CENTRE FOR ECONOMETRIC ANALYSIS

Transcription

1 CENRE FOR ECONOMERIC ANALYSIS Cass Business School Faculty of Finance 06 Bunhill Row London ECY Z esting for Instability in Covariance Structures Chihwa Kao, Lorenzo rapani, and Giovanni Urga CEA@Cass Working Paper Series WP CEA

2 esting for Instability in Covariance Structures Chihwa Kao y Syracuse University Lorenzo rapani z Cass Business School Giovanni Urga x Cass Business School and Università di Bergamo June 0, 204 Abstract We propose a test for the stability over time of the covariance matrix of multivariate time series. he analysis is extended to the eigensystem to ascertain changes due to instability in the eigenvalues and/or eigenvectors. Using strong Invariance Principles and Law of Large Numbers, we normalise the CUSUM-type statistics to calculate their supremum over the whole sample. he power properties of the test versus local alternatives and alternatives close to the beginning/end of sample are investigated theoretically and via simulation. We extend our theory to test for the stability of the covariance matrix of a multivariate regression model, and we develop tests for the stability of the loadings in a panel factor model context. he testing procedures are illustrated through two applications: we study the stability of the principal components of the term structure of US interest rates; and we investigate the stability of the covariance matrix of the error term of a Vector AutoRegression applied to exchange rates. JEL codes: C, C22, C5. Keywords: Covariance Matrix, Eigensystem, Factor Models, Changepoint, CUSUM Statistic. We wish to thank participants in the conference on High-Dimensional Econometric Modelling (London, 3-4 December, 200), in particular Elena Andreou, Jianqing Fan, Domenico Giannone, David Hendry and Soren Johansen, the New York Camp Econometrics VI (Syracuse University, 9-0 April, 20), the Rimini ime Series Workshop (Rimini, 2-3 June, 20), the Faculty of Finance Workshops at Cass Business School, the Glasgow Business School Seminar Series, the Lancaster University Management School Seminar Series, the LSE Department of Statistics Seminar Series, in particular Piotr Fryzlewicz and Myung Hwan Seo. Special thanks to Lajos Horvath for providing us with inspiring comments. he usual disclaimer applies. y Center for Policy Research, 426 Eggers Hall, Syracuse University, Syracuse, NY (USA). el , Fax , dkao@maxwell.syr.edu z Centre for Econometric Analysis, Faculty of Finance, Cass Business School, 06 Bunhill Row, London ECY Z (U.K.). el Fax , L.rapani@city.ac.uk x Centre for Econometric Analysis, Faculty of Finance, Cass Business School, 06 Bunhill Row, London ECY Z (U.K.) and Bergamo University (Italy). el Fax , g.urga@city.ac.uk

3 INRODUCION In this paper, we propose a testing procedure to evaluate the structural stability of the covariance matrix (and its eigensystem) of multivariate time series. A large amount of empirical evidence shows that the issue of changepoint detection in a covariance matrix is of great importance. A classical example is the application of Principal Component Analysis (PCA) to the term structure of interest rates, with the three main principal components interpreted as slope, level and curvature (Litterman and Scheinkman, 99). Bliss (997), Bliss and Smith (997) and Perignon and Villa (6) show that the principal components of the term structure change substantially over time. Similar ndings, using a di erent methodology, are in Audrino et al. (5). PCA is also widely used in macroeconometrics, for instance to forecast in ation (Stock and Watson, 999, 2, 5). he importance of verifying the stability of a covariance matrix is also evident in the context of Vector AutoRegressive (VAR) models. In the context of forecasting, Castle et al. (200) show that changes in the smallest eigenvalue of the covariance matrix of the error term have a large impact on predictive ability. Furthermore, the Choleski decomposition is routinely employed in the context of variance decomposition analysis, when examining how much of the variance of the forecast error of each variable in a VAR is due to exogenous shocks to the other variables (see e.g. Pesaran and Shin, 99). Despite the relevance of the topic, most studies either assume stability as a working assumption without testing for it, or the testing is carried out by splitting the sample, thus assuming knowledge of the break date a priori. his calls for a rigorous testing procedure to estimate the location of the changepoint when breaks are detected. Further, a typical requirement of classical PCA is that the data are i.i.d. and Gaussian (Flury, 94, 9; Perignon and Villa, 6), which is unsuitable for nancial data (see also Audrino et al., 5). he theoretical framework developed in this paper builds on a plethora of results for the changepoint problem available in statistics and in econometrics. Existing testing procedures (see e.g. the reviews by Csorgo and Horvath, 997, and Perron, 6) are typically based on taking the supremum (or some other metric - see Andrews and Ploberger, 994) of a sequence of CUSUM-type statistics, thus not requiring prior knowledge of the breakdate. In particular, Aue et al. (9) develop a test for the structural stability of a covariance matrix, based on minimal assumptions. However, a feature of this testis that, by construction, it has power p versus breaks occurring at least (respectively, at most) O time periods from the beginning (respectively to the end) of the sample. Lack of power versus alternatives close to either end of the sample is a typical feature in this literature (see also Andrews, 993), which somewhat limits the applicability of the test. Situations where breaks are due to recent events, like e.g. the recession, are left out of the analysis. Our contribution complements that of Aue et al. (9) by proposing a test that has power versus breaks occurring close to the beginning/end of the sample. he main contribution of this paper is twofold. First, testing for changepoints is extended 2

4 to PCA. In addition, the extension to testing for the stability of principal components is useful for the purpose of dimension reduction. Our simulations show that tests for the stability of the whole covariance matrix have severe size distortions in nite samples. Contrary to this, testing for the stability of eigenvalues is found to have the correct size and good power even for relatively small samples. As a second contribution, our testing procedure is able to detect breaks occurring up to O (ln ln ) periods to the end of the sample. his is achieved by using a Strong Invariance Principle (SIP) and a Strong Law of Large Numbers (SLLN) for the partial sample estimators of the covariance matrix, and by using these results to normalize the CUSUM-type test statistic, using a Darling-Erdos limit theory (see Csorgo and Horvath, 997; Horvath, 993). We also extend our results to the case of testing for the stability of the covariance matrix of the error term in a multivariate regression setting. he theory derived in our paper is illustrated through two applications, one to the US term structure of interest rates, and one to verifying the stability of the covariance matrix of the error term in a VAR model for exchange rates. In both cases, the span of the datasets is from the late nineties to the current date. As far as the former exercise is concerned, we nd (as expected) evidence of changes in the volatility and in the loading of the principal components of the term structure around the end of 7/beginning of. As regards the latter, although the sample covers roughly the same period, we nd very little evidence of any changes, suggesting that the covariance matrix of the error term has been stable even during the recession. he paper is organized as follows. eigensystem. Section 2 contains the SIP and its extension to the he test statistic and its distribution under the null (as well as its behaviour under local-to-null alternatives) is in Section 3. Monte Carlo evidence is in Section 4, while the applications to the term structure of interest rates and to exchange rates are in Section 5. Section 6 concludes. An Internet Appendix reports Lemmas (Appendix A), and Proofs of heorems and Propositions (Appendix B) in the paper. A word on notation. Limits are denoted as! (the ordinary limit); p! (convergence in probability); d! and (convergence in distribution). Orders of magnitude for an almost surely convergenth sequence (say s ) are denoted as O a:s: ( & ) and o a:s: ( & ) when, for some " > 0 and ~ <, P j & s j < " for all ~ i = and & s! 0 almost surely respectively. Orders of magnitude for a sequence converging in probability (say s 0 ) are denoted as O p ( & ) and o p ( & ) when, for some " > 0, " > 0 and ~ " <, P [j & s 0 j > "] < " for all > ~ " and & s 0! 0 in probability respectively. Standard Wiener processes and Brownian bridges of dimension q are denoted as W q () and B q () respectively; kvk denotes the Euclidean norm of a vector v in R n ; similarly, kak denotes the Euclidean norm of a matrix A in R n, and jj p the L p -norm; the integer part of a real number x is denoted as bxc. Constants that do not depend on the sample size are denoted as M, M 0, M 00, etc. 3

5 2 HEOREICAL FRAMEWORK Let fy t g t= be a time series of dimension n. We assume, without loss of generality, that y t has zero mean and covariance matrix E (y t yt) 0 - see also Section 3. on this. his section contains the asymptotics of the partial sample estimates of ; the results are used in Section 3 in order to construct the CUSUM-type test statistic to test for breaks in and its eigensystem. Speci cally, we report a SIP for the partial sample estimators of and an estimator of the long run covariance matrix of the estimated, say V ; and we extend the asymptotics to PCA. Strong Invariance Principle and estimation of V Let ^ be the sample covariance matrix, i.e. ^ = P t= y tyt. 0 For a given 2 [0; ], we de ne a point in time b c, and we use the subscripts and to denote quantities calculated using the subsamples t = ; :::; b c and t = b c+; :::; respectively. In particular, we consider the sequence of partial sample estimators ^ = ( ) P b c t= y tyt, 0 and similarly ^ = [ ( )] P t=b c+ y tyt. 0 Finally, henceforth we denote w t = vec (y t yt) 0 and w t = vec (y t yt 0 ). In the sequel, we need the following assumption. Assumption (i) sup t E ky t k 2r < for some r > 2; (ii) y t is L 2+ -NED (Near Epoch Dependent) for some > 0, of size 2 (; +) on a strong mixing base fv t g + t= of size P r= (r 2) and r > 2 ; (iii) letting V ; = E t= t w P 0 t= w t, V ; is positive de nite uniformly in, and as!, V ;! V with kv k < ; (iv) letting w it be the i-th element of w t and de ning S i;m P m+ t=m+ w it, there exists a positive de nite matrix = f$ ij g such that je [S i;m S j;m ] $ ij j M, for all i and j and uniformly in m, with > 0. Assumption speci es the moment conditions and the memory allowed in y t ; no distributional assumptions are required. According to part (i), at least the 4-th moment of y t is required to be nite, similarly to Aue et al. (9). As far as serial dependence is concerned, the requirement that y t be NED is typical in nonlinear time series analysis (see Gallant and White, 9) and it implies that y t is a mixingale (Davidson, 2a). Many of the DGPs considered in the literature generate NED series - examples include GARCH, bilinear and threshold models (see Davidson, 2b). Part (ii) illustrates the trade-o between the memory of y t (i.e. its NED size ), and its largest existing moment: as (the memory of y t ) approaches, r has to increase. Note that in our context, the data (y t ) undergo a non-lipschitz transformation (viz., they are squared), and therefore the relationship between moment conditions and memory is not the standard one (see e.g. the IP in heorem 29.6 in Davidson, 2a). In principle, moment conditions such as the one in part (ii) could be tested for, e.g. using a test based on some tail-index estimator - J. B. Hill (200, 20) extends the well-known Hill s estimator to 4

6 the context of dependent data. Other types of dependence could be considered, e.g. assuming a linear process for y t - an IP for the sample variance is in Phillips and Solo (992, heorem 3.). Part (iv) is a bound on the growth rate of the variance of partial sums of w t, and it is the same as equation (.5) in Eberlein (96); see also Assumption A.3 in Corradi (999). Although it is not needed to prove the IP for the partial sum process of w t, it is a su cient condition for the SIP. heorem contains the IP and the SIP for the partial sums of w t. heorem Under Assumptions (i)-(iii), as! b c X p t= w t d! [V ] =2 W n 2 () ; () uniformly in. Rede ning w t in a richer probability space, under Assumptions (i)-(iv) b c X t= w t = b c X t= X t + O a:s: b c 2 ; (2) uniformly in, where X t is a zero mean, i.i.d. Gaussian sequence with E (X t X 0 t) = V and > 0. Remarks. Equation () is an IP for w t (i.e. a weak convergence result), which is su cient to use the test statistics discussed e.g. in Andrews (993) and Andrews and Ploberger (994)..2 Equation (2) is an almost sure result, which also provides a rate of convergence. he practical consequence of (2) is that the dependent, heteroskedastic series w t can be replaced with a sequence of i.i.d. normally distributed random variables, with the same long run variance as w t. We now turn to the estimation of V. If no serial dependence is present, a possible choice is the full sample estimator ^V = P h i h ^i 0. t= w twt 0 vec ^ vec Alternatively, one could use the sequence of partial sample estimators ^V ; = X w t wt 0 t= h i h i 0 vec ^ vec ^ + ( h ) vec ^ i h vec ^ i 0 : o accommodate for the case l E w t w 0 t l 6= 0 for some l, we propose a weighted sum-ofcovariance estimator with bandwidth m: ~V = ^ 0 + mx l= 5 l h ^ l + m ^ i 0 l ; (3)

7 where ^ P h i h ^i l = 0; (l+) t=l+ w t vec ^ w t l vec or V; ~ = ^ 0;b c + ^ 0; b c + P m l= h l ^ m l;b c + ^ 0 l;b c + ^ l; b c + ^ i 0 l; b c, where ^ l;b c = (l+) h i h i 0, w t vec ^ w t l vec ^ and similarly for ^ l; b c. In order to derive the asymptotics of ^V ; and ~ V ;, consider the following assumption: P b c t=l+ Assumption 2. (i) either (a) l = 0 for all l 6= 0 or (b) P l=0n ls k lk < for some s ; (ii) sup t E ky t k 4r P < for some r > 2; (iii) letting = E t= vec [ w t w t 0 E ( w t w t)] 0 vec [ w t w t 0 E ( w t w t)] 0 0, is positive de nite uniformly in, and! with kk <. Assumption 2 encompasses various possible cases. Part (i)(a) considers the basic, non autocorrelated case, for which both ^V and ^V ; are valid choices. Part (i)(b) considers the possibility of non-zero autocorrelations. Intuitively, the assumption that the 4-th moment of y t exists, as in Assumption (i), entails, through a Law of Large Numbers (LLN), the consistency of ^V ;. Part (ii) supersedes Assumption (i), by requiring the existence of moments up to the -th. Intuitively, this implies that an IP holds for the partial sums of vec [ w t w 0 t he consistency of ^V ; and of ~ V ; is in heorem 2: heorem 2 Under H 0, as! : if Assumptions (i)-(iii) and 2(i)(a) hold: sup b c if Assumptions (i)-(iii) and 2(i)(b) hold: sup b c ~ V ; E ( w t w 0 t)]. ^V ; V = op ; (4) 0 V = Op m if Assumptions (i)-(iii) and 2(i)(b)-(ii)-(iii) hold: sup b c ~ V ; V = Op m where 0 > 0. he same rates hold for ^V or ~ V. Remarks + O p m 0 ; (5) + O p mp ; (6) 2. Equation (4) is based on a SLLN for the case of no autocorrelation in w t - see also Ling (7). heorem 2 provides a uniform rate of convergence for ^V ; and ~ V ;, as it is usually required in this literature (e.g. Lemma 2..2 in Csorgo and Horvath, 997, p. 76; see also the proof of heorem 3 below). 6

8 2.2 In case of serial dependence, (5) states that it is possible to construct an estimator of V with a rate of convergence. his can be re ned as in (6). Indeed, Assumptions 2(ii)- (iii) allow for an IP to hold for partial sums of vec w t w 0 t l E w t w 0 t l, whence the O p =2 convergence rate, uniformly in. Equation (6) also provides a selection rule for the bandwidth m: in addition to the conditions m! and m p! 0, the speed of convergence of sup b c ~ V; V can be maximised by setting m = O =4. Estimation of the eigensystem In this section, we extend the asymptotics for the partial sample estimates of to its eigensystem. Let the i-th eigenvalue/eigenvector couple be de ned as ( i ; x i ); the eigenvectors are de ned as an orthonormal basis, i.e. x 0 i x j = ij, where ij is Kronecker s delta. Since x i = i x i, a natural estimator for ( i ; x i ) is the solution to the system ( ^ ^X = ^X ^ ^X 0 ^X = I ; (7) where ^X = [^x ; :::; ^x n ], ^x i denotes the estimate of x i, and ^ is a diagonal matrix containing the estimated eigenvalues ^ i in decreasing order. Estimation of f( i ; x i )g n i= based on (7) is known as Anderson s Principal Component (PC) estimator. Similarly, the partial sample estimators of the eigenvalues and eigenvectors are the solutions to ^ ^x i; = ^ i; ^x i;. As we mention below (see Remark P.2), one disadvantage of Anderson s PC estimator is that the estimated eigenvectors have a singular asymptotic covariance matrix (see Kollo and Neudecker, 997). In order to avoid this issue, an estimator based on a di erent normalisation can be proposed, known as the Pearson-Hotelling s PC estimator; in this case, the estimated eigenvalues are the same as from (7), but the eigenvectors i are de ned (and estimated) as an eigenvalue-normed basis, viz. 0 i j = i ij. hus, i =2 i x i. A typical interpretation of the i s in the context of the term structure of interest rates (Litterman and Scheinkman, 99; Perignon and Villa, 6) is that i is the volatility of i, and x i represents its loading. he estimates of the eigensystem according to the Pearson-Hotelling approach are the solution to the system ( ^ ^X = ^X ^ ^X 0 ^X = ^ : () Upon calculating the solutions of (), it turns out that the eigenvectors are estimated by ^ i = ^ =2 i ^x i, i.e. by the same estimator for the eigenvector as in (7) multiplied by the square root of the corresponding estimate of the eigenvalue. Similarly, we de ne the partial sample estimator =2 of i as ^ i; = ^ i; ^x i;. 7

9 Consider the following assumption. Assumption 3. he matrix has distinct eigenvalues. Assumption 3 is typical of PCA and it allows to use Matrix Perturbation heory (MP); the assumption could be relaxed at the price of a more complicated analysis, still based on MP. In essence, the asymptotics of ^i; ; ^x i; is derived by treating ^ as a perturbation of, thus deriving the expressions for the estimation errors of ^ i; and ^x i;. he extension of the IP and the SIP to the eigensystem of is reported in Proposition : Proposition Under Assumptions and 3, as!, uniformly in h P i x where v x;i = k k6=i i k (x 0 k x0 i ) Remarks ^ i; i = x 0 i x 0 i vec ^ + O p ; (9) ^x i; x i = v x;i vec ^ + O p ; (0) ^ i; i = v ;i vec ^ + O p ; () and v ;i = 2 x i =2 i =2 x k (x 0 i x0 i ) + P i k6=i i k (x 0 i x0 k ). P. Proposition states that the estimation errors ^ i; i, ^x i; x i and ^ i; i are, asymptotically, linear functions of ^ ; thus, the IP and the SIP in heorem carry through to the estimated eigensystem. he results in Proposition, and the method of proof, can be compared to related results in Kollo and Neudecker (997). P.2 By (0), the asymptotic covariance matrix of p (^x i; x i ) is v x;i V vx;i 0. It can be shown (see e.g. Kollo and Neudecker, 997, p. 66) that v x;i V vx;i 0 is singular. hus, in order to carry out tests on the ^x i s, one would have to use a generalised inverse of the estimated v x;i V v 0 x;i. However, there is no obvious way to calculate the rank of v x;iv v 0 x;i. his makes it di cult to prove the consistency of the Moore-Penrose inverse for v x;i V v 0 x;i (see Andrews, 97). hus, we recommend to carry out tests on the eigenvectors using the i s. P.3 We show in the appendix that h E ^i; i i = X k6=i (x 0 i x0 k ) V (x k x i ) i k ; (2) as!. As far as the impact of n is concerned, V is an n 2 -dimensional matrix; thus, in general the quadratic form (x 0 i x0 k ) V (x k x i ) has magnitude of order O hn 2. Also, due i to the summation on the right hand side of (2) involving n elements, E ^i; i

10 = O n 3. hus, the asymptotic bias is of order O n 3, and it is always positive for the largest eigenvalue; a bias-corrected version is ~ i; = ^ i; P k6=i [^x0 i ^x0 k ] V ~ ^ i ^k [^x k ^x i ]. his result is of independent interest; it could be useful e.g. when measuring the percentage of the total variance of y t explained by each of its principal components. Similarly, we show that E [ (^x i; x i )] = X X x 0 k x0 j V (x j x i ) x k ; (3) ( i k ) ( i j ) k6=i j6=i which provides an expression to correct the bias of the ^x i; ; combining (2) and (3), a bias-corrected version of ^ i; s the can also be computed. P.4 In Appendix B, we show an ancillary result concerning the estimation of the eigenvalues of a matrix which undergoes a change over time. Suppose that the covariance matrix has a break at time k 0;, such that = 0 for t = ; :::; k 0; and = 0 + for t = k 0; + ; :::;. Assume further that the break a ects only a subspace of the matrix, i.e. if 0 = P n i= ix i x 0 i, it holds that = P ~ i2c i ~x i ~x 0 P i i2c ix i x 0 i, where C is a (nontrivial) family of indices, and ~ i and ~x i are perturbations of i and x i ; we assume that ~x 0 i ~x j = ij and that ~x 0 i x j = 0 for all i 2 C and j =2 C. By virtue of this, clearly 0 x i = i x i for all t = ; :::; for i =2 C. In the proof of Proposition, we show that ^ i; i = (x 0 i x0 i ^ ) vec 0 + O p and ^x i; x i = v x;i vec ^ 0 + O p for all i =2 C and uniformly in. his entails that as long as one eigenvalue/eigenvector couple is found to be unchanged over time, then it can be estimated consistently, with no need to test for the time stability of the rest of the eigensystem. Similar results, with a di erent method of proof and a focus on the eigenvalues only, have also been derived in the context of the so-called Google matrices (see Zhou, 20). his result can also be compared with the ndings in Bates et al. (203), who, in the context of panel factor models, show that factors can be estimated consistently even in presence of breaks in the loadings. De ne [ ; :::; n ] 0 as the n-dimensional vector containing the eigenvalues sorted in descending order, and [ ; :::; n ]; ^z ^ 0 ^ 0 0 ; vec with ^z z = D vec ^ + O p and D x x ; :::; x n x n ; v; 0 0. ; :::; ;n v0 he matrix D can be estimated as ^D = ^x ^x ; :::; ^x n ^x n ; ^v ; 0 ; :::; 0, ^v0 ;n with ^v;i = ^x i 2 (^x 0 ^ =2 i ^x0 i ) + P ^ =2 i ^x k k6=i (^x ^ 0 i i ^k i ^x0 k ). he asymptotics of ^z follows from heorem and Proposition, and we summarize it below. Corollary Under Assumptions and 3, as!, it holds that p (^z z)! d [V z ] =2 W n(2n+) (). Also, (^z z) = P b c t= ~X t + O a:s: b c 2, uniformly in, where V z = D V D 0 and ~X t is a zero mean, i.i.d. Gaussian sequence with E ~Xt X ~ 0 t = V z and > 0. 9

11 Corollary entails that p ^ d! [V ] =2 W n () ; p vec ^ d! [V ] =2 W n 2 () ; with: V a matrix with (i; j)-th element given by V ij = (x0 i x0 i ) V (x j x j ), and V is an n 2 n 2 -dimensional matrix whose (i; j)-th n n block is de ned as V ij = v ;i V v 0 ;j. 3 ESING his section studies the null distribution and the consistency of tests based on CUSUM-type statistics. Henceforth, we de ne the CUSUM process S () = P b c t= vec (y ty 0 t). In light of Corollary, test statistics for and its eigensystem can be based on ~S () = R D x S () b c S ( ) ; (4) with S ~ () = 0 for or, and R a p n (n + ) matrix. For example, when testing for the null of no changes in the rst eigenvalue, R is the matrix that extracts the rst element of D h S () with ~ V z; = RD ~ V; D 0 R0. b c S ( ) i. hence, testing is carried out by using s h () = b c b ( )c ~S () 0 V ~ ~ =2 z; S ()i ; (5) heorem 3 contains the asymptotics of sup b c () under the null. heorem 3 Under Assumptions -3, as (m; )! with m + m p! 0, sup ()! d b cb cb 2 c sup 2 kb p ()k p ( ) ; (6) where B p () is a p-dimensional standard Brownian bridge and [ ; 2 ] 2 (0; ). p q (m; )! with ln ln m + m ln ln! 0, P (a " sup nb c () n # x + b ) Also, as! e 2e x ; (7) where a = p 2 ln ln and b = 2 ln ln + p 2 ln ln ln ln p 2, with () the Gamma function. 0

12 Remarks 3. According to (6), the maximum is taken in a subset of [0; ], namely [ ; 2 ]. his approach requires an IP for S (), and the Continuous Mapping heorem (CM). As noted in Corollary in Andrews (993, p. 3), () is not continuous at f0; g and sup b c () p! under H 0. hus, trimming is necessary in this case. Further, in this case it sfuccies to have a consistent estimator of the long-run covariance matrix V which, in light of equation (6) in heorem 2, entails that both m! and m p! 0. he considerations in Remark 2.2 apply here. 3.2 As an alternative approach, the SIP can be used: sums of w t can be replaced by sums of i.i.d. Gaussian variables, with an approximation error. Upon normalising () with the appropriate norming constants, say a and b, an Extreme Value (EV henceforth) theorem can be employed. ests based on sup nb c n [a () b ] are designed to be able to detect breaks close to the end of the sample. Results like (7) have been derived by Horvath (993), for i.i.d. Gaussian data, and extended to the case of dependence by Ling (7), inter alia. As far as the long-run covariamce matrix estimator is concerned, in pln this case the theory requires a consistent estimator at a rate (at least) o p ln : q therefore, from (6), we need the restrictions! 0 and m! 0. Note that p ln ln m ln ln the optimal choice of the bandwidth m that maximizes the speed of convergence of sup b c ~ V; V is still m = O = heorem 3 allows to test for breaks in when n is nite. As n passes to in nity, Aue et al. (9) study the behaviour of ^ h 2 sup b c S ~ () 0 R V ~ i z; R 0 ~S (), showing that, as! followed by n!, 4^ 2 p n2 2n d! N (0; ). Based on Goetze and Zaitsev (202; see also the references therein), it is possible to extend (7) to the case of n!. Although this goes beyond the scope of this paper, we point out that, based on the results available in the literature, the theory developed here can only be extended to the case of n passing to in nity at a very slow rate compared with. his is also due to the fact that the maximum of the sequence of Wald-type statistics ought to be searched between n and n, so as not to have a singular matrix. herefore, in order to detect breaks close to either end of the sample, n cannot be too large, or too many initial observations would have to be trimmed away. In this respect, dimension reduction techniques such as PCA are bound to be advantageous. Consistency of the test We now turn to studying the behaviour of sup nb c n () under alternatives. As a leading example, we consider the case of testing for no change in in presence of one abrupt

13 change H ( ) a : vech( t ) = ( vech () for t = ; :::; k 0; vech () + for t = k 0; + ; :::; ; () where both the changepoint (k 0; ) and the size of the break ( ) could depend on. More general alternatives could be considered (see e.g. Andrews, 993; Csorgo and Horvath, 997): these include epidemic alternatives, and also breaks that occur as a smooth transition over time as opposed to abruptly as in (). Further, note that () does not rule out the possibility that only some series (i.e. only some of the coordinates of y t ) actually have a break. his entails that tests based on () are capable of detecting breaks that only a ect some of the series, and possibly at di erent points in time. heorem 4 illustrates the dependence of the power on and k 0;. heorem 4 Let Assumptions -3 hold, and de ne c ; such that, under H 0, P () c ; ] = for some 2 [0; ]. If, under H ( ) a, as! h sup nb c n ln ln ( k0; ) k 0; k k 2! ; (9) it holds that Remarks P " sup nb c # () > c ; = : (20) n 4. heorem 4 illustrates the impact of k 0; and on the power of tests based on sup nb c n (). Particularly, consider the two extreme cases: 4..a k k = O (), i.e. nite break size. In this case, the test has power as long as k 0; is strictly bigger than O (ln ln ). his can be compared with tests based on sup b c S ~ () 0 V ~ z; ~S (). Using similar algebra as in the proof of heorem 4, it can be shown that the noncentrality parameter of sup b c S ~ () 0 V ~ z; ~S () is proportional to k k 2 k2 0;. Under k k = O (), this entails that nontrivial p power is attained as long as k 0; = O. 4..b k 0; = O ( ) - i.e. the break occurs in the middle of the sample. he test is powerful q as long as the size of the break is strictly bigger than O. When using ln ln trimmed statistics such as in (6), the test is powerful versus mid-sample alternatives of size O p : when no trimming is used, there is some, limited loss of power versus mid-sample alternatives. 4.2 he presence of large breaks in is bound to a ect inference on the eigensystem - see e.g. Stock and Watson (2). Consider, as a leading example, testing for the stability of 2

14 i. From Proposition, the long-run variance of the estimated eigenvalues is estimated by (^x 0 i ^x0 i ) ~ V (^x i ^x i ), thus depending on ^x i. In presence of a mid-sample break of magnitude k k = O (), x i is estimated by ^x i with an error of magnitude O p (), as a consequence of Proposition ; similarly, V ~ V = Op () also. his entails that the estimated variance of ^ i i, (^x 0 i ^x0 i ) V (^x i ^x i ) is O p (). On the other hand, the numerator of the test statistic is proportional to ^ = O p ( ). his entails that the test based on sup nb c n () is consistent as!. 3. Applications to the residuals of a multivariate regression In this section, we extend the results developed above by proposing a test for the time stability of the covariance matrix (and its eigensystem) of the error term in a multivariate regression, such as a VAR. Consider the following multivariate regression model y t = x t + " t ; (2) where t = ; :::; and y t and " t are n vectors, x t is of dimension q and the matrix of regressors has dimensions n q. Of course, n can be equal to, thereby recovering the usual univariate regression; alternatively, the regressors x t can be the lags of y t, thus having a VAR. Bai (0) considers a test for structural changes in a VAR, extending it to test for changes in the covariance matrix of the error term. he theory developed in this section also extends naturally to the case of linear or polynomial trends among the regressors - see also Aue et al. (202). Let " = E (" t " 0 t) be the covariance matrix of the error term " t ; using the same notation as above, we denote its eigenvalues and eigenvectors as " i and x " i respectively, satisfying the relationship " x " i = " i x " i for i = ; :::; n. ests for the time stability of " and of its eigensystem are based on the residuals ^" t = y t ^xt, where ^ h P i h = t= y P i. tx 0 t t= x tx 0 t Hence, the full sample estimator of " is de ned as ^ " = P t= ^" t^" 0 t. As above, for a given point in time b c with 2 [0; ], we use the subscript to denote quantities calculated using the subsamples t = ; :::; b c. In particular, we extensively use the sequence of partial sample estimators ^ "; = ( ) P b c t= ^" t^" 0 t. We also employ the following de nitions: w t " = vec (" t " 0 t) and w t " = vec (" t " 0 t " ). We need the following Assumptions, which are closely related to Assumptions -3 above. Assumption R. (i) Assumptions (i)-(ii) hold for x t ; (ii) Assumptions (i)-(ii) hold for " t ; (iii) hlet wt x" = vec (" t x 0 t); it holds that: (a) E (wt x" ) = 0 for all t; (b) de ning V x" P P = E t= wx" t t= wt x" ) 0i, V x" is positive de nite uniformly in, and as! 3

15 , V x"! V x" with kv x" k < ; (c) letting wit x" be the i-th element of wt x" and de ning Si;m x" P m+ t=m+ wx" it, there exists a positive de nite matrix n o x" = $ x" ij such that i E hsi;m x" Sx" j;m $ x" ij M, for all i and j and uniformly in m, with M a constant and > 0; (iv) (a) letting V P " = P 0 E t= w" t t= w" t, V " is positive de nite uniformly in, and as!, V "! V " with kv " k < ; (b) letting w it " be the i-th element of w t " and de ning S i;m " P m+ t=m+ w" it, there exists a positive de nite matrix n o " = $ " ij such i that E hs i;m " S" j;m $ " ij M 0 0, for all i and j and uniformly in m, with 0 > 0. Assumption R2. Let " l = E w t " w t "0 P l ; it holds that (i) l=0 ls k " l k < for some s ; n (ii) sup t E k" t k 4r < for some r > 2; (iii) letting l" = P E t= vec w t " w t "0 l E w t " w t "0 l vec w t " w t "0 l E w t " w t "0 l" < for every l. l 0 o, l" Assumption R3. he matrix " has distinct eigenvalues. is positive de nite uniformly in, and l"! l" with Let ^w t " = vec ^" t^" 0 t. In order to estimate the long run covariance matrix V " de ned in Assumption R(iv)(b), we propose the weighted sum-of-covariance estimator V ~ " = ^ " 0;b c + ^ " 0; + P m l= h l ^ " m l;b c + ^ "0 l;b c + ^ " l; b c + ^ i "0 l; b c, where ^ " l;b c = (l+) h i h i 0, ^w t " vec ^" ^w t " l vec ^" and similarly ^ " l; b c. In order to set up the test statistic, consider the following notation. When estimating the b c P b c t=l+ eigensystem of ", we de ne the estimators of ( " i ; x " i ) as the solution to ^ "^x " i = ^ " i ^x " i, where ^ " i and ^x " i denote the estimates of q " i and x " i respectively; the Pearson-Hotelling estimator of the eigenvectors is de ned as ^ " i = ^ " i ^x " i. Similarly, the partial sample estimators of the eigenvalues and eigenvectors are the solutions to ^ "; ^x " i; = ^ " i; ^x " i;, and we de ne ^" i; = q^" i; ^x " i;. hence, let " [ " ; :::; " n] 0 be the n-dimensional vector containing the eigenvalues sorted in descending order; " [ " ; :::; " n]; and let D " [x" x" ; :::; x" n x " n; v "0 p " i x " k ; 0, ; :::; v"0 ;n with v " ;i = 2 (x "0 i x "0 i ) + P k6=i " i " (x "0 i x "0 k k ). Estimation of D" is based on ^ " i and ^x " i. Based on the notation spelt out above, we de ne the test statistic for the time stability of " and its eigensystem. Let ~S " () = R D " S " () b c S" ( ) ; where S " () is the CUSUM process de ned as P b c t= vec ^" t^" 0 t, S " () = 0 for or, and R is de ned in (4). he test statistic is de ned as with ~ V "; = RD " ~ V " D "0 R0. s " () = b c b ( )c ~S =2 " () 0 V"; ~ ~S () " ; 4 x p " i " i

16 " Let " sup nqb c nq " (). he following theorem summarizes the asymptotics of under the null and under the alternative. Similarly to heorem 4, we de ne the critical value c ; to be such that, under H 0, P [ " c ; ] = for some 2 [0; ]. heorem 5 Let Assumptions R-R3 hold, and let p be de ned as in heorem 3. Under the null, as (m; )! with p ln ln m + m p ln ln! 0, it holds that P fa " x + b g! e 2e x ; where a = p 2 ln ln and b = 2 ln ln + p 2 ln ln ln ln p 2, with () the Gamma function. Under the alternative () for ", if (9) holds, P [ " > c ; ] =. Remarks 5. he main feature of heorem 5 is that the same theory applies to residual-based tests as to observable data. his is due to the fact that, under our assumptions, an almost sure rate q of convergence can be derived for the OLS estimator ^, with ^ = O a:s:. By ln ln virtue of this result, a SIP and a SLLN can be shown for the partial sums of the squared residuals, so that the asymptotics of tests based on " () can be derived in the same way as for tests based on (). 5.2 heorem 5 can be readily extended to accommodate for the presence of linear or polynomial trends among the x t s; the only thing that changes is the rate of convergence of ^, which q in presence of linear trends is actually faster than O a:s:. A consequence of ln ln heorem 5 is that the test developed in Section 3 can be applied to data with nonzero mean and/or trends: it su ces to lter out such deterministics, and then apply the test to the resulting residuals. 5.3 On a similar note to Remark 5.2, the test in Section 3 is developed for the case of E (y t ) being constant over time. As a consequence of heorem 5, this does not need to be the case: in case of temporal shifts in the mean of y t, it su ces to demean y t by running (2) with time dummies corresponding to the dates at which E (y t ) changes, and then applying the test to the residuals. 4 MONE CARLO SIMULAIONS In this section, we discuss: (a) the calculation of critical values, and (b) size and power of the test. Our experiments refer to the small sample properties of the tests developed in Section 2, i.e. to the context of testing for changes in the eigensystem of a matrix of xed dimension. here are two possible approaches to the computation of critical values: either using the EV distribution in (7) or using an approximation similar to that proposed in Csorgo and Horvath 5

17 (997, Section.3.2); other approaches, such as bootstrap, are also possible (see Aue et al., 202). Direct computation of critical values c ; for a test of level is based on c ; = a b ln 2 ln ( )]g. hus, critical values only depend on p and. It is well known that convergence to the EV distribution is usually very slow, which hampers the quality of c ;. Alternatively, critical values can be simulated from < P : sup h n h n " px B;i 2 () # =2 c 0 ( ) i= ; 9 = = ; ; (22) where the B ;i ()s are independent, univariate, n squared o Brownian bridges, generated over a grid of dimension. We set h n = max n; ln 3=2. he time series part of this bound (i.e. the ln 3=2 part) is based on Csorgo and Horvath (997, p. 25), who show that computing the maxima over restricted intervals (speci cally, by truncating at h n = ln 3=2 ) yields tests with good size properties; in our simulations, we have tried other solutions to restrict the interval over which the maximum is taken, but truncating at ln 3=2 yielded the best size properties. In addition to this, due to the multivariate nature of the problem, we also need to truncate at n; this is in order to have full rank estimated covariance matrices. In view of this, critical values c 0 ; are to be simulated for a given combination of p, n and. 4. Finite sample properties of the test We evaluate size and power through a Monte Carlo exercise. As a benchmark, we consider a test for a change in the rst eigenvalue of the covariance matrix. Unreported simulations show that the nite sample performance of tests for changes in the other eigenvalues are very similar. In order to evaluate the impact of large n on nite sample properties, we also report a smaller Monte Carlo exercise applied to testing for changes in the whole covariance matrix. We also assess, by way of comparison, the size and power of Aue et al. s (9) test, which is based on sup () = b c sup b c p h ~S () 0 ~ V z; ~ S ()i =2 : (23) Data are generated as follows. In order to avoid dependence on initial conditions, + 0 data are generated, discarding the rst 0 observations. We carry out our simulations for = f; ; ; 0g and n = f3; 5; 7; 0; 5; 20g. Serial dependence in y t is introduced through an ARMA(,) process: y t = y t + e t + e t ; (24) where e t N (0; I n ) (see below for more details about simulations under the alternative). We conduct our experiments for the cases (; ) = f(0; 0) ; (0:5; 0) ; (0; 0:5) ; (0; 0:5) ; (0:5; 0:5)g. Evidence from other experiments shows that little changes when the covariance matrix of e t is 6

18 non-diagonal, or when it has di erent elements on the main diagonal. All experiments have been conducted using the long run variance estimator in (3), based on full sample estimation of the autocovariance matrices with m = 2=5. Other simulations show a heavy dependence of the results on m; in general, the larger m, the more conservative the test. Finally, the number of replications is 0; all routines are written using Gauss 0. Size We calculate the empirical rejection frequencies for tests of level 5%. We base the test on sup h n b c h n () : (25) his is in accordance with the approximation of the critical values discussed above. Whilst this procedure no longer yields power versus breaks occurring O (ln ln ) periods from the beginning or the end of the sample, however the test retains power versus breaks occurring O ( h n ) periods from the beginning/end of sample. [Insert able somewhere here] he test (see able ) is, in general, undersized in small samples; this tends to disappear as increases, with empirical rejection frequencies belonging, in general, to the interval [0:04; 0:06] with few exceptions. he test has the correct size for large samples. Considering the i.i.d. case, the nominal size level is attained for = or larger. In general, as far as the presence of time dependence is concerned, this does not seem to a ect the size properties of the test in a strong way, as it only makes the size slightly worse (with a tendency towards oversizement). he table also shows that higher values of n have a slight tendency to reduce the size. Similar results are found with Aue et al. s (9) test based on (23), which, in small samples, tends to be slightly more conservative. Power We conduct our simulations under alternatives de ned as ( for t = ; :::; k + for t = k + ; :::; (26) with k = 2 and = r ln ln I n : (27) We set = 2 3 ; 2 in ables 2a and 2b respectively. In able 2c, we also report power versus alternatives close to the beginning of the sample, with k = h n + and = I n. 7

19 [Insert ables 2a-2b somewhere here] Considering mid-sample alternatives (ables 2a and 2b), the test has nontrivial power versus local alternatives (represented here by the case = 2 3 ), and good power when = 2 ; the power becomes higher than %, in general, when is larger than. As n increases, the test becomes increasingly powerful for all the cases considered (sample size and dynamics in the error term); in general, the power of the test is not a ected by the presence of MA disturbances, although a reduction in power is seen in presence of autoregressive roots. As predicted by the theory, tests based on (23) have better power properties, at least for = or higher. Indeed, tests based on (25) have very similar (in fact, slightly higher) power for = ; in that case, however, the factor ln ln, by which the power of tests based on (25) is reduced with respect to tests based on (23), is negligible. [Insert ables 2c-2d somewhere here] ables 2c and 2d consider the power of the test versus nite alternatives that are close to the beginning of the sample. In particular, able 2c reports the power under the alternative that the breakdate is close to the boundary. As predicted by the theory, there is power versus such alternatives. Referring the i.i.d. case as a benchmark, the power becomes higher than % when = 0. As also observed in ables 2a and 2b, as n increases the power slightly increases. As is natural, tests based on (23) have very little power versus beginning of sample alternatives; p by construction, such tests do not have power versus changes that occur closer than O periods to the beginning (or the end) of the sample. Indeed, the presence of some, limited power is to be interpreted as p and k = h n + being of comparable magnitude. As far as the impact of serial correlation is concerned, the power deteriorates in presence of AR roots, which is even more evident in the ARMA case. A less dramatic power reduction is also observed in presence of MA roots. his is due to the use of the full sample estimator of the long run covariance matrix, V ~, which is consistent under the alternative but biased. Unreported simulations show that tests based on (23) and (25) exhibit a substantial power gain when using the partial sample estimates, V; ~. However, both tests become grossly oversized. herefore, as a guideline for empirical applications, we suggest that if an AR structure is found in the data, pre-whitening should be applied. As far as able 2d is concerned, by way of comparison we report the evolution of the power of both tests, based on (25) and (23), as the changepoint moves from the beginning of the sample towards the middle. Results are reported for i.i.d. data only; when considering serial correlation, no changes were noted in the results, save for a general decrease in power (see also the comments to able 2c), and we therefore omit them to save space. he alternatives considered have the following times of change: k = h n + (this is the same as in able 2c;

20 indeed, the rst column of able 2d is the same as the rst column of able 2c); k = 2 (ln )2 ; k = 2 (ln )5=2 and k = 3 p. We note little variation in the power of both tests between the experiments when comparing the rst and the second alternative. he third column, containing the case k = (ln ) 5=2, shows a di erent pattern. ests based on (23) are more powerful for small samples; in this case, however, 2 (ln )5=2 is usually larger than p, which is the threshold after which tests based on (23) become powerful. When increases (e.g. when = and, more noticeably, 0), tests based on (25) have more power than tests based on (23). he power of tests based on (25) becomes very similar, and in fact worse in some cases, to that of tests based on (23) when considering alternatives closer to the mid-sample, i.e. when k = 3 p. In this case, it can be expected that tests based on (23) have nontrivial power, which is indeed the case. However, it can be noted that tests based on (25) perform very well, indeed (marginally) better, than tests based on (23) when is large. Finally, as noted for able 2c, the power of tests based on (25) increases monotonically with across all experiments, and it also has a tendency to increase as n increases, albeit not monotonically. esting for the constancy of : the role of n We report a simulation exercise for the null of no change in. Data are generated as Gaussian, with the same scheme as in (24). We report results for n = f3; 4; 5; 6; 7g and = f; ; ; 0g. When generating data under the alternative, this is de ned as in (26) and (27). In the last column, k = h n + and = I n. [Insert ables 3a-3d somewhere here] able 3 illustrates the role played by n. As n increases, the test becomes increasingly conservative in nite samples; however, as!, the empirical rejection frequencies approach their nominal values. As far as the power is concerned, when considering mid-sample alternatives, the power increases monotonically with as expected, whereas it decreases with n. Such dependence on n is expected, since as n increases a larger trimming occurs, which entails a loss of information. However, for large samples ( = 0), we note that as n increases, the power of the test also increases. he results in able 3b and 3c could be contrasted with those in ables 2a-2b: under mid-sample alternatives, tests for the stability of the whole matrix are more powerful than tests based on the individual eigenvalues, at least in small samples; when = 0, the gures become comparable. A heuristic explanation of this is in Remark 4.2: under the alternative, the long run variance of the estimated eigenvalues is a ected by the eigenvectors, and thus it can be expected to be bounded in probability but biased. Di erent considerations apply to the case of breaks closer to the beginning of the sample, i.e. able 3d. In this case, as n increases, the power decreases. Again, this is likely to be due to the extent of the trimming, which always entails a loss of power. In addition to this, tests 9

21 based on (25) are decidedly more powerful than tests based on (23) for large samples ( = 0). Comparing able 3d with able 2c, it can be noted that the power of tests for the whole matrix is lower (substantially, in the i.i.d. case) than that of tests for changes in the eigenvalues, which is a reversal of the power performance under mid-sample alternatives. Moreover, the power of tests for changes in the eigenvalues is not negatively a ected by n; indeed, it tends to increase with n. 5 EMPIRICAL ILLUSRAIONS USING INERES RAES AND EXCHANGE RAES In this section, we report two empirical applications that illustrate the theory developed in Sections 2 and 3.. Firstly, we consider a classical application of PCA to nance - namely, we investigate the time stability of the covariance matrix of the term structure of interest rates - Section 5.. Secondly, we apply the theory developed in Section 3. to a VAR applied to exchange rates. 5. he time stability of the covariance matrix of interest rates In this section, we apply the theory developed above to test for the stability of the covariance matrix of the term structure of interest rates, and to infer the sources of instability if present. Our analysis is motivated by the study in Perignon and Villa (6), and follows similar steps. As a rst step, we investigate whether the volatility curve (i.e. the term structure of the volatility of interest rates) changes over time; this corresponds to testing for the stability of the main diagonal of the covariance matrix. Further, we verify whether the whole covariance matrix changes. his could be done by directly testing for the constancy of the matrix. Alternatively, in order to reduce the dimensionality of the problem, one could check whether the main three principal components (level, slope and curvature) are stable through time. We choose the latter approach, verifying separately, for each principal component, whether sources of time variation are in the loadings (i.e. the eigenvectors) or in the volatility (i.e. the eigenvalues), or both. Previous studies have found evidence of changes in the yield curve. Using a descriptive approach based on splitting the sample at some predetermined points in time, indicated by stylised facts, Bliss (997) nds that the eigenvectors of the covariance matrix of interest rates are quite stable, although the eigenvalues di er across subsamples. Perignon and Villa (6), under the assumption that data are i.i.d. Gaussian, nd evidence of changes in the volatilities (eigenvalues) of the principal components across four di erent subperiods (chosen a priori) in the time interval January December 999. We conduct a similar exercise to Perignon and Villa (6), relaxing the assumptions of i.i.d. Gaussian data, and avoiding the a priori selection of breakdates which could be rather arbitrary. We apply our test to US data, considering monthly and weekly frequencies, spanning 20

22 from April 997 to November 200 (monthly - the sample size is m = 64) and from the second week of February 997 to the last week of February 20 (weekly - the sample size is w = 733); the number of maturities which we consider is n =, corresponding to (m, 3m, 6m, 9m, 2m, 5m, m, 2m, 24m, 30m, 3y, 4y, 5y, 6y, 7y, y, 9y, 0y). Figure reports the term structure in the period considered. [Insert Figure somewhere here] Preliminary analysis shows that yields are highly persistent. herefore, we use returns, which are found to be much less autocorrelated (particularly, they have no autocorrelation pattern for higher maturities). able 4 shows some descriptive statistics for both monthly and weekly data; it is interesting to note how the large values of skewness and kurtosis of each maturity lead to reject the assumption of normality. [Insert able 4 somewhere here] As far as the notation is concerned, y t denotes, henceforth, the demeaned -dimensional vector of rst-di erenced maturities. Preliminary evidence based on the autocorrelation function of the squared returns shows that there is very little serial correlation, and, with higher maturities, no correlation at all. In light of this, we set the bandwidth, for the estimation of the long-run variance, as m = p ln (see equation (3)). he rst step of our analysis is an evaluation of the stability of the variances of the rst di erenced maturities, i.e. of the elements on the main diagonal of = E (y t yt). 0 Instead of checking for the stability of the whole main diagonal, we test the volatilities one by one; this approach should be more constructive if the null of no changes were to be rejected, in that it would indicate which maturity changes and when. In order to control for the size of this multiple comparison, we propose a Bonferroni correction. We calculate the critical values for each test as I = P n, where P is the size of the whole procedure. Using these critical values yields, approximately, a level P not greater than %, 5% and 0% corresponds to conducting each test at levels I = 0:056%, 0:2% and 0:56% respectively. Critical values for individual tests of levels I = %, 5% and 0%, and for procedure level P = %, 5% and 0%, are in able 5. [Insert able 5 somewhere here] 2

23 As a second step, we verify whether slope, level and curvature are constant over time. Particularly, we carry out separately the detection of changes in the volatility of the principal components (verifying the time stability of the three largest eigenvalues, say, 2 and 3 ), and in their loading (verifying the stability of the eigenvalue-normed eigenvectors corresponding to the three largest eigenvalues, denoted as, 2 and 3 ). As far as eigenvectors are concerned, (0) and () ensure that, when running the test, the CUSUM transformation of the estimated i s has the same sign for all values of, thus overcoming the issue of the eigenvectors being de ned up to a sign. Results for both experiments, at both frequencies, are reported in able 6. [Insert able 6 somewhere here] It is well known that controlling the procedure-wise error by a Bonferroni correction can be rather conservative. In our case, the values of test statistics (able 6) can be contrasted with the critical values to be used for single hypothesis testing (reported in able 5 as cv ), which is the least conservative approach. When using a 5% level, results are exactly the same. he only exception is the test for the stability of the second eigenvector, 2, when using weekly data, where the null of no change is now rejected at 5%. A marginal discrepancy can be observed in the rst panel of able 6, when testing for the constancy of the diagonal elements of with weekly data. When using cv as critical values, two maturities (the 30 months and the 3 years ones) now appear to have a break. he rest of the results (especially the absence of breaks in monthly data) is the same as when using a Bonferroni correction. able 6 shows an interesting discrepancy between monthly and weekly data. Monthly data, as a whole, have a stable covariance structure over time: no changes are present either in the volatilities of the maturities, or in any of the principal components. Such result can be ascribed also to the test being less powerful in small samples, but able 4 shows that the two datasets (monthly and weekly) do have di erent features, at least as far as descriptive statistics are concerned. As far as covariance instability is concerned, when monthly data are considered the only change is recorded in 3, the volatility of the curvature, which has a break signi cant at 0%. he second and third panel of the table show that the principal component structure has a change in the size of the curvature, signi cant at 5%. he corresponding estimated breakdate, selected as the maximizer of the CUSUM statistic, is January. able 7 reports the proportion of the total variance explained by each principal component before and after this date. [Insert able 7 somewhere here] As far as weekly data are concerned, there is evidence of instability in the covariance structure. At a macro level, the variances of longer maturities (from 4 years onwards) change, 22

24 whilst the variances of shorter maturities are constant. For most maturities, the breakdate is around the rst week of December 7. his is expected, since December 7 is generally associated with the deepening of the recent recession. It is interesting to note that the longest maturity, the 0-year one, has a break at around the last week of August. As far as principal components are concerned, the second panel of able 6 shows that whilst the volatility of slope and curvature does not change over time, the loading of the level changes at the rst week of December 7, consistently with the ndings for the variances. As the third panel of the table shows, the loadings of principal components are subject to change: the level and the curvature change signi cantly around the middle/end of March (possibly due to an attraction e ect of the variance of the 0-year maturity); the slope has a signi cant break also, a few weeks later. he presence of signi cant changes in the loadings of each principal component as a result of the 7-9 recession is a di erent feature to what Perignon and Villa (6) found in the time period they consider, when eigenvectors were not subject to changes over time. 5.2 A VAR model for exchange rates: the stability of the covariance matrix of the error term In this section, we apply the test developed in Section 3. to a VAR, checking whether the covariance matrix of the error term is stable over time. Following Carriero et al. (9), we consider a VAR consisting of 4 exchange rates vis-a-vis the US Dollar. Speci cally, we consider monthly averages of the following currencies: Euro, British Pound, Yen, Canadian Dollar; data are collected from January 999 to April 203, which entails a sample size (modulo the rst entry) of = 72. All data are taken from Datastream. wo tests are carried out in parallel. We verify whether the covariance matrix of the error term as a whole is subject to any changes, in a similar spirit to the simulations in Section 4 (ables 3a-3d). Furthermore, we also test for the stability of each of the eigenvalues of the covariance matrix. Since we nd very weak evidence of breaks (if any) in the whole matrix and in its spectrum, we do not carry out any analysis on the eigenvectors: the key nding in this section is that, despite a major event such as the 7-9 recession, the covariance matrix of the VAR model of exchange rates does not change over time. Carriero et al. (9) propose a Bayesian approach to model the exchange rates directly, imposing a prior on the VAR coe cients in order to shrink them towards a unit root representation. Preliminary analysis shows that, as expected, our series have a unit root. We therefore t a VAR to the returns of the exchange rates. We report descriptive statistics for the returns, and a plot of the original series, in able and Figure 2 respectively. [Insert able and Figure 2 somewhere here] As the evidence in Carriero et al. (9) shows, applying VAR models to exchange rates 23

25 could yield a good level of forecasting accuracy, also in light of the fact that exchange rates have a tendency to co-move. However, as pointed out in the Introduction, Castle et al. (200) nd that when the covariance matrix of the error term in a regression changes, this could have pernicious e ects on the forecasting ability of the model; thus, applying our test as discussed in Section 3. to the covariance matrix of the error term (and to its eigenvalues) could shed some light as to the forecasting ability of the VAR. Estimation results and mis-speci cation tests are reported in able 9: the best speci cation was found to be a VAR of order 5, using lags, 2 and 5. As can be seen from the able, the model is correctly speci ed, save for the lack of normality in the error term; in view of Assumption R, this is not an issue when applying our test. We also note that, as can be expected as regards the residuals of a correctly speci ed model, the error term is found to be serially uncorrelated, and we therefore estimate the long run variance without adjusting for serial correlation. [Insert able 9 somewhere here] he test is applied, preliminarily, to the (lower triangular part of the) whole covariance matrix: the results in Section 4 suggest that the dimension of the matrix (n = 4 in our context) a ords the test to have the correct size and power versus the alternative of one or more breaks. Given that we consider only the lower triangular part of the matrix, this entails applying our test with p = 0 constraints under the null, as opposed to p = 6 constraints which is the case in the Monte Carlo exercise in Section 4. We also verify the stability of each of the four eigenvalues of the matrix. As in Section 5., this procedure is implemented by controlling for the family-wise rejection rate though a Bonferroni correction. Critical values for individual tests of levels I = %, 5% and 0%, and for procedure level P = %, 5% and 0%, with the Bonferroni correction I = P 4 are in able 5. Results are reported in able 0. [Insert able 0 somewhere here] As can be seen from the rst panel of the able, the matrix, as a whole, is stable over time: as pointed out above, despite the presence of a major event such as the 7-9 recession, the covariance structure of the error term of the VAR does not change. his could be interpreted in light of the stylised fact that exchange rates tend to co-move (see the discussion in Carriero et al., 9). his nding is also reinforced by the second panel of able 0, where we nd that the rst three largest eigenvalues (denoted as 3 ) do not change over time. As far as the smallest one is concerned ( 4 ), some evidence of a change is indeed found. However, the test rejects the null of a stable eigenvalue only when applied to each eigenvalue separately. 24

26 Conversely, when adjusting for the family-wise error, no break is found at 5% level (the null of no break is rejected at 0% level for 4 ); as mentioned in Section 5., this is a consequence of the Bonferroni correction yielding relatively conservative results. However, this suggests very weak evidence of a break in the smallest eigenvalue; the estimated breakdate is found to be around June, which has the natural interpretation of being a consequence of the latest recession. 6 CONCLUSIONS In this paper, we propose a test for the null of no breaks in the eigensystem of a covariance matrix. he assumptions under which we derive our results are su ciently general to accommodate for a wide variety of datasets. We show that our test is powerful versus alternatives as close to the boundaries of the sample as O (ln ln ). Results are extended to testing for the stability of the eigensystem. We also derive a correction for the nite sample bias when estimating eigenvalues and eigenvectors, which can be relatively severe for large n or small. he theory is also extended to develop tests for the null of no change in the covariance matrix of the error term in a multivariate regression (including the case of VARs). As shown in Section 4, the properties of the test are satisfactory: the correct size is attained under various degrees of serial dependence, and the test exhibits good power. he results in this paper suggest several avenues for research. he test discussed here is a stability test for the null of no change in a covariance matrix. Although we only consider one alternative, the test could be extended to be applied sequentially, i.e. by splitting the sample around an estimated breakdate and test for breaks in each subsample. Also, results are derived under the minimal assumption that the 4-th moment exists. Aue et al. (9) provide a discussion as to how to proceed if this is not the case, which involves fractional transformations of the series, viz. yit for some 2 (0; ), although the optimal choice of is not straightforward. hese issues are currently under investigation by the authors. REFERENCES Andrews, D.W.K., 97. Asymptotic results for generalised Wald tests. Econometric heory 3, Andrews, D.W.K., 993. ests for parameter instability and structural changes with unknown change point. Econometrica 6, Andrews, D.W.K., Ploberger, W., 994. Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62, Audrino, F., Barone-Adesi, G., Mira, A., 5. he stability of factor models of interest rates. Journal of Financial Econometrics 3, Aue, A., Hormann, S., Horvath, L., Reimherr. M., 9. Break detection in the covariance structure of multivariate time series models. Annals of Statistics 37,

27 Aue, A., Horvath, L., Huskova, M., 202. Segmenting mean-nonstationary time series via trending regressions. Journal of Econometrics 6, Bai, J., 0. Vector autoregressive models with structural changes in regression coe cients and in variance-covariance matrices. Annals of Economics and Finance, Bates, B.J., Plagborg-Muller, M., Stock, J.H., Watson, M.W., 203. Consistent factor estimation in dynamic factor models with structural instability. Forthcoming, Journal of Econometrics. Bliss, R.R., 997. Movements in the term structure of interest rates. Economic Review, Federal Reserve Bank of Atlanta, Q4, Bliss, R.R., Smith, D.C., 997. he stability of interest rate processes. Federal Reserve Bank of Atlanta, Working Paper Carriero, A., Kapetanios, G., Marcellino, M., 9. Forecasting exchange rates with a large Bayesian VAR. International Journal of Forecasting 25, Castle, J.R., Fawcett, N., Hendry, D.F., 200. Forecasting with equilibrium-correction models during structural breaks. Journal of Econometrics 5, Corradi, V., 999. Deciding between I(0) and I() via FLIL-based bounds. Econometric heory 5, Csorgo, M., Horvath, L., 997. Limit heorems in Change-point Analysis. Chichester: Wiley. Davidson, J., 2a. Stochastic Limit heory. Oxford: Oxford University Press. Davidson, J., 2b. Establishing conditions for the functional central limit theorem in nonlinear and semiparametric time series processes. Journal of Econometrics 06, Davis, C., Kahan, W., 970. he rotation of eigenvectors by a perturbation. III. SIAM Journal of Numerical Analysis 7, -46. Eberlein, E., 96. On strong invariance principles under dependence assumptions. Annals of Probability 4, Flury, B.H., 94. Common principal components in k groups. Journal of the American Statistical Association 79, Flury, B.H., 9. Common principal components and related multivariate models. New York: Wiley. Gallant, A.R., White, H., 9. A uni ed theory of estimation and inference for nonlinear dynamic models. Oxford: Basil Blackwell. Götze, F., Zaitsev, A., 202. Estimates for the rate of strong approximation in Hilbert space. Mimeo. Hill, J.B., 200. On tail index estimation for dependent, heterogeneous data. Econometric heory, 26, Hill, J.B., 20. ail and non-tail memory with applications to extreme value and robust statistics. Econometric heory, 27, Horn, R.A., Johnson, C.R., 999. Matrix Analysis. Cambridge: Cambridge University Press. 26

28 Horvath, L., 993. he maximum likelihood method for testing changes in the parameters of normal observations. Annals of Statistics 2, Kollo,., Neudecker, H., 997. Asymptotics of Pearson-Hotelling principal-component vectors of sample variance and correlation matrices. Behaviormetrika 24, Ling, S., 7. esting for change points in time series models and limiting theorems for NED sequences. Annals of Statistics 35, Litterman, R., Scheinkman, J.A., 99. Common factors a ecting bond returns. Journal of Fixed Income, Perignon, C., Villa, C., 6. Sources of time variation in the covariance matrix of interest rates. Journal of Business 79, Perron, P., 6. Dealing with breaks. In Mills,.C. and Patterson, K. (eds.). Palgrave Handbook of Econometrics: Vol. Econometric heory. Basingstoke: Palgrave Macmillan. Pesaran, M.H., Shin, Y., 99. Generalized impulse response analysis in linear multivariate models. Economics Letters 5, Phillips, P.C.B., Solo, V., 992. Asymptotics for linear processes. Annals of Statistics 20, 97-. Stock, J. H., Watson, M.W., 999. Forecasting in ation. Journal of Monetary Economics 44, Stock, J. H., Watson, M.W., 2. Forecasting using principal components from a large number of predictors. Journal of the American Statistical Association 97, Stock, J. H., Watson, M.W., 5. Implications of dynamic factor models for VAR analysis. Manuscript. rapani, L., 203. On bootstrapping panel factor series. Journal of Econometrics 72, