Estimating and testing multiple structural changes in linear models using band spectral regressions

Econometrics Journal (2013), volume 16, pp. 400429.doi: 10.1111/ectj.12010

Estimating and testing multiple structural changes in linear modelsusing band spectral regressions

YOHEI YAMAMOTO AND PIERRE PERRONHitotsubashi University, Department of Economics, Naka 2-1, Kunitachi,

Tokyo 186-8601, Japan.E-mail: [email protected]

Department of Economics, Boston University, 270 Bay State Road, Boston, MA 02215, USA.E-mail: [email protected]

First version received: October 2012; final version accepted: April 2013

Summary We provide methods for estimating and testing multiple structural changesoccurring at unknown dates in linear models using band spectral regressions. We considerchanges over time within some frequency bands, permitting the coefficients to be differentacross frequency bands. Using standard assumptions, we show that the limit distributionsobtained are similar to those in the time domain counterpart. We show that when thecoefficients change only within some frequency band, we have increased efficiency of theestimates and power of the tests. We also discuss a very useful application related to contextsin which the data are contaminated by some low-frequency process (e.g. level shifts or trends)and that the researcher is interested in whether the original non-contaminated model is stable.All that is needed to obtain estimates of the break dates and tests for structural changes that arenot affected by such low-frequency contaminations is to truncate a low-frequency band thatshrinks to zero at rate log(T )/T . Simulations show that the tests have good sizes for a widerange of truncations so that the method is quite robust. We analyse the stability of the relationbetween hours worked and productivity. When applying structural change tests in the timedomain, we document strong evidence of instabilities. When excluding a few low frequencies,none of the structural change tests are significant. Hence, the results provide evidence to theeffect that the relation between hours worked and productivity is stable over any spectral bandthat excludes the lowest frequencies, in particular it is stable over the business-cycle band.

Keywords: Business-cycle, Band spectral regression, Hours-productivity, Low-frequencycontaminations, Multiple structural changes.

1. INTRODUCTION

This paper considers methods for estimating and testing multiple structural changes in linearmodels using band spectral regressions. Since the classic work by Hannan (1963), band spectralregressions have found wide applicability and have been useful for various problems when thecoefficients of linear regression models are suspected to be frequency-dependent. Engle (1974,1978) adopted Hannans insight to an econometric context and, for linear regression models,showed that the spectral least squares coefficients estimates and the associated test statistics have

C 2013 The Author(s). The Econometrics Journal C 2013 Royal Economic Society. Published by John Wiley & Sons Ltd, 9600 GarsingtonRoad, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA, 02148, USA.

JournalTheEconometrics

Breaks with band spectral regressions 401

the same properties as in the standard time domain regressions. He also considered the classicalChow test for a change in the coefficients across frequency bands.

Our paper tackles the problem of structural changes from a different angle. First, as hasbecome common now, we consider the possibility of multiple structural changes occurringat unknown dates. More importantly, instead of considering changes across frequencies, weconsider changes over time within some frequency bands, permitting the coefficients to bedifferent across frequency bands. We derive the appropriate methods to estimate the break datesand to construct the tests for structural changes. Using standard assumptions, we show that thelimit distributions obtained are similar to those in the time domain counterpart as derived byBai and Perron (1998) or Perron and Qu (2006); see Perron (2006) for a review. We show thatwhen the coefficients change only within some frequency band (e.g. the business-cycle), we canhave increased efficiency of the estimates of the break dates and increased power for the testsprovided, of course, that the user chosen band contains the band at which the changes occur. Ourframework can therefore be very useful in various empirical applications. For instance, using aninternational data set consisting of series covering a long span, Basu and Taylor (1999) documentthat the cyclical behaviour of the real wage (the relationship between aggregate output and realwages within some spectral band) may have been changing over time. Their analysis is based onchanges in the correlation coefficients across different spectral bands and different time segments.We provide a general framework to analyse such issues in a rigorous and systematic manner.

We also discuss a very useful application of testing for structural changes via a band spectralapproach. The framework we consider is one in which the data are contaminated by some low-frequency process and that the researcher is interested in whether the original non-contaminatedmodel is stable. For example, the dependent variable may be affected by some random levelshift process (a low-frequency contamination) but at the business-cycle frequency the model ofinterest is otherwise stable. We show that all that is needed to obtain estimates of the break datesand tests for structural changes that are not affected by such low-frequency contaminations isto truncate a low-frequency band that shrinks to zero at rate log(T )/T . Simulations show thatthe tests have good sizes for a wide range of truncations. The exact truncation does not reallymatter, as long as some of the very low frequencies are excluded. Hence, the method is quiterobust. We also show that our method delivers more precise estimates of the break dates and testswith better power compare to using filtered series obtained via a bandpass (BP) filter or from aHodrickPrescott (HP; 1997) filter.

Along this line, our method has enhanced potential applicability in wide range of problems inmacroeconomics, finance and other fields. Indeed, it has been shown for numerous problems thatestimates and tests are sensitive to the low-frequency components which are often driven by meanshifts or various types of trends. This feature also applies to issues related to structural changes.In a finance context, it has been documented that investigations of instabilities in stock returnspredictive regressions are largely driven by low-frequency components. For instance, mean shiftsin dividends can lead one to conclude that the dividend/price ratio no longer has predictivepower, e.g. Lettau and Van Nieuwerburgh (2008). Our framework allows one to draw conclusionsabout the stability of a relationship at some business-cycle frequency, say, without having tospecify the nature of the low-frequency movements. In a macroeconomic context, Fernald (2007)highlights the sensitivity of results about the effect of a productivity shock on hours worked basedon vector autoregressions identified from long-run restrictions to the specifications of the low-frequency components of hours worked and productivity. Our empirical application, reported inSection 5, sheds further light on this important issue. We analyse the stability of the relation

C 2013 The Author(s). The Econometrics Journal C 2013 Royal Economic Society.

402 Y. Yamamoto and P. Perron

between hours worked and productivity. When applying the structural change tests in the timedomain, or equivalently the full set of frequencies, we document strong evidence of instabilities.When excluding a few low frequencies, none of the structural change tests are significant. Hence,the results provide evidence to the effect that the relation between hours worked and productivityis stable over any spectral band that excludes the lowest frequencies, in particular it is stable overthe business-cycle band. This result has important implications for the analysis of the effect of atechnological shock on hours worked. It indicates that the various structural-based methods usedto assess the sign and magnitude of this effect should be carried using a frequency band thatexcludes the lowest frequencies or with a business-cycle band.

In view of this type of application of our methods, our work is related to a recent strandin the literature that attempts to deliver tests and estimates that are robust to low-frequencycontaminations. One example pertains to estimation of the long-memory parameter. It is bynow well known that spurious long memory can be induced by level shifts or various kindsof low-frequency contaminations. Perron and Qu (2007, 2010), McCloskey and Perron (2013)and Iacone (2010) exploit the fact that the level shifts or time trends will produce high peaksin the periodograms at a very few low frequencies, and suggests procedures that are robust byeliminating such low frequencies. Tests for spurious versus genuine long memory have beenproposed by Qu (2011) (see also Shimotsu, 2006). McCloskey and Hill (2013) provides a generalmethod applicable to the estimation of various time series models, such as ARMA, GARCH andstochastic volatility models.

The structure of the paper is the following. Section 2 introduces the framework adopted,the basic model, the assumptions imposed, the asymptotic distributions of the estimates ofthe break dates and of the tests for structural changes. Section 3 considers models with low-frequency contaminations and shows how the trimming of some low frequencies deliversestimates and tests having the same limit distribution as in the non-contaminated models.Section 4 presents simulation evidence showing that the procedures suggested have goodproperties in small samples and perform better than using filtered data. Section 5 illustrates theusefulness of our methods by considering the stability of the relation between hours workedand productivity. Section 6 provides brief concluding comments and the Appendix contains thetechnical derivations.

