Testing for Multiple Structural Changes with Non-Homogeneous Regressors

1 Introduction

This paper proposes tests for multiple structural changes with non-homogeneous regressors. In particular, we focus on trending regressors. Tests for structural changes have long been investigated in the econometric and statistical literature, and the most commonly used tests in empirical analysis for a one-time break are the supremum-type (sup-type) test by Andrews (1993) in the GMM framework and the exponential-type (exp-type) and the average-type (avg-type) tests by Andrews and Ploberger (1994) in linear regression models. The latter two tests have an optimal property, which was investigated by Andrews and Ploberger (1994) and Sowell (1996) under the Pitman-type alternative, while Kim and Perron (2009) compared these tests in a framework based on the Bahadur slope.

Although these tests are often used in practice to test for a one-time change, we need to take into account the possibility of multiple structural changes when economic data in long sample periods are available. Bai and Perron (1998) extended the sup-type test to the case of multiple structural changes in univariate regressions, while the multivariate case was considered by Qu and Perron (2007). Andrews, Lee, and Ploberger (1996) investigated the optimality of the exp-type and avg-type tests. Note that these tests are designed for the null hypothesis of no change against the alternative of the fixed number of breaks. On the other hand, Bai and Perron (1998) and Qu and Perron (2007) proposed double maximum tests against the alternative under which only the maximum number of breaks is prespecified, while Bai and Perron (1998), Bai (1999), and Qu and Perron (2007) considered tests for the null hypothesis of ℓ breaks against the alternative of ℓ+1 breaks. The multiple structural change tests have an advantage over the single structural change tests in that the former tests are more powerful than the latter when multiple breaks have actually occurred, as shown by Bai and Perron (2006).

The practical difficulty in the multiple structural change tests is that we need to take into account all permissible change points when constructing the test statistics. That is, for the sup-type, the exp-type, and the avg-type tests, we need to construct either the Wald, the likelihood ratio (LR), or the Lagrange multiplier test statistic for all permissible sets of change points, the number of which is proportional to Tm, where m indicates the number of breaks under the alternative. Then, the direct calculation of these test statistics is computationally very expensive when m is large. To overcome this problem, Bai and Perron (2003a) proposed an efficient algorithm for the sup-type test, which requires only the O(T2) calculations for any number of breaks. Critical values for the sup-type test are tabulated in Andrews (1993) for a one-time break and Bai and Perron (1998, 2003b) for multiple changes, and those for the exp-type and the avg-type tests are given in Andrews and Ploberger (1994) for a one-time change, while asymptotic p-values of these tests can be calculated by the method proposed by Hansen (1997). However, critical values for the exp-type and the avg-type tests with multiple breaks are not yet available.

Most of the above tests assume that regressors are homogeneous in the whole sample period, or at least in each regime under the null hypothesis. However, we sometimes include non-homogeneous regressors, such as trending variables. In this case, most of the above tests are not available in practical analysis. The exception is the LR test, denoted by sup F(ℓ+1|ℓ), for the null hypothesis of ℓ breaks against the alternative of ℓ+1 breaks proposed by Bai (1999). This test allows for polynomial trends, and hence the null hypothesis of no break can be tested using sup F(1|0). However, as pointed out by Bai and Perron (2006), this test may be less powerful than tests for multiple structural changes when multiple breaks have actually occurred.

In this paper, we develop tests for multiple breaks with non-homogeneous regressors, including trending regressors. We consider sup-type, exp-type, and avg-type tests, as in the literature, and derive the concise expressions of the limiting distributions. It is shown that in general, the limiting distributions depend on non-homogeneous regressors, and then we need to tabulate critical values depending on the case. For this reason, we focus on the linear trend case and tabulate critical values. Since we need to calculate the Wald test statistics for all permissible break points for the exp-type tests, which is computationally very expensive in the case of more than three breaks, we tabulate the critical values of the exp-type test for at most three breaks. On the other hand, we propose a computationally efficient method for obtaining critical values for the avg-type test, which requires only O(T2) operations for any given number of breaks. According to this efficient algorithm and Bai and Perron’s (2003a) method, critical values of the sup-type and the avg-type tests are calculated for up to five breaks. Finite sample properties are investigated by Monte Carlo simulations, and it is confirmed that the tests that assume the maximum number of breaks but not the specific number of breaks are useful in practical analysis.

