**4.3 Types of tests**

In what follows, three classes of testing useful during analysis of step-like change-points have been identified. The first two involve testing the segment of data within which a change-point is found. The third asks if a multiple change-point model is adequate.


*Severe Testing and Characterization of Change Points in Climate Time Series DOI: http://dx.doi.org/10.5772/intechopen.98364*

#### *4.3.1 Detection test*

Complex climate time-series data is almost certainly misspecified for *any* change-point detection test – thus the goal is adequate applicability to questions of interest. In testing for multiple change-points, many methods, including the MSBV, examine data only between presumptive lower and upper bounding points and restart estimation of the distribution parameters. The assumptions of the basic detection methods used must be considered.

Issues potentially arising include false detection, timing errors, and false negatives. Timing error includes misplacement and imprecision. The MSBV incorporates a resampling strategy [38] which reduces imprecision. False positives (deterministic and non-deterministic) can be uncovered by post-detection assessments, but false negatives introduce down-stream non-stationarities that interfere with detection of later change-points. Combining tests with differing assumptions and different nulls probes for both non-deterministic and deterministic causes of false results including sub-detection threshold events.

Analysis of covariance (ANCOVA) is used post-detection of a change-point by MSBV (which does not consider trend) to ensure that the presence of the changepoint provides explanatory power in an unconstrained disjoint linear statistical model which allows trend. It does not attempt to locate an alternative change-point – the Zivot-Andrews test however, see below, does this in passing. ANOVA tests for change of trend and change of level are obtained in passing but in R2019, final *p*values for change-points are obtained from only from ANCOVA.

### *4.3.2 Tests for heteroskedasticity for segmentation of data with change-points*

The full set of change-points in an entire sequence is tested here by the studentized Breusch-Pagan test (hereafter SBP test) for homoskedasticity of the residuals of the disjoint multi-segment model (JR2017 utilised the equivalent White's test [39]). An adequate model explanation of a time series, under the assumption of i.i.d. error, should have a featureless residual. This test has a null of homoskedasticity, rejected in favour of heteroskedasticity at low probabilities.

#### *4.3.3 Tests for stationarity in a segment*

Our detection test and the subsequent probability assignments by ANCOVA or ANOVA, and the further misspecification testing all assume serial independence either in the null or contrast hypothesis.

In these tests the segment containing a provisional change-point is tested for features that may deceive tests for shifts and trends. The MY test, ANCOVA, and where used ANOVA tests, have ruling assumptions of serial independence. The MSBV, and other multiple break tests assume some form of censorship between provisional data segments (determination of change-points within provisional bounds includes only the data within the bounds); but tests of the overall model assume homogeneity of error, thus of variance (e.g. the Akaike Information Criterion or AIC). The SBP also assumes this. All of these above tests are formalised as null hypothesis statistical tests (NHST) and as such they each are subject to their own ruling assumptions. The ruling assumptions are incorporated in the interpretation of the tests.

Autocorrelation in climate time-series is variously treated; some propose its estimation and removal [40], some warn against this idea [41]. Some treat it as a short term process and a cause of deception in change-point analyses [42], others have treated it as a persistent signal [43]. In climate signals, autocorrelation often appears to be time varying. Therefore we apply the MSBV without adjustment for autocorrelation and perform post-detection analysis to determine whether the detection test is likely to have been interfered with. In general, regression based statistical tests assume the absence of deterministic step-like changes and of unitroot, or red-noise progression.

Because multiple UR tests are performed, and because they each have differing ruling assumptions, the tests are interpreted in terms of evidence for and against stationarity in the underlying processes. In the econometric literature an exogenous change is one imposed upon a model from outside the model. We elected to retain this word where concepts were derived from economic papers as meaning an

Transient unit root behaviour, if it occurred, could indicate some sort of regime change, temporarily decoupled from normal forcings. If, in addition, measured noise was not persistent this would show I(0) behaviour; or, if it were fully persistent, as I(1) behaviour. In regional signals in which this occurs, the region may also have become coupled to other sub-systems [46]. This could indicate that the underlying physical model is incomplete and that a missing variable misspecification has resulted. On the other hand, persistent unit root behaviour means that a

