Let $\{{X_{t}}\}$ be a sequence of random variables on a given probability space $(\Omega ,\mathcal{F},\operatorname{P})$. We assume ${X_{t}}$ to be independently distributed according to a probability measure ${\operatorname{P}_{0}}$, which belongs to a parametric family $\operatorname{P}(\theta )$, i.e., ${\operatorname{P}_{0}}\in \{\operatorname{P}(\theta ),\theta \in \Theta \subset {\mathbb{R}^{p}}\}$, with p being the number of parameters. On an interval $\mathcal{I}=[a,b]$ with length ${n_{\mathcal{I}}}$, we aim to test the null hypothesis ${\mathrm{H}_{0}}$ of no change in θ. ${\mathrm{H}_{0}}$ may depend on all or just a subset of the elements in θ. The alternative hypothesis is that there is a break point $\tau \in (a,b)$, leading to distinct parameterizations for ${X_{t}}$ in the two intervals; specifically, one for ${X_{t\in {L_{\tau }}}}={\left\{{X_{t}}\right\}_{t\in {L_{\tau }}}}$ where ${L_{\tau }}=[a,\tau ]$, and another one for ${X_{t\in {R_{\tau }}}}={\left\{{X_{t}}\right\}_{t\in {R_{\tau }}}}$ where ${R_{\tau }}=(\tau ,b]$. Summarizing, we test against the alternative where ${\theta _{{L_{\tau }}}}\ne {\theta _{{R_{\tau }}}}$ and ${L_{\tau }}\cup {R_{\tau }}=\mathcal{I}$.

Following [20], we consider a set of candidate breakpoints $\mathcal{T}(\mathcal{I})\subset \mathcal{I}$ to test ${\mathrm{H}_{0}}$. For each candidate $\tau \in \mathcal{T}(\mathcal{I})$, we conduct a test belonging to the family of generalized likelihood ratio tests (LRT)—see [9]. More specifically, for every fixed τ, we employ the statistic where $L(\theta |{X_{t\in j}})={\textstyle\sum _{t\in j}}{\ell _{t}}(\theta |{X_{t}})$ is a log-likelihood function on the respective interval, for $j=\{{L_{\tau }},{R_{\tau }},\mathcal{I}\}$, and ${\ell _{t}}(\theta |{X_{t}})$ is the log-likelihood contribution of a single observation ${X_{t}}$ at time t. It is supposed to be clear from the context when the conditioning is on the random variable ${X_{t}}$ (e.g., for the computation of a probability) or its realization ${x_{t}}$ (e.g., for the computation of the value of a test statistic).

To test for the presence of a breakpoint jointly among the candidates, we employ the maximum statistic, as in [20], given by For a specified significance level α, we need to determine the critical value ${\mathfrak{z}_{\mathcal{I},\alpha }}$ such that implying that we reject ${\mathrm{H}_{0}}$ whenever ${T_{\mathcal{I}}}\gt {\mathfrak{z}_{\mathcal{I},\alpha }}$. For small samples or non-Gaussian distributions, this may prove to be difficult. Thus, we will compute ${\mathfrak{z}_{\mathcal{I},\alpha }}$ using the multiplier bootstrap.

Following [21] as well as [24], we use the multiplier bootstrap to approximate the unknown and non-asymptotic distribution of the test statistic. The multiplier bootstrap is carried out by re-weighting the log-likelihood function. The bootstrapped log-likelihood function is given by where $\{{u_{t}}\}$ is an $i.i.d$. sample from a sub-Gaussian distribution, independent from ${X_{t}}$, with $\operatorname{E}[{u_{t}}]=1$ and $\operatorname{Var}[{u_{t}}]=1$; here, $j=\{{L_{\tau }},{R_{\tau }},\mathcal{I}\}$ indexes the three samples involved.

From (2.3), we receive the bootstrap estimator As an important property, note that where ${\operatorname{E}^{\mathrm{\flat }}}[{L_{j}^{\mathrm{\flat }}}(\theta |{X_{t\in j}})]$ is the expectation conditional on the sample over the measure for $\{{u_{t}}\}$—the so-called bootstrap world.

As established in [21, p. 2660, Theorem 2.2], $\operatorname{P}\{{L_{j}}({\hat{\theta }_{j}})-{L_{j}}({\theta _{j}})\gt {\mathfrak{z}^{2}}/2\}$ is close in probability to the corresponding random quantity in the bootstrap world across a broad range of $\mathfrak{z}$-values. Thus, the test statistic in the bootstrap world corresponding to (2.1), which is given by where ${\hat{\theta }_{{L_{\tau }}}}$ and ${\hat{\theta }_{{R_{\tau }}}}$ are the maximum likelihood (ML) estimators on each subinterval, has a distribution close to ${T_{\mathcal{I},\tau }}$. Therefore, we use repeated bootstrap sampling to evaluate the critical value, in a data-dependent way.

