Bounds, Breaks and Unit Root Tests

The paper addresses the unit root testing when the range of the time series is limited and considering the presence of multiple structural breaks. The structural breaks can affect the level and/or the boundaries of the time series. The paper proposes five unit root test statistics, whose limiting distribution is shown to depend on the number and position of the structural breaks. The performance of the statistics is investigated by means of Monte Carlo simulations.


INTRODUCTION
The application of standard unit root and stationarity tests assumes that time series are not limited or bounded, that is, its range of variation is not constrained. However, there are some situations where time series are bounded below, above or both, so their range of variation is restricted and defines limited stochastic processes. Composition variables such as percentage of expenditure on different concepts, unemployment rates, participation in labour market ratios, income inequality measures, capacity utilization ratios and number of hours worked are some examples of time series that are limited either by construction or by definition. There are also variables, such as nominal interest rates, that cannot take negative values, so they are bounded below. Finally, variables such as the exchange rates of the currencies within the European monetary system have been controlled and moved inside bands of fluctuation. Cavaliere (2005) and Cavaliere and Xu (2014) show that the limit distributions of the Dickey-Fuller (DF) and the modified unit root tests of Ng and Perron (2001) -henceforth, M tests -shift to the left when a bounded time series takes values close to the boundaries, which causes severe over-rejection distortions of the null hypothesis of unit root for the standard unit root tests. Intuitively, bounded I(1) processes -hereafter, BI(1) -might be characterized as stationary I(0) processes because of the bounds, since the bounds prevent the processes from drifting away from the expected value. Consequently, standard unit root and stationarity tests might lead to conclude that a given process is mean-reverting, although mean reversion is due to the bounds, not to the I(0)-ness of the process.
The availability of long time series is a desirable feature in order to ensure the application of unit root tests with good properties. However, the longer the time, the higher the probability of having structural breaks affecting the behaviour of the time series. Perron (1989Perron ( , 1990) drew attention to the bias shown by the DF statistic towards the non-rejection of the null hypothesis of unit root when structural breaks are present but not accounted for. Practitioners working with limited time series might be facing two misspecification errors that bias the unit root test statistics in opposite directions.
t D 1; : : : ; T , where x t 2 h b t ; b t i almost surely for all t; DU i;t D 1 for t > T 0 i and 0 otherwise, for i D 1; : : : ; m structural breaks -the superscript 0 in T 0 i is used to denote the true break date -and initialized at y 0 D O p .1/. Further, define 0 i D T 0 i =T as the break fraction parameters, i D 1; : : : ; m, which are collected in the vector 0 D 0 1 ; : : : ; 0 m 0 . The presence of bounds requires that x t lies within the interval h b t x t 1 ; b t x t 1 i , where h b t ; b t i denote the boundaries that affect the time series. These boundaries are assumed to be defined by deterministic terms. Cavaliere (2005) focuses on the analysis of non-trending variables defining In this article, we restrict the analysis to non-trending variables and the changing boundaries can be specified where " t is a weakly dependent zero-mean process with infinite support and t and t are non-negative processes (regulators) such that t > 0 if and only if y t 1 C " t < b t d t and t > 0 if and only if y t 1 C " t > b t d t .