2. THE FRAMEWORK AND ASSUMPTIONS

2.1. The model

Consider a general multiple linear regression model with M breaks or M + 1 regimes. Thereare T observations and M is assumed known for now. The breaks occur at dates {T1, . . . , TM}.Let y = (y1, . . . , yT ) be the dependent variable and X a T by p matrix of regressors. DefineX = diag(X1, . . . , XM+1), a T by (M + 1)p matrix with Xm = (xTm1+1, . . . , xTm ) for m =

1, . . . ,M + 1, with the convention that T0 = 1 and TM+1 = T (each matrix Xm is a subsetof the regressor matrix X corresponding to regime m). Matrix X is a diagonal partition ofX, the partition being taken with respect to the set of break points {T1, . . . , TM}. It will alsobe convenient to define Ym = (yTm1+1, . . . , yTm ). The vector U = (u1, . . . , uT ) is the set ofdisturbances and = ( 1, . . . , M+1) is the (M + 1)p vector of coefficients. We consider the



general pure structural change model with restrictions on the coefficients, i.e.

Y = X + U, (2.1)where R = r with R a k by (M + 1)p matrix with rank k and r a k dimensional vector ofconstants. Note that this framework includes the case of a partial structural change model by anappropriate choice of the restrictions on the parameters (see Perron and Qu, 2006).

2.2. The band spectral regression

Band spectral regressions were early proposed by Hannan (1963) and have been adoptedsubsequently in the econometric literature, see in particular Engle (1974, 1978). The frameworkis useful in estimating linear regression models for which the coefficients are frequency-dependent. Many economic applications fit in this framework. For example, consider aconsumption function for which consumers are assumed to react to the transitory and permanentincome in different ways as the classical permanent income hypothesis suggests. Here, therelationship between income and consumption can be different for higher (transitory) andlower (permanent) frequency variations. More recently, the technique was found to be useful inestimating cointegrating relations by Phillips (1991). Also, Corbae et al. (2002) suggest that theremoval of time trends should be conducted in the frequency domain by estimating frequency-dependent coefficients using band spectral regressions.

We first provide a brief description of the basic principles underlying band spectralregressions. Consider the generic model for the mth segment where Ym is the dependentvariable and Xm is the matrix of regressors. The starting point is to apply a discrete Fouriertransformation to the data. Let W be an orthogonal Tm Tm matrix (Tm Tm Tm1)with wj,k = T 1/2m exp (ij (k 1)(2/Tm)) for its (j, k) component where, as usual,i = 1.1 Then the transformed data are, say, WYm and WXm. To have the analysis pertainto a particular band of interest, we follow the technique suggested by Corbae et al. (2002).Let the band of interest be BA = [l, h] (0 l < h ).2 It is often easier to describea certain frequency in terms of the position of the observation in the vector. Hence, we definejl = [lTm/ ] and jh = [hTm/ ] with [] returning the integer of the argument. Theband selection can then be applied with another linear operator consisting of a Tm Tmselection matrix A with ones for the j th diagonal elements for jl j jh and zeros for allother elements. The transformed dependent variable is now AWYm, the transformed regressorsare AWXm and the Ordinary Least Squares (OLS) estimate is

Am = (XmW AWXm)1(XmW AWYm),

=[

BA

Imx,T (j )]1 [

BA

Imxy,T (j )], (2.2)

1 Matrix W depends on the sample size of the segment m, but the subscript m is suppressed to ease notation since itshould cause no confusion. This applies to matrix A as well as to the quantities jl and jh that appear later.

2 Following the convention since Engle (1974), in practice we produce the observations in dual bands correspondingto [, ] and choose symmetric bands, i.e. at the same time. This way, the discrete Fourier transform willinclude both cos + i sin and cos i sin and we can avoid complex-valued quantities.



where Imx,T () is the matrix of sample cross-periodograms of xt in the mth regime and Imxy,T ()is a vector of cross-periodograms of xt and yt in the mth regime, both evaluated at frequency. If serial correlation in the errors is suspected, we can account for it using a matrix A definedwith an estimate of f mu ()1/2 in the diagonal of A instead of one, where f mu () is the powerspectrum of the error term at frequency . As a matter of notation, define nmA = jh jl + 1,the number of non-zero data points in the diagonal elements of A in the mth segment andNA =

M+1m=1 n

mA . Finally, we consider for now the case of fixed bands in the asymptotic analysis

so that l and h are fixed. This implies that jl and jh increase at the same rate as Tm so thatnmA remains a fixed portion of the sample size Tm. This framework is standard and does providea useful approximation in finite samples, as will be shown later.

With this background description, the estimates of the break dates in our model are definedas the solutions of the global least squares minimisation problem applied to band spectralregressions such that

( T1, . . . , TM ) = arg minT1,..,TM

SSRBAT (T1, . . . , TM ), (2.3)

where

SSRBAT (T1, . . . , TM ) =M+1m=1

(AWYm AWXm Am

)(AWYm AWXm Am), (2.4)with Am (m = 1, . . . ,M + 1) the band spectral least squares coefficient estimates for the selectedband BA defined by (2.2) with Ym and Xm, which contain the observations t = Tm1 + 1, . . . , Tm.

2.3. Assumptions

In order to derive the limit distribution of our estimates, we impose the following standardassumptions on the data, the errors and the break dates.

ASSUMPTION 2.1. For each segment m = 1, . . . ,M + 1, mt = {t }T0m

t=T 0m1+1=

{(x t , ut )}T0m

t=T 0m1+1is a jointly short-memory stationary time series, that is, mt =

j=0 c

mj

mtj ,

with mt i.i.d.(0, m) with finite fourth moments and coefficients cij satisfyingj=0 j

1/2cij < . Partitioned conformably, the spectral density matrix f m () of mt is

f m () =[f mx () 0

0 f mu ()

],

with f mx () a non-random positive definite matrix and f mu () a positive constant for any [l, h].ASSUMPTION 2.2. There exists an l0 > 0 such that for all l > l0, the minimum eigenvaluesof the sample periodogram matrix Ix,T () constructed by {xt }T

0m+l

t=T 0m+1 and that by {xt }T 0mt=T 0ml are

bounded away from zero (m = 1, . . . ,M) for some [l, h] .ASSUMPTION 2.3. The sample periodogram matrix associated with the spectral band [l, h],

h=l Ix,T (), constructed using {xt }lt=k for l k T , is invertible for some > 0.



ASSUMPTION 2.4. Let the Lr -norm of a random matrix Z be defined by Zr =(i j E|Zij |r )1/r for r 1. (Note that Z2 is the usual matrix norm or the Euclidian normof a vector.) With {Fi : i = 1, 2, . .} a sequence of increasing -fields, {xiui,Fi} forms a Lr -mixingale sequence with r = 2 + for some > 0. That is, there exist non-negative constants{i : i 1} and {j : j 0} such that j 0 as j and for all i 1 and j 0, we have:(a) E(xiui |Fij )r ij ; (b) xiui E(xiui |Fi+j )r ij+1; (c) maxi i K < ; (d)

j=0 j1+kj < ; (e) xi2r < C < and ui2r < N < for some K,C,N > 0.

ASSUMPTION 2.5. T 0m = [T 0m], where 0 < 01 < < 0M < 1.ASSUMPTION 2.6. Let T,m = m+1 m. Assume T,m = vTm for some m independentof T , where vT > 0 is a scalar satisfying vT 0 and T (1/2)vT for some (0, 1/2).In addition, E xt2 < K and E |ut |2/ < K for some K < and all t .ASSUMPTION 2.7. The minimisation problem defined by (2.3) is taken over all possiblepartitions such that Tm Tm1 T for some > 0.

These assumptions are standard in the literature. They follow Bai and Perron (1998) andPerron and Qu (2006) for the structural change problem and Corbae et al. (2002) for the bandspectral regression framework. It is important to note that these assumptions pertain to thetime domain model and are similar to those used in the standard structural change literature(Perron and Qu, 2006, for example). Doing so enables one to more easily verify the adequacyof the assumptions with respect to data in the same way as for standard methods. Assumption2.1 corresponds to Assumption 1 in Corbae et al. (2002), and it imposes stationarity withineach regime. It also implies Assumption A1 in Perron and Qu (2006). Assuming the cross-spectrum of ut and xt is 0 essentially rules out endogeneity. It can be relaxed by interpreting thecoefficients as the pseudo-true values, i.e. as the limits in probability of the inconsistent estimates.As shown in Perron and Yamamoto (2013), this still permits consistent estimation of the breakfractions and the confidence intervals for the estimates can be constructed in the usual manner.Assumptions 2.2 and 2.3 impose conditions that are the frequency domain analogues of A2 andA3 in Bai and Perron (1998). Assumption 2.4 imposes mild conditions on the regressors anderrors which permit a wide class of potential correlation structures in the errors and regressors. Italso allows lagged dependent variables as regressors when the errors are a martingale differencesequence. Assumption 2.5 imposes the break points to be asymptotically distinct, a standardcondition needed to have non-degenerate limit distributions. Assumption 2.6 is also standard inthe literature. It dictates an asymptotic framework whereby the magnitudes of the breaks decreaseas the sample size increases, a feature needed to derive a limit distribution of the estimates of thebreak dates that does not depend on the exact distribution of the errors.

