2.1. Basic Concepts and Notations

Let x_1t be a scalar I(1) process and x_2t be an (n − 1) vector I(1) process. Suppose that these variables have one of the following relations:

Define the innovation vector u_t = Δx_t or u_t = Δx_t − γ in the presence of some trending variable(s). Also, define its cumulative process

and its long-run variance Ω = lim_T T⁻¹ES_T S_T′. We use the following regularity conditions.

Assumption 1.

(a) {u_t} is mean-zero and strong mixing with mixing coefficient of size −rκ/(r − κ), and E|u_t|^r < ∞ for some

.

The conditions in Assumption 1 allow a broad class of weakly dependent time series and have been used by Phillips (1987), Phillips and Perron (1988), Phillips and Ouliaris (1990), and Gregory and Hansen (1996), among others, to study limiting behavior of some stochastic processes.

Now, consider the possibility that the error term ε_t is a unit root process in a subset of the data period, whereas it may follow a stationary process for the rest of the data period. We reserve notation for these two different types of periods,

, respectively. We assume that

consists of a single interval of periods. More specifically,

for two time periods k₀ and k₁, k₀ < k₁.

Assumption 2. Let

be the number of elements in

. We assume that

.

Assumption 2 rules out the case of a “trivial” segmented cointegration with

: for example, for n = 2 if

then any process forms a segmented cointegration by defining β = x_1h /x_2h and ε_h = 0 for each h = 1,…,T.

When

consists of more than one interval of periods, we need to determine the number of segmentations in practice. Formal analysis of the problem of multiple segmentations is beyond the scope of this paper. However, we briefly discuss this problem in Section 3.

Testing for cointegration for (2.1) is often based on testing for the null of no cointegration. Thus, writing the disturbance process as an AR(1):

the hypotheses are for the value of ρ: the null of ρ = 1 vs. |ρ| < 1. There are several different tests for testing such hypotheses, that is, Dickey–Fuller (1979) tests, Phillips (1987) tests, Phillips and Perron (1988) tests, Phillips and Ouliaris (1990) tests, and augmented Dickey–Fuller tests.

In the case of possible segmented cointegration we can consider the following hypotheses:

where

.

As an example of the situation implied in H₁ we can consider an economy that runs in two different regimes. For example, when the monetary authority changes the policy regime from targeting the interest rate to targeting money volume, the interest rate may be stationary only in the first regime.

A situation similar to that in H₁ is found in the following model of threshold cointegration studied in Balke and Fomby (1996), among others:

When the “threshold” θ is a sufficiently small constant, then the preceding process ε_t is effectively a stationary process as is the case in Balke and Fomby (1996). However, if the threshold θ is not a small number or is a random variable, which sometimes has high values, the process ε_t may effectively be nonstationary. On the other hand, there is an important difference between segmented cointegration and threshold cointegration: segmented cointegration assumes a deterministic interval of cointegration whereas threshold cointegration depends on the conditional variables.

2.2. Failure to Confirm a Cointegration Relation

In this subsection we briefly examine why usual tests fail to confirm cointegration in the presence of short-run nonstationary deviations. To see this consider an example of the system (2.1a) under H₁ where ε_t is such that

where

for two time periods k₀ and k₁, k₀ < k₁.

A trivial reason for the failure of confirming cointegration in the presence of short-run deviations is that the cointegrating vector may not be consistently estimated. This is because the short-run nonstationary deviations dominate in the estimate of the cointegrating vector, which very likely makes the regression spurious. If the cointegrating vector is not correctly estimated, the residual contains nonstationary regressors: in (2.1a) if

then the residual e_t becomes

In the presence of short-run nonstationary deviations, however, even when the cointegrating vector is known or is correctly estimated, the chance of failure of confirming cointegration of usual tests can be very high. Thus, suppose that β is known in (2.1a). Then, the simple Dickey–Fuller t-statistic for testing the null H₀ (2.3) is

where

. But, since

and

, the preceding t-statistic becomes

where

, the number of elements in

, replaces T in the t-statistic. Therefore, the behavior of the t-statistic in (2.8) under H₁ for the model (2.5) and (2.6) is arbitrarily close to that under H₀ (2.3). In fact, under the condition that E|ε_k0| < ∞ asymptotic behavior of the t-statistic (2.8) is the same as that under H₀. But, for t = k₀ in a stationary regime the condition E|ε_k0| < ∞ is not restrictive. As a consequence, the statistic (2.7) would very often fail to reject H₀ under H₁.

3.1. Detection of Segmented Cointegration

When the error term in a linear regression model such as (2.1) is heterogeneous across different periods, econometric theory suggests estimating the model by a weighted least square rather than by the ordinary least square:

where w_t is a weight given to the t-period residual and w = (w₁,…,w_T). For the model (2.1) with ε_t having the property in H₁ the weight series {w_t} should be such that

where w_t = 1 for

can be replaced by w_t = constant.

The specification (3.2) for w_t is justified by the following reasons: first, the data in

are not informative at all for estimating the coefficient β because in

there does not exist a β that yields stationary ε_t. Second, in the case of variance heterogeneity the weight w_t is the reciprocal of the “variance” of ε_t. The reciprocal of the “variance” is arbitrarily small for

because the sample “variance” of ε_t in

as the relation is a spurious regression in

, which is by virtue of Phillips (1986).

for a trial period for segmented cointegration and also for the true period of segmented cointegration, regarding which the reader should not be confused. Under H₀ the regression (2.1) is spurious for any choice of {w_t} or

. Also, under H₁ the regression yields a consistent estimator of the parameter for a proper choice of {w_t} or

. For this reason a test for segmented cointegration based on the estimation (3.1) with {w_t} (3.2) would have desirable properties.

Notice that the weighting process {w_t} depends on the selection of

. Thus, we denote it by

. Also, denote by

, respectively, the estimator of β and the residual from (3.1) based on the weight

for a segmentation

. Let

be the least square estimator of the first-order serial correlation coefficient ρ of e_t in e_t = ρe_t−1 + ν_t for

be the t-test of ρ = 1. That is,

is obtained from regressing

on

. Also, let

be the least square variance of

be the least square estimate of the variance of the second stage residual:

where

is the number of elements of

. Define

where

, respectively, are estimators of

.

Consistent nonparametric estimators of λ and γ₀ under H₁ are obtained as in the following expressions:

The approach given above is based on an AR(1) for the residual process e_t. Alternatively, we can consider the following variant of an augmented Dickey–Fuller expression for e_t:

where {ε_t} is an independent and identically distributed (i.i.d.) sequence with mean zero, finite variance σ_ε², and finite fourth moment. Notice that in (3.5) the error correction term (ρ − 1)e_t−1 is multiplied by the “weight” w_t. Let

, respectively, be the least square estimator of ρ in (3.5) and the t-test of ρ − 1 = 0.

The initial condition e_t0−j for j = 1,…,p for the starting period t₀ in

can be taken either from the last p observations of the previous noncointegration regime or from the first p observations of

depending on which of the two hypotheses, H₀ or H₁, is favored. To be in favor of H₀ it can be taken from the last p observations of the previous noncointegration regime:

. To be in favor of H₁ we can take the first p observations of

for the initial condition.

where

is consistently estimated by

where

is the least square estimator of ζ_j in (3.5).

Let

be a fixed subset of [0,1] such that

where [·] is the integer part of ·. Likewise,

. Since

consists of a single interval of periods, we may write it as

for some

where

.

Now, for a given

, the statistics

converge to some limits given in the following theorem.

THEOREM 1. Assume the conditions in Assumption 1. Under H₀ it is true that

where

are such that

where

where W₁(·) is a standard Wiener process and W_2,m(·) depends on the model. If the residuals are from ordinary least squares (OLS) estimation of the model (2.1a), then

where W₂(·) is the (n−1)-dimensional standard Wiener process. If the residuals are from OLS estimation of the model (2.1b), then

If the residuals are from OLS estimation of the model (2.1c), then

In practice, the period

is not known. In this case we adopt the following approach: we get values of the statistic

for all the possible candidates of

and take the smallest value of the statistic. The smallest value constitutes evidence against the null hypothesis.