The Earth system is constrained so that the overall temperature cannot solely follow a pure random walk – at worst it would follow a Brownian bridge (i.e. sequences where the end-points are meaningful and accepted as deterministic but the path is apparently a random walk [47]). However the composition of summary deterministic signals, such as the GMST, involves manipulations that can produce data that existing unit root tests will identify as containing unit roots, and furthermore deceive deterministic tests in much the same way as random walk data. This

Random walk progression may be present in climate data because of transient physical conditions, or because the data is unrelated to the physical processes assumed (M-S due to irrelevant variables). Additionally there may be features in the data that do not correspond to any of a shift, a trend change, or unit root behaviour (M-S due to missing variables), and UR tests are potentially sensitive to this. This source of deception must also be dealt with. Other features may be present in the data but not detected. For instance, a step-like shift well above a detectability threshold may be present together with a number of small, deterministic shifts below detectability, and this latter may be taken to be evidence of stochastic drift by

The unit root based tests used here all inherit in one form or another the Dickey

*ρ* represents the portion of the signal (*Yt*�1) carried forward by autocorrelation, *β* represents the (deterministic) linear trend, *μ* represents the intercept, and *et* is the i.i.d. error with zero mean and a constant variance *<sup>σ</sup>*2. If *<sup>ρ</sup>* <sup>¼</sup> 0 this describes a deterministic trend with no autocorrelation, if 0>*ρ* <1 there is a deterministic trend with a degree of autocorrelation, and if *ρ* ¼ 1, regardless of other parameters it contains a unit root. If all other parameters are zero and *ρ* equals one, then there is no deterministic trend, no offset, and *Yi* is a random walk. This formulation is modified and sometimes rearranged in different ways by the three UR tests used

*Yt* ¼ *μ* þ *βt* þ *ρYt*�<sup>1</sup> þ *et* (6)

abrupt and deterministic change in a deterministic time-series.

*Severe Testing and Characterization of Change Points in Climate Time Series*

issue was extensively examined in R2019 and is addressed later.

*4.3.7 Unit roots, non-stationarity, and climate*

*DOI: http://dx.doi.org/10.5772/intechopen.98364*

deterministic change-point analysis is suspect.

*4.3.8 Detecting unit root presence*

a UR test.

here.

**219**

Fuller (DF) model [48].

The term unit-root refers to processes with a characteristic equation that has a value of one. If a unit characteristic is moving average, the error is integrated order zero or I(0), if it is auto-regressive, it is integrated order one, I(1). I(0) processes tend to revert to a mean, I(1) processes follow a martingale [44], and is dominated by red-noise. The integration order defines the number of successive differencing operations required to produce a trend-stationary series.

#### *4.3.4 Residuals compared to initial data*

In our work, both the raw data, and the residuals after removal of internal steps and trends, are tested. The rationale for testing both derives from the formulation of the tests themselves, since in these tests, the deterministic and non-deterministic components are separately parameterised. The set of tests chosen are from the econometric literature, and each is framed as a null hypothesis significance test (NHST) with its own specific assumptions. Each test poses either *H0* or *H1* as presence of an assumed non-deterministic unit root progression (see Chapter 2) in data, and the alternatives are chosen from a small range of deterministic features. Crucially, each must be interpreted in the light of its own ruling assumptions.

### *4.3.5 The full process applied to a single time-series*


The program of tests thus sharpens the error-statistical reasoning component of the TM/SI framework.

#### *4.3.6 Deceptive features detectable with unit root tests*

The application of deterministic methods such as OLS to non-deterministic data progression such as a random walk is a misspecification; the results may be deceptive with meaningless shifts and/or trends. Unit root (UR) tests probe the data for features that can superficially imitate deterministic structural changes by cumulative random walks, a red or near red progression. It has been shown by Monte Carlo methods that a test for deterministic trends will find deterministic trends in about 85% of realizations that contain only a stochastic (UR) trend [45]. However combinations of UR tests may also be used to detect both stochastic and deterministic non-stationary sequences, due their varied ruling assumptions and constructions.

