A.1. Preliminary Lemmas.

LEMMA A.1. The following magnitude orders hold:

(iii) if E_P|ε₁|^ν+1 < ∞, ν ≥ 0, then

.

Proof. As ε_t, Y_−k−1, and (u_−2k,…,u_−k−1) are assumed independent of δ_s for all t and s, conditioning on N_T ≥ 1 in the definition of P is irrelevant for this proof. Under P_c we have that

(see (2) for the definition of B_c); hence (i). In (ii) the evaluation of max_t≤T|u_t| (and under P_f, also those of max_t≤T|Y_t| and max_t≤T|ΔY_t|) holds since u_t (resp. Y_t and ΔY_t) has a moving average representation with exponentially decreasing coefficients. Under P_c, from ΔY_t = (−c/T)Y_t−1 + u_t and (i) it follows that max_t≤T|ΔY_t| inherits the magnitude order of max_t≤T|u_t|.

As to part (iii), because u_t is the stationary solution of

, (u_t)^ν+1 is stationary and ergodic by Theorem 3.35 of White (2001). Then E_P|ε₁|^ν+1 < ∞ implies, by the Marcinkiewicz–Zygmund inequality, that E_P|u_t|^ν+1 < ∞. By an ergodic law of large numbers (LLN) (White, 2001, Thm. 3.34),

. Under P_f a similar argument applies to

. Under P_c, the evaluation

follows from the inequalities

and

; see (i). █

LEMMA A.2. If y_t = O_P(1) is a sequence of random variables independent of δ_s for all t,s under P, then

and max_t:δt=1|y_t| = O_P(1).

Proof. This is direct from the definitions of the measure P (by Assumption

) and of boundedness in probability. █

A.2. Proof of Theorem 1 and Related Results.

To avoid confusion with the conditional measure P, we use Q for the measure in Theorem 1.

A.2.1. Proof of Theorem 1 (Part I).

Here we derive the weak limit of the process T^−1/2X_{[lfloor ]T·[rfloor ]}. The limiting distributions of the ADF statistics will be obtained at the end of Section A.2.2.

Recall that X_t = Y_t + μ_t and define

. To obtain the weak limit of T^−1/2μ_{[lfloor ]T·[rfloor ]}, we start from the convergence of

and then use the representation

.

First, we show that the serial dependence of {η_t*} is lost in the limit of μ_{[lfloor ]T·[rfloor ]}*. Consider, possibly upon enlarging the underlying probability space, a doubly infinite matrix

of i.i.d. random variables, with ω_1t := ω_t, and which is independent of {δ_t}_t=1^T and {Y_t}_t=−k−1^T. Denote by r_t the time of the last shift occurrence before time t (r_t := −∞ if there are no shifts before time t) and for t = 1,…,T define

where

. For a fixed T the vector {η_t⁺}_t=1^T is distributed as T independent draws from the distribution of η₁* (no ω_it is present in two different η_t⁺'s) and is independent of {δ_t}_t=1^T (direct by writing its distribution conditional on {δ_t}_t=1^T and observing that it does not depend on {δ_t}_t=1^T). Thus, the weak limit of

obtains from Theorem 3.1 of Leipus and Viano (2003). Because E|η_t⁺| = E|η₁*| < ∞ (as implied by E|ω_t,i| = E|ω₁| < ∞ and

, we have that

in

, where

is the Poisson process defined in Theorem 1.

Next, we argue that also μ_{[lfloor ]T·[rfloor ]}* has

as its weak limit. For s ∈ [0,1] we have that

By the independence of ω's and δ's, and with m_T := min_1≤t≤T{t − r_t} denoting the minimum distance between two consecutive shift dates (∞ if at most one shift occurs),

Because

by assumption and

, it follows that

and, further, that

by dominated convergence. Thus, by taking expectations in the preceding display,

because EN_T² → G(1) + G(1)² by a direct calculation (G denoting the Poisson intensity), so that max_s∈[0,1]|ϑ_s| = o_Q(1) by Markov's inequality.

The last relation is equivalent to

, and hence μ_{[lfloor ]T·[rfloor ]}* has the same weak limit as

in

. Because of the equality

, the convergence

is expected. For brevity and consistency with further references, we note that it follows, e.g., from Theorem 2.7 of Kurtz and Protter (1991). On the other hand, using standard local-to-unity asymptotics,

in

. Because of the stochastic independence of μ_t and Y_t, the joint convergence

in the product space

obtains. The map (x,y) [map ] x + y is continuous on the support

of

, and therefore,

by the continuous mapping theorem (CMT).