In the last term of (2.6), an additive bias correction in the form ${\hat{\theta }_{{R_{\tau }}}}-{\hat{\theta }_{{L_{\tau }}}}$ is used to ensure that the simulation is carried out under ${\mathrm{H}_{0}}$ [24]. It adjusts the parameters estimated on the interval ${R_{\tau }}$ to align with those estimated on ${L_{\tau }}$. The following section introduces a multiplicative bias correction, a natural choice when testing for variance.

From this section on, we specialize the discussion to the particular case, where we assume ${X_{t}}\sim \mathcal{N}\left(\mu ,{\sigma ^{2}}\right)$. Besides testing for complete homogeneity, involving the mean and variance simultaneously, we are also interested in proposing a test solely for homogeneity in variance. Notably, even in the case of model misspecification, the bootstrap validity continues to apply and becomes conservative under significant deviations [21]. Therefore, the use of the normal distribution in this context carries the characteristics of a quasi-maximum likelihood approach.

In the first step, we discuss the test for homogeneity in variance. We consider a left sample ${X_{t\in {L_{\tau }}}}$, a right sample ${X_{t\in {R_{\tau }}}}$, and their combination ${X_{t\in \mathcal{I}}}$. We wish to test whether ${X_{t\in \mathcal{I}}}\sim \mathcal{N}\left(\mu ,{\sigma ^{2}}\right)$ against the alternative that ${X_{t\in {L_{\tau }}}}\sim \mathcal{N}\left(\mu ,{\sigma ^{2}}\right)$ and ${X_{t\in {R_{\tau }}}}\sim \mathcal{N}\left(\breve{\mu },{\breve{\sigma }^{2}}\right)$ with ${\sigma ^{2}}\ne {\breve{\sigma }^{2}}$ and arbitrary means.

Under these assumptions, for $j=\{{L_{\tau }},{R_{\tau }},\mathcal{I}\}$ with sample size ${n_{j}}$, the log-likelihood function is given by and the LRT for homogeneity in variance becomes

It is, of course, well-known that where ${\hat{\sigma }_{j}^{2}}={n_{j}^{-1}}{\textstyle\sum _{t\in j}}{\left({x_{t}}-{\bar{x}_{j}}\right)^{2}}$ and ${\bar{x}_{j}}={n_{j}^{-1}}{\textstyle\sum _{t\in j}}{x_{t}}$, for $j=\{{L_{\tau }},{R_{\tau }}\}$. The ML estimator over the entire interval $\mathcal{I}$ is obtained as the pooled variance of both subintervals weighted by the sample sizes, i.e., ${\hat{\sigma }_{\mathcal{I}}^{\circ 2}}={n_{\mathcal{I}}^{-1}}({n_{{L_{\tau }}}}{\hat{\sigma }_{{L_{\tau }}}^{2}}+{n_{{R_{\tau }}}}{\hat{\sigma }_{{R_{\tau }}}^{2}})$.

Thus, the test statistic for homogeneity in variance can be written as In the case of a single break point and under correct model specification, this corresponds to the classical likelihood ratio test, which follows a ${\chi ^{2}}(1)$-distribution due to [28].

Additive bias correction. Following (2.3), the multiplier bootstrap log-likelihood function obtains as Now, ${\hat{\sigma }_{j}^{\ast 2}}={\textstyle\sum _{t\in j}}{w_{t}}{({x_{t}}-{\bar{x}_{j}^{\ast }})^{2}}$ and ${\bar{x}_{j}^{\ast }}={\textstyle\sum _{t\in j}}{w_{t}}{x_{t}}$, for $j=\{{L_{\tau }},{R_{\tau }}\}$, are the ML estimators of the variance and the mean in the bootstrap world, respectively. We define ${w_{t}}={u_{t}}{n_{j}^{-1}}$, where ${\left\{{u_{t}}\right\}_{t\in j}}$ are drawn as discussed in Section 2.2. Note that for the second line in (2.11) to hold, we assume that ${\textstyle\sum _{t\in j}}{u_{t}}={n_{j}}$, because it helps us keep the formulae more transparent. During simulation, this property can be ensured by normalizing ${\left\{{u_{t}}\right\}_{t\in j}}$ in each bootstrap iteration. While this standardization introduces a slight linear dependence within the sampled $\left\{{u_{t}}\right\}$, to our experience, it enables the test to better keep its size because the variance of the bootstrap estimators is reduced.3

The bootstrapped generalized LRT becomes where ${\hat{\sigma }_{a}^{2}}={\hat{\sigma }_{{R_{\tau }}}^{2}}-{\hat{\sigma }_{{L_{\tau }}}^{2}}$ is the additive bias correction introduced in (2.6).