Because multiple UR tests are performed, and because they each have differing ruling assumptions, the tests are interpreted in terms of evidence for and against stationarity in the underlying processes. In the econometric literature an exogenous change is one imposed upon a model from outside the model. We elected to retain this word where concepts were derived from economic papers as meaning an abrupt and deterministic change in a deterministic time-series.

## *4.3.7 Unit roots, non-stationarity, and climate*

Transient unit root behaviour, if it occurred, could indicate some sort of regime change, temporarily decoupled from normal forcings. If, in addition, measured noise was not persistent this would show I(0) behaviour; or, if it were fully persistent, as I(1) behaviour. In regional signals in which this occurs, the region may also have become coupled to other sub-systems [46]. This could indicate that the underlying physical model is incomplete and that a missing variable misspecification has resulted. On the other hand, persistent unit root behaviour means that a deterministic change-point analysis is suspect.

The Earth system is constrained so that the overall temperature cannot solely follow a pure random walk – at worst it would follow a Brownian bridge (i.e. sequences where the end-points are meaningful and accepted as deterministic but the path is apparently a random walk [47]). However the composition of summary deterministic signals, such as the GMST, involves manipulations that can produce data that existing unit root tests will identify as containing unit roots, and furthermore deceive deterministic tests in much the same way as random walk data. This issue was extensively examined in R2019 and is addressed later.

#### *4.3.8 Detecting unit root presence*

Random walk progression may be present in climate data because of transient physical conditions, or because the data is unrelated to the physical processes assumed (M-S due to irrelevant variables). Additionally there may be features in the data that do not correspond to any of a shift, a trend change, or unit root behaviour (M-S due to missing variables), and UR tests are potentially sensitive to this. This source of deception must also be dealt with. Other features may be present in the data but not detected. For instance, a step-like shift well above a detectability threshold may be present together with a number of small, deterministic shifts below detectability, and this latter may be taken to be evidence of stochastic drift by a UR test.

The unit root based tests used here all inherit in one form or another the Dickey Fuller (DF) model [48].

$$Y\_t = \mu + \beta t + \rho Y\_{t-1} + \mathbf{e}\_t \tag{6}$$

*ρ* represents the portion of the signal (*Yt*�1) carried forward by autocorrelation, *β* represents the (deterministic) linear trend, *μ* represents the intercept, and *et* is the i.i.d. error with zero mean and a constant variance *<sup>σ</sup>*2. If *<sup>ρ</sup>* <sup>¼</sup> 0 this describes a deterministic trend with no autocorrelation, if 0>*ρ* <1 there is a deterministic trend with a degree of autocorrelation, and if *ρ* ¼ 1, regardless of other parameters it contains a unit root. If all other parameters are zero and *ρ* equals one, then there is no deterministic trend, no offset, and *Yi* is a random walk. This formulation is modified and sometimes rearranged in different ways by the three UR tests used here.

It is important to note that time-series of successive differences of a step-change in an otherwise stationary time series will contain only one out of range difference. Hence the DF model is intrinsically insensitive to deterministic step changes. Another important property of a unit root process is that the variance of the process increases over time, whereas the variance of a stationary process is constant. This gives a second strategy for determining unit root like behaviour – testing for diverging variance. The Kwiatkowski-Phillips-Schmidt-Shin test (KPSS), [49] examines the properties of the variance rather than the fitted parameters, and it primarily focussed on determination of stationarity. As a result it is more sensitive to exogenous changes.

Eq. (6) is expanded to allow for multiple lags in the case of the Augmented Dickey Fuller (ADF) test, taking advantage of the recursive nature of the formula.