Note next that (1) implies the following representation of ΔX_t:

Let ∇X_t−1 := (ΔX_t−1,…,ΔX_t−k)′ and similarly for ∇Y_t−1 and ∇μ_t−1. The numerator of the

statistic based on X_t can be expressed as

, where

see Chang and Park (2002). We write the limits of these and of related quantities using stochastic integrals. The limits follow from Theorem 2.7 of Kurtz and Protter (1991), in view of the relation N_T = O_Q(1) and of the convergence

on the space

of 5 × 1-vector càdlàg functions, endowed with the Skorohod topology. Convergence (A.2) holds on the product space

similarly to (A.1) and can be extended to

by Proposition VI.2.2(b) of Jacod and Shiryaev (2003). Thus, Kurtz and Protter's theorem delivers that, jointly with (A.2),

Another useful limit is that of

and its continuous transformations. As (i)

for i ≠ j, (ii) δ_T−i = 0 with probability approaching one for i = 0,…,k, and (iii) max_t:δt=1|η_t| = O_Q(1) by Lemma A.2, it holds that

the limit as

by (A.2), (A.3), and the CMT. Convergence is joint with (A.2) and (A.3) again by the CMT. █

A.2.2. Proof of Theorem 1 (Part II).

To complete the proof of Theorem 1, we need the following lemma.

LEMMA A.3. Let Ω_∇∇ denote the probability limit of

and let Ω_1∇ be the nonrandom vector defined through

(see, e.g., Hansen, 1992; Phillips, 1987, Lem. 1), where 1_k is a k × 1 vector of ones. Then, as T → ∞, the following converge jointly:

(i)

;

(ii)

;

(iii)

;

(iv)

;

(v)

;

(vi)

.

Proof. Convergence (i) obtains from (6) and the CMT. Further, the left side of (ii) equals A₁ + A₂ + A₃, with the following A_i's. First,

by (A.3) and the CMT, because σ = σ_ε/Γ(1). Second,

, which can be written, with γ₀ := −1, as

The first summation in brackets contributes to A₂ with

. The contribution of the second summation is a continuous transformation of M in (A.4) and tends to

. Because

by Lemma A.2, we can conclude that

. Third,

by (A.2) and the CMT. Combining the limits of the A_i's, and recalling that B_c(s) satisfies dB_c(s) = −cB_c(s) ds + dB(s), gives the limit in (ii).

With

, convergence (iii) follows from the identity

Here B = o_Q(1) by Lemma A.2 applied elementwise, whereas the other two terms have limits Ω_∇∇ and

, respectively, the latter by (A.4). For (iv), note that

where the limit of the first term on the right side is given in the hypothesis, the second term equals

by Lemma A.2 and converges weakly to

by partial summation and (A.3), and, again by Lemma A.2, the third term equals

and can be seen to tend to

by (A.3), (A.4), and the CMT.

For items (v) and (vi) we have

because two kinds of o_Q(1) terms appear in these expressions. First,

,

, and

are o_Q(1) by Lemma A.2. Second,

by an LLN for

and by (A.3) for

, and similarly for F and H, which contain overnormalized contributions of cT⁻¹Y_t−1. The terms with nonzero limits are

by an LLN,

by (A.4) and the CMT, and

by (A.4) and the CMT again.

Convergence is joint because, up to terms with constant probability limits, the left sides (i)–(vi) can be collected in a continuous transformation of the left sides of (A.2) and (A.3). █

Let

and

, which can be written more shortly as

and

. It follows from Lemma A.3 that

Because

, it follows that

, and hence,

From Lemma A.3 we find that

. Together with

, these results yield the limit of

in (7) with

. We have used that Γ_∞ > 0 since, for each outcome in the sample space, it is the value at unity of the characteristic polynomial of a stationary AR(k) process with covariance of order i = 0,…,k equal to

; cf. Prob. 1.3 in Brockwell and Davis (1991).

The proof for the t-statistic is similar. Specifically, as A_T²/T² = O_Q(1) (see the preceding discussion), the following equalities hold:

with (Chang and Park, 2002)

From Lemma A.3 it obtains, after rearranging terms, that

. Hence, (A.5) and (A.6) ensure that

which is the limit in (7) with ϰ₁ := C_∞Γ_∞⁻²σ⁻². In the special case k = 0, we have Γ_∞ = 1,

, and σ² = σ_ε², from where the simpler expressions for ϰ₀ and ϰ₁. █

A.3. Proof of Theorem 3 and Related Results.

A.3.1. Uniform Evaluations Related to Dejumping.

LEMMA A.4. Suppose that

is a sequence of random variables such that

. Then

and

is bounded away from zero in P-probability.