It is straightforward that for $j=\{{L_{\tau }},{R_{\tau }}\}$. Thus, the first two summands in (2.12) are In contrast, the third term in (2.12) is more involved. We can still concentrate out the mean, after which we obtain the following function involving ${\sigma ^{2}}$ only: As we discuss in Appendix A.1, we find the maximizer of ${f_{HVa}^{\mathrm{\flat }}}$ numerically as a root to a cubic polynomial in ${\sigma ^{2}}$. Our simulations show that for a sufficiently large sample size the discriminant of the cubic polynomial is always negative, implying the existence of a single real root. However, the discriminant can sometimes be positive for very small sample sizes. In such instances, we skip the iteration and draw a new set of weights ${u_{t}}$ until the desired number of bootstrap samples is obtained.

Denote the maximizer of ${f_{HVa}^{\mathrm{\flat }}}$ by ${\hat{\sigma }_{HVa}^{\mathrm{\flat }\ast 2}}$. Then, the bootstrapped test statistic is given by

Multiplicative bias correction. We now consider the proposed multiplicative bias correction, which takes the form ${\hat{\sigma }_{m}^{2}}=$ ${\hat{\sigma }_{R\tau }^{2}}/{\hat{\sigma }_{{L_{\tau }}}^{2}}$. Instead of (2.12), we utilize As before, we have $\{{\bar{x}_{j}^{\ast }},{\hat{\sigma }_{j}^{\ast 2}}\}=\arg {\max _{\mu \in \mathbb{R},\hspace{0.1667em}{\sigma ^{2}}\gt 0}}{L_{j}^{\mathrm{\flat }}}\left(\mu ,{\sigma ^{2}}\mid \right.\left.{X_{t\in j}}\right)$, for $j=\{{L_{\tau }},{R_{\tau }}\}$. Thus, the first two summands in (2.16) are as in (2.13). Proceeding as above, we can rewrite the third term in (2.16) as which notably has the unique maximizer Unlike the additive bias correction, which lacks a closed-form solution, this estimator offers a compelling and intuitive interpretation: it is a pooled variance estimator incorporating the bias correction. Thus, the multiplicative bias correction provides significant computational advantages compared to the additive bias correction, which requires numerical root-finding methods (see Appendix A.1 for details).

Summarizing, the bootstrapped test statistic is

We now discuss testing for complete homogeneity. More specifically, we test whether ${X_{t\in \mathcal{I}}}\sim \mathcal{N}\left(\mu ,{\sigma ^{2}}\right)$ against the alternative that ${X_{t\in {L_{\tau }}}}\sim \mathcal{N}\left(\mu ,{\sigma ^{2}}\right)$ and ${X_{t\in {R_{\tau }}}}\sim \mathcal{N}\left(\breve{\mu },{\breve{\sigma }^{2}}\right)$ where $\mu \ne \breve{\mu }$ and/or ${\sigma ^{2}}\ne {\breve{\sigma }^{2}}$. Of course, this test is closely related to the test for homogeneity in variance. Because (2.8) represents the log-likelihood function for all $j=\{{L_{\tau }},{R_{\tau }},\mathcal{I}\}$, the test statistic for complete homogeneity is where ${\hat{\sigma }_{\mathcal{I}}^{2}}$ is the usual ML estimator of the variance on $\mathcal{I}$. Under correct model specification, the test with a single break point corresponds to the classical likelihood ratio test which follows a ${\chi ^{2}}(2)$-distribution.

Additive bias correction. The bootstrapped generalized LRT now takes the form where ${\hat{\sigma }_{a}^{2}}={\hat{\sigma }_{{R_{\tau }}}^{2}}-{\hat{\sigma }_{{L_{\tau }}}^{2}}$ and ${\hat{\mu }_{a}}={\bar{x}_{{R_{\tau }}}}-{\bar{x}_{{L_{\tau }}}}$ are the additive bias corrections.