THEOREM 2. Denote by

, etc. Assume the conditions in Assumption 1. Then, under H₀ it is true that

Asymptotic critical values of these statistics can be obtained by simulating test statistics for a large sample size for a large number of replications. Because computation in the simulation for the present example for large samples is very time consuming we adopt a procedure due to MacKinnon (1991): using 10,000 replications for each of the sample sizes T = 50, 70, 100, 120, 150, 170, 200, 220, 250, 270, and 300 we obtain critical values Crt(T,p,n), where p is the percentile. We then estimate by OLS for each p and n the response surface

Various polynomials were tried (involving T⁻³, T⁻⁴, etc.), but the one given above appeared to be the best fit overall (R²'s were generally over 0.98). The asymptotic critical value is taken to be the OLS estimate

. Although the above specification seemed to be the best overall, estimates of ψ₀ changed on occasion by a factor of 2 under alternative specifications. Also, we get the critical values of

for some given upper limit of the length of

denoted by

.

This is again to reduce the computational requirements. In practice the length of the noncointegration period

is variable, and we need critical values for each

, which are almost impossible to obtain. Fortunately, however, for a conservative test we can get an upper limit

to get the critical values (Lemma 1 in Sect. 3).

where

is the length of the interval

. In Tables 1 and 2 asymptotic critical values of the minimum statistics in (3.13) and (3.14) are provided for

. In Tables 1 and 2, Case I is for the model (2.1a), Case II is for the model (2.1b), and Case III is for the model (2.1c).

Asymptotic critical values of Zρ*(C) and ADFρ*(C)

Asymptotic critical values of Zt*(C) and ADFt*(C)

In practice, critical values depend on the value of

chosen for simulation. We can show that the critical value for a given size of the test is a decreasing function of

.

LEMMA 1. Let

. Denote by

the lower α-size critical value of

obtained based on

. Then, it is true that

.

Therefore, the test (3.13) or (3.14) based on critical values

for a given [ell ]₂ is a conservative test when the actual test statistic

is computed for [ell ]₁ such that [ell ]₁ < [ell ]₂.

In general, as the assumed length of the noncointegration period is smaller the distribution of the test statistic is more concentrated, which makes the test more conservative. This is true with respect to the absolute length of the noncointegration period as well as the relative length of the noncointegration period

. Thus, for a small sample analysis the test based on asymptotic critical values would lead to conservative inference.

As was explained in Section 2, we consider in this paper the case when

consists of a single interval of periods. In practice a researcher would have situations where the number of such intervals is possibly more than one. In this case the researcher would have to determine the number of such intervals. Formal analysis of the problem of multiple segmentations including determination of the number of such intervals is beyond the scope of this paper. However, in this case we could suggest adopting the idea of detecting one by one as in Bai (1997). To describe it briefly, the approach is to divide the sample into several successive subsamples and detect the existence of segmented cointegration in each subsample based on the approach discussed in this section. This approach would save us a lot of computation time for detecting multiple intervals of noncointegration periods when the number of such intervals is unknown.

The tests for segmented cointegration

are consistent tests, which is shown by the following lemma.

LEMMA 2. Let {x_t}, x_t = (x_1t, x_2t′)′, be an n-vector I(1) process where x_2t, an (n − 1) vector I(1) process, has a positive definite long-run variance-covariance matrix

. Suppose that ε_t = x_1t − β′x_2t is I(0) in a subset of data

where ε_t has a continuous spectral density f_εε with f_εε(0) > 0 in

. Then,

A Monte Carlo result in Section 4 shows that empirical power values of the tests (3.13) and (3.14) are quite high for a 5% test and for T = 200.

