Perron et al. (

2019) provided a comprehensive treatment for the problem of testing jointly for structural changes in the regression coefficients and the variance of the errors. Here, we consider two versions of their tests to illustrate how they solve the size and power problems. The first version investigates whether a given number (

$m$) of structural changes in the coefficients are present when a given number (

$n$) of structural changes in the error variance are accounted for. The structural change dates for both the coefficients and the variance are unknown and occur at the same or different times. The second version considers whether n structural changes in the error variance are present when

m structural changes in the regression coefficients are allowed. Following their labels, we call the former testing problem TP-3 and the latter TP-2. The hypotheses are

${H}_{0}:\left\{m=0,n={n}_{a}\right\}$ versus

${H}_{1}:\left\{m={m}_{a},n={n}_{a}\right\}$ for TP-3 and

${H}_{0}:\left\{m={m}_{a},n=0\right\}$ versus

${H}_{1}:\left\{m={m}_{a},n={n}_{a}\right\}$ for TP-2, where

${m}_{a}$ and

${n}_{a}$ are pre-selected values. The break dates for the coefficients are denoted by

$\left\{{T}_{1}^{c},\dots ,{T}_{m}^{c}\right\}$, those for the error variance by

$\left\{{T}_{1}^{v},\dots ,{T}_{n}^{v}\right\}$ and the break fractions by

$\left\{{\lambda}_{1}^{c},\dots ,{\lambda}_{m}^{c}\right\}$ and

$\left\{{\lambda}_{1}^{v},\dots ,{\lambda}_{n}^{v}\right\}$, respectively. We also let the number of the union of the coefficients and the variance breaks be

$K$.

The test statistics are the quasi-likelihood ratio tests assuming i.i.d. Gaussian disturbances. For TP-3, the log-likelihood function under

${H}_{0}$ is

where

${\tilde{\sigma}}_{i}^{2}={\left({T}_{i}^{v}-{T}_{i-1}^{v}\right)}^{-1}{\displaystyle \sum}_{t={T}_{i-1}^{v}+1}^{{T}_{i}^{v}}{\left({y}_{t}-\tilde{\mu}\right)}^{2}$ for

$i=1,\dots ,{n}_{a}+1$ with

$\tilde{\mu}={T}^{-1}{\displaystyle \sum}_{t=1}^{T}{y}_{t}$. Under

${H}_{1}$,

where

${\widehat{\sigma}}_{i}^{2}={\left({T}_{i}^{v}-{T}_{i-1}^{v}\right)}^{-1}{\displaystyle \sum}_{t={T}_{i-1}^{v}+1}^{{T}_{i}^{v}}{\left({y}_{t}-{\widehat{\mu}}_{t,j}\right)}^{2}$ for

$i=1,\dots ,{n}_{a}+1$ with

${\widehat{\mu}}_{t,j}={({T}_{j}^{c}-{T}_{j-1}^{c})}^{-1}{\displaystyle \sum}_{t={T}_{j-1}^{c}+1}^{{T}_{j}^{c}}({y}_{t}/{\widehat{\sigma}}_{i})$. Because the break dates are unknown, the supremum type LR test over all the permissible break dates is given by

where

${\mathsf{\Lambda}}_{\epsilon}$ is the union of the set of permissible break fractions for the coefficients and variance and

${\mathsf{\Lambda}}_{v,\epsilon}$ is a set of the permissible variance break fractions.