A unit root exists if *ρ*<sup>1</sup> ¼ 1. The number of lags can be specified by the user or, as

The ADF test implementation used is programmed in R, available in the package 'urca' [57], and estimate and removes auto-correlation then applies a DF test. The code allows for three variants, are available, (a) a unit root, (b) a unit root with drift, and (c) a unit root with drift and a deterministic time trend – which corresponds to the model of Eq. (7) (above) and which we use. We select suitable autocorrelation lags on the basis of an information criterion, using the call "ur.df (ys, type = "trend", lags = 7, selectlags = "AIC")" following Hacker [58]. The resulting possible reduction in power in the test (inability to distinguish unit root from near unit root) is compensated by other tests in the suite. The test assumes no exogenous change, and *H0* may be accepted in the presence of one

There are two variant of the KPSS test used here to test for level and trend stationarity. These tests invert the sense of the testing with respect to the ADF test, rejecting an *H0* of stationarity in favour of *H1*, a presumption of a unit root. In this case a regime shift may well appear as *H1*, with a step change being non level stationary and a trend change being non trend stationary. We use the R package 'tseries' [60] and invoke the two tests as kpss.test(ys), to test for level stationarity (henceforth KPSS-L) and kpss.test(ys,null = "Trend") to test for trend stationarity

KPSS tests are designed to give weight to stationarity. Assuming that the timeseries can be decomposed into the sum of a deterministic trend, a random walk and a stationary error, the model of Eq. (6) is re-parameterised as follows with *rt*

*Yt* ¼ *rt* þ *βt* þ *u*1*<sup>t</sup>*

*rt* ¼ *rt*�<sup>1</sup> þ *u*2*<sup>t</sup>*

Where *u*1*<sup>t</sup>* is a stationary process, and *u*2*<sup>t</sup>* is an i.i.d. process with zero mean and a

If *<sup>σ</sup>*<sup>2</sup> <sup>¼</sup> 0 then *rt* is constant and the stationary process*u*1*<sup>t</sup>* dominates. If not, then a unit root enters via *u*2*<sup>t</sup>* and *rt* is a random walk. Under a random walk, variance increases with time. Therefore this expectation is tested by estimating the variance

series (f g *e*1*::en* ) is given by residuals of an OLS linear regression (f g *e*1*::en* ).). To test for level stationarity the residual series is replaced by *et* ¼ *yt* � *y*. Then for both

<sup>2</sup> . To test for trend stationarity, a residual

*<sup>i</sup>*¼<sup>1</sup>*ei* and for *<sup>T</sup>* samples, the test

*k*

*j*¼2

*<sup>j</sup>*¼<sup>2</sup>*<sup>ρ</sup> <sup>j</sup>*

*<sup>ρ</sup> <sup>j</sup>*Δ*yt*�*j*þ<sup>1</sup> <sup>þ</sup> *et* (7)

<sup>Δ</sup>*yt*�*j*þ<sup>1</sup>.

(8)

This is more explicit below where *k* multiple lags are included as P*<sup>k</sup>*

*Severe Testing and Characterization of Change Points in Climate Time Series*

<sup>Δ</sup>*Yi* <sup>¼</sup> *<sup>b</sup>*<sup>0</sup> <sup>þ</sup> *<sup>b</sup>*1*<sup>t</sup>* <sup>þ</sup> ð Þ *<sup>ρ</sup>*<sup>1</sup> � <sup>1</sup> *Yi*�<sup>1</sup> <sup>þ</sup><sup>X</sup>

The difference series is then computed,

*DOI: http://dx.doi.org/10.5772/intechopen.98364*

([59], page 76).

(henceforth KPSS-T).

variance *σ*2.

statistic is given as

**221**

representing the random walk

using the Newey-West estimator [61] *s*

cases, partial sums of residuals are defined as *St* <sup>¼</sup> <sup>P</sup>*<sup>i</sup>*

**5.2 KPSS**

here, selected by using an information criterion.

## **5. Proposed tests and strategies**

The unit root methods used are all coded in R and are, (a) a development of the DF test, the Augmented Dickey-Fuller test (ADF), which takes *H0* of a I(1) unit root against an alternative *H1* of a presumption of no unit root (in this implementation trend and multiply lagged autocorrelation is allowed for), (b) two variants of the KPSS, which takes a *H0* of stationarity (or trend-stationarity) rejecting it in favour of an alternative *H1* of a presumption of unit root, and (c) the Zivot-Andrews test (ZA) [50], which takes a *H0* of I(1) unit root behaviour with a possible endogenous drift against an alternative *H1* of trend-stationarity with exogenous structural change. A trend change or a step change would constitute an exogenous structural change.