2.4. Asymptotic properties

We now establish the consistency, rate of convergence and asymptotic distribution of theestimates of the break dates defined by (2.3) and (2.4). We start with the following importantlemma.

LEMMA 2.1. For the full spectrum case, that is A = I , the following equivalence holds:SSRBAT ( T1, . . . , TM ) SSRT ( T1, . . . , TM ),



where SSRT ( T1, . . . , TM ) is the overall sum of squared residuals when the structural changemodel is applied using a standard time domain procedure for model (2.1), viz.

SSRT ( T1, . . . , TM ) =M+1m=1

(Ym Xm m)(Ym Xm m),

with m = (XmXm)1(XmYm).This lemma shows that the global minimisation problem (2.3) applied to the full spectrum

reduces to the standard time domain structural change problem for model (2.1). This is anintuitive and useful property and a short proof is given in the Appendix. This equivalence willbe useful in deriving the asymptotic results when the analysis is restricted in a certain bandspectrum. To see this, consider the following time domain data-generating process for the mthsegment instead of (2.1):

yt = xt m + ut , t = Tm1 + 1, . . . , Tm, (2.5)where t =

{xt , u

t

}is a process with the same spectral density as that of t at the Fourier

frequencies BA and has no variation for / BA. In matrix notation,Y m = Xmm + U m,

where Xm = W1AWXm,U m = W1AWUm and Y m = W1AWYm. Note first that pre-multiplying by W1 applies an inverse Fourier transform to the variables so that we are back tothe time domain and, second, that the coefficient vector m is not affected by this transformation.As discussed in the Appendix, the asymptotic properties of the series xt are investigated usingthe following structure sometimes called ideal (but infeasible) BP filter:

xt =

j= bjxtj ,

with b0 = [(h l)/2 ] and bj = [(sin(hj ) sin(lj ))/(2j )]. By applying Lemma 2.1 to(2.5), we obtain an equivalence of the global sum of squared residuals pertaining to model (2.5)in the time domain and that pertaining to (2.3) with an arbitrary selector matrix A . This impliesthat the asymptotic properties of the estimates of the structural change model involving a bandspectral regression can be analysed by investigating its time domain counterpart (2.5). To thiseffect, we now state a lemma applicable to the variables in the time domain model (2.5).LEMMA 2.2. Let T 0m = T 0m T 0m1 and suppose Assumptions 2.12.5 hold. With t defined for any non-empty band BA, the following holds, uniformly in s:(a) (T 0m)1

T 0m1+[sT 0m]t=T 0m1+1

xt xt

p sQm; (b) (T 0m)1T 0m1+[sT 0m]

t=T 0m1+1u2t

p s 2m ; (c) (T 0m)1T 0m1+[sT 0m]r=T 0m1+1

T 0m1+[sT 0m]t=T 0m1+1

E(xr xt ur ut )psm; (d) (T 0m)1/2

T 0m1+[sT 0m]t=T 0m1+1

xt ut Bm(s);

where Qm and m are p p positive definite matrices, 2m is a positive scalar andBm(s) is a multivariate Gaussian process on [0,1] with mean zero and covarianceE[Bm(s)B m(u)] = min {s, u}m.

Lemma 2.2 plays an essential role in establishing the asymptotic distribution of the estimatesof the break dates and test statistics. The main result is stated in the following theorem.



THEOREM 2.1. Let Tm be the estimates defined by (2.3) and m = Tm/T for m = 1, . . . ,M .Then, under Assumptions 2.12.7, we have for any non-empty choice of the band BA: (a) Forevery > 0, there exists a C < , such that for all large T , P (|T v2T (m 0m)| > C) < . (b)(

mQmm

)v2T(

Tm T 0m) arg max

sZ(m)(s) (m = 1, . . . ,M),

where

Z(m)(s) ={m,1W

(m)1 (s) |s| /2, for s 0,

mm,2W(m)2 (s) m |s| /2, for s > 0,

with m = mQm+1m/mQmm, 2m,1 = mmm/mQmm, 2m,2 = mm+1m/mQ

m+1m, and W

(m)j (j = 1, 2) are independent Wiener processes defined on [0,).

REMARK 2.1. Note that Qm and m can also be expressed as Qm = hl

f mx ()d and m = hl

f mx ()f mu ()d, which are fixed matrices under Assumption 2.1 for any l and h, 0 l < h .

2.5. Testing for structural changeWe now consider the problem of testing the null hypothesis of no break versus a fixed number(M) of breaks and show that the conventional SupF test applied to a band spectral regression,has the same limit distribution as in the standard time domain setup (see Andrews, 1993, and Baiand Perron, 1998). Note that, as pointed out by Engle (1974), the number of degrees of freedomis NA and not T . For the pure structural change case, the SupF test is then defined by

SupFT = sup(1,...,M )M

FT (1, . . . , M ),

where

FT (1, . . . , M ) =(NA (M + 1)p k

Mp

)AH (H ( X X)1H )1H A

SSRBAT (T1, . . . , TM ), (2.6)

with H the usual matrix such that (H) = (1 2, . . . , M M+1), SSRBAT (T1, . . . , TM ) isas defined in (2.4) and M = {(1, . . . , M ) : |m+1 m| , 1 , M 1 } for somesmall trimming value . Also X = diag(X1, . . . , XM+1) with Xm = W1AWXm.3 The limitingdistribution of the SupFT statistic is described in the following theorem.

THEOREM 2.2. With Assumptions 2.12.5, 2.7 and m = 2m Q2m (i.e. no serial correlationin the errors) and under the null hypothesis of no break, M = 0, we have SupFT sup(1,...,M )M F (1, . . . , M ) where

F (1, . . . , M ) = 1Mp

Mm=1

mWp(m+1) m+1Wp(m)2m+1m(m+1 m) ,

3 W is a Tm by Tm matrix of discrete Fourier transformation and A is the corresponding selector matrix defined inSection 2.2 Since W W = I , we can equivalently use Xm = AWXm to construct X

X.



and Wp(s) is a p vector of independent Wiener processes with p the number of parameterssubject to change.

When the model exhibits a partial structural change with some regressors having constantcoefficients, the standard results of Bai and Perron (1998) apply by following Perron and Qu(2006) with an appropriate change of notations. Note that the sequential tests for l versus l +1 breaks (which permit estimating the number of breaks M) and the double maximum testsinvestigated in Bai and Perron (1998) can also be constructed with appropriate changes for theregressors, residuals, coefficient estimates and the number of observations as described earlier.They have the same limit distributions as those stated in Bai and Perron (1998). Serial correlationin the errors is accounted for using heteroscedastic robust standard errors in the frequency domainas pointed out by Engle (1974).

3. ESTIMATING AND TESTING STRUCTURAL CHANGES WITHCONTAMINATED MODELS

In this section, we discuss a very useful application of testing for structural changes via a bandspectral approach. The framework we consider is one in which the data are contaminated bysome low-frequency process and that the researcher is interested in whether the original non-contaminated model is stable. For example, the dependent variable may be affected by somerandom level shift process (a low-frequency contamination) but at the business-cycle frequenciesthe model of interest is otherwise stable.

Let {dt } be an unobservable contaminating component whose exact form is not known to theresearcher. The specification of the data-generating process is then.

yDt = xtm + dt + ut (3.1)

for m = 1, . . . ,M + 1, or equivalently

YD = X + D + U

in vector form with D = (d1, . . . , dT ) a T 1 vector. The interest is in testing whether thecoefficient vector is stable over time without requiring a particular model for the contaminatingcomponent dt . The only requirement is that the contaminating component is dominant at lowfrequency and that it is uncorrelated with the regressors and the errors, which are standardconditions in this literature and reasonable given the types of contaminations analysed (seebelow). The specific conditions required are stated in the following assumptions.ASSUMPTION 3.1. The cross-spectral density f md () of t and dt is 0 for any [l, h].ASSUMPTION 3.2. Imd,T (j ) = Op(Tmj2) for all j = 1, . . . , [Tm/2].

Assumption 3.1 ensures the strict exogeneity of the process dt in the model. Assumption 3.2states that the contaminating component has a periodogram that is divergent for j < (Tm)1/2but is negligible for j > (Tm)1/2. Hence, by restricting the analysis to a set of frequenciesthat exclude a neighbourhood around zero, one can obtain results that are not affected bythe contamination. Many processes of interest satisfy Assumption 3.2. The following is a



non-exhaustive list: (a) a random level shift process of the form

dt =t

j=1T,j , T ,j = T,jj , (3.2)

where j i.i.d.(0, 2 ) with finite moments of all orders, T,j i.i.d. Bernoulli(p/T , 1) forsome p 0, and with the components T,j and j being mutually independent; (b) deterministiclevel shifts of the form

dt =N

n=1cnI (Tn1 < t Tn), (3.3)

where N is a fixed positive integer and I () is the indicator function; (c) deterministic trends ofthe form

dt = (t/T ), (3.4)where () is a deterministic non-constant function on [0,1] that is either Lipschitz continuousor monotone and bounded. The fact that Assumption 3.2 is satisfied for the random level shiftprocess (3.2) was shown in Perron and Qu (2007), for the deterministic level shifts process it wasshown in McCloskey and Perron (2013), while Qu (2011), building on results by Kunsh (1986),showed it for the general deterministic trend function. To have more generality and methodswith increased efficiency, we allow jl to approach 0 at some rate so that what is excluded isonly a shrinking frequency band near zero. Recall that the lower bound of the truncation isjl = [lTm/ ]. We start with a result that states the relationship between the global sums ofsquared residuals from the band spectral regressions obtained from the original and contaminatedmodels.

LEMMA 3.1. Consider model (3.1) with {dt } satisfying Assumptions 3.1 and 3.2. With h > 0,let jl and jl/ log(Tm) as T . Then,

SSRD,BAT (T1, . . . , TM ) = SSRBAT (T1, . . . , TM ) + T + op(1),with T = op(T ), where

SSRD,BAT (T1, . . . , TM ) =M

m=1

(AWYDm AWXm D,Am

) (AWYDm AWXm D,Am

),

with D,Am = (XmW AWXm)1(XmW AWYDm ) and YDm = [yDTm1+1, . . . , yDTm ].Since SSRBAT (T1, . . . , TM ) = Op(T ), the lemma shows that, under the stated assumptions,

one obtains the asymptotic equivalence of the sum of squared residuals given a set of breakdates between the models with and without the contaminating term. What is required is that acertain low-frequency band that shrinks to zero at rate l log(T )/T is truncated, the bandspectrum estimates of the break dates are then not affected, at least in large samples, by anunknown contaminating component {dt } specified by Assumptions 3.1 and 3.2. If one restrictsthe analysis to a fixed band BA = [l, h] with l any fixed positive number, then jl = O(T )and the requirement is automatically satisfied. This provides a method to obtain estimates andtests that are robust to such misspecifications. The results are formally stated in the followingproposition.



PROPOSITION 3.1. Consider the contaminated model (3.1) with Assumptions 2.12.7, 3.13.2holding. With h > 0, let jl and jl/ log(Tm) as T , then the band spectrumestimates of the multiple structural change dates

( T1, . . . , TM ) = arg minT1,...,TM

SSRD,BAT (T1, . . . , TM )

satisfy the properties stated in Theorem 2.1. Define the SupF test statistic by

FT =(NA (M + 1)p k

Mp

)D,AH (H ( X X)1H )1H D,A

SSRD,BAT (T1, . . . , TM ),

where D,A = ( D,A1 , . . . , D,AM+1) with D,Am = (XmW AWXm)1(XmW AWYDm ) and A is aselection matrix with zeros in the first jl diagonal elements. Then, under the null hypothesisof no break, M = 0, and with m = 2m Qm (i.e. no serial correlation in the errors), the SupFtest has the limiting distribution as stated in Theorem 2.2.

4. MONTE CARLO SIMULATIONS

In this section, we present simulation results about the properties of the estimates of the breakdates obtained from the band spectral regression (bias, standard deviation, coverage rate of theasymptotic distribution) and the size and power of the test for structural change. We start inSection 4.1 with the case of no low-frequency contamination and in Section 4.2 we considermodels with such contaminations. In Section 4.3 we compare the proposed band spectralapproach with the standard Bai and Perron (1998) method using filtered data, via a BP filteras suggested by Baxter and King (1999) or after applying a HP filter.4

4.1. Models without contamination

The data-generating process used is

yt = xtt + ut , t = 1, . . . , T ,where the regressor xt is a stationary ARMA(1,1) process with a constant mean :

xt = + zt ,zt = zt1 + et + et1,

with et and ut sequences of i.i.d. N (0, 1) random variables independent of each other. Weconsider a single break model

t ={ c, for t Tb,c, for t > Tb.

We consider three cases for the type of regressors: for case 1, xt is uncorrelated so that (, ) =(0, 0); for case 2, xt is MA(1) with = 0 and = 0.5; for case 3, xt is an AR(1) process with

4 To have increased computational efficiency and to avoid potential problems associated with complex numbers, weadopted the finite Fourier transforms in real term proposed by Harvey (1978).



Table 1. Finite sample properties of the estimates of the break dates.(T = 100)

l 0 0 0 /2 /16 /4 /2h /4 /2 /2+ /2

bias DGP-1 for xt 0.002 0.002 0.002 0.001 0.003 0.011 0.002DGP-2 for xt 0.012 0.002 0.004 0.006 0.004 0.012 0.001DGP-3 for xt 0.005 0.006 0.003 0.003 0.005 0.011 0.007

s.d. DGP-1 for xt 0.198 0.204 0.189 0.208 0.195 0.207 0.201DGP-2 for xt 0.193 0.195 0.189 0.205 0.189 0.209 0.204DGP-3 for xt 0.207 0.205 0.190 0.208 0.193 0.212 0.210

cov DGP-1 for xt 0.884 0.892 0.904 0.900 0.932 0.932 0.925DGP-2 for xt 0.886 0.886 0.910 0.903 0.947 0.924 0.910DGP-3 for xt 0.859 0.877 0.897 0.891 0.933 0.928 0.926

(T = 200)l 0 0 0 /2 /16 /4 /2h /4 /2 /2+ /2

bias DGP-1 for xt 0.007 0.015 0.005 0.001 0.005 0.004 0.004DGP-2 for xt 0.000 0.002 0.001 0.009 0.000 0.016 0.005DGP-3 for xt 0.002 0.003 0.000 0.001 0.000 0.015 0.010

s.d. DGP-1 for xt 0.176 0.189 0.180 0.201 0.208 0.197 0.203DGP-2 for xt 0.169 0.182 0.170 0.194 0.197 0.205 0.208DGP-3 for xt 0.182 0.181 0.183 0.204 0.199 0.207 0.209

cov DGP-1 for xt 0.873 0.883 0.905 0.920 0.932 0.932 0.918DGP-2 for xt 0.872 0.899 0.900 0.926 0.924 0.929 0.919DGP-3 for xt 0.860 0.884 0.902 0.931 0.915 0.937 0.943

Notes: s.d. and cov denote the standard deviations and coverage ratios, respectively. is set at .15 .

= 0.5 and = 0. We set = 1 in all cases. For the choice of the bands, we consider severalcases which are popular in empirical applications. The first group pertains to low-frequencybands with (l, h) = (0, /4) and (0, /2). The second group corresponds to typical seasonaland business-cycle bands with quarterly data given by (l, h) = ([1/2 0.15], [1/2 +0.15] ) and (/16, /2), respectively. The third group consists of high-frequency bands with(l, h) = (/4, ) and (/2, ). We tried several other types of bands and the results weresimilar. We set T = 100 and 200 and the break date is at mid-sample, so that Tb = 50 forT = 100 and Tb = 100 for T = 200. The number of replications is 1000.

We first consider the properties of the estimates of the break fraction Tb/T with c = 0.1.Table 1 reports results for the bias and standard deviation of the break fraction estimates.The bias is very close to zero for all cases considered, which supports the consistency resultfor the break fractions in Theorem 2.1(a). Table 1 also reports the coverage rates obtained fromthe asymptotic distribution of the break date estimates for 90% nominal confidence intervals. Theresults show the exact coverage rates to be very close to the nominal level in all cases confirmingthe adequacy of the limiting distribution as an approximation to the finite sample distribution. The



Table 2. Exact size of the SupF test: 5% nominal size.(T = 100)

l 0 0 0 /2 /16 /4 /2h /4 /2 /2+ /2 DGP-1 for xt 0.07 0.08 0.05 0.11 0.07 0.05 0.06DGP-2 for xt 0.07 0.08 0.05 0.10 0.06 0.05 0.06DGP-3 for xt 0.07 0.08 0.05 0.12 0.08 0.06 0.07

(T = 200)l 0 0 0 /2 /16 /4 /2h /4 /2 /2+ /2 DGP-1 for xt 0.05 0.06 0.05 0.09 0.06 0.04 0.05DGP-2 for xt 0.06 0.06 0.05 0.08 0.07 0.05 0.06DGP-3 for xt 0.06 0.06 0.05 0.07 0.05 0.05 0.07

Note: is set at .15 .

results with T = 200 are similar with, as expected, a reduction in the variance of the estimates.We also computed the size of the heteroscedastic robust SupF test when the errors follow anAR(1) process and obtained broadly similar results.

Table 2 shows the finite sample size properties of the SupF test with c = 0 for the 5%nominal size. The results show that the exact size is very close to 5% in all cases. We nextconsider its power. Figure 1 shows the rejection frequency as a function of the magnitude of thebreak c. The three panels correspond to the cases with low, middle and high-frequency bands.In all cases, the results show good power, which approaches one quickly. As expected, using thefull spectrum gives tests with the highest power. This is due to the fact that the data are generatedwith coefficients that are the same across frequencies. Of more interest are cases for which thecoefficients change only in some particular frequency band, a problem we address next.

We now consider the power of the SupF test when the true data-generating process has astructural break only in a particular spectral band B0A. In such cases, we expect that power wouldbe highest when the band used in constructing the test BA is the same as B0A, showing that testsfor structural change based on our band regression framework can yield higher power. As weshall see, this is indeed the case. To this end, the data-generating process used is

yt = xAt t + xCt + ut , t = 1, . . . , T ,where

xAt ={xt , if B0A,0, otherwise,

xCt ={xt , if / B0A,0, otherwise,

and t is the same as in the previous experiments but is a constant (set at = 0) so thata structural change is present only in the frequency band B0A. The following four bands areconsidered for B0A: (a) a low-frequency band (0, /4), (b) a high-frequency band (/4, ), (c)



Figure 1. Finite sample power of the SupF test with a break common to all frequencies.



Figure 2. Finite sample power of the SupF test with a break in a particular spectral band.

a seasonal frequency band ([1/2 0.15], [1/2 + 0.15] ) and (d) a business-cycle frequencyband (/16, /2). In Figure 2, results are presented for the power of the SupF test using thetrue spectral band B0A and the full spectrum since the latter is equivalent to the standard timedomain structural break test. The results show that important power gains can be achieved usingtests based on a band spectral regression if one uses the correct band in which the change occurs.Note that the power gains are more important when the band in which the change occurs consistsof higher frequencies. As expected, if the band considered is one in which no break occurs, thenpower is equal to the size of the test, see panels (a) and (b).

4.2. Models with contaminating components

We now consider models with a contaminating component and evaluate how the truncation ofthe low frequencies helps in obtaining tests with good size and power properties. The data aregenerated by

yt = xtt + dt + ut .



Table 3. Exact size of the SupF test with contaminated models: 5% nominal size.Case D1: deterministic level shift

jl 0 1 5 logT logT 2 T 0.5 T 0.6

DGP-1 for xt 1.00 0.04 0.04 0.05 0.05 0.05 0.05DGP-2 for xt 1.00 0.03 0.04 0.03 0.04 0.03 0.05DGP-3 for xt 0.99 0.06 0.07 0.07 0.07 0.07 0.07

Case D2: random level shiftsjl 0 1 5 logT logT 2 T 0.5 T 0.6


Case D3: linear trend with a breakjl 0 1 5 logT logT 2 T 0.5 T 0.6


Case D4: quadratic trendjl 0 1 5 logT logT 2 T 0.5 T 0.6


We consider the following four cases for the contaminating component dt , which all satisfyAssumptions 3.1 and 3.2:

CASE 1. deterministic level shifts: dt = c1I (1 t < TD) + c2I (TD t T );CASE 2. random level shifts: dt =

nj=1 T,j j , where j i.i.d. N (0, 1) and T,j i.i.d.

B(p/T , 1), j and T,j are independent;CASE 3. linear trend with a break: dt = 1tI (1 t < TD) + 2tI (TD t T );CASE 4. quadratic trend: dt = t2.

The parameter values were selected in order to have the long-run variance of all fourprocesses be of similar magnitude. To that effect, we set (c1, c2) = (1, 1), p = 5, (1, 2) =(0.02, 0.01) and = 0.01. Although these values are arbitrary, simulations using other valuesyielded qualitatively similar results. The break date of the contaminating processes D1 andD3 was set to TD = 50. We only report the results for T = 100 (those for T = 200 werequalitatively similar). The specifications for the other components xt , t and ut are as in theprevious subsection. We consider the following truncations jl : the integer values of 1, 5, log(T ),log(T 2), T 0.5 and T 0.6.

Table 3 provides the exact sizes of the SupF test for a nominal 5% size test according tothe pattern of {dt }, the truncations and the DGP for xt . Of importance is the fact that for allcases serious size distortions are present when no truncation is applied. However, the exact sizeis much closer to the nominal level when a truncation is applied. For power, Table 4 presents the



Table 4. Finite sample power of the SupF test with contaminated models: non-size-adjusted.Case D1: deterministic level shift

jl

c 0 1 5 logT logT 2 T 0.5 T 0.6

0.0 1.00 0.04 0.04 0.05 0.05 0.05 0.050.1 1.00 0.07 0.09 0.09 0.10 0.10 0.100.2 1.00 0.29 0.31 0.31 0.31 0.30 0.300.3 1.00 0.60 0.60 0.61 0.60 0.59 0.580.4 1.00 0.86 0.84 0.85 0.84 0.83 0.810.5 1.00 0.97 0.96 0.97 0.95 0.95 0.950.6 1.00 1.00 1.00 1.00 1.00 1.00 0.990.7 1.00 1.00 1.00 1.00 1.00 1.00 1.000.8 1.00 1.00 1.00 1.00 1.00 1.00 1.000.9 1.00 1.00 1.00 1.00 1.00 1.00 1.001.0 1.00 1.00 1.00 1.00 1.00 1.00 1.00

Case D2: random level shiftsjl

c 0 1 5 logT logT 2 T 0.5 T 0.6

0.0 0.65 0.04 0.05 0.05 0.05 0.05 0.050.1 0.67 0.08 0.12 0.12 0.13 0.12 0.130.2 0.71 0.25 0.29 0.29 0.29 0.29 0.270.3 0.78 0.52 0.60 0.60 0.60 0.59 0.560.4 0.82 0.74 0.84 0.83 0.82 0.82 0.800.5 0.84 0.93 0.96 0.96 0.96 0.96 0.950.6 0.87 0.97 0.99 0.99 0.99 0.99 0.990.7 0.90 0.99 1.00 1.00 1.00 1.00 1.000.8 0.94 1.00 1.00 1.00 1.00 1.00 1.000.9 0.96 1.00 1.00 1.00 1.00 1.00 1.001.0 0.96 1.00 1.00 1.00 1.00 1.00 1.00

Case D3: linear trend with a breakjl

c 0 1 5 logT logT 2 T 0.5 T 0.6

0.0 0.70 0.03 0.05 0.05 0.06 0.06 0.070.1 0.92 0.09 0.09 0.10 0.09 0.10 0.090.2 0.99 0.26 0.28 0.28 0.27 0.28 0.260.3 1.00 0.61 0.61 0.63 0.61 0.62 0.580.4 1.00 0.85 0.84 0.85 0.84 0.84 0.81

Continued



Table 4. Continued.Case D3: linear trend with a break

jl

c 0 1 5 logT logT 2 T 0.5 T 0.6

0.5 1.00 0.97 0.97 0.98 0.97 0.97 0.960.6 1.00 1.00 1.00 1.00 1.00 1.00 1.000.7 1.00 1.00 1.00 1.00 1.00 1.00 1.000.8 1.00 1.00 1.00 1.00 1.00 1.00 1.000.9 1.00 1.00 1.00 1.00 1.00 1.00 1.001.0 1.00 1.00 1.00 1.00 1.00 1.00 1.00