As discussed in the context of (2.11), we have $\big\{{\bar{x}_{j}^{\ast }},{\hat{\sigma }_{j}^{\ast 2}}\big\}=\arg {\max _{\mu \in \mathbb{R},{\sigma ^{2}}\gt 0}}{L_{j}^{\mathrm{\flat }}}\left(\mu ,{\sigma ^{2}}\mid {X_{t\in j}}\right)$, for $j=\{{L_{\tau }},{R_{\tau }}\}$. Hence, as in the test for homogeneity in variance, the first two summands in (2.21) are given by The third term in (2.21) is now more complicated. To tackle it, introduce the mean-corrected right-hand sample ${\tilde{x}_{t\in {R_{\tau }}}}={\left\{{x_{t}}-{\hat{\mu }_{a}}\right\}_{t\in {R_{\tau }}}}$, and let ${\hat{\tilde{\sigma }}_{{R_{\tau }}}^{\ast 2}}$ and ${\bar{\tilde{x}}_{{R_{\tau }}}^{\ast }}$ be the ML estimators of the respective subsample in the bootstrap world. Maximizing (2.11) with respect to μ, we obtain Note that $\mu ({\sigma ^{2}})$ is a function in ${\sigma ^{2}}$.

Inserting (2.23) into (2.11), we can reformulate the third term in (2.21) involving only ${\sigma ^{2}}$. This yields In Appendix A.2, we maximize ${f_{CHa}^{\mathrm{\flat }}}$, which leads to a quintic polynomial in ${\sigma ^{2}}$. Our simulations indicate that the quintic polynomial’s discriminant is positive with adequately large samples, implying a single real root. If the discriminant is negative during a bootstrap draw, we redraw ${u_{t}}$ until we have the desired number of bootstrap samples. We denote the solution by ${\hat{\sigma }_{CHa}^{\mathrm{\flat }\ast 2}}$.

In summary, the bootstrapped test statistic is given by

Multiplicative bias correction. Alternatively, we propose using a multiplicative bias correction for the variance, which results in a more efficient procedure. Nonetheless, we continue to use the additive correction for the mean. Thus, we consider where ${\hat{\sigma }_{m}}=$ ${\hat{\sigma }_{R\tau }^{2}}/{\hat{\sigma }_{{L_{\tau }}}^{2}}$ is the proposed multiplicative bias correction for the variance, while ${\hat{\mu }_{a}}={\bar{x}_{{R_{\tau }}}}-{\bar{x}_{{L_{\tau }}}}$ is the additive correction for the mean, used already in (2.21).

We have $\left\{{\bar{x}_{j}^{\ast }},{\hat{\sigma }_{j}^{\ast 2}}\right\}=\arg {\max _{\mu \in \mathbb{R},{\sigma ^{2}}\gt 0}}{L_{j}^{\mathrm{\flat }}}\left(\mu ,{\sigma ^{2}}\mid {X_{t\in j}}\right)$ for $j=\{{L_{\tau }},{R_{\tau }}\}$ as before. As a benefit of the multiplicative bias correction for the variance, we can explicitly solve the maximization problem implied by the third term in (2.26). This yields which can be viewed as a weighted estimator for the mean, with weights applied across the two subsamples.

Together with (2.27), we receive a maximization problem solely in ${\sigma ^{2}}$: Let ${n_{{L_{\tau }}}}{\hat{\sigma }_{{L_{\tau }}}^{\ast \ast 2}}={\textstyle\sum _{t\in {L_{\tau }}}}\left[{u_{t}}{\left({x_{t}}-\hat{\mu }\right)^{2}}\right]$ and ${n_{{R_{\tau }}}}{\hat{\tilde{\sigma }}_{{R_{\tau }}}^{\ast \ast 2}}={\textstyle\sum _{t\in {R_{\tau }}}}\left[{u_{t}}{\left({\tilde{x}_{t}}-\hat{\mu }\right)^{2}}\right]$ be the sum of the weighted squared deviations from $\hat{\mu }$, where we recall that ${\tilde{x}_{t\in {R_{\tau }}}}={\left\{{x_{t}}-{\hat{\mu }_{a}}\right\}_{t\in {R_{\tau }}}}$ stands for the mean-corrected sample.

Surprisingly, (2.28) has a unique maximizer in ${\sigma ^{2}}$ given by Therefore, in the bootstrap context, the variance estimator resembles (2.18) seen in the homogeneity in variance test. It represents as a pooled variance estimator, but corrects the bias in variance, accommodates the bias-induced mean correction of the right-hand sample, and incorporates the weighted mean from (2.27). Consequently, by utilizing closed-form estimates, the bootstrap with multiplicative bias correction secures a substantial computational advantage compared to the case with additive bias correction, which requires numerical solutions (see Appendix A.2 for details).

In summary, the bootstrapped test statistic for complete homogeneity reads as

Central to LCP detection is the notion that there exists an interval, called a ‘homogeneous interval’ or an ‘interval of homogeneity’, on which the process is governed by a fixed probability measure ${\operatorname{P}_{0}}$. In line with this understanding, for a given point in time t, one tries to find—backward looking—the longest interval of homogeneity for local estimation [15, 20, 2, 24, 13].