Note that this paper is related to another strand of the time series literature, which deals with tests for trend breaks robust to the stationary/unit root property in the stochastic errors. See, for example, Roy, Falk, and Fuller (2004), Harvey, Leybourne, and Taylor (2009), Perron and Yabu (2009; 2012), Kejriwal and Perron (2010), Saiginsoy and Vogelsang (2011) and Chun and Perron (2013). These papers focus on testing for only the trend coefficient with possibly integrated errors, whereas our paper considers tests for all (a part) of the regression coefficients associated with both trends and stationary regressors, although we do not allow for integrated regressors/errors. Thus, the main purpose of this paper is different from that of those robust tests.

For example, the international terms of trade between primary and manufactured commodities have been investigated in the literature and it has been pointed out that they follow a downward secular trend (the Prebisch–Singer hypothesis). However, this effect has recently been lessened by the growing demand from emerging market countries and then this hypothesis has been re-investigated by taking structural change into account in such as Kellard and Wohar (2006), Harvey et al. (2010), Ghoshray (2011) and Arezki et al. (2012), among others. These papers focus only on the coefficient associated with a linear trend, but as modelled by Bloch and Sapsford (1996), the relative prices are explained by such as manufacturing output and relative wages as well as a linear trend and thus our framework with a multiple regression is useful for the full investigation of the relative prices.^[1]

The rest of this paper is organized as follows. Section 2 explains a model and assumptions. The test statistics are given in Section 3, and their limiting distributions are derived. The computational problem of the test statistics is discussed in Section 4. A model is extended to partial structural changes and serially correlated errors in Section 5, and the finite sample properties are investigated in Section 6. Section 7 gives concluding remarks.

2 Model and assumptions

Let us consider the following regression with m structural changes (m+1 regimes):

where xt is p-dimensional regressors, including a constant, and εt is an error term. We set T0=0 and Tm+1=T so that the total number of observations is T. The testing problem we consider is given by

and we then consider the null of no structural change. The following assumptions are made throughout the paper.

Assumption A1For some normalizing matrixDT, (a) DT−1∑t=klxtx′tDT−1is invertible forl−k>k0for some0<k0<∞. (b) DT−1∑t=1[Tr]xtx′tDT−1→pΩruniformly over0<r≤1, whereΩris ap×ppositive definite matrix for0<r≤1withΩ0=0, [k]signifies the largest integer less than k, and→psignifies convergence in probability. (c) Ωs−Ωris positive definite for all0≤r<s≤1.

Assumption A2(a) {εt}is a martingale difference sequence with respect toFt=σ(εt,εt−1,…,xt+1,xt,…)withE[εt2|Ft−1]=σ2for all t. (b) suptE|εt|2+δ<∞for someδ>0. (c) For the same normalizing matrixDTin Assumption A2, DT−1∑t=1[Tr]xtεt⇒σG(r)for0≤r≤1, whereG(r)is a p-dimensional Gaussian process with mean zero andE[G(r)G′(s)]=Ωr∧s, and⇒signifies weak convergence of the associated probability measures.

Assumption A1(a) is made for the identification of the coefficient. A1(b) allows for non-homogeneous regressors because the second moment of xt is not asymptotically proportional to the sample fraction r, but possibly depends on r in a complicated way. Note that integrated regressors are not allowed for because Ωr is non-stochastic. A1(c) implies that the probability limit of DT−1∑t=Tj−1+1Tjxtx′tDT−1 is positive and invertible for (Tj−Tj−1)/T>0, so that the limiting distribution of DT(βj−β) is well defined. Assumption A2 is standard in linear regressions, but we do not allow for serial correlation for a while. The case with serially dependent errors will be discussed in the later section.

Exactly speaking, Assumptions A1 and A2 are required only under the null hypothesis in order to derive the null limiting distributions of the test statistics and they can be relaxed under the alternative in order for the tests to be consistent. See, for example, Assumptions made in Bai and Perron (1998) for the case of (regime-wise) stationary regressors.

One of the interesting non-homogeneous regressors is a polynomial trend. For example, when xt is given by

where x1t,…,xqt are stationary regressors, we can choose DT=diag{T1/2,T3/2,T5/2,…,Td+1/2,T1/2Iq} and Assumptions A1(b) and A2(c) then become

where μx and Γx are q×1 and q×q and consist of the first and second moments of x1t,…,xqt, respectively, W1(r) and W2(r) are 1- and q-dimensional standard Brownian motions on [0,1] and [0,1]q, respectively, and they are independent of each other. Apparently, the second moment of xt is not proportional to r, and hence we cannot use the exiting results for multiple structural changes. We also note that a model considered by Vogelsang (1997) is a special case of our model with no stationary regressors (q=0) and thus his tests are equivalent to ours if q=0.

3 Tests for multiple structural changes

In this section, we define the test statistics for multiple structural changes and derive their limiting distributions. Let β^=[β^′1,…,β^m+1]′ be the least squares estimator of the coefficients for a given number of breaks m with change points {T1,…,Tm}, Σˆ=diag{Σˆ1,…,Σˆm+1} be an (m+1)p×(m+1)p block-diagonal matrix where Σ^j=(∑t=Tj−1+1Tjxtx′t)−1, and σˆ2 be a consistent estimator of σ2. Typically, σˆ2=T−1∑t=1Tεˆt2, where εˆt is the regression residual. Then, the Wald test statistic for the null hypothesis of H0 is given by

and Λm={λ1,…,λm} with λj=Tj/T for j=1,…,m being break fractions. We set λ0=0 and λm+1=1 for convention because T0=0 and Tm+1=T. Using WT(Λm), we construct the exp-type, the sup-type, and the avg-type tests, as in the literature.

where Λm∈={(λ1,…,λm):λj−λj−1≥∈forj=1,…,m+1} for a given trimming parameter ∈ and T∗ is the number of permissible sets of break fractions included in Λm∈. The trimming parameter ∈ should be small; ∈=0.05, 0.1, and 0.15 have often been considered in the literature. See Bai and Perron (2006) for the discussion on how to choose ∈; roughly speaking, the empirical sizes of the tests become closer to the nominal one for larger ∈, and the large value of the trimming parameter is required for complicated models. As discussed in Andrews, Lee, and Ploberger (1996), the exp-type test is optimal against the alternative of the large magnitude of structural changes, whereas the avg-type test is asymptotically most powerful against the alternative of small changes.

The above three tests require the specific number of breaks m under the alternative before constructing the test statistics, but, if we do not want to prespecify the number of breaks, then we may set only the maximum number of breaks given by M and consider the following weighted exp-type and avg-type tests^[2] suggested by Andrews, Lee, and Ploberger (1996) as well as the weighted double maximum test proposed by Bai and Perron (1998):

where ci(p,α,m) for i=exp, sup, and avg are the critical values of eqs [2]–[4] for a given m with significance level α. These weights are suggested by Bai and Perron (1998).

The limiting distributions of these test statistics are given by the following theorem.

Theorem 1Assume that Assumptions A1 and A2 hold. Then, under the null hypothesis,

where

Remark 2When the regressors are homogeneous withΩr=rΩ, we haveG(r)=Ω1/2B(r), whereB(r)is a p-dimensional standard Brownian motion on[0,1]p, and it is then not difficult to show that

Remark 3Another important aspect of Theorem 1 is that when obtaining the critical values by simulations, we need to invert onlyp×pmatricesΩλjfrom the expression ofW(Λm), whereas the original definition of the Wald test statistic requires the inverse of themp×mpmatrixRΣˆR′. As a result, we can save the computational time by using expression [5]. We also develop a computationally efficient method for the avg-type test in the next section.

As we can see from Theorem 1, the limiting distributions of the test statistics depend on the structure of Ωr, and then, we need to calculate critical values for a given regressor xt. The dependency of critical values on xt has sometimes been observed in different situations in the literature. For example, the critical values for unit root tests depend on whether a linear trend is included as a regressor, while the LR tests for cointegrating rank are known to have different distributions depending on the structure of the deterministic term.

