A.1. Proof of Theorem 3.1.

Instead of the series y_t it will be convenient to use the mean adjusted series

Solving the preceding equation for y_t and inserting the result into (3.1) yields

Here

Note that the true values of ν₀⁽⁰⁾ and ν₁⁽⁰⁾ are zero.

It will also be convenient to use the transformation Πx_t−1 = α⁽⁰⁾u_t−1⁽⁰⁾ + ρ⁽⁰⁾v_t−1⁽⁰⁾, where u_t−1⁽⁰⁾ = β_o′x_t−1, v_t−1⁽⁰⁾ = β_o⊥′x_t−1, α⁽⁰⁾ = αβ′β_o(β_o′ β_o)⁻¹, and ρ⁽⁰⁾ = αβ′β_o⊥(β_o⊥′ β_o⊥)⁻¹. Clearly, the true values of α⁽⁰⁾ and ρ⁽⁰⁾ are α_o and zero, respectively. With this transformation the preceding error correction form can be expressed as

Denote q_tτ = [d_tτ : d_tτ′]′ and

With this notation (A.2) becomes

where Φ = [ν₀⁽⁰⁾ : Tν₁⁽⁰⁾ : T^1/2ρ⁽⁰⁾ : α⁽⁰⁾ : Γ₁ : ··· : Γ_p−1], Ξ = [δ₁ : γ], and Ξ⁽⁰⁾ = [δ₁⁽⁰⁾ : γ⁽⁰⁾].

Let Θ = [Φ : Ξ] contain the freely varying parameters in (A.3) or (A.2) (Ξ⁽⁰⁾ is not a freely varying parameter because it is determined by α⁽⁰⁾, ρ⁽⁰⁾, and Γ₁,…,Γ_p−1). Set

Then

is −2 times the (conditional) Gaussian log-likelihood function of the parameters in (A.3). Minimizing this function yields Gaussian ML estimators of the parameters Θ, τ, and Ω. It is not difficult to see that the resulting estimators of Θ and τ can alternatively be obtained by minimizing the concentrated counterpart of l_T(Θ,τ,Ω), that is,

The definition of ε_tτ(Θ) (and the fact that Ξ⁽⁰⁾ is not a freely varying parameter) makes it clear that the value of τ that minimizes the function l_T^(c)(Θ,τ) is identical to

defined by (3.2). Thus, (asymptotic) properties of

can be studied by using the Gaussian ML estimator of τ discussed previously. Before turning to this issue we note that the preceding discussion also makes clear that a minimizer of l_T(Θ,τ,Ω) exists (for every T larger than some constant).

The proof of Theorem 3.1 consists of several steps. In the first one we consider a subset of the parameter space of (Θ,Ω) defined by

and

Note that here M does not depend on T although Φ and δ₁⁽⁰⁾ do. We now prove the following lemma.

LEMMA A.1. Let B₁ = B₁(M,ω,ω) be the part of the parameter space of (Θ,τ,Ω) in which conditions (A.4) and (A.5) hold. Then there exist choices of M, ω, and ω such that

with probability approaching one.

Proof. First note that

where the latter equality is justified by the weak law of large numbers.

Next, because [Tλ] ≤ τ, τ_o ≤ [Tλ], we find from the definitions that

Hence,

where Φ⁽⁰⁾ = Φ + [δ₁ − δ₁⁽⁰⁾ : 0]. Here

where ε_* > 0 is a suitable real number and the inequality holds with probability approaching one. This fact can be justified in the same way as Lemma A.4 of Saikkonen (2001). A similar result is also obtained by changing the range of summation on the l.h.s. of (A.8) to t = [Tλ] + p,…,T. When these two eigenvalue conditions are assumed, arguments entirely similar to those in Saikkonen (2001, pp. 320–321) show that, with suitable choices of M, ω, and ω, the r.h.s of (A.7) can be made arbitrarily large whenever (Θ,τ,Ω) ∉ B₁(M,ω,ω). The assertion of the lemma follows from this and (A.6). █

Lemma A.1 implies that a minimizer of l_T(Θ,τ,Ω) will asymptotically satisfy inequality restrictions of the form (A.4) and (A.5). In what follows, the set B₁ is always assumed to be defined in such a way that the conclusion of Lemma A.1 holds. We shall now proceed in the same way as in Saikkonen (2001) and express the function l_T(Θ,τ,Ω) as a sum of two components. To this end, define

Then w_t⁽⁰⁾ = [w_1t⁽⁰⁾′ : w_2t⁽⁰⁾′]′, and we also partition the parameter matrix Φ conformably as Φ = [Φ₁ : Φ₂] where Φ₁ = [ν₀⁽⁰⁾ : Tν₁⁽⁰⁾ : T^1/2ρ⁽⁰⁾] and Φ₂ = [α⁽⁰⁾ : Γ₁ : ··· : Γ_p−1]. With these definitions,

where ε_1tτ(Θ) = −Φ₁ w_1t⁽⁰⁾ − Ξq_tτ + Ξ⁽⁰⁾q_tτo and ε_2t(Φ₂) = Δx_t − Φ₂ w_2t⁽⁰⁾. Clearly, ε_1tτo(Θ_o) = 0 and

where

For l_2T(Φ₂,Ω) we have the following result.

LEMMA A.2.

where the infimum is over unrestricted values of Φ₂ and Ω > 0.

Proof. Because we can treat Δx_t as a zero mean stationary process and because l_2T(Φ₂,Ω) can be interpreted as −2 times the logarithm of the Gaussian likelihood function associated with the regression model Δx_t = Φ₂ w_2t⁽⁰⁾ + ε_t, the stated result follows from standard regression theory (cf. Saikkonen, 2001, p. 321). █

Next consider the function l_1T(Θ,τ,Ω). Our treatment will be divided into several steps in which the time index t is suitably restricted. This means considering the function l_1T(Θ,τ,Ω) with the sample size T replaced by appropriate quantities smaller than T. Most of the subsequent results will explicitly be formulated for τ ≤ τ_o and only briefly discussed in the case τ ≥ τ_o (cf. Bai et al., 1998).

In the following results about the function l_1T(Θ,τ,Ω), c₁,c₂,… denote positive constants and a_1T,a_2T,… are nonnegative random variables that depend on the sample size but not on the parameters Θ, τ, or Ω. First we prove the following lemma.

LEMMA A.3. There exists a constant c₁ > 0 such that, with probability approaching one and uniformly in [Tλ] ≤ τ ≤ τ_o and (Θ,τ,Ω) ∈ B₁,

where a_1T ≥ 0 and a_1T = O_p(1).

Proof. For t ≤ τ − 1, ε_1tτ(Θ) = −Φ₁ w_1t⁽⁰⁾ and, consequently,

For L₁ we have

For (Θ,τ,Ω) ∈ B₁, the first eigenvalue in the last expression is bounded away from zero. That the same holds with probability approaching one and uniformly in [Tλ] ≤ τ ≤ τ_o for the second eigenvalue can be seen by using an analog of (A.3) of Gregory and Hansen (1996, p. 118). Thus, we have shown that L₁ ≥ c₁∥Φ₁∥², c₁ ≥ 0, with probability approaching one.

It remains to show that L₂ ≥ −a_1T∥T^1/2Φ₁∥ with a_1T having the properties stated in the lemma. To demonstrate this, notice that

Here we have used the definition of ε_2t(Φ₂), the Cauchy–Schwarz inequality, and the norm inequality. By an analog of (A.4) of Gregory and Hansen (1996, p. 118), the norm in the middle of the last expression is of order O_p(1) uniformly in [Tλ] ≤ τ ≤ τ₀ and for any fixed value of Φ₂. Thus, because the parameters Φ₂ and Ω belong to bounded sets when (Θ,τ,Ω) ∈ B₁, it can similarly be shown that the last expression as a whole has an upper bound a_1T∥T^1/2Φ₁∥ with a_1T as required. This completes the proof. █

Our next result deals with the contribution of l_1,τo−1(Θ,τ,Ω) − l_1,τ−1(Θ,τ,Ω) to l_1T(Θ,τ,Ω). Here the relevant expression of ε_1tτ(Θ) is

where Ψ₁ = Φ₁ + [δ₁ : 0].