Case D4: quadratic trendjl

c 0 1 5 logT logT 2 T 0.5 T 0.6

0.0 0.46 0.04 0.04 0.04 0.05 0.05 0.050.1 0.85 0.09 0.11 0.11 0.11 0.12 0.110.2 0.97 0.29 0.30 0.31 0.30 0.30 0.290.3 1.00 0.62 0.62 0.64 0.60 0.60 0.570.4 1.00 0.87 0.87 0.87 0.86 0.85 0.830.5 1.00 0.98 0.97 0.97 0.95 0.95 0.940.6 1.00 1.00 0.99 1.00 0.99 0.99 0.990.7 1.00 1.00 1.00 1.00 1.00 1.00 1.000.8 1.00 1.00 1.00 1.00 1.00 1.00 1.000.9 1.00 1.00 1.00 1.00 1.00 1.00 1.001.0 1.00 1.00 1.00 1.00 1.00 1.00 1.00

non-size-adjusted powers and Table 5 displays the size-adjusted ones. We only report the casewith a white noise regressor (case 1) given that the results were qualitatively similar for the othercases. First, the size-adjusted power of the test without truncation is comparatively very small.Secondly, when a truncation is applied, the power is improved considerably. Thirdly, in general,power is not much sensitive to the particular choice for the truncation rule.

The size and power results are comforting since any reasonable choice of the truncation rule,say greater than or equal to log(T ) and less than T 0.6, will lead to test with similar properties.What is important is that some truncation be done, even truncating a single frequency yieldsdramatic improvements over the full sample-based tests.

4.3. Comparisons with filtered seriesAn issue of interest is how our method compares to simply using filtered data prior to estimatingand testing for structural changes. To provide some answers to this question, we compare theproperties of the break date estimates and the structural change tests based on our band spectralregression (BSR) approach with standard methods applied to filtered series. For the latter, BP



Table 5. Finite sample power of the SupF test with contaminated models: size-adjusted.Case D1: deterministic level shift

jl

c 0 1 5 logT logT 2 T 0.5 T 0.6

0.0 0.05 0.05 0.05 0.05 0.05 0.05 0.050.1 0.19 0.11 0.11 0.10 0.11 0.10 0.100.2 0.49 0.37 0.34 0.31 0.32 0.31 0.300.3 0.77 0.68 0.63 0.61 0.61 0.60 0.580.4 0.90 0.91 0.87 0.85 0.85 0.83 0.820.5 0.98 0.99 0.97 0.97 0.95 0.95 0.950.6 0.99 1.00 1.00 1.00 1.00 1.00 0.990.7 1.00 1.00 1.00 1.00 1.00 1.00 1.000.8 1.00 1.00 1.00 1.00 1.00 1.00 1.000.9 0.99 1.00 1.00 1.00 1.00 1.00 1.001.0 1.00 1.00 1.00 1.00 1.00 1.00 1.00

Case D2: random level shiftsjl

c 0 1 5 logT logT 2 T 0.5 T 0.6

0.0 0.05 0.05 0.05 0.05 0.05 0.05 0.050.1 0.05 0.09 0.13 0.13 0.13 0.12 0.130.2 0.10 0.28 0.31 0.32 0.29 0.29 0.270.3 0.13 0.55 0.61 0.64 0.60 0.59 0.550.4 0.24 0.75 0.85 0.85 0.82 0.82 0.800.5 0.30 0.94 0.96 0.96 0.96 0.96 0.950.6 0.42 0.98 0.99 0.99 0.99 0.99 0.990.7 0.53 0.99 1.00 1.00 1.00 1.00 1.000.8 0.63 1.00 1.00 1.00 1.00 1.00 1.000.9 0.71 1.00 1.00 1.00 1.00 1.00 1.001.0 0.78 1.00 1.00 1.00 1.00 1.00 1.00

Case D3: linear trend with a breakjl

c 0 1 5 logT logT 2 T 0.5 T 0.6

0.0 0.05 0.05 0.05 0.05 0.05 0.05 0.050.1 0.21 0.13 0.10 0.10 0.07 0.09 0.070.2 0.53 0.32 0.28 0.29 0.22 0.26 0.220.3 0.84 0.68 0.62 0.65 0.55 0.61 0.520.4 0.96 0.89 0.85 0.85 0.80 0.83 0.770.5 1.00 0.99 0.98 0.98 0.96 0.97 0.94

Continued



Table 5. Continued.Case D3: linear trend with a break

jl

c 0 1 5 logT logT 2 T 0.5 T 0.6

0.6 1.00 1.00 1.00 1.00 1.00 1.00 0.990.7 1.00 1.00 1.00 1.00 1.00 1.00 1.000.8 1.00 1.00 1.00 1.00 1.00 1.00 1.000.9 1.00 1.00 1.00 1.00 1.00 1.00 1.001.0 1.00 1.00 1.00 1.00 1.00 1.00 1.00

Case D4: quadratic trendjl

c 0 1 5 logT logT 2 T 0.5 T 0.6

0.0 0.05 0.05 0.05 0.05 0.05 0.05 0.050.1 0.22 0.11 0.13 0.13 0.11 0.12 0.110.2 0.56 0.33 0.32 0.34 0.30 0.30 0.290.3 0.89 0.67 0.66 0.66 0.61 0.61 0.570.4 0.99 0.91 0.88 0.88 0.86 0.86 0.830.5 1.00 0.98 0.97 0.98 0.95 0.95 0.940.6 1.00 1.00 1.00 1.00 0.99 0.99 0.990.7 1.00 1.00 1.00 1.00 1.00 1.00 1.000.8 1.00 1.00 1.00 1.00 1.00 1.00 1.000.9 1.00 1.00 1.00 1.00 1.00 1.00 1.001.0 1.00 1.00 1.00 1.00 1.00 1.00 1.00

filters as well as the HP filter are considered. For an original series yt , the filtered series obtainedusing Baxter and Kings (1999) approximate BP filter with frequency band [l, h], denotedyBPt , is defined by

yBPt = (L)yt = [h(L) l(L)]yt ,where

h(L) =K

k=KbhkL

k, bh0 =h

(k = 0), bhk =

sin(kh)k

(k = 0),

l(L) =K

k=KblkL

k, bl0 =l

(k = 0), blk =

sin(kl)k

(k = 0).

For the truncation parameter, we consider K = 4 and 12. The HP filtered series, yHPt , is definedby

yHPt =[

(1 L)2(1 L1)21 + (1 L)2(1 L1)2

]yt .



Table 6. Comparisons with filtered series using an approximate bandpass filter.Band 1 Band 2 Band 3

xt BSR BP4 BP12 BSR BP4 BP12 BSR BP4 BP12

bias DGP1 0.004 0.001 0.002 0.010 0.012 0.002 0.011 0.002 0.000DGP2 0.004 0.007 0.004 0.004 0.004 0.002 0.002 0.000 0.008DGP3 0.002 0.002 0.007 0.002 0.002 0.014 0.004 0.005 0.006

s.d. DGP1 0.192 0.225 0.224 0.204 0.222 0.211 0.214 0.231 0.234DGP2 0.187 0.223 0.220 0.203 0.220 0.206 0.207 0.224 0.228DGP3 0.190 0.228 0.220 0.201 0.220 0.211 0.207 0.226 0.233

cov DGP1 0.904 0.688 0.602 0.825 0.878 0.612 0.869 0.718 0.548DGP2 0.895 0.687 0.613 0.812 0.849 0.621 0.892 0.759 0.566DGP3 0.894 0.680 0.618 0.811 0.829 0.624 0.877 0.743 0.555

Notes: s.d. and cov denote the standard deviations and coverage ratios, respectively. BP4 and BP12 denote the bandpassfilters with the truncation parameter K = 4 and K = 12, respectively.

For the parameter , we consider two popular choices, namely 1600 and 6.25.In Table 6, we present the bias and standard deviation of the break fraction estimates, and

the exact coverage ratios of the asymptotic 90% confidence intervals of the break date when theDGPs are the non-contaminated models of Section 4.1 Throughout, 1000 replications are used.For brevity, we only consider three spectral bands: [/16, /2] (band 1), [0, /16] (band 2)and [/2, ] (band 3). Here, our comparisons are only with the BP filtered series. The resultsshow that when using the BP filtered-based estimates the bias remains small but the standarddeviations are larger than when using the band spectral approach. Also, the exact coverage rateof the asymptotic confidence intervals are near 90% in all cases using the band spectral approachbut there is severe undercoverage when using the BP filtered approach.

The next experiment pertains to a comparison of our log(T ) truncation method with standardmethods applied to HP filtered series in the case of the contaminated processes considered inSection 4.2 The results presented in Table 7 show that using HP filtered series leads to estimateswith larger variance and exact coverage rates below the 90% nominal level.

The last experiment pertains to the power of the SupF test for a single structural change.We use the non-contaminated models with a break in all frequencies when comparing withthe BP filter. Figure 3 reports the power functions for 5% nominal size tests. In all cases, theband spectral method provides tests with higher power. Next we compare the band spectralapproach with the HP filter. In doing so, we consider data with a break in the frequencyband [/16, /2]. Unlike the HP filter, the band spectral approach enables researchers toconsider an arbitrary frequency band, but given that one does not know the true band wherethe break occurs, we consider the power functions with the following (mis)specifications: (a)[/16, /2], (b) [/16 l, /2 h], (c) [/16 + l, /2 + h], (d) [/16 l, /2 + h]and (e) [/16 + l, /2 h] with l = /32 and h = /8. Experiment (a) corresponds to thecase where one correctly specifies the frequency band in which the break occurs, whereas thecases (b)(e) introduce moderate misspecifications of the true band. Table 8 shows that, in case(a), the power of BSR is higher than the HP filter as expected. If we consider bands that are not



Figure 3. Finite sample power of the SupF test with a break common to all frequencies.



Table 7. Comparisons with filtered series using the HodrickPrescott filter.Case D1 Case D2

BSR HP BSR HP

xt logT l = 1600 l = 6.25 logT l = 1600 l = 6.25bias DGP1 0.001 0.001 0.002 0.001 0.005 0.005

DGP2 0.001 0.003 0.001 0.002 0.005 0.001DGP3 0.001 0.006 0.002 0.002 0.005 0.001

s.d. DGP1 0.123 0.212 0.219 0.200 0.219 0.222DGP2 0.124 0.206 0.221 0.194 0.218 0.222DGP3 0.126 0.206 0.222 0.196 0.218 0.226

cov DGP1 0.926 0.812 0.737 0.928 0.783 0.720DGP2 0.916 0.811 0.746 0.918 0.776 0.743DGP3 0.907 0.813 0.754 0.915 0.773 0.746

Case D3 Case D4

BSR HP BSR HP

xt logT l = 1600 l = 6.25 logT l = 1600 l = 6.25bias DGP1 0.019 0.000 0.001 0.002 0.002 0.004

DGP2 0.012 0.003 0.003 0.002 0.001 0.005DGP3 0.016 0.001 0.002 0.002 0.001 0.003

s.d. DGP1 0.189 0.220 0.222 0.197 0.218 0.221DGP2 0.188 0.215 0.218 0.195 0.218 0.221DGP3 0.191 0.217 0.225 0.195 0.216 0.223

cov DGP1 0.922 0.779 0.725 0.908 0.787 0.738DGP2 0.915 0.777 0.743 0.898 0.774 0.746DGP3 0.897 0.779 0.737 0.895 0.787 0.756

Note: s.d. and cov denote the standard deviations and coverage ratios, respectively.

exactly the actual one in which the break occurs, the band spectral method still leads to tests withhigher power.5

These simulations illustrate the relative efficiency and flexibility of our proposed method overstandard methods based on filtered series.

5. EMPIRICAL EXAMPLE

At the core of real business-cycle theories is the prediction that labour supply rises followinga technological shock. A large body of literature has tackled this problem empirically. One ofthe first and most influential is the study by Gal (2001) who found that, if hours worked and

5 We also compared the power of the test for structural change using our truncation method with the HP filteredapproach when the models are contaminated as in Section 4.2 In this case, the power functions are comparable.



Table 8. Finite sample power of the SupF tests: comparisons with the HodrickPrescott filtered series.HP BSR

c 1600 6.25 a b c d e

0.0 0.05 0.09 0.07 0.07 0.06 0.05 0.110.2 0.11 0.13 0.19 0.17 0.13 0.14 0.200.4 0.30 0.21 0.53 0.43 0.36 0.38 0.460.6 0.54 0.36 0.84 0.72 0.64 0.67 0.750.8 0.76 0.50 0.97 0.90 0.85 0.88 0.921.0 0.90 0.65 1.00 0.97 0.95 0.97 0.981.2 0.96 0.76 1.00 0.99 0.98 0.99 1.001.4 0.98 0.84 1.00 1.00 0.99 1.00 1.001.6 0.99 0.90 1.00 1.00 1.00 1.00 1.001.8 1.00 0.93 1.00 1.00 1.00 1.00 1.002.0 1.00 0.96 1.00 1.00 1.00 1.00 1.00

productivity are specified as integrated processes, hours worked instead fall after a technologicalshock. On the other hand, Christiano et al. (2004) argued that hours worked should be consideredstationary, in which case hours worked do increase after a technological shock. However, asargued by Fernald (2007), the results appear largely driven by low-frequency components suchas types of time trend and structural breaks in the data. The aim here is to assess whether therelation between hours worked and productivity is stable over time when allowing for possiblelow-frequency contaminations and also whether it is stable over the business-cycle frequencies.

Note that we are not concerned about addressing the issue about whether technologicalshock have a positive or negative impact on hours worked. This would require a full structuralmodel that is well identified. Our concern is on the stability of the relationship between the twovariables, which is a valuable starting point to analyse the structural issues of interest. To thateffect, one does not have to specify a structural model. One can indeed simply use a reducedform estimated by OLS even if it involves correlation between the errors and the regressors, asshown in Perron and Yamamoto (2013).

The data used are the same as in Gal (2001) and were downloaded directly from the USDepartment of Labor website. Labour productivity is output per hour of all persons, the hoursworked series is hours of all persons in the business sector and the population is measured by thecivilian non-institutional population over 16 years. All series are transformed into their naturallogarithms. The data used are from 1948Q1 to 2009Q4. We consider the following reduced-formequation:

nt =4

j=1ajptj +

4k=1

bkntk + ut , (5.11)

where nt stands for hours worked per capita and pt is productivity. When we use thelevel specification, nt is linear detrended. When we use the difference specification, the first



Table 9. Empirical results.(a) Model with lagged dependent variables

Hours in levels Hours in first differences

Full Truncated Cycle Full Truncated CycleSupF 23.84*** 7.58 2.77 14.94* 0.69 3.48SupF (2|1) 10.27 1.12 3.66 19.72** 8.80 6.17SupF (3|2) 8.12 1.55 11.03 7.95 6.34 5.28SupF (4|3) 3.49 1.12 0.47 7.13 5.09 1.64UDmax 23.84*** 7.70 5.03 14.94* 7.03 6.57Dates 1986:Q1 1967:Q1

1981:Q2 (b) Model without lagged dependent variablesHours in levels Hours in first differences

Full Truncated Cycle Full Truncated CycleSupF 5.75 6.10 11.29 18.92** 0.34 5.93SupF (2|1) 7.70 13.99 1.42 10.59 8.68 2.33SupF (3|2) 29.36*** 1.92 4.33 8.78 7.94 20.58**SupF (4|3) 0.30 0.00 0.00 4.87 1.24 4.52UDmax 7.91 6.61 11.29 18.92** 6.36 7.00Dates 1976:Q1 Notes: *, ** and *** denote significance at the 10%, 5% and 1% levels, respectively. For the results without laggeddependent variables, we use a heteroscedasticity and autocorrelation robust covariance estimate with a Bartlett kerneland the bandwidth chosen using Andrews (1991) AR(1) approximation method for the full frequency results. For thetruncated and the cycle results, Whites (1980) heteroscedasticity robust covariance matrix estimate is used.

differences nt are used for the regression.6 Note that this is a part of the system estimated inShapiro and Watson (1988). We consider possible breaks in the autoregressive coefficients ajand report the Sup F , double maximum (UD max), and SupF (l + 1|l) tests. If tests for breaksare significant, the estimated break dates resulting from the sequential procedure are reported.The trimming used for the permissible break dates is = 0.15 and the maximum number ofbreaks allowed is 5. Table 9(a) reports the results of the specification (5.11) and Table 9(b)is for the model without lagged nt (or nt ) but allowing for possible serial correlations inut using a heteroscedasticity and autocorrelation robust covariance matrix estimate. Each tablepresents results for the full spectrum (0 ), the truncated spectrum (LT = log T ),and the business-cycle band (/16 /2), the latter including frequencies correspondingto periods ranging from 1 to 8 years.

Consider first the results in Table 9(a) using the full spectrum (the usual time domain tests).With the level specification, the SupF test is significant at the 1% level suggesting strong

6 We use demeaned data instead of including a constant in the regression. If a constant is included, this will imply arank deficiency of the regressor matrix given that applying a discrete Fourier transform and truncating the zero frequencyimplies that the constant becomes zero.



evidence of at least one break in the coefficients. Since the SupF (2|1) test is insignificant,we conclude that there is a one-time structural change at 1986:Q1. With the differencespecification, the results are similar although now the SupF test is significant at the 10%level and the SupF (2|1) test is significant at the 5% level. The two estimated break dates are1967:Q1 and 1981:Q2, quite different from those obtained with the level specification. Theresults make it difficult to give a relevant economic interpretation. A possibility is that thetests are significant because of some low-frequency components in the series, suggesting theneed to apply the tests with a truncation and within the business-cycle band. When doingso, the results are very different. None of the structural change tests are significant usingeither the level or difference specification for hours. Hence, the results provide evidence tothe effect that the relation between hours worked and productivity is stable over any spectralband that excludes the lowest frequencies, in particular it is stable over the business-cycleband.

Table 9(b) provides the results using a model without the lagged dependent variables. Here,we find no break with the level specification and one break with the difference specification. Thebreak date is estimated at 1976:Q1, which is not consistent with the previous specification withlagged dependent variables. What is noteworthy is how different the results are between the twospecifications when using a full frequency regression. The differences can be ascribed to possiblelow-frequency components in the hours and productivity series, which has been extensivelydiscussed in the literature (e.g. Fernald, 2007, Francis and Ramey, 2009, and Gospodinov et al.,2011). However, once we exclude the low-frequency contamination using a small trimming,there is no evidence of structural changes in the relation between hours and productivity in eithermodels. Furthermore, there is no significant change when considering a business-cycle band.These results have important implications for the analysis of the effect of a technological shockon hours worked. It indicates that the various structural-based methods used to assess the signand magnitude of this effect should be carried using a frequency band that excludes the lowestfrequencies or within a business-cycle band.

6. CONCLUSION

We investigated methods for estimating and testing multiple structural changes using bandspectral regressions. We showed that all standard results in Bai and Perron (1998) and Perronand Qu (2006) continue to hold with appropriate modifications. We documented the fact thatthe tests have good size in finite samples and that the estimates of the break dates obtainedhave good properties, including the adequacy of the limit distributions as approximations to thefinite sample distributions of the estimates of the break dates. Structural change tests using bandspectral regressions were shown to be more powerful than their time domain counterparts whenbreaks occur only within some frequency band, provided of course that the user-chosen bandcontains the appropriate subset. An important advantage of using a band spectral framework isthat tests and estimates that are robust to low-frequency contaminations can easily be obtained.We have shown that inference can be made robust to various contaminations (trends, randomlevel shifts, etc.) by simply excluding a few frequencies near zero. We illustrated our methodsby showing that the relationship between hours worked and productivity is stable if one usesestimates that are robust to such low-frequency contaminations but not otherwise. This examplesheds light on the importance of a careful consideration of the frequency band in estimating and



testing multiple structural changes and highlights the usefulness of the methods developed in thispaper.

ACKNOWLEDGMENT

We are grateful to an anonymous referee for useful suggestions.

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APPENDIX: PROOFS

Proof of Lemma 2.1: Denote SSRBAT with A = I by SSRfullT . Since

SSRT =m

[Ym Xm m][Ym Xm m],

SSRfullT =m

[WYm WXm m][WYm WXm m],

=m

[Ym Xm m][Ym Xm m],

by using W W = I , all we need to show is m = m. The result follows from the fact that

m = (XmW WXm)1(XmW WYm),= (XmXm)1(XmYm) = m.



Proof of Lemma 2.2: For parts (a) and (b), it is easy to show from Assumption 2.1 that the process tconstructed with any band has a constant spectral density at any . This implies the covariance stationarityof t from which the results follow applying standard law of large number. For parts (c) and (d), given thefact that the series t can be represented as a bandpass version of t , the former can be expressed as aninfinite order moving average of the latter such that t =

j= bj tj =

j= c

j tj and

cj =1

2

h

l

cj eij d

= cj

2ij(eihj eil j ) = cj sin(hj ) sin(lj )2j .

Following Phillips and Solo (1992), we need to show that j=0 j 1/2 cj < for the invariance principleto hold. This follows given that

j 1/2cj

(sin(hj ) sin(lj )

2j

) j 1/2 cj sin(hj ) sin(lj )2j

,and

sin(hj ) sin(lj )2j

h l2 1.

Proof of Theorems 2.1 and 2.2: We show that the assumptions for the original model with t are alsosatisfied with the model involving the series t . In particular, we need to show that Assumptions 2.12.4 hold. It is obvious that Assumption 2.1 holds for t since f m () = f m () for [l, h]. ForAssumptions 2.2 and 2.3, we know that

xt x

t =

jBA I

mx,T (j ). Since

xt x

t is symmetric, for any

non-zero p 1 vector ,

min [

xt xt

] = min

[BA

Imx,T ()],

minBA

Imx,T (),

(0 + 1)

with 0 and 1 the minimum eigenvalue of Imx,T (0) and Imx,T (1). By Assumption 2.2, these are boundedaway from zero. For Assumption 2.3, the minimum eigenvalue of

lt=k x

t x

t is shown to be strictly

positive using the same argument. Since it is symmetric and the minimum eigenvalue is strictly positive, theinvertibility is guaranteed. For Assumption 2.4, the fact that cj cj for all j implies that propertiesfrom (a)(e) in Assumption 2.4 are satisfied with {xt , ut }. The time domain estimates of the break datesbased on model (2.5) can be expressed as

( T 1 , . . . , T M ) = arg max SST T (T1, . . . , TM )

with

SST T (T1, . . . , TM ) =m

[W1AWYm W1AWXm m][W1AWYm W1AWXm m].



The equivalence SST BAT SST T is then shown using Lemma 2.1 so that Theorem 2.1 followsgiven that the conditions for Proposition 5 of Perron and Qu (2006) are satisfied. Similarly, Theorem 2.2follows since the conditions for Proposition 6 of Perron and Qu (2006) are satisfied.

Proof of Lemma 3.1: Let = A D,A where D,A and A are the band spectral regression estimate of obtained from model (3.1) and model (2.1), respectively. Then for the mth segment

SSRD,BAT,m = [AWUm + AWXm(m Am

)+ AWXm m + AWDm][AWUm + AWXm

(m Am

)+ AWXm m + AWDm],= SSRBAT,m + DmDm + mXmXm m + 2U mXm m + 2

(m Am

)XmXm m

+2U mDm + 2(m Am

)XmDm + 2mXmDm,

= SSRBAT,m + DmDm + I + II + III + IV + V + V I,where = W AW . By Assumption 2.7, XmDm = op(1) and U mDm = op(1). We also have

m = (XmXm)1(XmDm) = op(T 1).Hence, we obtain I = op(T 1) Op(T ) op(T 1) = op(T 1), I I = Op(T 1/2) op(T 1) = op(T 1/2),III = Op(1) Op(T ) op(T 1) = op(1), IV , V = op(1) and V I = op(T 1) op(1) = op(T 1). Considernow the term DmDm. We have DmDm =

jhjlImd,T (j ). Given Assumption 3.1, this is bounded in

probability by CTmjh

jl(1/j 2) for some large enough C > 0. Now, jhjl (1/j 2) (1/jl)jhjl (1/j ) =

O(log(Tm)/jl) since jh = [hTm/ ] = O(T ). Hence, DmDm = Op(Tm log(Tm)/jl) = op(T ) iflog(Tm)/jl 0 as T .


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Estimating and testing multiple structural changes in linear models using band spectral regressions