Proof. Recall that θ_t = T^1/2η_t. The evaluation of

follows from the inequality

by applying Lemma A.2 to the first two maxima on the right side and the hypothesis of the lemma to the last one. The fact that

is bounded away from zero in P-probability obtains from

. The first minimum on the right side equals (max_t:δt=1 η_t⁻²)⁻¹ = (O_P(1))⁻¹ by Lemma A.2 and the assumption that η_t⁻¹ = O_P(1), whereas the second one is 1 + o_P(1). Indeed, because

it follows that, with P-probability approaching one,

Remark A.1. In the discussion of joint dejumping and parameter estimation, instead of a sequence of random variables

, a sequence of random functions

of a vector variable with domain

(say) will appear. If we suppose that

, the conclusions of Lemma A.4 hold for

and

. The proof carries over with only notational changes.

The next lemma provides conditions for uniformly consistent detection of level shifts.

LEMMA A.5. Let

and ζ(·) be as in Lemma 1,

be as in Lemma A.4, and

be as in (10). Then

and

, provided that

.

Proof. Recall that for functions f with appropriate domain we write

for

and similarly for suprema involving the components of ζ = (ζ^λ,ζ^η,ζ^σ)′. As φ_ν(e;a²) of (10) equals a⁻¹(1 + e²/(νa²))^−(ν+1)/2 up to a multiplicative constant, and 0 < ζ^λ < T on

with P-probability tending to one, we find that

Let

and

. As

, it follows that

Summing over t, and accounting for the assumption that N_T = O_P(1), yields the evaluation

Because

and, by Lemma A.4,

is bounded away from zero in P-probability, we have that

uniformly on

. Further,

again by Lemma A.4, and as

is bounded away from zero in P-probability, it follows that M_T,2 ≤ O_P(1) uniformly on

. Combining the two conclusions yields

.

The magnitude order of

is addressed next. By evaluating from below the denominator of (10), it can be concluded that

The contribution of the factors not containing

is O_P(T^−3/2) uniformly on

, whereas the last factor does not exceed unity. Further, by applying to

the inequality (1 + x²)^(ν+1)/2 < 2^(ν+1)/2(1 + |x|^ν+1), it is seen that

, because

. Hence,

Provided that

, we get that

uniformly on

. This result and the first part of the lemma give

on

, again uniformly. █

Remark A.2. If

are random functions satisfying the condition of Remark A.1, and if also

, then the conclusions of Lemma A.5 hold for

and

. Again, the proof is only notationally more involved.

Next, let d = (d₁,…,d_T)′ be a (continuous)

-valued random function with domain

(say). For instance, in the proof of Lemma 1 later in this Appendix, d will have components

and domain

. In the next lemma we consider as a function on

the dejumped series

and, among other things, evaluate the distance between {Y_t} and {X_t^d} depending on how close d is to δ = (δ₁,…,δ_T)′. The notation

is used.

LEMMA A.6. Let max_t≤T|ε_t| = O_P(T^τ), τ ≥ 0. If d is defined on a set

such that

for some β ∈ [0,τ), with β + τ < ½, then

(i)

and

;

(ii)

,

with a_T = O_P(T^β) and b_T = O_P(T^β+τ−1/2), and

;

(iii)

with

and

,

, and

;

(iv)

,

, and if, additionally,

, then

, where

.

Proof. For t = 1,…,T we have

, and for such t we find

and for t = −k,…,0, X_t^d − Y_t = 0. The first relation in (i) follows using (A.8):

with magnitude orders from Lemmas A.2 and A.1(ii) and from the hypothesis. Together with max_t≤T|Y_t| = O_Pc(T^1/2) (= O_Pf(T^τ)), (A.10) implies the other relations in (i).

The right side of (A.10) is a uniform upper bound for

(see equation (A.9)), hence, the first relation in (ii). Again from (A.9), |ΔX_t^d − ΔY_t| for t = 1,…,T does not exceed

from where, upon taking suprema, the definitions of a_T and b_T in (ii) are obvious. Their magnitude orders follow as in (i). Further,

by the preceding evaluation, Lemma A.1(ii), and the inequality β < τ.

A vector version

obtains readily and, together with (i), gives for Y_t := (Y_t,(∇Y_t)′)′ and X_t^d := (X_t^d,(∇X_t^d)′)′ the first relation in (iii). As max_0≤t≤T∥D_TY_t∥ ≤ max_t≤T|T^−1/2Y_t| + k max_t≤T|ΔY_t| = O_P(T^τ), it holds further that

which is O_P(T^τ) for β < τ. Similarly, max_t≤T|Y_t| = O_Pf(T^τ) leads to max_0≤t≤T∥Y_t∥ = O_Pf(T^τ) and

.