Use of a combination of UR tests is not new. The combination of ADF and of KPSS testing has been used before in order to add precision to an analysis of monthly inflation expectations (e.g. [51] Appendix B).

The tests are being applied to data within which a single presumptive deterministic, exogenous, step-like changes was detected. No such change is allowed for in the KPSS and ADF tests, the presumption of unit-root in *H0* or *H1* of the above tests is reinterpreted as evidence of non-stationarity. Evidence of unit-root like behaviour is then sought by examination of the residuals after the removal of the deterministic internal trends and shifts detected in the data.

In general, where evidence of a unit-root is detected, it may be due to undetected deterministic features, and hence will be initially treated as evidence of either deterministic non-stationarity or stochastic non-stationarity.

For all of the above tests, the R implementations take published critical values of the test statistic at the 0.01, 0.05, and 0.1 levels. The KPSS implementation interpolates the test statistic against these values to give probabilities between 0.01 and 0.1, the ADF and ZA implementations simply give the critical values and the test statistic.

None of the tests proposed consider unit root presence or absence when possible structural breaks (such as shifts or trend changes) exist under both the null and alternate hypotheses. The problem is under active consideration [52–54].

#### **5.1 ADF**

The ADF test is a variation of the Dickey Fuller test for trend stationarity in the possible presence of unit root. It has a null hypothesis of unit root against an alternative of stationarity after compensation for auto-correlation [48, 55]. The ADF test has relatively low power, and in this type of application a finding of a UR may be because of a single deterministic permanent shift or trend-change [56], as noted above.

*Severe Testing and Characterization of Change Points in Climate Time Series DOI: http://dx.doi.org/10.5772/intechopen.98364*

Eq. (6) is expanded to allow for multiple lags in the case of the Augmented Dickey Fuller (ADF) test, taking advantage of the recursive nature of the formula. This is more explicit below where *k* multiple lags are included as P*<sup>k</sup> <sup>j</sup>*¼<sup>2</sup>*<sup>ρ</sup> <sup>j</sup>* <sup>Δ</sup>*yt*�*j*þ<sup>1</sup>. The difference series is then computed,

$$
\Delta Y\_i = b\_0 + b\_1 t + (\rho\_1 - 1)Y\_{i-1} + \sum\_{j=2}^k \rho\_j \Delta y\_{t-j+1} + \mathbf{e}\_t \tag{7}
$$

A unit root exists if *ρ*<sup>1</sup> ¼ 1. The number of lags can be specified by the user or, as here, selected by using an information criterion.

The ADF test implementation used is programmed in R, available in the package 'urca' [57], and estimate and removes auto-correlation then applies a DF test. The code allows for three variants, are available, (a) a unit root, (b) a unit root with drift, and (c) a unit root with drift and a deterministic time trend – which corresponds to the model of Eq. (7) (above) and which we use. We select suitable autocorrelation lags on the basis of an information criterion, using the call "ur.df (ys, type = "trend", lags = 7, selectlags = "AIC")" following Hacker [58]. The resulting possible reduction in power in the test (inability to distinguish unit root from near unit root) is compensated by other tests in the suite. The test assumes no exogenous change, and *H0* may be accepted in the presence of one ([59], page 76).

#### **5.2 KPSS**

There are two variant of the KPSS test used here to test for level and trend stationarity. These tests invert the sense of the testing with respect to the ADF test, rejecting an *H0* of stationarity in favour of *H1*, a presumption of a unit root. In this case a regime shift may well appear as *H1*, with a step change being non level stationary and a trend change being non trend stationary. We use the R package 'tseries' [60] and invoke the two tests as kpss.test(ys), to test for level stationarity (henceforth KPSS-L) and kpss.test(ys,null = "Trend") to test for trend stationarity (henceforth KPSS-T).

KPSS tests are designed to give weight to stationarity. Assuming that the timeseries can be decomposed into the sum of a deterministic trend, a random walk and a stationary error, the model of Eq. (6) is re-parameterised as follows with *rt* representing the random walk

$$\begin{aligned} Y\_t &= r\_t + \beta t + u\_{1t} \\\\ r\_t &= r\_{t-1} + u\_{2t} \end{aligned} \tag{8}$$

Where *u*1*<sup>t</sup>* is a stationary process, and *u*2*<sup>t</sup>* is an i.i.d. process with zero mean and a variance *σ*2.

If *<sup>σ</sup>*<sup>2</sup> <sup>¼</sup> 0 then *rt* is constant and the stationary process*u*1*<sup>t</sup>* dominates. If not, then a unit root enters via *u*2*<sup>t</sup>* and *rt* is a random walk. Under a random walk, variance increases with time. Therefore this expectation is tested by estimating the variance using the Newey-West estimator [61] *s* <sup>2</sup> . To test for trend stationarity, a residual series (f g *e*1*::en* ) is given by residuals of an OLS linear regression (f g *e*1*::en* ).). To test for level stationarity the residual series is replaced by *et* ¼ *yt* � *y*. Then for both cases, partial sums of residuals are defined as *St* <sup>¼</sup> <sup>P</sup>*<sup>i</sup> <sup>i</sup>*¼<sup>1</sup>*ei* and for *<sup>T</sup>* samples, the test statistic is given as

$$LM = \frac{\sum\_{i=1}^{T} \mathbb{S}\_i^2}{s^2 T^2} \tag{9}$$

**5.4 Empirical quantification of false determination rates**

*Severe Testing and Characterization of Change Points in Climate Time Series*

both UR and non-UR separately, for each test.

*DOI: http://dx.doi.org/10.5772/intechopen.98364*

method hold for the segment of data and to what extent?

tional deterministic change-points below detection thresholds.

discontinuous trend stationary data fitted appropriately.

the residuals supports the existence of a change-point.

Monte Carlo method.

**5.5 Applying these UR tests**

low *p*-values by ANCOVA.

*5.5.1 Level stationarity*

*5.5.2 Trend stationarity*

showing non-stationarity.

**223**

All of these tests are posed as null hypothesis tests. As such they only reject the null hypothesis at a particular level once sufficient evidence is found against it, and when the data size is limited, the power (the probability of correctly rejecting the null hypothesis) is similarly reduced. Therefore, in R2019 (page 100) the four tests were each tested separately for their false positive and false negative rates using a

This aids interpretation since data segments vary in length. Before proceeding further, one may ask how meaningful the nominal *p-*values are, or as in R2019, one can determine the minimum data length required to allow acceptance of a finding of

Assuming an objective change-point method has been used bounded between two objectively determined change-points. Do the assumptions of the detection

These tests are all applied to the segments of data within which a single changepoint has already been provisionally identified. The change-point itself is not otherwise considered. However, since the climate data being tested provisionally contains a deterministic change and only the ZA test is formulated with this as a ruling assumption, findings of non-stationarity may be caused by the presence of addi-

Level stationarity is not simply a zero trend, since data with zero trend may be either deterministically or stochastically level, and even if deterministic may not be linear. A deterministic change-point detection method may return indeterminate change-points given non-linear trend. The residuals around stochastic trend will retain a UR characteristic. Trend stationary data has level stationary residuals, as do

A segment of data with a valid change point should not be found to be level stationary, it should not be in a segment with unit root behaviour, and if it shows trending behaviour this should not be due to a drifting unit root. It should also have

The KPSS-L test is used here with an expectation that segments of climate data in which a change-point occurs contain a step-like shift but may also contain a change of trend. Hence it is used as a cross check. Further, once the deterministic internal shift and trend components are removed the residual should be both level and trend stationary. Level non-stationarity in the segment and level stationarity in

Data with a provisional change of trend is expected to be non-trend-stationary.