Remark. A model related to the one in our alternative hypothesis (2.4) has been considered by Leybourne, McCabe, and Tremayne (1996) and McCabe and Tremayne (1995):

where ρ_t is assumed to be an independent process. McCabe and Tremayne (1995) study how to determine whether or not the null of nonstationarity, ρ_t = 1, is rejected against the alternative of the root being independently changing, and Leybourne, McCabe, and Tremayne (1996) provide some applications of McCabe and Tremayne (1995). In our approach the model for alternative hypothesis is a special case of that in McCabe and Tremayne. However, the model for our alternative hypothesis allows richer statistical inference than that of McCabe and Tremayne: First, our alternative allows us to analyze a cointegration relation as well as observable individual series whereas the alternative of McCabe and Tremayne is adequate only for analyzing the latter as ρ_t in McCabe and Tremayne has to be an independent process. Second, our approach allows us to identify the two distinct periods, the period of noncointegration (nonstationarity) and the period of cointegration (stationarity), whereas the approach in McCabe and Tremayne, by its nature, is not appropriate for identifying or estimating the varying root.

3.2. Identifying the Period of Noncointegration

When a segmented cointegration is detected by the procedure discussed previously, we are interested in the location of the period of noncointegration. In this section we study how to identify the location of the period of noncointegration.

As in Section 3.1 (before Theorem 1), let

for some

, and [·] is the integer part. Notice that we allow the possibility that the nonintegration period lies at the beginning or at the end of the sample. Now, let

be the estimator of β from (3.1) based on

in (3.2) for

. Construct a residual process

from the estimate

for all t = 1,…,T. For a given τ = (τ₀,τ₁), let Λ_T(τ) be such that

Also, let

be such that

where

.

We assume the following condition.

Assumption 3. Under H₁ let e_s+1,e_s+2,…,e_s+q for s ∈ {0,…,T − 1} and q ≤ T − s be a sequence of residuals of (2.1) in the stationary regime

. Assume that

for E [e²] < ∞.

Assumption 3 describes a law of large numbers for a sequence of stationary e's, which usually holds for least square residuals such as {e_t} in the previous section.

THEOREM 3. Suppose that there exists

such that ε_t ∼ I(0) for

whereas ε_t ∼ I(1) for

. Assume that the condition of Assumption 3 is satisfied. Then, it is true that for i = 0,1

Another approach to identifying

is to find the set

for which the p-value of the test statistic

is the smallest: denoting T = {1,2,…,T},

This approach is well approximated by

when the value of the Z-statistic does not heavily depend on the size of

. The term Z_t or ADF_t is relatively less dependent on the sample size than Z_ρ or ADF_ρ. In an empirical example in Kim (2001), we find that the two approaches (3.17) and (3.18) yield similar results but with the tendency of

is the estimate of τ from (3.18).

Kim (2000) considers change in persistence of linear time series, considers change from stationarity to nonstationarity and vice versa, and studies how to detect the existence of such change and how to identify the period of such change. Chong (2001) considers structural change in AR(1) models including change from stationarity to a unit root and vice versa and studies how to locate the period of such change. The subjects of these papers are related to the work in this paper. However, there is difference in some important aspects: first, whereas this paper studies change in persistence of the (unobservable) error of a cointegration relation, the two previous works study similar change in an observable series. Statistical treatment in practice for the former case is fundamentally different from that for the latter case. Second, whereas the main interest of this paper is the existence of a stable relation in the majority of the data period (segmented cointegration), the two previous works are interested in a one-time break in the associated property of a series.

Another related work found in the literature is Bai, Lumsdaine, and Stock (1998). Bai et al. study how to test for the existence of and date a structural break in a cointegration relation. Their analysis includes a vector error correction model as well as a single-equation cointegration relation. The subject of Bai et al. is related to that of this paper in that both are discussing inference on a cointegration relation with possible turbulence. However, the subjects of the two papers are different in an important aspect: whereas Bai et al. study the problem of a structural break (parameter change) assuming that cointegration relation(s) hold in both periods before and after the break, this paper considers the case when there exists an interval of periods in which a (stable) cointegration relation does not exist. Statistical inference on the two systems, testing for the existence of such systems and dating the period(s) of break(s), as well as economic implications of the two systems are quite different. The main reason for the difference in inference is that the disturbance of the system considered in this paper undergoes change from stationarity to nonstationarity (and vice versa later) whereas there does not exist such change in the system considered in Bai et al.