For part (iv), note that

, so that

by the weak convergence of

under P_f and by Lemma A.1(ii). Using (A.8), we find the decomposition

, where, first,

satisfies

by Assumption

, Lemma A.2, and (A.11). Second, as

on

, we see that

satisfies

and similarly,

satisfies

using Lemma A.1(ii) and again (A.11). The first relation in (iv) follows by combining the preceding magnitude orders. The second relation obtains upon partial summation; with

, so that

, we find that

uniformly on

by (i) and (ii). Finally,

as shown earlier, whereas, using (A.8), we can see that

because

,

using Lemma A.2 and the assumed magnitude orders of

and

, whereas the magnitude orders of cross-products in the expansion of (X_t^d − Y_t)², summed over t, follow from the preceding three evaluations and the Cauchy–Schwartz inequality. █

We conclude Section A.3.1 with an important lemma on the properties of OLS estimators in the ADF regression based on X_t^d, with X_t^d as in Lemma A.6.

LEMMA A.7. Let

and

be the OLS estimators from the regressions X_t^d = (α,γ′)X_t−1^d + error_t and Y_t = (α,γ′)Y_t−1 + error_t, and let

and

be the respective residual variances. Under the assumptions of Lemma A.6, with β = 0 and τ < ½, it holds that

(i)

.

Furthermore, if additionally

, then

(ii)

and

where (α_*,γ_*′) under

is defined through the representation Y_t = (α_*,γ_*′)Y_t−1 + ε_t implied by (1),

, and i := (1,0_1×k)′.

Proof. To prepare jointly the proof of (i) and (ii), we define (α_*,γ_*′) := (1,γ_*′) under

and

, and c_* := 0 under

and

. With this notation,

where

and

. The representation of ΔX_t^d obtains by inserting ΔY_t = ΔX_t^d + (ΔY_t − ΔX_t^d) and Y_t−1 = X_t−1^d + (Y_t−1 − X_t−1^d) into that of ΔY_t.

Define D_T := D_T, B_T := diag(T,T^1/2), h := ½ under

and

, and D_T := I_k+1, B_T := I₂, h := 1 under

. With

,

,

,

, and

, we have

We show that

, j ∈ {1,e,∇}, are O_Pc(T^τ−1/2) and that

, and

are O_Pf(T^τ−1/2). Then we use the standard facts that, under P, S₁₁^T has a positive definite and S_1e^T has a finite (zero, under P_f) weak limit.

First,

under

and

, where the maxima are O_Pc(T^τ) and the summation is bounded by

, all uniformly on

; see Lemmas A.1(ii) and A.6 (with β = 0). Hence, r₁₁^T = O_Pc(T^τ−1/2). Under

, we have the following uniform evaluations on

equals

by Lemmas A.1(ii) and A.6(iii) and (iv), applied in this order;

equals

by Lemma A.6(iii) and (iv), and

equals

by Lemma A.6(i) and (ii). From the last three evaluations,

on

, and from Lemma A.6(iv),

under the assumption that

.

Second,

, where

by partial summation. Therefore,

is bounded by

on

, because the factor in parentheses is O_P(1) by Lemma A.6(i) and (ii), and

with U = O_P(1) denoting a contribution from initial values. Still by Lemma A.6(ii), and because

, we find that

uniformly on

. By combining the preceding evaluations,

.

An evaluation of

can be obtained similarly:

which is O_P(T^τ) by Lemma A.6. Under

and

this shows that

. Under

, it gives together with Lemma A.6(iv) that

.

We proceed with the proof of (i). From (A.12), (A.13), and the identity (S₁₁^T + R₁₁^T)⁻¹ = (S₁₁^T)⁻¹ − (S₁₁^T + R₁₁^T)⁻¹R₁₁^T(S₁₁^T)⁻¹ it follows that

equals

To evaluate this matrix, define the event

. As r₁₁^T = O_Pc(T^τ−1/2) and S₁₁^T converges weakly to an a.s. positive definite matrix under P_c, it holds that

, and it suffices to study outcomes in

. For such outcomes ∥(S₁₁^T + R₁₁^T)⁻¹∥ ≤ ∥(S₁₁^T)⁻¹∥(1 − ∥R₁₁^T∥∥(S₁₁^T)⁻¹∥)⁻¹ (this follows from ∥(S₁₁^T + R₁₁^T)⁻¹∥ = ∥(S₁₁^T)⁻¹ − (S₁₁^T + R₁₁^T)⁻¹R₁₁^T(S₁₁^T)⁻¹∥ ≤ ∥(S₁₁^T)⁻¹∥ + ∥(S₁₁^T + R₁₁^T)⁻¹∥∥R₁₁^T∥∥(S₁₁^T)⁻¹∥ by solving the inequality for ∥(S₁₁^T + R₁₁^T)⁻¹∥), so that

is dominated by

which is O_Pc(T^τ−1/2) by the first part of the proof and because (S₁₁^T)⁻¹ and S_1e^T are O_Pc(1). With regard to residual variances,

As

uniformly on

(see Lemmas A.1 and A.6(ii)), and because

is a well-known fact, from the previous argument it follows that

. This completes the proof of (i).