$\epsilon $ is a small positive trimming value so that

and

${\mathsf{\Lambda}}_{v,\epsilon}=\left\{\left({\lambda}_{1}^{v},\dots ,{\lambda}_{n}^{v}\right);\left|{\lambda}_{i+1}^{v}-{\lambda}_{i}^{v}\right|\ge \epsilon \text{}\left(i=1,\dots ,{n}_{a}-1\right),\text{}{\lambda}_{1}^{v}\ge \epsilon ,\text{}{\lambda}_{n}^{v}\le 1-\epsilon \right\}$. Note that we denote the estimates of the break dates in coefficients and variance by a “

$\tilde{}$” when these are obtained jointly, and by a “

$\widehat{}$” when obtained separately. For TP-2, the sup-LR test is

where

$\mathrm{log}{\tilde{L}}_{T}\left({T}_{1}^{c},\dots ,{T}_{{m}_{a}}^{c}\right)=-\left(\frac{T}{2}\right)\left(\mathrm{log}2\pi +1\right)-\left(T/2\right)\mathrm{log}{\tilde{\sigma}}^{2},$ with

${\tilde{\sigma}}^{2}={T}^{-1}{\displaystyle \sum}_{t=1}^{T}{\left({y}_{t}-{\widehat{\mu}}_{t,j}\right)}^{2}$,

${\widehat{\mu}}_{t,j}={\left({T}_{j}^{c}-{T}_{j-1}^{c}\right)}^{-1}{\displaystyle \sum}_{t={T}_{j-1}^{c}+1}^{{T}_{j}^{c}}{y}_{t}$ and

${\mathsf{\Lambda}}_{c,\epsilon}=\left\{\left({\lambda}_{1}^{c},\dots ,{\lambda}_{m}^{c}\right);\left|{\lambda}_{i+1}^{c}-{\lambda}_{i}^{c}\right|\ge \epsilon \text{}\left(j=1,\dots ,m-1\right),\text{}{\lambda}_{1}^{c}\ge \epsilon ,\text{}{\lambda}_{m}^{c}\le 1-\epsilon \right\}$.

Perron et al. (

2019) showed that the asymptotic distributions of these tests are bounded by limit distributions obtained in

Bai and Perron (

1998). Hence, somewhat conservative tests are possible using their critical values. They also show that the distortions are very minor via Monte Carlo simulations.

We implemented the sup-

${\mathrm{LR}}_{3,T}$ and sup-

${\mathrm{LR}}_{2,T}$ tests for the same DGP as above. We concentrated on testing for the presence of breaks rather than determining their number. Hence, we used the true values

${n}_{a}=1$ and

${m}_{a}=1$ as applicable when breaks were present. The size of the sup-

${\mathrm{LR}}_{3,T}$ test for a change in

$\mu $ given one break in the error variance is presented in

Figure 7. As explained in

Perron et al. (

2019), the exact size is slightly smaller than the nominal level; the distortions due to the variance break are however, minor. The size is more distorted as

${\delta}_{1}$ becomes larger for the case of

${T}^{v}=\left[.25T\right]$ but there are no evident distortions for the cases with

${T}^{v}=\left[.5T\right]$ and

$\left[.75T\right]$. The power of the sup-

${\mathrm{LR}}_{3,T}$ test is presented in

Figure 8 for the cases

${T}^{c}=\left[.3T\right]$ and

${T}^{v}=\left[.5T\right]$ as well as

${T}^{c}=\left[.3T\right]$ and

${T}^{v}=\left[.75T\right]$. For the first case, the power function decreases somewhat as the variance break increases. However, and more importantly, all power functions are higher than those presented in

Figure 2, indicating a reliable power performance. For the second case, the magnitude of the variance break has no evident effect on the power function, which remains high.

To test for structural breaks in the error variance,

Figure 9 shows the size of the sup-

${\mathrm{LR}}_{2,T}$ test which accounts for a change in the mean. Again, the exact size is slightly conservative, as expected, but not affected by the magnitude of

${\delta}_{2}$.

Figure 10 shows the power functions of the sup-

${\mathrm{LR}}_{2,T}$ test for the cases

${T}^{c}=\left[.5T\right]$ and

${T}^{v}=\left[.3T\right]$ as well as

${T}^{c}=\left[.75T\right]$ and

${T}^{v}=\left[.3T\right]$. Here, the power functions are not affected by

${\delta}_{2}$ for both cases and are higher than those in

Figure 6. The results overall illustrate significant improvements of the size and power properties when using the conditional tests.