The stochastic processes involved in (4) satisfy Assumption 1. " t D C.L/ t , where C.L/ D P 1 iD0 C i L i with P 1 iD0 i s jC i j < 1 for some s 1 and t is a martingale difference sequence adapted to the filtration F t D field¹ t i I i 0º. The long- two consecutive regimes is given by c i c i 1 D 1 T 1=2 Â i D 1 Ä i . Note that the same applies for the lower bound, that is, c i c i 1 D 1 Ä i . Case D: the boundaries change, and the magnitude of the structural breaks is non-negligible, Â i D Ä i T 1=2 ; i D 1; 2; : : : ; m . In this case, the bounds are defined as in (5), and the change of c i between two consecutive regimes is given by Some remarks are in order. First, in Case A, the limits that affect the time series do not change because Â i is fixed 8i , so that the change in the bounds is c i c i 1 D c i c i 1 D o.1/. This does not happen for the other cases. Second, we have shown different ways in which the bounds that affect the time series can change, that is, Cases B to D, although these specifications lead to the same result as for the unit root testing procedure. However, there is a qualitative difference between the changes driven by the boundaries and those caused by the structural breaks with non-negligible effects. On the one hand, the boundaries are usually known a priori so we do not need to estimate them. Furthermore, in most cases, the boundaries remain constant by the definition of the time series that is analysed; for instance, the boundaries for the unemployment rate are [0, 100]. On the other hand, the presence of structural breaks with non-negligible effects can relax the assumption of known break points since, in this case, it is possible to obtain a consistent estimate of the break fractions -further details are given in the succeeding text. When the effect of the structural breaks is negligible, we need to return to the known break points assumption, given that it is not possible to obtain a consistent estimation of the break fractions. Finally, it is worth noticing that hybrid specifications can be defined where, in some of the regimes, the change in the limits is driven by changes in the boundaries, whereas in other regimes, the change in the limits is caused by the presence of structural breaks with non-negligible effects.

UNIT ROOT TEST STATISTICS FOR BROKEN BOUNDED PROCESSES
We present in this section five unit root test statistics that can be applied to test for the presence of a unit root when working with bounded integrated processes with multiple structural breaks. First, we have the three M-type test statistics by Ng and Perron (2001), that is, the modified Sargan-Bhargava MSB. 0 / D x t denotes the OLS detrended variable and O 2 is a consistent estimate of the LRV. The estimation of the LRV 2 can be obtained using either a non-parametric estimator s 2 NP , as in Phillips and Perron (1988), or a parametric estimator s 2 AR , as in Ng and Perron (2001). The non-parametric estimator is given by where D T 0 i t D 1 for t D T 0 i C 1 and 0 otherwise, and k.j; M T / is some kernel that weights the sample autocovariances and M T is a bandwidth that acts as a truncation lag parameter when k.j; M T / D 0 for jj j > M T . The parametric estimator is based on the augmented Dickey-Fuller (ADF) regression equation: so that s 2 tk are obtained from the OLS estimation of (7) with the lag length selected as suggested by Perron and Qu (2007). 1 The fourth unit root test that can be computed is the pseudo t-ratio ADF statistic ADF. 0 / to test the null hypothesis thatˇ0 D 0 in (7). Finally, we also consider the variance ratio VR. 0 / statistic in Breitung (2002) The limit distribution of the unit root test statistics is given in Theorem 1.
Theorem 1. Let ¹x t º T tD1 be a stochastic process with its DGP given in (1)-(3). Further, let T 0 i ! 1 and T ! 1 in such a way that T 0 i =T ! 0 i ; 0 < 0 i < 1, where 0 i denotes the break fraction parameter, i D 1; 2; : : : ; m. Following Chang and Park (2002), let k in (7) be chosen in a way that 1=k C k 2 =T ! 0 as T ! 1. Under the null hypothesis that˛D 1 in (3), the MSB. 0 /; MZ˛. 0 /; MZ t . 0 /; ADF. 0 / and VR. 0 / statistics converge, for Cases B-D to i D 1; 2; : : : ; m C 1, are independent demeaned regulated Brownian motions and ) denotes weak convergence of the associated measure of probability, with the convention that 0 0 D 0 and 0 mC1 D 1. The limiting distribution for the MZ t . 0 / test can be obtained from the fact that MZ t . 0 / D MZ˛. 0 / MSB. 0 /, a limiting distribution that is equivalent to that of the ADF. 0 / test statistic.
The proof is given in the appendix. The critical values have been obtained through Monte Carlo simulations for the pair of values c i ; c i given by c i D c i D ¹0:3; 0:5; 0:7; 0:9; 1:1; 1:5º with all possible combinations of break fraction parameters D ¹0:1; 0:2; : : : ; 0:9º taken in groups of up to m D 4 elements, that is, we allow for up to m D 4 structural breaks. We have used 300 steps to approximate the Brownian motion functionals and 10,000 replications. 2; 3 It is worth mentioning that the effect of c i ; c i on the asymptotic critical values depends on how narrow the limits are set. As expected, the asymptotic critical values do not change significantly when the range of variation is wide, although this is not the case for narrower ranges. It would be also possible to compute (case specific) critical values using bootstrap methods as in Cavaliere and Xu (2014), which is expected to improve finite sample performance of the statistics. Some remarks are in order. First, the limit distribution of the statistics depends both on the number .m/ and position . 0 / of the structural breaks, although this dependence is not only due to the effect that the level shifts has on the deterministic component of the time series, that is, the specification given by Case C, but also on the bounds that limit the Brownian motion, that is, the specification given by Case B. Second, these limiting distributions admit, as a particular case, the situation where the time series is unbounded, that is, c i D c i D 18i; i D 1; 2; : : : ; m C 1, so the regulated Brownian motion becomes a standard Brownian motion. Third, note that the limiting distribution of the MSB. 0 / statistic has been expressed as a weighted sum of independent functionals of demeaned regulated Brownian motions. Fourth, although we have considered that the bounds are symmetric, it would be possible that the time series is only limited below or above. In this case, the results that have been obtained are still valid, if the unconstrained limit is replaced by either c i D 1 (bounded above) or c i D 1 (bounded below), depending on the case. Finally, the limit distribution of the test statistics when the structural breaks do not change the limits (Case A) is given in Corollary 1.
tD1 be a stochastic process with its DGP given in (1) where 0 i denotes the break fraction parameter, i D 1; 2; : : : ; m. Following Chang and Park (2002), let k in (7) be chosen in a such way that 1=k C k 2 =T ! 0 as T ! 1. Under the null hypothesis that˛D 1 in (3) The proof follows from the one for Theorem 1 specifying c i ; c i D OEc; c 8i; i D 1; 2; : : : ; m C 1. Note that, although the presence of structural breaks does not affect the limits of the process, they affect the limiting distribution of the unit root tests, so test statistics for bounded time series without structural breaks cannot be used in this case. Finally, the limiting distribution of the MZ˛. 0 /; MZ t . 0 / and ADF. 0 / test statistics with c i D c i D 18i coincides with the ones obtained by Perron (1990), for the one break case, and, implicitly, by Clemente et al. (1998), for the two breaks case.

UNKNOWN BREAKS CASE
So far, we have assumed that both the number and the position of the structural breaks are known. However, it is important to design a procedure to estimate the break points when they are unknown. As mentioned earlier, consistent estimation of the break fractions is only possible when the structural breaks have non-negligible effects on the time series, structural breaks that are modelled as in (6). The approach that we propose in this article is based on the minimization of the sum of squared residuals under the null hypothesis of unit root .S.1; //, which implies the OLS estimation of the model:  The proof is given in the appendix. In order to estimate m, we could think of defining a maximum number of structural breaks .m max / and select m by minimizing the Bayesian information criterion (BIC) as proposed by Yao (1988) or the modified BIC information criterion by Liu, Wu and Zidek (1997) -henceforth, LWZ information criterion. 4 The following theorem establishes the properties of using these information criteria to estimate m.
