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Asymptotic Theory for Regressions with Smoothly Changing Parameters

  • Eric Hillebrand EMAIL logo , Marcelo C. Medeiros and Junyue Xu

Abstract: We derive asymptotic properties of the quasi-maximum likelihood estimator of smooth transition regressions when time is the transition variable. The consistency of the estimator and its asymptotic distribution are examined. It is shown that the estimator converges at the usual -rate and has an asymptotically normal distribution. Finite sample properties of the estimator are explored in simulations. We illustrate with an application to US inflation and output data.

JEL Codes: C22

Acknowledgments

Parts of the research for this article were done while the first author was visiting the Department of Economics at the Pontifical Catholic University of Rio de Janeiro, Brazil, and while the second author was visiting CREATES, Aarhus University, Denmark. Their hospitality is gratefully appreciated. EH acknowledges support from the Danish National Research Foundation. MCM’s research is partially supported by the CNPq/Brazil.

Appendix A: proof of consistency

Proof of Theorem 1. We establish the conditions for consistency according to Theorem 4.1.1 of Amemiya (1985). We have , if the following conditions hold: (1) is a compact parameter set; (2) is continuous in and measurable in ; (3) converges to a deterministic function in probability uniformly on as ; and (4) attains a unique global maximum at .

Item (1) is given by Assumption 1. Item (2) holds by definition of the QMLE (5) from the definition of the normal density. For item (3), we refer to Theorem 4.2.1 of Amemiya (1985): This holds for i.i.d. data if and is continuous in for each . The extension to stationary and ergodic data using the same set of assumptions is achieved in Ling and McAleer (2003, Theorem 3.1). We have by Jensen’s inequality and , where denotes the normal density function. The finiteness of the last expression follows from the assumption that for some constant c. The log density is continuous in for every .

Consider Item (4). By the Ergodic Theorem, . Rewrite the maximization problem as . Now, for a given number ,

[9]
[9]

We show that and that [9] attains an upper bound at uniquely. Consider . Substituting for , we obtain .

The inequality holds from Assumption 2(3). We have established that for any given , the objective function [9] attains its maximum of

at , , . Define , then

attains its maximum of 0 at , therefore the maximizer is . This shows that, under Assumption 4, is uniquely maximized at . □

Appendix B: proof of asymptotic normality

In this proof, terms will sometimes involve expectations of cross–products of the type , where X and Y are correlated random variables. Note that by the Cauchy–Schwarz inequality, we have , and thus in order to show that the cross-product has finite expectation, it suffices to show that both random variables have finite second moments. By the same token, if both X and Y have finite second moments,

for some .

In the outline of the proof, we follow Theorem 4.1.3 of Amemiya (1985). Therefore, we have to establish the conditions

  1. exists and is continuous in an open neighborhood of .

  2. for all sequences .

where is standard Brownian motion on the unit interval.

Item (1) is shown in Lemma 3. Item (2) needs consistency of for , which we established in Theorem 1. It further needs . We use Ling and McAleer (2003, Theorem 3.1) to establish this. We show the uniform convergence in Lemma 4.

Item (3) uses Billingsley (1999, Theorem 18.3) and needs (a) that is a stationary martingale difference sequence and (b) that exists. Both will be proved in Lemma 3. The first two lemmas show a few technical properties of that are needed in the following.

Lemma 1

The transition function given by eq. [3] is bounded, and so are its first and second derivatives with respect to and .

Proof. We will use shorthand notation f for below unless otherwise stated. Defining , it is easy to verify that . Since the transition function has the range , it is clearly bounded. For the first derivative of f with respect to , ,

The first inequality follows from the fact that . The second inequality holds because both and f are bounded. For the second derivative of f with respect to , ,

The second inequality follows from the fact that , the last inequality holds because both and f are bounded. The proof of the boundedness of the first and second derivatives of f with respect to is almost identical to the one above and is omitted for brevity.□

Lemma 2

Let , then

  1. .

  2. , where denotes the standard vector and matrix norms.

Proof. We will prove the statements element by element. For statement (1),

by Assumption 3 (2). As ,

By Lemma 1, Assumption 1, and Assumption 3(2),

Similarly,

For statement (2),

For the second inequality, we use the fact that is bounded from Lemma 1.

Similarly,

Lemma 3

  1. The sequence is a stationary martingale difference sequence. is the sigma-algebra given by all information up to time t.

Proof. For part (1) of the proof, all derivatives are evaluated at . The nought-subscript is suppressed to reduce notational clutter. Let , as before.

since is independent of and its derivatives are bounded (Lemma 2).

since has mean zero and variance .

For part (2) and (3) of the proof, the expressions are evaluated at any , if not otherwise stated. The data-generating parameters will be explicitly denoted by a nought-subscript. The process is data and thus evaluated at throughout.

We first consider the gradient vectors of ,

The finiteness of the second factor follows from Lemma 2 (1). For the first factor, note that

Therefore, there exists such that

where L is some positive constant. The existence of such L is guaranteed by the compactness of the parameter space and the fact that f is bounded. Using Assumption 3 (2), it is clear that is bounded.

For ,

This shows statement (2) of Lemma 3. Statement (3) uses similar techniques in the proof. We will only show the case of , which requires most work. The rest of the proof will be omitted for brevity.

The finiteness of follows from Assumption 3 (2). is finite due to Assumption 1. Lemma 1 ensures that the last factor is bounded. To see the finiteness of the first factor, recall in part (2) we have shown that . It follows that . Therefore, by Assumption 3.□

Lemma 4

The function

where
is such that , it is continuous in and has zero mean: .

Proof. From the triangular inequality,

If , exists and by the Ergodic Theorem, there is pointwise convergence. Thus, showing absolute uniform integrability reduces to showing that

Proving finiteness of the expected value of the supremum consists of repeated application of the Lebesgue Dominated Convergence Theorem (Shiryaev 1996, 187), Ling and McAleer (2003), Lemmas 5.3 and 5.4). We will show the statement for second derivatives, element by element, starting with ,

According to Assumption 2 (1) there exists a constant c such that , therefore

By Assumption 3 (3),

For ,

The last inequality follows from the fact that . Therefore,

We next examine the second derivatives of the log likelihood with respect to ,

In order to show , it is sufficient to show that . Recall, we have already proved in Lemma 3 (2) that . It follows that .

To show that , consider

where L is some positive constant. The second term on the right side can be written as

where K is some positive constant. Again, the compactness of the parameter space, boundedness of f, and stationarity of ensures the existence of K and L. It follows that

The finiteness of the derivatives of f was shown in Lemma 1. Thus,

The proof that closely resembles the proof above and is omitted for brevity.□

Proof of Theorem 2. The proof establishes the conditions of Theorem 4.1.3 of Amemiya (1985) with a generalization due to Ling and McAleer (2003, Theorem 3.1). We need consistency of for , which was shown in Theorem 1. Then, we show

where is N-dimensional standard Brownian motion on the unit interval. This is condition (C) in Theorem 4.1.3 of Amemiya (1985). The convergence follows from Theorem 18.3 in Billingsley (1999) if (a) is a stationary martingale difference, and (b) exists. Both conditions were shown in Lemma 3.

To satisfy condition (B) of Theorem 4.1.3 of Amemiya (1985), we have to establish

for any sequence ,

is non-singular. Conditions for the double stochastic convergence can be found in Theorem 21.6 of Davidson (1994). We need to show

  1. consistency of for (Theorem 1), and

  2. uniform convergence of to in probability, i.e. .

We prove uniform convergence of using Theorem 3.1 of Ling and McAleer (2003), who generalize Theorem 4.2.1 of Amemiya (1985) from i.i.d. data to stationary and ergodic data. This allows the immediate invocation of the Ergodic Theorem without having to check finiteness of third derivatives of as in Andrews (1992, Theorem 2). To apply Theorem 3.1 of Ling and McAleer (2003), we need that

is continuous in (this also establishes condition (A) of Theorem 4.1.3. of Amemiya (1985) along the way), has expected value , and . This was shown in Lemma 4. Thus, we have established all conditions for asymptotic normality according to Theorem 4.1.3 of Amemiya (1985).□

Proof of Proposition 1. The proof of uniform convergence in probability of to is given in Lemma 4 and Theorem 2. We need to show uniform convergence of to . We employ Theorem 3.1 of Ling and McAleer (2003) again and show that

is absolutely uniformly integrable, continuous in , and has expected value . The detailed proof is in complete analogy to Lemma 4 and is omitted for brevity.□

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  1. 1

    Note that, contrary to the work on multiple structural breaks, for each break there are two nuisance parameter ( and c) instead of one.

Published Online: 2013-04-30

©2013 by Walter de Gruyter Berlin / Boston

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