Next, the identity (S₁₁^T + R₁₁^T)⁻¹ = [I_k+1 − (S₁₁^T + R₁₁^T)⁻¹(R₁₁^T − ii′ϒ_T)](S₁₁^T + ii′ϒ_T)⁻¹ and (A.13) yield

, where

Using that ∥(S₁₁^T + ii′ϒ_T)⁻¹∥ ≤ ∥(S₁₁^T)⁻¹∥, we find that

does not exceed

as S₁₁^T converges to a positive definite matrix in P_f-probability, S_1e^T = O_Pf(T^−1/2) by a central limit theorem, and the suprema were found to be O_Pf(T^τ−1/2). Further, for outcomes in the set

, with

, we have

so that

is bounded by O_Pf(1) times

similarly to

. This proves the first part of (ii). Because

can be written as

it is not hard to see that

, where, with

,

As

and

, we conclude that

. █

A.3.2. Proof of Lemma 1.

We check that, for 3 ≤ ν ≤ r − 1,

satisfies the hypotheses of Lemmas A.4 and A.5. First, max_t≤T|ε_t| = O_P(T^τ) with τ = r⁻¹ ≤ ¼ in view of Assumption

, and max_t≤T|ΔY_t| = O_P(T^τ) by Lemma A.1(ii). Hence, max_1≤t≤T|ΔX_t − ε_t − δ_tθ_t| = max_1≤t≤T|ΔY_t − ε_t| = o_P(T^1/2) as required in Lemma A.4 for

(from here and Lemma A.2, it is seen that max_1≤t≤T|ΔX_t| = O_P(T^1/2), which will be used subsequently). Second, (1 − δ_t)|ΔX_t|^ν+1 = (1 − δ_t)|ΔY_t|^ν+1 and

by Lemma A.1(iii), because ν + 1 ≤ r, and hence also Lemma A.5 is applicable to

. Recalling that

, from Lemma A.5 we obtain Lemma 1(i).

To prove (ii), we use (i) jointly with max_1≤t≤T|ΔX_t| = O_P(T^1/2):

uniformly on

. As (1 − δ_t)(ΔX_t)² = (1 − δ_t)(ΔY_t)² and δ_t ΔX_t = δ_t(ΔY_t + T^1/2η_t), we can insert

in the preceding display. Combining the result with the relation

, which is direct from (i), we obtain (ii).

Part (i) and Lemma A.7 with

and

, t = 1,…,T, imply (iii).

Consider now the Jacobian

, where

and w_t := (T,(ΔX_t)²,−(ΔX_t)²)′. It can be evaluated as follows:

We discuss the factors on the right side separately, starting from v_t. First,

where

. A similar expression (with a different

) obtains for ∂[ln φ_ν(ΔX_t;ς^σ)]/∂ς^σ. Using that

, it follows that

Upon evaluation at (ς^η,ς^σ) = (ζ^η(z),ζ^σ(z)), all right sides in (A.15) are O_P(1) uniformly on

. Also

. Hence,

. The other terms on the right side of (A.14) are max_1≤t≤T∥T⁻¹w_t∥ ≤ 1 + 2T⁻¹ max_1≤t≤T(ΔX_t)² = O_P(1) (see paragraph one of the proof) and

, which upon evaluation at ζ(z) is O_P(T^−1/2) uniformly on

(see part (i)). Part (iv) follows by inserting the obtained orders of magnitude into (A.14). █

A.3.3. Proof of Theorem 2.

(i) Existence. To be able to use a standard fixed-point theorem, we set up an auxiliary fixed-point problem for a self-map of a compact. For the auxiliary map, defined by centering, normalizing, and truncating Φ_T, existence of a random fixed point is classical, and this point turns out to be a fixed point of Φ_T, with P-probability tending to one.

Let

,

,

, and

. With

acting on z = (z^λ,z^η,z^σ)′ according to ζ(z) = (z^λ + N_T,H_T z^η,z^σ)′, let Θ_T := ζ⁻¹ ○ Φ_T ○ ζ. As H_T ≠ 0 P-a.s., Θ_T is well-defined P-a.s. Further, let

be defined componentwise: for v ∈ {λ,η,σ}, (Θ_T^tr)^v(z) := Θ_T^v(z) if

,

if

, and

if

. As Θ_T^tr is continuous and

is a convex compact, Θ_T^tr admits a random fixed point

. This is guaranteed by Theorem 10 in Bharucha-Reid (1976).

For outcomes such that

, we have z_T = Θ_T^tr(z_T) = Θ_T(z_T), i.e., ζ_T = ζ(z_T) is a fixed point of Φ_T. Because, in view of Lemma 1(ii), Θ_T(z_T) = ζ⁻¹((N_T,H_T,σ_Y²)′ + o_P(1)) = (0,1,σ_Y²)′ + o_P(1), and because (0,1,σ_Y²)′ is an interior point of

, it follows that

. Hence, ζ_T is a random fixed point of Φ_T with P-probability approaching one, and by the choice of

, ζ_T is bounded and bounded away from zero in P-probability.

(ii) Uniqueness. We use a contraction argument. Let ς_T have the properties of ζ_T from the existence part of the theorem. Fix ε > 0. Because N_T, H_T (by Lemma A.2 for η_t and η_t⁻¹), ζ_T and ς_T are bounded and bounded away from zero in P-probability, there exist constants q_h > 2q_l > 0 and a set

with

such that, for outcomes in

, N_T ≤ q_h, 2q_l ≤ H_T ≤ q_h, q_l1₃ ≤ ζ_T ≤ q_h1₃, and q_l1₃ ≤ ς_T ≤ q_h1₃, where the inequalities for ζ_T and ς_T are componentwise. Define the compact

. Then, for outcomes in

, first,

, and second, the point (N_T,H_T,σ_Y²)′ is interior for

and at a distance from the boundary of

bounded from below by a positive constant. From the latter and Lemma 1(ii), with

in place of

and with ζ(·) equal to the identity function, it follows that on some

, with

, and for T > T₂ (say),

. By Lemma 1(iv), there exist T₃ > T₂ and

, with

, such that for T > T₃ and on

,

. Because (Φ_T)_z′ is of dimension 3 × 3, under the same conditions,

, where the ith row of (Φ_T)_z′ is evaluated at z_i (i = 1,2,3). By the mean-value theorem for Φ_T on

, we see that for outcomes in

and for large T, Φ_T is a contraction on

and has a unique (per outcome) fixed point on

by Banach's fixed-point theorem.

Consider finally T₄ and

, with

, such that for T > T₄ and outcomes in

, Φ_T(ζ_T) = ζ_T and Φ_T(ς_T) = ς_T. This is possible by the choice of ζ_T and ς_T. Then, for outcomes in

and T > max(T₃,T₄),

and

must be equal to the unique fixed point of Φ_T on

. Because

and ε is arbitrary, P(ζ_T = ς_T) → 1.

(iii) Computability. A contraction argument is used again. Let

. As in the proof of uniqueness, on some event

, with

, and for T > T₁ (say), Φ_T is a contraction of

onto

, and

. Further, from the proof of existence, Φ_T(ζ_T) = ζ_T on some

with

and for T > T₅ (say). By Banach's fixed-point theorem, for T > max(T₁,T₅) and outcomes in

, with

, the sequence of iterates of Φ_T converges to the unique (per outcome) fixed point ζ_T of Φ_T on

.

(iv) Consistency. This part of Theorem 2 follows from Lemma 1(i) and (ii) by evaluating the functions there at

, with z_T, ζ(·), and

as in the proof of existence. █

A.3.4. Proof of Theorem 3.

Under

and

, the result for

is direct from (2) and Lemma 1(iii), with the functions there evaluated at

and with ζ(·) and z_T as in the proof of existence in Theorem 2. For ADF_t^δ, in addition to these results, also the conclusion that S₁₁^T + R₁₁^T(ζ(z_T)) = S₁₁^T + o_Pc(1) is invoked to ensure asymptotic equivalence of the standard errors based on

and on Y_t; see the proof and the notation of Lemma A.7.

Let the OLS estimators of α, γ, and σ_ε² from the ADF regression for

be

,

, and

and let ϑ_T := 1/(1 + ϒ_Ti′(S₁₁^T)⁻¹i) ∈ (0,1]. After some algebra, we obtain from Lemma A.7(ii) (upon evaluation of the functions there at z_T, with d_t(·) = δ(ΔX_t;ζ(·)) and

) that, under the hypothesis

,

where S_0∇^∞ and S_∇∇^∞ are the P_f-probability limits, respectively, of

and

. Because, under

, α_* = (α_* − 1)Γ_*(1) + 1 < 1 and ϒ_Ti′(S₁₁^T)⁻¹i = O_Pf(1), we find that

. Because

and (S₁₁^T + R₁₁^T(ζ(z_T)))⁻¹ = O_Pf(1) by Lemma A.7(ii) and its proof, it follows that

. Similarly, because

(see equation (A.16)), also

. █

A.4. Proof of Theorem 4 and Related Results.