Theorem 3. Let ¹x t º T tD1 be a stochastic process with its DGP given in (1)-(3), with˛D 1; m > 0 and Â i D Ä i T 1=2 ¤ 0; i D 1; 2; : : : ; m, being m max the maximum number of structural breaks that are allowed. Let us denote by O m BIC and O m LWZ the estimated number of structural breaks obtained by minimizing the BIC and LWZ information criteria respectively. Thus, as Further, under the assumption that e t i:i:d: The proof is given in the appendix. 5 Theorem 3 states that the use of the BIC and LWZ information criteria might lead, in general, to an overestimation of m. However, there might be situations in which the LWZ information criteria leads to a consistent estimation of m, which will depend on the extreme value behaviour of e t in (8). Thus, under the restrictive assumption that e t i:i:d: N.0; 1/ in (8), it can be shown that the LWZ information criterion provides a consistent estimation of m but that the BIC information criterion does not -see the appendix for further details. With this potential risk, it should be borne in mind that an overestimation of m might imply a power loss of the unit root test statistics. In order to overcome this drawback, we also explore the use of the approach by Burridge and Taylor (2006) -hereafter, BT -who design a procedure for additive outlier detection using extreme value theory. Burridge and Taylor (2006) show that their procedure works reasonably well for different probability distributions of the disturbance term, preventing us from making assumptions on the probability distribution of e t in (8). 6 The performance of the information criteria and the algorithm proposed by Burridge and Taylor (2006) in finite samples are investigated in the Monte Carlo section in the succeeding text.
Once consistent estimates of the break fractions are obtained, the bounds can be computed as in Cavaliere and Xu (2014) In order to compute these bounds, we propose using the deterministic component estimated under the null hypothesis that˛D 1, which defines ı i denotes the OLS parameter estimates of (8) and O indicates a consistent estimate of the LRV -either s 2 NP or s 2 AR . 7

FINITE SAMPLE PERFORMANCE
We investigate in this section the performance of the statistics that have been proposed in the article. The DGP is given in (1) to (4), In principle, it would be possible to design a testing procedure to estimate the number of structural breaks following the approach in Harvey et al. (2010). However, this testing procedure will depend on the true number of structural breaks and the values of the bounds, which makes the approach very cumbersome from an implementation point of view. The use of information criteria avoids these complications and, as will be shown by the simulation experiments below, produce satisfactory results in terms of empirical size and power of the unit root test statistics. 5 We are indebted to an anonymous reviewer for providing insightful help in the proof of this theorem. 6 We have computed the weights that are used in the BT approach taking into account the presence of bounds, although the differences are negligible. Therefore and provided that no gains are obtained from the use of bound-specific weights, we suggest using the weights in Burridge and Taylor (2006). 7 Note that in practice, we can compute either a parametric or a non-parametric estimation of the bounds, depending on whether we use a parametric or a non-parametric estimation of the LRV. In what follows, we will make abuse of language and talk about the parametric and non-parametric versions of the ADF and VR unit root test statistics, where this distinction will come from the different estimates that can be used to estimate the LRV, which defines the estimate of the bounds. and T 0 2 D T . Note that this set-up corresponds to Case C earlier. The Monte Carlo experiment considers the presence of one unknown structural break where the true break point is located at 0 1 D 0:5. 8 The bounds for the first regime are c 1 ; c 1 D OE 0:3; 0:3; whereas for the second regime, they are defined as c 2 ; c 2 D c 1 Ä; c 1 C Ä with Ä D ¹0:2; 0:4; 0:6; 0:8º. The empirical size is investigated by setting˛D 1 in (3), whereas for the empirical power, we specify˛D 0:8 in (3). The selection of the order of the model used in the estimation of the parametric LRV is based on the modified Akaike information criterion proposed by Ng and Perron (2001) with a maximum of 4.T =100/ 1=4 lags. The non-parametric LRV is estimated with the Newey and West (1994) automatic procedure using the quadratic spectral window. The sample sizes are T D ¹100; 200; 300º and 1000 replications are used. In all cases, the 5% level of significance asymptotic critical values are used. Table I presents the performance of the BIC, LWZ and BT statistics when used to estimate m following the procedure described in Section 4. The true number of structural breaks is either m D 0 or m D 1, with the maximum number of m set at m max D 2. As can be seen, the BIC overestimates m in all cases, tending to select m max . The use of the LWZ statistic shows better performance. When m D 0, the selection of the true m improves as T gets large, reaching values around 0.85 for T D 300, regardless of the order of integration of the process. When m D 1, the LWZ tends to underestimate m for Ä D 0:2 but renders good results for the other values of Ä. These results confirm the theoretical analysis earlier. The BT statistic outperforms the BIC and LWZ when m D 0, where the probability of selecting the true m is close to one in all cases. In general, the superiority of BT over the BIC and LWZ statistics is also evident when m D 1 and Ä > 0:4, although the LWZ statistic outperforms the BT when Ä Ä 0:4, in these cases, the BT statistic underestimates m.

Estimation of the Number of Structural Breaks
In summary, the BIC is liberal in the selection of m, whereas the BT statistic outperforms the other statistics when m D 0 but underestimates m when Ä is small. The behaviour of the LWZ statistic lies between these extremes. Thus, LWZ presents mild overestimation problems of m when m D 0, but its performance is similar to that of BT when m D 1 with large T and Ä. Moreover, LWZ outperforms the BT statistic when m D 1 with small Ä.

Empirical Size and Power of the Unit Root Tests
We investigate the performance of the MSB. O / and ADF. O / statistics when the BIC, LWZ and BT statistics are used to estimate m, with m max D 2 -note that we are focusing on the case where both the number and position of the structural breaks are unknown. The results for the other unit root test statistics, which are available in the companion appendix, are similar. Let us first focus on the empirical size. Table II shows that, in general, the MSB. O / statistic presents under-rejection problems, which are more noticeable when m is estimated using the BT approach. Under-rejection is less pronounced when the BIC is used and when the LRV is non-parametrically estimated. The ADF. O / statistic also shows mild size distortions (under-rejection) although, in most cases, the empirical size is close to the nominal one. Note that the performance of the ADF. O / statistic does not depend on the estimator that is used to estimate the LRV.
The estimation of m reduces the empirical power of the unit root test statistics when compared with the (unreported) results for known m. As expected, the empirical power of the statistics increases with T and Ä, so that, for large values of T and Ä, the empirical power is the same as if m is known. If we compare the results based on the BIC, LWZ and BT statistics, we can conclude that, for small values of Ä, the BIC-based unit root test statistics are the most powerful. This is a direct consequence of the overestimation of m that is observed for the BIC -it has Table I been shown that the LWZ and BT statistics tend to underestimate m when Ä is small. As Ä increases, the LWZbased unit root test statistics tend to improve their empirical power, outperforming the BIC-based ones -see, for instance, the results for Ä D 0:4. Finally, when Ä is large, the empirical power of both the LWZ and BT-based unit root test statistics is similar. The method that is used in the estimation of the LRV affects the empirical power of the test statistics. Thus, for a given method of estimating m, we observe that the empirical power of the MSB. O / statistic is higher when using the non-parametric LRV. This is to be expected, given the size distortion (under-rejection) observed for this statistic when the parametric LRV is used. For the ADF. O / statistic, the use of the parametric LRV gives higher empirical power.
The simulations that have been conducted in this section reveal that the overestimation of m leads to unit root tests statistics with both an empirical size that is controlled -in some cases, under-rejection problems are observed -and decent empirical power figures. The underestimation of m can result in important reductions of the empirical power of the unit root test statistics, a situation that can appear for small values of Ä. Therefore, in such cases, it is preferable to overestimate m.

EMPIRICAL ILLUSTRATION
There are several studies in the economic literature that assess the properties of the unemployment rate as a way to test whether it behaves in accordance with the (perfect) hysteresis hypothesis or in accordance with the natural unemployment rate theory. In this section, we use the monthly US unemployment rate from January 1948 through   Note: a; b and c indicate rejection of the null hypothesis at 1%, 5% and 10% respectively. P, parametric; NP, non-parametric.
November 2014 taking into account the existence of bounds and structural breaks when testing the unit root hypothesis (Figure 1). 9 Table III reports the M-type, VR and ADF test statistics. 10 When bounds are ignored, the null hypothesis of unit root is rejected by the M-type and ADF test statistics at different levels of significance, regardless of the LRV estimate that is used. However, if we consider the limited nature of the variable, the evidence against the unit root hypothesis is weaker. In order to assess whether this conclusion is robust to the introduction of structural breaks, our analysis has specified up to two structural breaks, with the LWZ selecting one structural break and the BIC and BT statistics selecting two. The fact that the BT statistic detects more structural breaks than the LWZ, together with its tendency to either under (narrow bands) or correctly (wide bands) estimate m, would suggest specifying m D 2 -as shown in the succeeding text, wide bands are estimated. However, in order to check the robustness of the conclusions, we proceed considering both m D 1 and m D 2. The first structural break is placed in December 1974, which reflects the increase of unemployment as a result of the 1970s oil crises - Perron (1990) fixed the exogenous break in the fourth quarter of 1973. The second structural break is detected in March 1958, capturing the decrease of unemployment from the expansion during the Kennedy-Johnson administrations until the macroeconomic turmoil of the 1970s (McNown and Seip, 2011). 9 We have used the information provided by the U.S. Bureau of Labor Statistics: Current Population Survey, for the seasonally adjusted total unemployment rate. 10 The parametric LRV has been estimated using up to the integer value of h 12 .T =100/ 1=4 i lags.
Let us first focus on analysis for m D 1. When the parametric LRV estimate is used, the M-test statistics leads to the rejection of the null hypothesis of BI(1) at the 1% level of significance, whereas the VR and ADF tests reject it at the 10% and 5% respectively. Evidence against the null hypothesis is reduced for the M-test statistics when the non-parametric LRV estimate is used, although the null hypothesis is still rejected at the 10% level of significance. The converse situation is found for the VR and ADF tests, where evidence against the null hypothesis increases. In general, the picture is not modified when we specify m D 2. Thus, the same conclusion is reached for the M-type tests. The ADF statistic still rejects the null hypothesis at the 5% level of significance, regardless of the LRV estimate that is used. The exception is the VR statistic, where the null hypothesis is not rejected. Finally, Table III also presents the estimated bounds for each specification, which define wide bands. For the parametric LRV-based results, the interval ranks are between 6.4 and 9.5, depending on the model, whereas for the nonparametric LRV-based results, they are between 9.6 and 10. This feature reinforces the specification of m D 2, as argued earlier.
To sum up, evidence against the unit root hypothesis in the postwar US unemployment rate is found when structural breaks are considered, a conclusion that is in accordance with earlier results such as those by Perron (1990). Furthermore, contrary to previous analyses, our results are robust to the presence of theoretical bounds and structural breaks.

CONCLUSIONS
The article extends the concept of bounded integrated processes to cover situations where the time series is affected by the presence of multiple structural breaks. Structural breaks enter into the model in two different ways, either affecting the deterministic component or the boundaries component. It has been shown that the limiting distribution of the proposed unit root tests depends not only on the number and position of the structural breaks but also on the way that the structural breaks affect the (deterministic and boundaries) components of the model. The article has also shown that consistent estimates of the break fractions are possible if the magnitude of the level shifts is non-negligible. This allows us to relax the assumption of known breaks in some situations. The Monte Carlo simulations that have been carried out allow us to compare the different unit root tests that have been designed. Simulation results evidence that the M-type tests show a better overall performance. In some cases, they are conservative, but they show good empirical power.
The proposal is illustrated analysing the US unemployment rate. Taking into account both the presence of multiple structural breaks and the limited nature of this time series lead to robust conclusions of the statistical properties of the unemployment rate, which is characterized as a broken bounded I(0) process.
Our approach has assumed that the boundaries are known, something that is not a limitation if we are dealing with variables that are bounded by definition or by construction. However, it would be of interest to consider our framework in cases where it is suspected that the variable is regulated but the boundaries are not known. In this case, Cavaliere and Xu (2014) suggest that a reasonable range of bounds can be inferred from historical observations or from the economic theory. Although the unknown bounds case is outside the scope of this article, we think that this topic is very interesting from an empirical point of view, so we plan to address it in future research.
Á 2 ds i so that the limit distribution of the MSB. 0 / test can be expressed as a weighted sum of independent functionals of demeaned regulated Brownian motions, i.e., The limit distribution of the MZ t . 0 / D MZ˛. 0 / MSB. 0 / statistic follows from the derivations above. For the ADF . 0 / statistic, note that the ADF regression equation in (7) can be expressed as:  (2002), we can define O t e tk Á , so that Ǒ 0 D B 1 T . /A T . /, with the variance of the error term given by O 2 k D T 1 C T . / A 2 T . /B 1 T . / and the variance of the estimatedˇ0 parameter given by s 2 Using these elements and following Chang and Park (2002), it is straightforward to see that as T ! 1; Note that, as shown in Chang and Park (2002) and Cavaliere and Xu (2014), Ǒ .1/ p !ˇ.1/ and where O 2 is defined here using the parametric version of the LRV estimator s 2 AR -the same result can be established using the non-parametric version of the LRV estimator. Then, the ADF . 0 / statistic is computed as ADF . 0 / D O 1 B 1=2 T . /A T . / , which in the limit converges to which coincides with the limiting distribution of the MZ t . 0 / derived above.

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Finally, the numerator of the VR. 0 / statistic for a generic T i 1 < t Ä T i involves the square term . Therefore, we can see that the last two terms are asymptotically negligible compared to the first two so that, in the limit, the test statistics using the estimated break dates converge to the limiting distribution of the test statistics where the break dates are known.

A.3 Proof of Theorem 3
The S.1; / of the OLS estimated model defined in (8) can be written as S.1; / D x 2 t P m iD1 x 2 T i C1 D x 2 t P m iD1 Â 2 i C O p .1/. The BIC and LWZ information criteria that we propose to use in the paper are given by BIC D log .S.1; /= .T 1 m// C m log .T 1 m/ = .T 1 m/ and LW Z D log .S.1; /= .T 1 m// C 0:299m log .T 1 m/ 2:1 = .T 1 m/. Let us focus on the BIC. As can be seen, the first element of the right hand side is log 1 T 1 m x 2 t P m iD1 Â 2 i C O p .1/ D log 1 T 1 m x 2 t P m iD1 Ä 2 i T C O p .1/ provided that the magnitude of the structural breaks is nonnegligible, i.e., Â i D Ä i T 1=2 . As T ! 1, this element converges to log 2 x P m iD1 Ä 2 i , where 2 x denotes the variance of x t , whereas the second element of the right hand side converges to zero for fixed m. Therefore, accounting for the presence of structural breaks reduces the sum of the squared residuals in the limit, leading to the detection of the structural breaks when using the BIC information criterion -the same applies to the LWZ information criterion.
However, this does not necessarily imply a consistent estimation of the number of structural breaks, i.e., it might be the case that the use of information criteria leads to over-detection of the number of structural breaks. In order to investigate this issue, let us consider the case where there is one structural break, m D 1, and we compute the BIC information criterion for the models that consider one .BIC 1 / and two structural breaks .BIC 2 /. Let us denote by S.1; 1 / and S.1; 2 / the sum of squared residuals of the model (8) for the one and two breaks cases, respectively. It can be seen that BIC 2 BIC 1 ln T 1 S.1; 2 / C 2T 1 ln.T / ln T 1 S.1; 1 / T 1 ln.T / D ln 1 C S.1; 2 / S.1; 1 / S.1; 1 / Á C T 1 ln.T /, where S.1; 2 / S.1; 1 / D x 2 T2C1 , with T 2 denoting the second break date. We need BIC 2 BIC 1 > 0 if the correct number of structural breaks is going to be selected. The order of S.1; 2 / S.1; 1 / is determined by the extreme value of e t in (8), which depends on the underlying assumptions that we make about its distribution. In the simplest case where e t i idN.0; 1/ in (8), max t je t j D O p p ln.T / Á so that BIC 2 BIC 1 O p .T 1 ln.T // C T 1 ln.T /, which is not guaranteed to be positive in the limit. Therefore and even in this simple case, the BIC information criterion might lead to an over-specification of the number of structural breaks, i.e., O m BIC m. This situation is not found when using the