To find the longest possible homogeneous interval, we sequentially test for homogeneity on $K+1$ nested intervals, where ${\mathcal{I}_{0}}\subset {\mathcal{I}_{1}}\dots \subset {\mathcal{I}_{K}}$. We choose the interval length to increase arithmetically, i.e., ${N_{k}}=\left\lceil {N_{0}}+ck\right\rceil $, where $c\in \mathbb{N}$, with c representing the increment per iteration. Denoting the most recent index by t (point of estimation), we set ${\mathcal{I}_{k}}=[t-{N_{k}},t]$, for $k=0,\dots ,K$ and $k\in \mathbb{N}$. The initial interval ${\mathcal{I}_{0}}$ is assumed to be homogeneous, which requires its length ${N_{0}}$ to be sufficiently large for reliable estimation. At the same time, ${N_{0}}$ should not be so long that the homogeneity assumption becomes implausible. The collection of nested intervals is referred to as the grid $\mathcal{G}$.

Figure 1 illustrates the intervals involved in the LCP procedure for a given iteration k. In more detail, for each k, every candidate point τ of the interval ${\mathcal{J}_{k}}={\mathcal{I}_{k}}\setminus {\mathcal{I}_{k-1}}$ is regarded as a potential break point. In other words, we test on ${\mathcal{I}_{k}}$ for homogeneity against the alternative of a breakpoint in ${\mathcal{I}_{k}}$ at an unknown point $\tau \in {\mathcal{J}_{k}}\subset {\mathcal{I}_{k}}$. Define the two intervals ${L_{k,\tau }}=[t-{N_{k+1}},\tau ]$ and ${R_{k,\tau }}=(\tau ,t]$, which are subintervals of ${\mathcal{I}_{k+1}}$. Given k, we evaluate either (2.15) and (2.19), or (2.25) and (2.30) on ${L_{k,\tau }}$ and ${R_{k,\tau }}$, for all $\tau \in {\mathcal{J}_{k}}$, depending on the test case as well as the version of bias correction employed.

The kth interval is rejected if ${T_{{\mathcal{I}_{k}}}}={\max _{\tau \in {\mathcal{J}_{k}}}}{T_{\tau }^{(k)}}\gt {\mathfrak{z}_{{\mathcal{I}_{k}},\alpha }}\hspace{0.2778em}$, where ${\mathfrak{z}_{{\mathcal{I}_{k}},\alpha }}$ is found using the multiplier bootstrap as described in Section 2.2. Note that to reduce the computational costs, one can run τ only on a subset of ${\mathcal{J}_{k}}$, e.g., on every second observation.

If homogeneity cannot be rejected, we increment k and continue with ${\mathcal{I}_{k+1}}$ until a rejection occurs, or the largest possible interval ${\mathcal{I}_{K-1}}$ is accepted. As the final estimate of the adaptive estimation procedure, we choose where ${\mathcal{I}_{\hat{k}}}$ corresponds to the largest non-rejected interval of homogeneity with ${\hat{\theta }_{{\mathcal{I}_{\hat{k}}}}}$ as its estimate. The procedure is repeated for all t in the sample, such that ${N_{K}}\lt t\le T$.

Simulation framework. To evaluate the size and power of the proposed homogeneity tests, we consider two DGPs. In case $(i)$, we sample from a normal distribution $\mathcal{N}\left(0,1\right)$; in case $(ii)$, we draw from a standardized $t(5)$-distribution. The standardization helps maintain comparability across both DGPs. Case $(i)$ represents our baseline setting, where the parametric assumptions are correctly specified, while case $(ii)$ represents a case, where the model is misspecified. We benchmark our tests against the ${\chi ^{2}}$-test with one (homogeneity in variance) or two degrees of freedom (complete homogeneity). We expect similar results in case $(i)$, especially for moderately large sample sizes, whereas the inference based on the ${\chi ^{2}}$-distribution will be invalid in case $(ii)$.

We consider three setups with sample sizes ${n_{{L_{\tau }}}}={n_{{R_{\tau }}}}=\{5,25,50\}$, which further results in total interval lengths of ${n_{\mathcal{I}}}=\{10,50,100\}$. The tests are evaluated on three significance levels, $\alpha =\{0.025,0.05,0.1\}$. To compute the coverage probabilities for assessing the size and power of the test, we repeat each test case $M=1000$ times across all scenarios.

To study the power, we use a parameter $\lambda \in (-0.8,2)$ to control the magnitude of a breakpoint. We start by assessing the power of each test given a change in mean, where the samples are generated from: where $\mathcal{Z}$ is drawn independently either from (i) $\mathcal{N}\left(0,1\right)$ or ($ii$) a standardized $t(5)$-distribution. ${\mathrm{H}_{0}}$ is obtained for $\lambda =0$.

To examine power, when there is a break in the variance, the samples are generated according to: When generating the alternative by multiplying by $\sqrt{1+\lambda }$, note that the sample mean inflates. To ensure comparability across samples in terms of their empirical means, we first demean each sample, apply the multiplication, and then add back its empirical sample mean. This prevents artificial inflation of the empirical means as we range over λ. When $\lambda =0$, we simulate under ${\mathrm{H}_{0}}$.

Size. Figure 2 illustrates the size discrepancies, which represent the differences between the actual coverage probabilities and theoretical target values (type I error). We find very similar results for the ${\chi ^{2}}$-test relative to the multiplier bootstrap, when we draw $\mathcal{N}\left(0,1\right)$ sequences, no matter which bias correction is used. For standardized $t(5)$ sequences, the ${\chi ^{2}}$-test fails. In comparing the two bias corrections, we observe that the additive version performs slightly more accurately under model misspecification, given a moderate sample size ${n_{\mathcal{I}}}\in \{50,100\}$. However, the multiplicative correction is markedly superior in the extreme case of ${n_{\mathcal{I}}}=10$. Generally, the discrepancies in size are larger under misspecification, as one may expect. Overall, the results indicate that the multiplicative bias correction is a suitable substitute while providing significant computational advantages as pointed out in Sections 2.3 and 2.4.

Power: break in μ. In Figure 3, we investigate breaks in the mean parameter. Of course, the test of homogeneity in variance is expected to display a constant coverage probability, independent of $\lambda \in (-0.8,2)$, which we confirm. In contrast, the test for complete homogeneity detects the changes in the mean, and panels (c) and (d) exhibit the power (type II error). As with size, power decreases under model misspecification and smaller sample sizes. Comparing both bias correction methods, we find similar power across both settings, except for small samples, where the multiplicative correction is superior.

Power: break in ${\sigma ^{2}}$. For changes in the variance parameter, the power is illustrated in Figure 4. The results suggest that the test for homogeneity in variance consistently achieves a marginally better performance in terms of power than the test for complete homogeneity across both DGPs. This is evidenced by lower frequencies of type II errors across all test configurations, including bias correction, sample size, and significance level. Once again, lower power is observed under the misspecified model and with small sample sizes. Consistent with previous findings, the simulations suggest that the multiplicative bias correction is a suitable alternative, particularly in tiny samples.

Unconditional changes in the mean and variance. We continue to consider two different underlying families of distributions for drawing segments of non-stationary sequences: $(i)$ $i.i.d$. $\mathcal{N}\left(0,1\right)$ sequences and $(ii)$ $i.i.d$. standardized $t(5)$ ones. We draw a cross-section of $M=1000$ processes in both settings, comprising 1000 realizations each. The processes are modeled as follows: where $\mathcal{Z}$ is drawn as discussed above. Furthermore, as described in Section 3.1, we ensure a pure change in variance scenario for $t\gt 670$, consistent with the mean-preserving transformation outlined earlier.

We employ $\mathcal{G}=\{50,100,150,200,250,300\}$ as the grid of interval sizes for the sequential LCP detection algorithm. To ease the computational burden, we let τ run on every second observation, which doubles the speed of the simulations. When testing, we consider a significance level of $\alpha =0.025$. Figure 5 presents the results for each test configuration. At each time point t, for a given test and DGP, the plot displays the cross-sectional mean and median of the estimated index $\hat{k}$, computed using (2.31), where the corresponding interval length is given by ${\mathcal{I}_{\hat{k}}}$.

The test for homogeneity in variance excels the one for complete homogeneity in detecting changes in the second moment. Moreover, under model misspecification, the tests incorporating the multiplicative adjustment detect changes in more instances in the cross-section of simulated data sequences. For both DGPs, in the case of testing for homogeneity in variance, we find a clustering of false rejections due to changes in the mean. The reasons for these false alarms are heterogeneous subintervals: Whenever a subinterval (left or right) is rolled over a change point in the mean, the estimated variance in the respective subinterval can be affected, which leads to rejections of the hypotheses. Notably, the additive bias correction mitigates this behavior in the presence of misspecification while providing lower detection power.

