argument will work for the lower bound, this suffices to prove the result. █
References
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