LEMMA A.4. Let ε be any real number with the property 0 < ε < λ_o − λ. Then, for λ ≤ λ ≤ λ_o − ε there exists a constant c₂ > 0 such that, with probability approaching one and uniformly in [Tλ] ≤ τ ≤ [T(λ_o − ε)] and (Θ,τ,Ω) ∈ B₁,

where a_iT ≥ 0 (i = 2,3), a_2T = O_p(1), and a_3T = o_p(T^η) with 1/b < η < ¼.

Proof. By the definitions,

First consider L₃ and for simplicity denote Ψ₁ = [Ψ₁ : γ] and z_1tτ⁽⁰⁾ = [w_1t⁽⁰⁾′ : d_tτ′]′. Then

where D_1T = diag[T^−1/2I_n−r+2 : I_p].

Next note that

uniformly in [Tλ] ≤ τ < τ_o. Because w_1t⁽⁰⁾ = [1 : t/T : T^−1/2v_t−1⁽⁰⁾′]′ this is obvious for the first and second components of w_1t⁽⁰⁾. For the third component the same is true because T^−1/2max_1≤t≤τo∥v_t−1⁽⁰⁾∥ ≤ T^−1/2max_1≤t≤T∥β_o⊥′x_t−1∥ = O_p(1), where the equality follows from the fact that T^−1/2β_o⊥′x_[Ts] obeys an invariance principle. Thus, we can conclude that

uniformly in [Tλ] ≤ τ < τ_o − p.

The next step is to observe that

where M₁₁(λ) is the weak limit of

(cf. (A.3) of Gregory and Hansen, 1996, p. 118). It is straightforward to check that the difference M₁₁(λ_o) − M₁₁(λ) is positive definite and its smallest eigenvalue is bounded from below by a positive constant when λ ≤ λ ≤ λ_o − ε.

The preceding discussion implies that, with probability approaching one, the smallest eigenvalue of the matrix on the l.h.s. of (A.10) is bounded away from zero uniformly in [Tλ] ≤ τ ≤ [T(λ_o − ε)]. Thus, with probability approaching one and in the required uniform sense,

where c₂ > 0 is a (small) constant. This implies that it only remains to show that L₄ ≥ −a_2T∥T^1/2Ψ₁∥ − a_3T∥γ∥ with a_2T and a_3T as stated in the lemma.

To show the previously mentioned inequality about L₄, conclude from the definitions that

Arguments similar to those already used in the proof of Lemma A.3 show that

where a_2T = O_p(1) in the required uniform sense.

Regarding L₄₂, one similarly obtains

where a_3T = o_p(T^η), 1/b < η < ¼, in the required uniform sense. The latter inequality follows if the last norm in the preceding expression can be replaced by o_p(T^η). To justify this, recall that Δx_t and w_2t⁽⁰⁾ are stationary processes with finite moments of order b > 4 and that Φ₂ can be assumed to belong to a bounded set. Thus, it suffices to show that max_1≤t≤T∥Δx_t∥ = o_p(T^η) and similarly with Δx_t replaced by w_2t⁽⁰⁾. This, however, can be done by using an argument entirely similar to that in (A.14) of Saikkonen and Lütkepohl (2002). The inequalities obtained for |L₄₁| and |L₄₂| show that L₂ has the required lower bound, and the proof is complete. █

Our next result describes the contribution of l_1,τo+p−1(Θ,τ,Ω) − l_1,τo−1(Θ,τ,Ω) to l_1T(Θ,τ,Ω). We introduce the notation

In the following lemma the relevant values of ε_1tτ(Θ) can then be written as

where Ψ₂ = Φ₁ + [δ₁ − δ₁⁽⁰⁾ : 0]. Note that here the first term in the definition of ζ_tτ⁽⁰⁾ vanishes, but the general definition is convenient in later derivations. Now we can formulate the following lemma.

LEMMA A.5. There exists a constant c₃ > 0 such that, with probability approaching one and uniformly in [Tλ] ≤ τ ≤ τ_o and (Θ,τ,Ω) ∈ B₁,

where a_iT ≥ 0 and a_iT = O_p(1) (i = 4,5).

Proof. By the definitions,

Assuming (Θ,τ,Ω) ∈ B₁ we find that

Because we can here assume that Ψ₂ is bounded (see (A.5)), an application of the triangle inequality and the Cauchy–Schwarz inequality shows that the absolute value of the third term in the last expression can be bounded from above by

Here the latter square root is of order O_p(1) (see the argument leading to (A.10)). Hence, we can conclude that

where c₃ = ω⁻¹ > 0 and a_41T = O_p(1) in the required uniform sense.

Now consider L₆. Arguments similar to those used in previous derivations combined with the present definition of ε_1tτ(Θ) yield

It is easy to see that the first term on the r.h.s. can be used to define the term a_5T in the lemma. The arguments needed are similar to those used to obtain (A.11), and they can also be applied to the second term so that we can write

where also a_42T = O_p(1) in the required uniform sense. The result of the lemma now follows from (A.11) and (A.12) by defining a_4T = a_41T + a_42T. █

The next lemma is concerned with the contribution of l_1T(Θ,τ,Ω) − l_1,τo+p−1(Θ,τ,Ω) to l_1T(Θ,τ,Ω). Here ε_1tτ(Θ) is given by

LEMMA A.6. There exists a constant c₄ > 0 such that, with probability approaching one and uniformly in [Tλ] ≤ τ ≤ τ_o and (Θ,τ,Ω) ∈ B₁,

where a_6T ≥ 0 and a_6T = O_p(1).

Proof. The proof is similar to that of Lemma A.3. █

Our next lemma is used as an alternative to Lemma A.4 in some of the subsequent derivations. The formulation of this lemma makes use of the notation ζ_tτ⁽⁰⁾ employed in Lemma A.5.

LEMMA A.7. There exists a constant c₅ > 0 such that with probability approaching one and uniformly in [Tλ] ≤ τ ≤ τ_o − 1 and (Θ,τ,Ω) ∈ B₁,

where 1/b < η < ¼, a_iT ≥ 0, and a_iT = O_p(1) (i = 7,8,9).

Proof. By the definitions,

Recall that Ψ₁ = Φ₁ + [δ₁ : 0] and Ψ₂ = Φ₁ + [δ₁ − δ₁⁽⁰⁾ : 0]. For t = τ,…,τ_o − 1, we thus have ε_1tτ(Θ) = −Ψ₁ w_1t⁽⁰⁾ − γd_tτ = −Ψ₂ w_1t⁽⁰⁾ − ζ_tτ⁽⁰⁾. Hence,

Assume that (Θ,τ,Ω) ∈ B₁. An application of the Cauchy–Schwarz inequality, the norm inequality, and the triangle inequality yields

Because max_1≤t≤T∥w_1t⁽⁰⁾∥ = O_p(1) (see the arguments leading to (A.10)), the second square root in the last expression is of order O_p(1) uniformly in [Tλ] ≤ τ < τ_o. Hence,

where a_8T = O_p(1) in the required uniform sense.

Next note that L₇₁ ≥ 0 and λ_min(Ω⁻¹) ≥ ω⁻¹ for (Θ,τ,Ω) ∈ B₁. Consequently,

Now consider L₈, for which we have

Arguments similar to those used for L₇₃ show that

where a_9T = O_p(1) in the required uniform sense. The latter inequality is obtained because, for (Θ,τ,Ω) ∈ B₁, the last norm in the second expression can be replaced by O_p(1) by an analog of (A.4) of Gregory and Hansen (1996, p. 118).

As for L₈₂, assume first that τ < τ_o − p and use the Cauchy–Schwarz inequality to conclude that

Here the second inequality is based on the definitions and the triangle inequality, whereas the third one also makes use of the Cauchy–Schwarz inequality and the norm inequality.

In the last expression

uniformly in [Tλ] ≤ τ < τ_o − p and (Θ,τ,Ω) ∈ B₁. Here the latter result can be concluded from the Hájek–Rényi inequality given in Proposition 1 of Bai (1994). The former can be obtained by an argument similar to that used to prove (A.14) of Saikkonen and Lütkepohl (2002).

Combining the preceding discussion of L₈₂ shows that

where a_71T = O_p((τ_o − τ)^η) in the required uniform sense and the equality follows from definitions. Because for any real numbers a ≥ 0 and b ≥ 0 we have

, it follows that

In the proof of this result it was assumed that τ < τ_o − p, but it also holds for τ_o − p ≤ τ < τ_o. In that case arguments similar to those used for L₇₃ give

and (A.16) holds with a_71T = O_p(1). The result of the lemma is obtained from the definitions of L₇ and L₈ in conjunction with (A.13)–(A.16) by defining

and a_9T as done in (A.13) and (A.15), respectively. █

In the proof of the next lemma and also in subsequent proofs, frequent use will be made of the elementary inequality

which holds for a₀,a₁ ≥ 0, and a₂ > 0.

LEMMA A.8. Let ε > 0 and B₂ = {(Θ,τ,Ω) : ∥T^1/2−ηΦ₁∥² + ∥T^1/2−ηΨ₂∥² ≤ ε²}, where 1/b < η < ¼ is the same as in Lemma A.7. Then,

with probability approaching one and uniformly in [Tλ] ≤ τ ≤ τ_o.

Proof. By the definitions and Lemma A.2,

Thus, it suffices to show that, for some ε_* > 0,

with probability approaching one and uniformly in [Tλ] ≤ τ ≤ τ_o.

From Lemma A.1 it follows that we only need to prove (A.19) with the set B₂^c replaced by B₁ ∩ B₂^c. Let 0 < ε₁ ≤ λ_o − λ and define the sets

According to what was said previously, it suffices to establish (A.19) separately with B₂^c replaced by B₂₁ and B₂₂. Here we are free to choose the value of ε₁. Whatever our choice, Lemma A.4 can be applied on the set B₂₁, on which we shall first concentrate.

From Lemmas A.4 and A.5 and (A.17) we first find that, uniformly in B₂₁,

Combining these inequalities with those obtained from Lemmas A.3 and A.6 shows that, uniformly in B₂₁,

Denote

. Then the preceding inequality implies that, uniformly in B₂₁,

For simplicity, denote φ_T² = ∥T^1/2−ηΦ₁∥² + ∥T^1/2−ηΨ₂∥² and note that the sum of the two norms in the last expression is at most

. Thus, uniformly in B₂₁,

Because φ_T > ε on B₂₁ and a_T* = O_p(1) uniformly in B₂₁, this shows that (A.19) holds with B₂^c replaced by B₂₁.

Now consider proving (A.19) with B₂^c replaced by B₂₂. Here we can use Lemmas A.3, A.5, A.6, and A.7 to conclude that, with probability approaching one and uniformly in B₂₂,

Here it is understood that a_9T and the last two terms on the r.h.s. are deleted if τ = τ_o because then Lemma A.7 becomes redundant. By (A.17) the sum of the fifth, sixth, and seventh terms on the r.h.s. is of order o_p(1) uniformly in B₂₂, and the sum of the last two terms can be bounded from below by −(1/4c₅)[a_7T((τ_o − τ)/T)^η + a_8T((τ_o − τ)/T)^1/2∥T^1/2−ηΨ₂∥]². Thus, expanding the square and inserting the result on the r.h.s. of the preceding inequality yields, uniformly in B₂₂,

where

Note that here a_6T,…,a_9T are of order O_p(1) uniformly in B₂₂ and that, on B₂₂, (τ_o − τ)/T ≤ 2ε₁, say. Because we are free here to choose the value of ε₁ we can choose it so small that the following two conditions hold with probability approaching one and uniformly in B₂₂: (i) c_4T(τ) ≥ c₄ /2 and (ii) a_10T(τ) and a_11T(τ) become smaller than any preassigned positive number. Taking these facts into account and comparing the inequality (A.23) with (A.20) shows that there are only two points that make the previous proof based on inequality (A.20) directly inapplicable in the present context. These points are that instead of the terms T^−ηa_6T = o_p(1) and o_p(1) we have in (A.23) a_10T(τ) and a_11T(τ) + o_p(1), respectively, which are not of order o_p(1) but can only be replaced by an arbitrarily small positive number independent of parameters. However, this is sufficient for the application of essentially the same proof as previously. Indeed, we can conclude that, uniformly in B₂₂, an analog of (A.21) holds except that in the last expression T^η is replaced by a fixed positive number that can be assumed as large as we wish and o_p(1) is replaced by a fixed negative number that, in absolute value, can be assumed as small as we wish. In particular, we can assume that T^η and o_p(1) in (A.21) are replaced by M/ε and −ε/M, respectively, where M can be chosen arbitrarily large. This shows that we can make the r.h.s. of the present version of (A.21) larger than some ε_* > 0 with probability approaching one. Thus, there is a choice of ε₁ such that (A.19) holds with B₂^c replaced by B₂₁ and B₂₂. This completes the proof. █

The next lemma is similar to Lemma A.8 except that it deals with the short-run parameter Φ₂.

LEMMA A.9. Let ε > 0 and B₃ = {(Θ,τ,Ω) : ∥T^1/2−η(Φ₂ − Φ_2o)∥ ≤ ε}, where 1/b < η < ¼ is the same as in Lemma A.7. Then,

with probability approaching one and uniformly in [Tλ] ≤ τ ≤ τ_o.

Proof. By Lemma A.1 it suffices to prove the result with B₃^c replaced by B₁ ∩ B₃^c. First consider the break dates [Tλ] ≤ τ ≤ [T(λ_o − ε₁)] and note that the derivation of the inequality in (A.21) is valid for these break dates and for all (Θ,τ,Ω) ∈ B₁ ∩ B₃^c. It is also valid for every ε₁ > 0. Thus, an application of (A.17) shows that in this part of the parameter space T^−2ηl_1T(Θ,τ,Ω) ≥ o_p(1) holds uniformly. Next note that the inequality (A.23) is valid for [T(λ_o − ε)] < τ ≤ τ_o and for all (Θ,τ,Ω) ∈ B₁ ∩ B₃^c. Moreover, as the discussion after that inequality reveals, we can, with a suitable (small) choice of ε₁, use (A.17) to obtain an analog of (A.21) from which we conclude that, with probability approaching one and uniformly in the considered part of the parameter space, T^−2ηl_1T(Θ,τ,Ω) ≥ −ε₂, where ε₂ > 0 can be chosen arbitrarily small. From the preceding discussion and the first equality in (A.18) it thus follows that we need to show that, for some ε_* > 0,

with probability approaching one. Arguments needed to show this are similar to those used in previous proofs and also very similar to those used to prove the consistency of the LS estimators of the parameters Φ₂ and Ω in the standard regression model Δx_t = Φ₂ w_t⁽⁰⁾ + ε_t. Details are straightforward and are omitted. █

The next lemma again makes use of the notation ζ_tτ⁽⁰⁾ introduced for Lemma A.5.

LEMMA A.10. Let

, where τ < τ_o and 1/b < η < ¼ is the same as in Lemma A.7. Then, there exists a real number M₀ > 0 such that, for all M ≥ M₀,

with probability approaching one and uniformly in [Tλ] ≤ τ ≤ τ_o − 1. If the quantity (τ_o − τ)^−2η in the definition of the set B₄ is replaced by T^−2η the same conclusion holds.