Let ρ > 0 be a real number that we will choose as small as convenient. The compact

from the proof of Theorem 2 is replaced here by

, defined as

, with

A.4.1. More Lemmas.

The point at which Ψ_T is evaluated is specified as

, where

is the random function with components ξ^δ(x^δ) = x^δ + (δ₁,…,δ_T)′, ξ^λ(x^λ) = x^λ + N_T, ξ^η(x^η) = H_T x^η and ξ^a,ξ^γ,ξ^σ are the identity functions, respectively, on

,

, and

. It will be shown that there exists a random sequence

such that P_c((Ψ_T ○ ξ)(x_T) = ξ(x_T)) → 1. The proof uses properties in the spirit of Lemma 1 that are worked out next. The argument of ξ and its components is subsumed; the notation

is employed for

and similarly for the components.

Upon substitution of d = (d₁,…,d_T)′ by ξ^δ in the definition of X_t^d (t = 1,…,T), the dejumped series

obtains; we treat it, and also the associated residual series e_t^ξ := ΔX_t − (ξ^a,(ξ^γ)′)D_TX_t−1^ξ, as a function on

. Similarly, the updated dejumped series

and residuals e_t^Ψ := ΔX_t − Ψ_T^a,γ(ξ)D_TX_t−1^Ψ are functions on

. Thus, so is

; see (10).

LEMMA A.8. If 3 ≤ ν ≤ r − 1, 2ρ(ν + 1) < 1, and max_t≤T|ε_t| = O_Pc(T^τ), then

(i)

and

;

(ii)

, with

as in Lemma A.7;

(iii)

and

;

(iv)

.

Proof. Without loss of generality, we take τ ≤ ¼, because E|ε₁|⁴ < ∞ under P_c. Note also that ρ < ⅛ under the hypothesis of the lemma (this represents no restriction on the model).

For part (i), we check that Lemma A.5 (as stated in Remark A.2) is applicable to

. First, we evaluate e_t^ξ − ε_t − δ_tθ_t − (γ_* − ξ^γ)′∇Y_t−1 on

. Under

and

, they equal (−T⁻¹c_* − T^−1/2ξ^a)Y_t−1 − (ξ^a,(ξ^γ)′)D_T(X_t−1^ξ − Y_t−1), where

,

, max_t≤T|Y_t| = O_Pc(T^1/2) and Lemma A.6(iii) with d(·) = ξ^δ(·),

, and β = ρ is applicable to

. By collecting the magnitude orders, we obtain that

where

. Thus, because

, max_1≤t≤T∥∇Y_t−1∥ = O_Pc(T^τ) by Lemma A.1(iii), δ_t ≤ k, and ρ + τ < ½, it follows that

, as required in Remarks A.1 and A.2.

Second, we verify that

. Indeed, in consequence of (A.17) and the inequalities ρ(ν + 1) < ½, ρ + τ < ½, it holds that

uniformly on

. Further, for ν + 1 ≤ r, we have E|ε_t|^ν+1 < ∞, so that

and

by Lemma A.1(iii), whereas

and

. These imply the asserted uniform order of

. Having checked the conditions of Remark A.2, we obtain part (i).

Part (ii) follows from (i) and Lemma A.7(i) with d(·) = Ψ_T^δ(ξ(·)),

. Further, (ii) together with the standard properties

and

implies that

,

, and

. These are useful in a derivation similar to that of (A.17) but this time invoking for

Lemma A.6(iii) with d(·) = Ψ_T^δ(ξ(·)),

, and β = 0, the latter in view of (i):

because max_1≤t≤T∥∇Y_t−1∥ = O_Pc(T^τ) by Lemma A.1(ii). From here, Lemma A.2, and the assumption that max_t≤T|ε_t| = O_Pc(T^τ), (iii) follows.

Finally, part (iv) can be derived starting from

by (i) and (iii). Next, by applying to u = e_t^Ψ and v = ε_t + T^1/2δ_tη_t the inequality |u² − v²| ≤ |u − v|² + 2|u − v| |v|, and then (A.19) to evaluate |u − v|, we see that

Inserting

and combining with the previous display completes the proof of part (iv). █

Next, for use in contraction arguments, we evaluate the Jacobian matrix of Ψ_T. For a matrix A = (a_ij), introduce ∥A∥₁ := [sum ]_i,j|a_ij| and ∥A∥_h := max_j [sum ]_i|a_ij|.