Data with constant trend and a step-like change may show as trend stationary depending on the assumptions of the specific test. The KPSS-T test and the ADF test as formulated here may return different results in the presence of a step-like shift and no trend change, with the ADF test showing trend stationarity and the KPSS

Both the ADF test and the ZA test below, perform by estimating an autoregression parameter by OLS, whereas the KPSS tests examine the properties of the variance of the time series (KPSS-L) or of the difference series (KPSS-T).

#### **5.3 Zivot-Andrews test**

The previous tests are confounded by deterministic/exogenous change (steps or shifts), and additionally a combination non-deterministic and deterministic change must be detected.

The Zivot-Andrews test (ZA) [50] tests for the presence of a unit root (with a possible deterministic/exogenous change) against an alternative of stationarity with at most one exogenous change. An advantage is that the test also returns a time of a possible exogenous change [62] – but note that an exogenous change can be any of step, transient or trend change.

The code is in the R package "urca", called as "ur.za(ys, model = "both")", which allows for changes in trend or steps. *H0* is UR without exogenous change. *H1* is trend-stationary with a possible exogenous change at an unknown time.

The ruling assumptions are (a) that there is at most one exogenous structural change (b) in a multivariable model, that only one exhibits unit root. In either of these cases other tests are preferred [54]. Here, we are testing a single variable with intervals bounded by breaks within which we have already detected exactly one break, whilst others may be below a detectability threshold. It has been previously shown that rejection of the null of a unit root could be due to a structural break even in the presence of unit root [63], whilst the presence of more than one break in the absence of a unit root may lead to the acceptance of the *H0* of UR [64].

Acceptance of *H0* does not imply merely UR, but rather, UR without exactly one deterministic break, [56], and thus *H1* means not UR or not a single break. Given we know there is a break (detected by MSBV, confirmed by ANCOVA), *H1* is reinterpreted as not UR, or more than one break.

The model used here is that documented by Zivot and Andrews (50) as Model (C). The model follows the ADF approach and its equation contains more complex parameters for: intercept and change of intercept (a step-like change), *<sup>μ</sup>*^ <sup>þ</sup> *<sup>θ</sup>DUt* ^*<sup>λ</sup>* � �; and trend and change of trend, *<sup>β</sup>*^*<sup>t</sup>* <sup>þ</sup> ^*γDT*<sup>∗</sup> *<sup>t</sup>* ^*<sup>λ</sup>* � �. The remaining parameters are similar to the ADF; autocorrelation with lags, *<sup>α</sup>*^*yt*�<sup>1</sup> <sup>þ</sup> <sup>P</sup>*<sup>k</sup> <sup>j</sup>*¼<sup>1</sup>^*cj*Δ*yt*�*<sup>j</sup>* and the presumed i.i.d. error …

$$\mathcal{Y}\_t = \hat{\mu} + \theta \mathcal{D} U\_t(\hat{\lambda}) + \hat{\beta}t + \hat{\gamma} \mathcal{D} T\_t^\* \left(\hat{\lambda}\right) + \hat{\alpha} \mathcal{Y}\_{t-1} + \sum\_{j=1}^k \hat{c}\_j \Delta \mathcal{Y}\_{t-j} + \mathcal{e}\_t \tag{10}$$

Circumflexes above represent estimates of parameters. ^*λ* is a value that is minimised during the search for the most likely time of a break, *DUt* ^*λ* � � <sup>¼</sup> 1 if *<sup>t</sup>*<sup>&</sup>gt; *<sup>T</sup>λ*, the time of change, 0 otherwise, and *DT*<sup>∗</sup> *<sup>t</sup>* ^*<sup>λ</sup>* � � <sup>¼</sup> *<sup>t</sup>* � *<sup>T</sup><sup>λ</sup>* if *<sup>t</sup>*>*Tλ*, 0 otherwise. Parameters estimated include the time of change and each of the parameters of the above model. ^*<sup>λ</sup>* is estimated so as to minimise the one side t-statistic for *<sup>α</sup>* <sup>¼</sup> 1, which in turn leads to rejection of the null. One should note that in the absence of any deterministic change-point the test functioned as a stationarity test when empirically assessed.