Proof. From (A.18) it follows that it suffices to show that there exists a real number M₀ > 0 such that, for all M ≥ M₀ and any M₁ > 0,

with probability approaching one and uniformly in [Tλ] ≤ τ ≤ τ_o − 1. From Lemmas A.1, A.8, and A.9 it further follows that here the set B₄^c can be replaced by B₁ ∩ B₂ ∩ B₃ ∩ B₄^c. From (A.19) it can be seen that the value of ε in the definition of B₂ can be chosen arbitrarily small.

We wish to apply Lemmas A.3, A.5, A.6, and A.7 to obtain a lower bound for l_1T(Θ,τ,Ω). This lower bound can be obtained by multiplying both sides of the inequality (A.22) by T^2η. By (A.17) the contribution of the first four terms to the r.h.s. of the resulting inequality can be replaced by O_p(1). This is also the case for the seventh term. Hence, we can write

This holds uniformly in B₁ ∩ B₂ ∩ B₃ ∩ B₄^c and [Tλ] ≤ τ ≤ τ_o − 1. In this part of the parameter space we also have

and a_4T ≤ a_4T(τ_o − τ)^η (see Lemma A.8). Denote c* = min(c₃,c₅), a_T* = max(a_4T, a_7T + εa_8T), and for simplicity,

. From the lower bound obtained for l_1T(Θ,τ,Ω) previously we can then further obtain

Again, this holds uniformly in B₁ ∩ B₂ ∩ B₃ ∩ B₄^c and [Tλ] ≤ τ ≤ τ_o − 1. Now, on B₄^c, ξ_τ > M(τ_o − τ)^η so that, for all M large enough and with probability approaching one, we can make the r.h.s. of (A.25) larger than any preassigned number M₁ > 0. Thus, we have established (A.24) and thereby the first assertion of the lemma. The second assertion is obvious by (A.25) and the discussion thereafter. █

Before proceeding to new proofs we discuss how Lemmas A.3–A.10 are formulated when τ ≥ τ_o.

The counterpart of Lemma A.3 is concerned with the time points t = p + 1,…,τ_o − 1 and break dates τ_o ≤ τ ≤ [Tλ] but is otherwise similar to Lemma A.3.

The next time points of interest are now t = τ_o,…,τ_o + p − 1 so that we need to consider a counterpart of Lemma A.5. Here we write

where Ψ₂ = Φ₁ + [δ₁ − δ₁⁽⁰⁾ : 0] as before and ζ_tτ = (d_tτ − d_tτo)δ₁ + γd_tτ − γ⁽⁰⁾d_tτo. In other words, in place of ζ_tτ⁽⁰⁾ we now use an analogous variable defined by using the parameter δ₁ instead of δ₁⁽⁰⁾. However, replacing ζ_tτ⁽⁰⁾ in Lemma A.5 by ζ_tτ is clearly possible, as can be seen from the given proof.

Instead of the time points t = τ_o + p,…,τ − 1 it is next reasonable to consider the time points t = τ_o + p,…,τ + p − 1. Then the number of time points is the same as in Lemmas A.4 and A.7. Changes in parameters have to be made, though. Now

where Ψ₁⁽⁰⁾ = Φ₁ − [δ₁⁽⁰⁾ : 0]. Thus, we now have the matrix Ψ₁⁽⁰⁾ in place of Ψ₁ used in Lemma A.4, and, as before, the former is defined by using δ₁⁽⁰⁾ instead of δ₁ in Ψ₁. The parameter γ used in Lemma A.4 is also changed by adding δ₁ to its columns. With these replacements the counterpart of Lemma A.4 applies with [T(λ_o + ε)] ≤ τ ≤ [Tλ].

Next consider the counterpart of Lemma A.7, which is also concerned with time points t = τ_o + p,…,τ_o + p − 1. Here the preceding expression of ε_1tτ(Θ) is modified to the form

where Ψ₂ is as defined in the proof of Lemma A.7. In the counterpart of Lemma A.7 we then have ζ_tτ in place of ζ_tτ⁽⁰⁾ and τ_o + 1 ≤ τ ≤ [Tλ]. The proof can again be obtained basically by following the previous proof.

The counterpart of Lemma A.6 is straightforward. The relevant time points are t = τ,…,T, and the obtained lower bound is as before except for the obvious change in the values of τ, which become τ_o ≤ τ ≤ [Tλ]. The proof is similar to the proof of Lemma A.3.

It is not difficult to check that the modified versions of Lemmas A.3–A.7 can be used to show that the results of Lemmas A.8 and A.9 also apply for τ_o ≤ τ ≤ [Tλ]. Regarding Lemma A.10, when τ_o + 1 ≤ τ ≤ [Tλ], the set B₄ is defined as

but otherwise the same result obtains.

Now we can turn to our next lemma, which is central in studying asymptotic properties of the break date estimator. Recall that δ_1o = −Π_oδ_o = −α_o β_o′δ_o, where δ_o = T^aδ_*. Thus, δ_1o ≠ 0 if and only if β_o′δ_o ≠ 0. Note also that we shall use the convention that the infimum over an empty set is ∞.

LEMMA A.11. Let M > 0. Assume that δ_1o ≠ 0 and define B₅ = {(Θ,τ,Ω) : (|τ_o − τ| − p)∥δ_1o∥^2/(1−2η) ≤ M}, where 1/b < η < ¼ is the same as in Lemma A.7 or its counterpart when τ > τ_o. Then there exists a real number M₀ > 0 such that, for all M ≥ M₀,

with probability approaching one. If δ_1o = 0 the same result holds with the set B₅ replaced by

.

Proof. Assume first that τ < τ_o − p and δ_1o ≠ 0. From Lemmas A.1, A.8, and A.9 it follows that we can replace the set B₅^c by B₁ ∩ B₂ ∩ B₃ ∩ B₅^c.

By the definitions, δ₁⁽⁰⁾ = −Πδ_o = −α⁽⁰⁾β_o′δ_o − ρ⁽⁰⁾β_o⊥′δ_o, where β_o′δ_o ≠ 0. On B₃, ∥α⁽⁰⁾ − α_o∥ ≤ εT^η−1/2 and, on B₂, ∥ρ⁽⁰⁾∥ ≤ εT^η−1 (see Lemmas A.8 and A.9). Thus, because δ_1o = −α_o β_o′δ_o and δ_o = T^aδ_*,

for some positive and finite constant c. Hence, because ζ_tτ⁽⁰⁾ = (d_tτ − d_tτo)δ₁⁽⁰⁾ = δ₁⁽⁰⁾ for t = τ + p,…,τ_o − 1, we have on B₁ ∩ B₂ ∩ B₃ ∩ B₅^c,

Here the fourth relation makes use of the triangle inequality. For all T and M large enough the last expression can be made larger than the real number M₀ in Lemma A.10. Thus, the stated result follows from Lemma A.10.

Now consider the case τ > τ_o + p but maintain the assumption δ_1o ≠ 0. Then, using the counterparts of Lemmas A.8 and A.9 we can proceed in the same way as in the case τ < τ_o − p until the relations (A.26), which start now as

Thus, in place of δ₁⁽⁰⁾ we now have δ₁. However, from the counterpart of Lemma A.8 we find that, on B₂, ∥δ₁ − δ₁⁽⁰⁾∥ ≤ εT^η−1/2 and a straightforward modification of the arguments in the latter part of (A.26) combined with the present version of Lemma A.10 give the desired result.

Next assume that δ_1o = 0 and τ ≤ τ_o. In this case we use the inequality

where

, that is, the summation on the r.h.s. is over the values of t for which Δd_tτo ≠ 0 or Δd_tτ ≠ 0. Clearly the number of such time points is at most 2p.

From the definitions it follows that

Notice that here Γ_joδ_o = −γ_jo (j = 1,…,p − 1) and, because now δ_1o = 0, δ_o = γ_0o. Thus, the sum of the last three terms equals γd_tτ − γ_o d_tτo, and we wish to show that the contribution of the first three terms to the r.h.s. of (A.27) can be ignored. To this end, note that now δ₁⁽⁰⁾ = −ρ⁽⁰⁾β_o⊥′δ_o so that, on B₂, ∥δ₁⁽⁰⁾∥ ≤ cεT^η+a−1 for some 0 ≤ c < ∞ (see Lemma A.8). Furthermore, on B₃, ∥(Γ_j − Γ_jo)δ_o∥ ≤ ∥Γ_j − Γ_jo∥∥δ_o∥ ≤ εT^η+a−1/2∥δ_*∥ (j = 1,…,p − 1) (see Lemma A.9). Using these facts and the triangle inequality we find that

On the r.h.s. the summation can be extended to all t = p + 1,…,T. This means that on B₁ ∩ B₂ ∩ B₃ ∩ B₅₀^c the last expression becomes larger than the real number M₀ in Lemma A.10 for all T and M large enough. Thus, the stated result follows from the latter part of Lemma A.10.

Finally, assume that δ_1o = 0 and τ > τ_o. In place of (A.27) we then have a similar inequality with t = τ_o,…,τ + p − 1 and ζ_tτ⁽⁰⁾ replaced by ζ_tτ. However, using the fact that ∥δ₁ − δ₁⁽⁰⁾∥ ≤ εT^1/2−η on B₂ it is straightforward to show that the proof can be reduced to a form entirely similar to that in the case τ ≤ τ_o. This completes the proof of the lemma. █

Now we can prove Theorem 3.1. As discussed earlier, the estimator

can also be obtained by minimizing −2 times the Gaussian log-likelihood function l_T(Θ,τ,Ω). First consider the case a > 0 and δ_1o ≠ 0. By Lemma A.11 we can then concentrate on the break dates τ_o − p ≤ τ ≤ τ_o + p. First consider the case τ_o − p ≤ τ ≤ τ_o. If γ_jo = 0 for all j = 0,…,p − 1, Lemma A.11 shows that, asymptotically,

, as required. Next suppose that γ_j0,o ≠ 0 and consider the break dates τ_o − p ≤ τ ≤ τ_o − p + j₀. For any of these break dates we have

Suppose first that j₀ > 0. Then, because γ_j0⁽⁰⁾ − γ_j0,o = −(Γ_j0 − Γ_j0,o)δ_o, we have for (Θ,τ,Ω) ∈ B₃,

Because γ_j0,o = −T^aΓ_j0,oδ_* ≠ 0, the last quantity tends to infinity as T → ∞. Hence, we can conclude from Lemmas A.9 and A.10 that asymptotically the function l_T(Θ,τ,Ω) is not minimized for τ ≤ τ_o − p + j₀. Now consider the case j₀ = 0. From the definitions it follows that γ_o⁽⁰⁾ − γ_0o = δ_1o − δ₁⁽⁰⁾, where ∥δ_1o − δ₁⁽⁰⁾∥ ≤ cT^η+a−1/2ε on B₂ ∩ B₃ (see the beginning of the proof of Lemma A.11). Hence, because γ_0o = T^a(δ_* + α_o β_o′δ_*) ≠ 0, the proof given in the case j₀ > 0 applies with obvious changes and shows that asymptotically

cannot occur.

To complete the proof of the first assertion, consider the case τ_o + 1 ≤ τ ≤ τ_o + p. By the definitions we then have ζ_τoτ = −δ₁ − γ₀⁽⁰⁾ = −δ_o + (δ₁⁽⁰⁾ − δ₁), where δ_o ≠ 0 and ∥δ₁⁽⁰⁾ − δ₁∥ ≤ εT^η−1/2 for (Θ,τ,Ω) ∈ B₂ ∩ B₃ (see Lemma A.8 and the definition of Ψ₂ given before Lemma A.5). In the same way as in the preceding case we can thus conclude from Lemmas A.8–A.10 that asymptotically

cannot occur. This completes the proof of the first assertion in the case a > 0 and δ_1o ≠ 0.

Next assume that a > η > 1/b and δ_1o = 0. Then, if τ ≤ τ_o − p + j₀ and j₀ > 0,

Because the last quantity tends to infinity as T → ∞ it follows from the latter part of Lemma A.11 that asymptotically

cannot occur. If j₀ = 0, we have γ_0o = δ_o − δ_1o = δ_o ≠ 0, and (A.28) holds with Γ_j0,oδ_* replaced by δ_*. Hence the same conclusion also obtains for j₀ = 0.

If τ > τ_o the l.h.s. of (A.28) can be bounded from below by T^−2η∥γ_0o∥² = T^−2η∥δ_o∥² = T^2a−2η∥δ_*∥², and the situation is similar to the case j₀ = 0.

Finally, the second part of the theorem follows directly from the first part of Lemma A.11. This completes the proof of Theorem 3.1.

A.2. Proof of Theorem 3.2.

The break date estimator

can also be obtained by minimizing the objective function l_T(Θ,τ,Ω) over the relevant restricted part of the parameter space. Compared to the previous unrestricted estimation, the parameters δ₁ and γ in (A.2) are no more freely varying but (smooth) functions of the parameters δ, ρ⁽⁰⁾, α⁽⁰⁾, and Γ₁,…,Γ_p−1. Specifically, δ₁ = −Πδ = −α⁽⁰⁾β_o′δ − ρ⁽⁰⁾β_o⊥′δ, γ₀ = δ − δ₁, and γ_j = −Γ_jδ (j = 1,…,p − 1). Unlike with the unconstrained estimation it is not quite obvious that these restricted estimators exist. This fact will therefore be justified first. After that the proof follows straightforwardly from the results used to prove Theorem 3.1.

Define

Using y_t^(τ) in place of x_t we can obtain an analog of (A.2) in which d_tτo and d_tτo are replaced by d_tτ and d_tτ, respectively, and u_t−1⁽⁰⁾ and v_t−1⁽⁰⁾ are replaced by analogs defined in terms of y_t^(τ) instead of x_t. In other words, in place of u_t−1⁽⁰⁾ and v_t−1⁽⁰⁾ we use u_t−1^(τ) = β_o′y_t−1^(τ) and v_t−1^(τ) = β_o⊥′y_t−1^(τ), respectively. In place of (A.3) we then have

where w_t^(τ) is an obvious modification of w_t⁽⁰⁾.

Clearly, we can express the vector ε_tτ(Θ) as

and use this expression in the previous definition of l_T(Θ,τ,Ω). To demonstrate the existence of a minimizer of the objective function l_T(Θ,τ,Ω) it also appears convenient to use the reparameterization Θ → Θ⁽⁰⁾ = [Φ : Ξ − Ξ⁽⁰⁾]. Thus, if for simplicity we denote z_t^(τ) = [Δy_t^(τ)′ : w_t^(τ)′ : q_tτ′]′ and R(Θ⁽⁰⁾) = [I_n : − Φ : Ξ − Ξ⁽⁰⁾] we can write the relevant objective function as

Note that in the present context the parameter Θ has the same meaning as before except that it is treated as a (smooth) function of the parameters ν₀⁽⁰⁾, ν₁⁽⁰⁾, δ, ρ⁽⁰⁾, α⁽⁰⁾, and Γ₁,…,Γ_p−1. Because the parameter Ξ⁽⁰⁾ is also a (smooth) function of (some of) these parameters the same is true for the parameter Θ⁽⁰⁾. All these parameter restrictions are taken into account when the minimization of the objective function l_T(Θ⁽⁰⁾,τ,Ω) is considered. Notice that, because the objective function is expressed as a function of the “reduced form” parameter Θ⁽⁰⁾, the role of the parameter restrictions is to define the permissible space of Θ⁽⁰⁾. A similar idea, of course, applies to the previous parameterization of the objective function, that is, to l_T(Θ,τ,Ω) (cf. Saikkonen, 2001, and the references therein for a similar approach).

A useful consequence of the fact that we can still interpret the objective function l_T(Θ,τ,Ω) as a function of the “reduced form” parameter Θ and only restrict its permissible space is that results obtained to prove Theorem 3.1 can be applied straightforwardly even here. In particular, we wish to apply Lemma A.11 to conclude that, when the existence of a minimizer of the objective function l_T(Θ,τ,Ω) is studied in the present setup, values of the break date parameter τ can be restricted as implied by this lemma. Of course, this conclusion also holds when the objective function is parameterized as l_T(Θ⁽⁰⁾,τ,Ω).

To justify the application of Lemma A.11, we first discuss how Lemmas A.1–A.10 have to be modified to match the present setup. Notice that the existence of a minimizer of the objective function l_T(Θ,τ,Ω) is not needed to prove Lemmas A.1–A.11 and the same is also true for their modified versions to be discussed subsequently.

First note that Lemma A.2 is still used in its previous form and, because it is concerned with unrestricted values of Φ₂ and Ω, it obviously applies in the present context. Lemma A.1 is simply modified by replacing B₁ by the intersection of the restricted parameter space of (Θ,τ,Ω) and values for which the inequality constraints in (A.4) and (A.5) hold. This restricted version of the parameter space B₁ is then used to replace B₁ in Lemmas A.3–A.7. It is straightforward to check that the previous proofs of these lemmas apply in essence despite the differences in parameter spaces.

Next consider Lemmas A.8–A.10, where, in addition to B₁, also the parameter spaces B₂, B₃, and B₄ are redefined to allow for the employed restrictions. Again, it is not difficult to check that the previous proofs carry over. It is also easy to see that the modifications needed for Lemmas A.3–A.10 can be done in the case τ ≥ τ_o.

Because analogs of Lemmas A.1–A.10 hold in the present context, it is further straightforward to show that the result of Lemma A.11 also holds with the parameter space B₅ redefined to account for the employed restrictions. Thus, we can conclude that when searching for a minimizer of the objective function l_T(Θ⁽⁰⁾,τ,Ω), the value of the break date parameter τ can be restricted as implied by Lemma A.11. Specifically, if δ_1o ≠ 0, Lemma A.11 directly shows that τ_o − p ≤ τ ≤ τ_o + p can be assumed. If δ_1o = 0 and a > b, we can even assume τ_o − p + 1 ≤ τ ≤ τ_o + p − 1, as the argument used to prove the corresponding case of Theorem 3.1(i) readily shows.

We shall now show that the function l_T(Θ⁽⁰⁾,τ,Ω) and hence l_T(Θ,τ,Ω) have a minimizer with probability approaching one. In what follows, reference to Lemmas A.1–A.11 will be understood to mean the present restricted setup. We first show the following intermediate result, where the matrix D_T = diag[T^−1/2I : I_p] is used. Its dimension equals the dimension of the vector z_t^(τ).

LEMMA A.12. There exists an ε_* > 0 such that

with probability approaching one and uniformly in τ, when the value of the break date parameter τ can be restricted as implied by Lemma A.11.

Proof. The values of τ can be restricted depending on the value of a and whether δ_1o = 0 or not. Different cases will therefore be discussed separately.

Case (i). a > 0 and δ_1o ≠ 0 or a > η > 1/b.

From Lemma A.11 we can then conclude that, if a minimizer of l_T(Θ⁽⁰⁾,τ,Ω) exists, in large samples it must be such that the corresponding τ is in the interval [τ_o − p,τ_o + p]. If δ_1o ≠ 0 this follows directly from the first part of Lemma A.11. If δ_1o = 0 (and a > η > 1/b) the same conclusion can be drawn from the second part of the lemma by the argument used in the proof of Theorem 3.1 to obtain (A.28).

To justify (A.31), assume first that a < ½. Then the moment matrix in (A.31) behaves asymptotically in the same way as in the proof of Theorem 3.1 in that the vectors Δy_t^(τ) and w_t^(τ) in the definition of z_t^(τ) can be replaced by analogs defined in terms of x_t. This follows by observing that, when |τ_o − τ| ≤ p, the latter term on the r.h.s. of (A.29) satisfies

When a < ½ the last quantity converges to zero, and the desired conclusion is readily obtained.

If a = ½ the latter term on the r.h.s. of (A.29) has an impact, but (A.31) still obtains. To see this, suppose first that δ_1o ≠ 0. Then, as |τ − τ_o| ≤ p, the latter term on the r.h.s. of (A.29) behaves like an impulse dummy. Because now δ_o = T^1/2δ_* this term affects the asymptotic behavior of the moment matrix in (A.31), but, as can be readily seen, it only affects the diagonal and off-diagonal elements related to u_t−1^(τ) and Δy_t−j^(τ) (j = 0,…,p − 1). Moreover, the impact is such that asymptotically the moment matrix in (A.31) only differs from that obtained in the previous case by an additive positive semidefinite matrix. Thus, from this fact and the result of the previous case one again obtains (A.31).

Next assume that δ_1o = 0 and a = ½. Here the situation is similar to the preceding case except for being simpler because now u_t−1^(τ) = β_o′x_t−1 = u_t−1⁽⁰⁾. Thus, we again get (A.31), and, thus, we have justified (A.31) in the case of the first part of the theorem. It remains to consider the second part, for which the following assumption is made.

Case (ii). a ≤ 0 and δ_1o ≠ 0.

If a = 0 it follows from the first part of Lemma A.11 that we can assume |τ − τ_o| to be bounded, and arguments similar to those in the case 0 < a < ½ and δ_1o ≠ 0 show (A.31). If a < 0 we cannot restrict the values of τ. However, from (A.32) it can be seen that the vectors Δy_t^(τ) and w_t^(τ) in the definition of z_t^(τ) can be replaced by analogs defined in terms of x_t. Arguments similar to those used in the proof of Theorem 3.1 then show that (A.31) also holds in the present case. (In particular, analogs of (A.3) and (A.4) of Gregory and Hansen, 1996, and (A.14) of Saikkonen and Lütkepohl, 2002, can be used to handle sums of cross products between [Δx_t′ : w_t^(0)′]′ and q_tτ.) █

We have now shown that when searching for a minimizer of the function l_T(Θ⁽⁰⁾,τ,Ω) we can in both parts of Theorem 3.2 restrict the values of the break date τ in such a way that (A.31) holds with probability approaching one and uniformly in τ.

Using Lemma A.12 we can analyze the function l_T(Θ⁽⁰⁾,τ,Ω) in the same way as in the proof of Proposition 3.1 of Saikkonen (2001, pp. 320–321) and conclude that it suffices to search for a minimizer of l_T(Θ⁽⁰⁾,τ,Ω) in that part of the parameter space where, in addition to the restrictions on τ, we also have 0 < ω ≤ λ_min(Ω) ≤ λ_max(Ω) ≤ ω < ∞ and ∥Θ⁽⁰⁾∥ ≤ M < ∞.

We shall demonstrate that the parameter space defined by all these restrictions is compact. To this end, note first that the restrictions imposed on Θ⁽⁰⁾ are of the form h(Θ⁽⁰⁾) = 0, where h(·) is a continuous function. Thus, because the unrestricted parameter space of Θ⁽⁰⁾ is the whole euclidean space, it follows that the restricted space is closed and its intersection with parameter values restricted by 0 < ω ≤ λ_min(Ω) ≤ λ_max(Ω) ≤ ω < ∞ and ∥Θ⁽⁰⁾∥ ≤ M < ∞ is compact. The continuity of the function l_T(·,τ,·) therefore ensures that, for every relevant value of τ, a minimizer exists with probability approaching one. This proves the (asymptotic) existence of the nonlinear LS estimators of Θ⁽⁰⁾, τ, and Ω and hence also that of Θ.

To prove part (i) of Theorem 3.2, first consider the case δ_1o ≠ 0 and assume that τ ≤ τ_o − 1. As noted previously, we can also assume that τ_o − p ≤ τ. Using the definitions we can express the vector ζ_tτ⁽⁰⁾ as

Taking the assumed restrictions into account we can write this further as

Here we have also made use of the facts that δ₁ = −Πδ and δ₁⁽⁰⁾ = −Πδ_o with Π = α⁽⁰⁾β_o′ + ρ⁽⁰⁾β_o⊥′.

To show that asymptotically the function l_T(Θ,τ,Ω) cannot be minimized for τ_o − p ≤ τ ≤ τ_o − 1, we consider two cases separately. In the first case it is assumed that δ ≥ T^aε_*, where ε_* > 0 is arbitrary. The second case will then assume that δ < T^aε_*.

Now consider parameter values for which τ_o − p ≤ τ ≤ τ_o − 1 and δ ≥ T^aε_* hold for some ε_* > 0. By Lemma A.8 we can also assume that ∥δ₁ − δ₁⁽⁰⁾∥ ≤ εT^η−1/2. Using this, (A.33), and the previously mentioned parameter restrictions, we find that

Because the last quantity tends to infinity with T, it follows from Lemma A.10 that asymptotically

cannot occur.

For parameter values τ_o − p ≤ τ ≤ τ_o − 1 and δ < T^aε_* we can also use (A.33) and Lemma A.10. First note that, by Lemma A.8, the norm of the first term on the r.h.s. of (A.33) can be bounded by εT^η−1/2. Next, from Lemmas A.8 and A.9 it follows that the term in front of δ in the second term on the r.h.s. of (A.33) can be assumed bounded, and so the norm of the whole term can be bounded by a quantity of the form c₁ε_*T^a, where 0 < c₁ < ∞. Similar arguments can also be used to show that, at least for t = τ_o, the norm of the third term on the r.h.s. of (A.33) can be bounded from below by a quantity of the form c₂∥δ_*∥T^a, where 0 < c₂ < ∞ and ∥δ_*∥ ≠ 0. Thus, because ε_* can be chosen arbitrarily small, the asymptotic behavior of

is dominated by the third term on the r.h.s. of (A.33), and the preceding discussion implies that this sum tends to infinity with T. From this and Lemma A.10 we can conclude that asymptotically

cannot occur.

Thus, we have shown that, when δ_1o ≠ 0, asymptotically

cannot occur. A similar argument with ζ_tτ⁽⁰⁾ replaced by ζ_tτ and with Lemma A.10 replaced by its corresponding counterpart shows that asymptotically

cannot occur either.

Now suppose that δ_1o = 0 and consider the break dates τ_o − p ≤ τ ≤ τ_o − 1. Instead of (A.33) we use a slightly different representation of ζ_tτ⁽⁰⁾ given by

This representation can be obtained from the definitions (cf. the similar representation used in the proof of Lemma A.11). As with the case δ_1o ≠ 0, our treatment will be divided into two separate cases.

In the first one the parameter δ is restricted as δ ≥ T^aε_*, where ε_* > 0 is arbitrary and a > η > 1/b. From the preceding representation of ζ_tτ⁽⁰⁾ it then follows that

Here the last inequality makes use of the fact that ∥δ₁ − δ₁⁽⁰⁾∥ ≤ εT^η−1/2 can be assumed by Lemma A.8. Because the last quantity tends to infinity with T, it follows from the latter result of Lemma A.10 that asymptotically

cannot occur.

When δ < T^aε_* (a > η > 1/b) is assumed, (A.34) and Lemma A.11 give the desired result much in the same way as in the case δ_1o ≠ 0, where (A.33) was used instead of (A.34). First note that the norm of the first four terms on the r.h.s. of (A.34) can be bounded by a quantity of the form εcT^η+a−1/2, where 0 < c < ∞. This follows from Lemma A.8 and arguments used to prove Lemma A.11 for δ_1o = 0. Next, in the same way as in the case δ_1o ≠ 0 one can show that the term in front of δ in the fifth term on the r.h.s. of (A.34) can be assumed bounded and, hence, the norm of the whole term can be bounded by a quantity of the form c₁ε_*T^a, where 0 < c₁ < ∞. By similar arguments we finally find that, at least for t = τ_o, the norm of the last term on the r.h.s. of (A.34) can be bounded below by a quantity of the form c₂∥δ_*∥T^a, where 0 < c₂ < ∞ and ∥δ_*∥ ≠ 0. Thus, because ε_* can be chosen arbitrarily small, the asymptotic behavior of

is dominated by the last term on the r.h.s. of (A.34), and it follows from the latter result of Lemma A.10 that asymptotically

cannot occur.

Thus, we have shown that, when δ_1o = 0, we asymptotically cannot have

. Again a similar proof with ζ_tτ⁽⁰⁾ replaced by ζ_tτ and Lemma A.10 replaced by its corresponding counterpart shows that asymptotically

cannot occur either. This completes the proof of part (i) of the theorem in the case δ_1o = 0. Part (ii) is a consequence of the (asymptotic) existence of

and Lemma A.11. Hence, the proof of Theorem 3.2 is complete.

A.3. Proof of Theorem 4.1.

For simplicity we will denote the break date estimator by

. This estimator can be either of the two estimators considered in Section 3 unless explicit distinctions are made. From the assumptions δ₁ ≠ 0 and 0 < a < ½ and Theorems 3.1 and 3.2 it follows that asymptotically

can be assumed. This fact will be used in several arguments of the proof without explicit notice.

Properties of RR Estimators. We shall first show that the RR estimators of the parameters based on equation (2.7) with the unknown break date τ_o replaced by the estimator

satisfy appropriate consistency properties. This replacement changes the VECM (2.7) to

where

Write

where

. Using the transformation

we can transform the preceding VECM to the form

where ν⁽⁰⁾ = ν + αβ′μ_0o − Ψμ_1o, φ⁽⁰⁾ = φ − β′μ_1o, θ⁽⁰⁾ = θ − β′δ_o, γ₀*⁽⁰⁾ = δ − δ_o, and γ_j*⁽⁰⁾ = γ_j* + Γ_jδ_o (j = 1,…,p − 1). Note that the true values of these parameters are zero. RR estimators of the parameters in (A.38) are obtained by transforming the RR estimators based on (A.35) in the same way as the corresponding parameters (e.g.,

). Asymptotic properties of these transformed estimators are derived subsequently. We denote by

normalized versions of the estimators

, respectively, such that

.

LEMMA A.13. Under the conditions of Theorem 4.1,

,

.

Proof. We first note that the result of Lemma A.13 also holds when the break date is assumed known. A formal proof of this can be obtained by following the proof of Lemma 2.1 of Saikkonen and Lütkepohl (2000a) and observing that the omission of some impulse dummies from the model considered by Saikkonen and Lütkepohl is of no significance and that the same is true for the dependence of the parameter δ on the sample size. The latter fact is clear because the results of Lemma A.13 are formulated by using the transformed model (A.38) in which the true values of the deterministic parameters are zero.

Because the result of Lemma A.13 holds when the break date is assumed known it also holds when the break date can be consistently estimated, that is, when

. Indeed, then the analysis can be restricted to that part of the sample space where

holds and the probability of this can be made arbitrarily close to unity for all T large enough. This proves the results of the lemma for the constrained estimator

.

If j₀ = p − 1 in Theorem 3.1(i) the preceding argument also applies to the unconstrained estimator

. For other values of j₀ further arguments are needed. By Theorem 3.1(i) it suffices to consider any value of the break date such that τ_o − p + 1 + j₀ ≤ τ ≤ τ_o. For simplicity, consider the case j₀ = p − 2 and τ = τ_o − 1. It is easy to see that even though the break date is misspecified by one we can still consider (2.7) a correctly specified model if we only redefine the parameters γ₀*,…,γ_p−1* as γ₀* = αβ′δ, γ₁* = δ, and γ_j* = −Γ_j−1δ, j = 2,…,p − 1. By assumption we then have γ_p−1* ≠ 0 whereas Γ_p−1δ = 0. With these new definitions the error term of model (2.7) is still ε_t, and the analysis given in the case of a known break date can be used. Because the other parameters of the model are not affected by the redefinition of the parameters γ_j* (j = 0,…,p − 1) the obtained consistency results will be the same as in the case where the true break date is known. The same argument can clearly be extended to other values of j₀. This completes the proof. █

Properties of the new estimators of the deterministic parameters. We shall now consider asymptotic properties of the estimators

by assuming that the break date τ in (2.7) is replaced by one of the estimators

.

LEMMA A.14. Under the conditions of Theorem 4.1, the estimators

have the following properties:

Proof. We start with the results (A.39) and (A.40). Recall the definitions ν = −αβ′μ₀ + Ψμ₁, ν⁽⁰⁾ = ν + αβ′μ_0o − Ψμ_1o, and φ⁽⁰⁾ = φ − β′μ_1o, which imply that

Here the latter equality is obtained by arguments similar to those used to define the estimator

. These arguments further show that β_⊥′(μ₁ − μ_1o) = β_⊥′C(ν⁽⁰⁾ − Ψ_βφ⁽⁰⁾), and the same relation applies to estimators. Thus, we have

Here and in what follows the subscript 0 is omitted from the estimators of α and β to simplify the notation. By Lemma A.13, one obtains from the previous equality

Note that the estimator

can be viewed as the LS estimator of the parameter ν⁽⁰⁾ in the auxiliary regression model obtained by replacing

in (A.38) by its observed analog

. This implies that

can be obtained by LS from the auxiliary regression model

where

, Λ is a conformable coefficient matrix, and the error has the representation

. By the definition of C and Lemma A.13,

. Using this fact, Lemma A.13, and the assumptions, it is straightforward to show that the asymptotic properties of the LS estimator of the parameter Λ in the auxiliary regression model (A.43) can be obtained by assuming that the error equals

. The same arguments and the definition of

(see (A.36)) further show that the error can be assumed to be

or even β_⊥′Cε_t. Because it is also straightforward to show that the estimation of the intercept term in (A.43) is asymptotically independent of the estimation of the other regression coefficients we can conclude that

This and a standard central limit theorem yield

To obtain (A.40) we need to show that

on the l.h.s. can be replaced by β_⊥. To see this, write

By the consistency of the estimator

and the result just obtained the latter term on the r.h.s. is of order o_p(T^−1/2), and the same is true for the former because

by Lemma A.13. From this last result one can obtain (A.39) because

can be replaced by β using an argument similar to that used in (A.44).

Now consider the estimator

. From its derivation we get the identity

. By the definitions, this is equivalent to

Because the same relation applies to estimators, arguments similar to those used to define the estimator

yield

Lemma A.13 implies that the r.h.s. of this equality is of order O_p(1). Moreover,

. Thus, (A.41) and (A.42) follow because in these results

can be replaced by β and β_⊥, respectively, by using an argument similar to that in (A.44). This completes the proof of Lemma A.14. █

Proof of the limiting distribution of LR^PAR. The structure of our proof of the limiting distribution of LR^PAR(r₀) is similar to that of Theorem 11.1 of Johansen (1995). Therefore we just outline the arguments in the following discussion.

First note that the limiting distribution of the test statistic LR^PAR(r₀) can be derived by assuming that the true value of the parameter μ₀ is zero. Thus, we can write equation (4.1) as

Using this representation, the assumption a < ½, and the asymptotic properties of the estimators

obtained in Lemma A.14, we can now mimic the proof given in Johansen (1995, pp. 158–160) and see that all the quantities that therein converge in probability to constants will here converge in probability to the same constants. However, quantities that in Johansen (1995, pp. 158–160) converge weakly to functionals of a Brownian motion will here converge weakly to different functionals of a Brownian motion. Here these weak limits are determined by the weak limit of

. We have

where W(s) is an (n − r₀)-dimensional Brownian motion with covariance matrix Ω and hence the limit is a linear transformation of the Brownian bridge W₊(s) = W(s) − sW(1). The error term in the equality is understood to hold in the Skorohod topology.

To justify (A.46), first consider the equality. Because

by (A.42) of Lemma A.14, it is clear that the contribution of the third and fourth terms on the r.h.s. of (A.45) to

is asymptotically negligible. The same argument also applies to the fifth term on the r.h.s. of (A.45) because a < ½. As for the weak convergence in (A.46), it can be justified by a standard functional central limit theorem and (A.40) of Lemma A.14 by observing that the limit of the second expression is determined by the process ε_t (see Johansen, 1995, eqn. (B.24), and the proof of (A.40)).

The preceding discussion implies that the limiting distribution of the test statistic LR^PAR(r₀) can be derived by ignoring the last three terms on the r.h.s. of (A.45). This means that in the same way as in Saikkonen and Lütkepohl (2000a) we have reduced the problem to that of no break studied by Saikkonen and Lütkepohl (2000b). From Lemma A.14 and the proof of Theorem 3 of Saikkonen and Lütkepohl (2000b) it can be seen that, when μ_0o = 0 is assumed, the trace test statistic in that theorem is asymptotically equivalent to a similar test statistic based on an analog of (4.2) defined by replacing

. It is straightforward to show that the use of

instead of

changes the limiting distribution of the test statistic as stated in the theorem. In other words, because the vector

is obtained from

by augmenting with unity, the same augmentation results in one of the two Brownian bridges in the limiting distribution obtained in Theorem 3 of Saikkonen and Lütkepohl (2000b). Technical details, which are similar to the corresponding two cases in Johansen (1995, Sect. 11.2), are straightforward and will be omitted.

Asymptotic properties of the GLS estimators of the deterministic parameters. Because the break date estimator is asymptotically between τ_o − p and τ_o it is straightforward to follow the proof of Theorem 2.1 of Saikkonen and Lütkepohl (2000a) (case a₁ < 1) and obtain asymptotic properties of the GLS estimators of the parameters μ₀, μ₁, and δ. Denoting these GLS estimators by

, it can be demonstrated that (A.39)–(A.42) of Lemma A.14 hold for

except that in (A.42) O_p(T^a) replaces O_p(1). For

it follows that

.

Limiting distribution of LR^GLS. The test statistic LR^GLS(r₀) is defined as LR^PAR(r₀) except that now

. Instead of (A.45) we therefore have

where the estimators on the r.h.s. satisfy the rates of convergence obtained previously. It is straightforward to check that, under the conditions of Theorem 4.1, the rates of convergence obtained for

are sufficient for the fifth term on the r.h.s. of (A.47) to have no effect on the asymptotic properties of the second sample moments on which the test statistic is based, and the same is true for the last term, which is as in (A.45). Thus, the problem reduces to that of a known break date studied by Saikkonen and Lütkepohl (2000a), and the limiting distribution of the test statistic is obtained from Theorem 3.1 of that paper. Here it suffices to note the following points. First, the dependence of the break size on the sample size has no effect because the needed arguments only involve the difference

. Second, it is not difficult to check that the rate of convergence

suffices instead of

, which could be used in Saikkonen and Lütkepohl (2000a) and the same is true for

. Thus, we have demonstrated that, under the conditions of Theorem 4.1, the test statistic has the same limiting distribution as in Saikkonen and Lütkepohl (2000a).