LEMMA A.9. Let each row of the Jacobian matrix (Ψ_T)_ς′ = (Ψ_T(ς))_ς′ be evaluated at some point of the form ξ(x), with x possibly varying across rows, and let (Ψ_T)_□′ denote the resulting matrix as a function on

. Suppose that Assumption

is satisfied for some r ≥ 5 and that 4 ≤ ν ≤ r − 1. Then there exists a ρ > 0 such that

.

Proof. The proof is rather long but mechanical and hence is omitted from brevity. It is available in the accompanying working paper, Cavaliere and Georgiev (2006), or from the Web page http://www2.stat.unibo.it/cavaliere/lshifts. █

A.4.2. Proof of Theorem 4.

The argument is similar to that for Theorem 2. For ξ(·) defined in Section A.4.1, let Ξ_T := ξ⁻¹ ○ Ψ_T ○ ξ; it is well-defined P_c-a.s.

Let ρ > 0 be chosen such that the conclusions of Lemma A.8 are applicable; this choice is possible under the conditions of Theorem 2. Further, let

be obtained from Ξ_T by truncating it as follows: (Ξ_T^tr)^δ(x) := Ξ_T^δ(x) if

and (Ξ_T^tr)^δ(x) := T^ρ−1/2Ξ_T^δ(x)/Σ_t=1^T|Ξ_Tt^δ(x)| otherwise; (Ξ_T^tr)^γ(x) := Ξ_T^γ(x) if

and (Ξ_T^tr)^γ(x) := γ_* + (Ξ_T^γ(x) − γ_*)/∥Ξ_T^γ(x) − γ_*∥ otherwise; (Ξ_T^tr)^λ, (Ξ_T^tr)^a, (Ξ_T^tr)^η, (Ξ_T^tr)^σ are defined similarly to the components of Θ_T^tr in the proof of Theorem 2. Then Ξ_T^tr has a random fixed point ϰ_T on

by Theorem 10 in Bharucha-Reid (1976). Define ξ_T := ξ(ϰ_T). Similarly to the proof of Theorem 2, but invoking Lemma A.8(i), (ii), and (iv) instead of Lemma 1, we conclude that P_c(Ψ_T(ξ(ϰ_T)) = ξ(ϰ_T)) → 1, i.e., ξ_T has property (i).

Because

, from Lemma A.8 it follows that ξ_T also has property (ii). Property (iii) follows once it is noticed that the ADF statistics based on the dejumped series are

and ADF_t^Ψ = T^−1/2Ψ_T^a(ξ_T)[σ_Ψ²(ξ_T)v₁₁]^−1/2, where σ_Ψ²(ξ_T) is the residual variance from (13) and v₁₁ is the first entry of the matrix

, both evaluated at ξ_T. As

, Lemma A.8 allows us to invoke Lemma A.7(i) and to obtain the limiting distributions of the statistics as in the proof of Theorem 3.

In what follows we assume that r ≥ 5, and we choose ρ such that Lemma A.9 is applicable.

To discuss uniqueness, fix an ε ∈ (0,½). Because

is convex, for every

, by the mean-value theorem Ξ_T(x₁) − Ξ_T(x₂) = (Ξ_T)_□′(x₁ − x₂), where (Ξ_T)_□′ is the Jacobian matrix (Ξ_T)_x′ with each row evaluated at some point in

, possibly varying across rows and also with x₁,x₂. We have

in view of Assumption

and Lemmas A.2 and A.9, where (Ψ_T)_□′ is as in Lemma A.9. Therefore, we can find a set

, with

, such that for T > T₁ (say) and outcomes in

, ∥(Ξ_T)_□′∥_h ≤ ½ on

. Further,

; i.e., Ξ_T is Lipschitz on

with modulus ½. On the other hand, in view of Lemma A.8(i), (ii), and (iv), there exist a set

, with

, and an integer T₂ ≥ T₁ such that, on

and for T > T₂,

. Thus, for T > T₂ and outcomes in

, Ξ_T is a contraction on

, and by Banach's fixed-point theorem, Ξ_T has a unique (per outcome) fixed point on

.

Further, if ξ_T and ς_T satisfy part (i), there exists a set

, with

, such that on

and for T > T₃ (say), Ψ_T(ξ_T) = ξ_T and Ψ_T(ς_T) = ς_T. If they also satisfy parts (ii) and (iii), so that Ψ_T(ξ_T) = ξ(ϰ_∞) + o_Pc(1) and Ψ_T(ς_T) = ξ(κ_∞) + o_Pc(1), with ϰ_∞ := (0_T+1,γ_*′,0,1,σ_ε*²)′ and with convergence rate faster than the shrinkage rates of