In the following, we focus on the case where xt includes a linear trend, which is widely used in practical analysis. More precisely, let us consider the case where

with x1t,…,xqt being stationary variables and lagged dependent variables. In this case, we have the following corollary.

Corollary 1Assume that Assumptions A1 and A2 hold. Then, under the null hypothesis withxtgiven by eq. [7], Theorem 1 holds withW(Λm)=Q1,m+Q2,m, whereQ1,mis given by eq. [5] with

The result in Corollary 1 is similar to that given by Bai (1999) for testing the null hypothesis of ℓ breaks against the alternative of ℓ+1 breaks; the limiting distribution is the sum of the two independent distributions corresponding to (constant plus) a linear trend and stationary regressors. We can see that the limiting distribution of Bai’s (1999) test with ℓ=0 is the same as ours with m=0.

4 Computation of critical values

Since the limiting distributions of the test statistics are nonstandard, we obtain the critical values by simulations with G(r) approximated by 1,000 partial sums of the appropriate pseudo-random variables. For example, if G(r) is a standard Brownian motion, then we approximate G(r) using the normalized partial sums of i.i.d.N(0,1) pseudo-random variables. However, the computation of the critical values is not necessarily easy for large values of m because the number of permissible sets of breaks is proportional to Tm, so the direct calculation of all permissible Wald test statistics is computationally too expensive when m≥4. For the sup-type test, Bai and Perron (2003a) gives an efficient algorithm for the computation of the test statistics, which requires operations of order O(T2) for any given number of breaks; we can use this in our case.

For the avg-type test, we can also calculate the critical values computationally efficiently.^[3] Let Q1(Ta,Tb) be the summand of eq. [5] approximated by the above method with T observations given Ta=λaT and Tb=λbT. Since the distance between two consecutive break points must be at least h=∈T, the permissible ranges of T1,T2,…,Tm are T1=h,h+1,…,T−hm, T2=T1+h,T1+h+1,…,T−h(m−1), …, Tm=Tm−1+h,Tm−1+h+1,…,T−h; then, the limiting distribution of the avg-type test statistic can be approximated by

However, eq. [8] requires the summation operators of order O(Tm), which is computationally expensive as explained above. Instead, we calculate the limiting distributions by noting that each of Q1(Tj,Tj+1) appears in eq. [8] many times; if we count them, we can save computational time. For example, Q1(T1,T2) appears as many times as the permissible number of allocations of T3,…,Tm in [T2,T]. Since, in general, the permissible number of combinations of ℓ breaks in [Ta,Tb] with two consecutive breaks’ distance being larger than h is given by

which is obtained by direct calculations, we can see that Q1(T1,T2) appears kh(T2,T,m−2) times in eq. [8]. Similarly, we observe Q1(Tm,Tm+1) as many times as the number of allocations of T1,…,Tm−1 in [1,Tm], which is given by kh(1,Tm,m−1). For the case of Q1(Tj,Tj+1) for j=2,…,m−1, there are j−1 and m−j−1 breaks allocated in [1,Tj] and [Tj+1,T], respectively. Then, we can see that

where kh(Ta,Tb,0)=1 for convention. We can see that the number of summation operators on the right-hand side of eq. [9] is proportional to O(T2) for any given number of m.

On the other hand, we cannot find an efficient computational method for the exp-type test. Therefore, we consider the exp-type test only up to m=3.

The critical values in the case of a linear trend are given in Tables 1–3 for ∈=0.05, 0.10, and 0.15 and q=0 to 9, where q is the number of homogeneous regressors. They are obtained by approximating Brownian motions by 1,000 partial sums of i.i.d.N(0,1) pseudo-random variables with 10,000 replications. Because of the above reason, the critical values for the exp-type test are given for only up to m=3 and M=3, whereas those for the sup-type and avg-type tests are obtained for up to m=5 and M=3 and 5. As in the case of homogeneous regressors, the critical values get larger as q and/or m increase.