#### 5.1 A Test of No Break versus a Fixed Number of Breaks

We consider the supF type test of no structural break (*m* = 0) versus *m* = *k* breaks. Let (*T*_{1}, …, *T*_{k}) be a partition such that *T*_{i} = [*T*λ_{i}] (*i* = 1, …, *k*). Let *R* be the conventional matrix such that (*R*δ)′ = (δ_{1}′ − δ_{2}′, …, δ_{k}′ − δ_{k + 1}′). Define

- (10)

where is an estimate of the variance covariance matrix of that is robust to serial correlation and heteroscedasticity; i.e. a consistent estimate of

- (11)

Following Andrews (1993) and others, the test is sup where minimize the global sum of squared residuals which is equivalent to maximizing the F-test assuming spherical errors. This is asymptotically equivalent to, and yet much simpler to construct than, maximizing the F-test (10) since the estimated break dates are consistent even in the presence of serial correlation. The asymptotic distribution depends on a trimming parameter via the imposition of the minimal length *h* of a segment, namely ϵ = *h*/*T*.

Various versions of the tests can be obtained depending on the assumptions made with respect to the distribution of the data and the errors across segments. These relate to different specifications in the construction of the estimate of given by (11). In the case of a partial structural change model (*p* ≠ 0), we consider three specifications.

Serially uncorrelated errors, different distributions for the data across segments and the same distribution for the errors across segments. In this case, which can be estimated using .

In the case of a pure structural change model, we consider more possible specifications on how to estimate the relevant asymptotic covariance matrix. They are the following:

No serial correlation, different distributions for the data and identical distribution for the errors across segments. In this base case, the estimate is .

Serial correlation in the errors, same distribution for the errors across segments. In this case the limiting covariance matrix is where (using the convention that λ_{0} = 0 and λ_{m + 1} = 1) Λ = *diag*(λ_{1} − λ_{0}, …, λ_{m + 1} − λ_{m}). This can be consistently estimated using and a HAC estimator of *Z*′Ω*Z* based on {*z*_{t} û_{t}} constructed using the full sample.

In the construction of the tests we do not consider imposing the restriction that the distribution of the regressors *z*_{t} be the same across segments even if they are (except as they enter in the construction of a HAC estimate involving the pair {*z*_{t} û_{t}}). This might seem surprising since imposing a valid restriction should lead to more precise estimate. This is, however, not the case. Consider the case with no serial correlation in the errors and the same distribution for the errors across segments. Imposing the restriction that the distribution of the regressors *z*_{t} be the same across segments leads to the asymptotic covariance matrix , where . Note that a consistent estimate can be obtained using , and constructed using (*i* = 1, …, *m*). Suppose that the *z*'s are exogenous and the errors have the same variance across segments. Then, for a given partition (*T*_{1}, …, *T*_{m}), the exact variance of is . Using the asymptotic version may imply an inaccurate approximation especially if small segments are allowed, in which case the exact moment matrix of the regressors may deviate substantially from its full-sample analogue.

The relevant asymptotic distribution has been derived in BP and critical values were provided for a trimming ϵ = 0.05 and values of *k* from 1 to 9 and values of *q* from 1 to 10. As discussed in Bai and Perron (2000), a trimming as small as 5% of the total sample can lead to tests with substantial size distortions when allowing different variances of the errors across segments or when serial correlation is permitted. This is because one is then trying to estimate various quantities using very few observations; for example, if *T* = 100 and ϵ = 0.05, one ends up estimating, for some segments, quantities like the variance of the residuals using only 5 observations. Similarly, with serial correlation a HAC estimator would need to be applied to very short samples. The estimates are then highly imprecise and the tests accordingly show size distortions. When allowing different variances across segments or serial correlation, a higher value of ϵ should be used. Hence, the case with no serial correlation and homogenous errors should be considered the base case in which the tests can be constructed using an arbitrary small trimming ϵ. For all other cases, care should be exercised in the choice of ϵ and larger values should be considered. For that purpose, we supplemented the critical values tabulated in BP with similar ones for ϵ = 0.10, 0.15, 0.20 and 0.25. The results are presented in Bai and Perron (‘Additional critical values for multiple structural changes tests’, unpublished manuscript, 2001). Note that when ϵ = 0.10 the maximum number of breaks considered is 8 since allowing 9 breaks impose the estimates to be exactly , up to . For similar reasons, the maximum number of breaks allowed is 5 when ϵ = 0.15, 3 when ϵ = 0.20 and 2 when ϵ = 0.25.

Note that the asymptotic theory for these tests in BP is valid only for the case of non-trending data. The case with trending data, discussed in Bai (1999), yields different asymptotic distributions. However, the asymptotic distributions in the two cases are fairly similar, especially in the tail where critical values are obtained. Hence, one can safely use the same critical values. Using simulations, we found the size distortions to be minor.

#### 5.2 Double Maximum Tests

Often, an investigator wishes not to pre-specify a particular number of breaks to make inference. To allow this BP have introduced two tests of the null hypothesis of no structural break against an unknown number of breaks given some upper bound *M*. These are called the *double maximum tests*. The first is an equal weighted version defined by , where (*j* = 1, …, *m*) are the estimates of the break points obtained using the global minimization of the sum of squared residuals. The second test applies weights to the individuals tests such that the marginal *p*-values are equal across values of *m* and is denoted *WD max F*_{T}(*M*, *q*); see BP for details. Critical values were provided for *M* = 5 and ϵ = 0.05 in BP. A value *M* = 5 should be sufficient for most empirical applications. In any event, the critical values vary little for choices of the upper bound *M* larger than 5. Bai and Perron (2001) provide additional critical values for ϵ = 0.10 (*M* = 5), 0.15 (*M* = 5), 0.20 (*M* = 3) and 0.25 (*M* = 2).

#### 5.3 A Test of **ℓ** versus *ℓ* + 1 Breaks

BP proposed a test for ℓ versus ℓ + 1 breaks, labelled sup*F*_{T}(ℓ + 1|ℓ). The method amounts to the application of (ℓ + 1) tests of the null hypothesis of no structural change versus the alternative hypothesis of a single change. The test is applied to each segment containing the observations *T̂*_{i−1} to *T̂*_{i}(*i* = 1, …, ℓ + 1). We conclude for a rejection in favour of a model with (ℓ + 1) breaks if the overall minimal value of the sum of squared residuals (over all segments where an additional break is included) is sufficiently smaller than the sum of squared residuals from the ℓ breaks model. The break date thus selected is the one associated with this overall minimum. The estimates *T̂*_{i} need not be the global minimizers of the sum of squared residuals, one can also use sequential one at a time estimates which allows the construction of a sequential procedure to select the number of breaks (see Bai, 1997b).

Asymptotic critical values were provided by BP for a trimming ϵ = 0.05 for *q* ranging from 1 to 10, and Bai and Perron (2001) present additional critical values for ϵ = 0.10, 0.15, 0.20 and 0.25. Note that, unlike for the sup*F*_{T}(*k*; *q*) test, we do not need to impose similar restrictions on the number of breaks for different values of the trimming ϵ.4 Of course, all the same options are available as for the previous tests concerning the potential specifications of the nature of the distributions for the errors and the data across segments.