Because empirical practice often raises concerns about potential misspecification regarding serial dependence, we also consider sequences generated by the autoregressive moving average (ARMA) process where we have ${\phi _{1}}=0.2$, ${\gamma _{1}}=0.2$ and ${\sigma _{1}^{2}}=1$ initially, and ${\phi _{2}}=0.6$, ${\gamma _{2}}=0.6$ and ${\sigma _{2}^{2}}=2.5$ after the change. This setup enables us to examine two distinct jumps in the unconditional variance of the ARMA($1,1$) sequences: one driven by changes in the residual variance and the other caused by shifts in the ARMA coefficients. Notably, changes in γ and ϕ do not affect the first moments.

We use the LCP configuration described above, including the values for $\mathcal{G}$, τ, and α. Figure 6 presents the results for both tests. Again, at each time point t, the plot shows the cross-sectional mean and median of the estimated index $\hat{k}$.

In line with previous findings on variance changes, our results demonstrate that the test for homogeneity in variance exhibits superior detection power compared to the test for complete homogeneity. Moreover, it offers an additional advantage by showing greater robustness to time-dependent dynamics in the mean. This is particularly evident in the final segment, where we have increased magnitudes for both ARMA coefficients, ϕ and γ. In such cases, the test for complete homogeneity frequently produces rejections, whereas the test for homogeneity in variance more effectively mitigates the time-variation in the conditional mean.

Conditional changes in the variance. So far, we have examined the tests’ ability to detect abrupt transitions in the DGPs, including both independent and serially dependent sequences. Next, we evaluate the performance of our tests in capturing conditional variance. A widely used model for this purpose is the generalized autoregressive conditional heteroskedasticity (GARCH) model, which allows the conditional variance to evolve over time as a function of past errors and past conditional variances [1]. This exercise compares the LCP detection algorithm, which uses data-driven interval lengths, with a naive rolling window approach employing a fixed window width for variance estimation, under a GARCH process. Rolling window estimators for variance are commonly used when no specific model is available.

We sample $M=1000$ GARCH(1,1) processes for six different parametric configurations. Each process comprises 1000 realizations and is given by: where ${\varepsilon _{t}}\sim \mathcal{N}\left(0,1\right)$. We set $\omega ={10^{-5}}$ in each configuration. To work with meaningful magnitudes for α and β, we collect daily closing prices for each 503 constituents of the S&P 500 as of May 3, 2023, where the earliest starting date is January 1, 1970. We fit the GARCH(1,1) model to the log return sequences of each asset and record the estimated coefficients—see Table 1 for the descriptive statistics on the estimates of α and β across all 503 time series.

The persistence of the conditional variance process is $\rho =\alpha +\beta $. If $\rho \lt 1$, the process is strictly stationary [6]. As reported in Table 1, U.S. large-cap equity returns exhibit notable persistence. For our simulations, we select five configurations from the 503 candidates based on their proximity to the minimum (Min), first quartile (Q1), median (Median), third quartile (Q3), and maximum (Max) of all persistence parameters estimated.

We ‘forecast’ the 1-day ahead conditional variance of the GARCH(1,1) sequence by estimating the realized variance over varying interval lengths. We examine the forecasting accuracy in terms of the mean squared forecasting error (MSFE) and calculate this measure for $(i)$ a locally adaptive estimation procedure incorporating the proposed test for homogeneity in variance; and $(ii)$ various rolling window estimators relying on a fixed interval length. We use $\mathcal{G}=\{25,50,75,100,125,150\}$ as the grid, run τ on each observation in ${\mathcal{J}_{k}}={\mathcal{I}_{k}}\setminus {\mathcal{I}_{k-1}}$, and set a significance level of $\alpha =0.025$ for testing.

In Figure 7, we present the ratios of the MSFE for the LCP algorithm relative to a specific fixed window estimator. A ratio below one indicates that the LCP method outperforms the fixed window approach. As is visible, there is no clear relationship between the persistence of the process and the predictive performance of the models considered. Longer interval lengths improve accuracy in environments with either very low or very high persistence (“Min” and “Max”), while shorter ones excel in “moderate” persistence settings (“Q1”, “Median” and “Q3”). Interestingly, in the “Max” scenario, which features the highest persistence and the largest magnitude of β, neither the estimators with the smallest ($k=1$) nor those with the largest window lengths ($k=5$) rank among the most accurate predictive models. Although our locally adaptive approach never outperforms all benchmarks (based on the median, it beats at least three out of five benchmarks in four out of five cases), no benchmark constantly outperforms the LCP approach across all scenarios. Because persistence in practice is unknown or may even be subject to structural breaks, these findings underscore the benefit of locally adaptive estimation over fixed-window methods.

In recent years, the variance of consumer prices changes has made price stability a central topic. Therefore, our first analysis focuses on month-over-month inflation rates in the U.S., computed from the seasonally adjusted core consumer price index ($CP{I_{t}}$), excluding energy and food. The data spans from January 1957 to September 2023, comprising 802 observations.

We fit ARIMA(p, d, q) models to monthly log inflation rates ${r_{t}}$ and extract the residuals ${\varepsilon _{t}}$. Both the original series and the residuals are treated as data sequences for the application of our LCP procedure. Model selection based on various information criteria (AIC, AICc, and BIC) suggests an ARIMA($0,1,1$) specification. The presence of a unit root is further supported by an augmented Dickey–Fuller test.

Figure 8 displays the results. For both sequences (${r_{t}}$ and ${\varepsilon _{t}}$), we present the estimated index ${\hat{k}_{t}}$ and the corresponding locally adaptive estimates for the log variance. The LCP identifies a break in the late 70s and early 80s for both series, associated with high inflation levels. Since then, the variance has appeared fairly constant. Additionally, we observe a break in variance between 2016 and 2018, coinciding with a reduction in variance. Another break in the variance is detected around the start of the COVID-19 pandemic. Comparing the results across both sequences, the LCP procedure on ${r_{t}}$ identifies an additional break in variance at the beginning of our sample. The results demonstrate the effectiveness of our LCP detection procedure, whether applied to the raw signal or the residuals of the ARIMA fit, in identifying changes in the variance of month-over-month inflation rates.

The Great Moderation, characterized by reduced variance in the growth rates of industrial production from the mid-1980s until at least 2007, is a widely discussed phenomenon in macroeconomic literature [22]. In our second analysis, we apply the LCP to monthly growth rates of U.S. industrial production (monthly seasonally adjusted index INDPRO), covering the period from February 1919 to September 2023, with 1,257 observations. We calculate monthly log growth rates, again denoted here by ${r_{t}}$, and also extract, as previously, the residuals from an ARIMA($2,0,3$) model, following a same search methodology as above.

Figure 9 presents the results for both data sequences, which are qualitatively very similar. The LCP procedure identifies numerous structural breaks in the variance, particularly before 1960. After 1965, another break in variance is observed, followed by a prolonged period of homogeneity with historically low variance lasting until the mid-80s. At this point, the procedure detects a new change. The subsequent period, characterized by even lower variance, extends until 2007. Thus, the locally adaptive approach not only delivers results that align precisely with the Great Moderation but, remarkably, also identifies the period from 1970 to the mid-1980s as a time of relatively low variance.

In our last application, we analyze the historical variance of Bitcoin (BTC) using daily closing prices from January 1, 2016, to November 30, 2023 (2,891 observations). For this purpose, we calculate the daily log returns ${r_{t}}$ of BTC.

Figure 10 presents the outcomes for ${r_{t}}$. Initially, we observe a high number of breaks in variance in the first half of the sample, whereas breaks become considerably less frequent in the latter half. Several variance breaks are detected during 2017/18, coinciding with sharp price fluctuations in BTC. Additional variance breaks are noted in 2018/19. Further, the LCP procedure indicates a growing period of homogeneity until the rapid price decline triggered by the recession induced by the COVID-19 pandemic. Since 2022, the variance has been fairly constant. As Figure 10 shows, breaks in variance tend to occur during extreme price movements, both positive and negative. The fewer change points over time possibly indicate the maturation of the cryptocurrency market.

Figure 11 provides another perspective, focusing on the variance of BTC and its volatility. To this end, we compute $\log |{r_{t}}|$ as a (rough) measure of daily volatility. Consequently, the mean estimator, applied to $\{\log |{r_{t}}|\}$ becomes an estimator of variance, while the variance estimate is interpretable as a measure of volatility of variance. Therefore, it is now of interest to apply both tests, the test for homogeneity in variance as well as the test for complete homogeneity.

The comparison of the estimated intervals of homogeneity offers telling insights: Until mid-2017, our procedures detected changes only in the mean (here meaning: the variance), as the test for homogeneity in variance (here meaning: volatility of variance) does not identify any breaks. Likewise, another period of instability occurred in 2019. These findings align well with the previous analysis of the returns. Besides these two periods, the sequences of the estimated intervals of homogeneity are quite similar, suggesting that changes in the mean (here: variance) might also correspond to breaks in the variance (here: volatility of variance), see for instance, at the end of 2017, the beginning of 2020 (COVID-19), and the end of 2022. When comparing the results across both characteristics, we see that the volatility of variance tends to have fewer structural breaks than the variance. Overall, these observations suggest that a thorough understanding of both the variance and the volatility of variance is essential for effectively modeling the rapidly evolving BTC market.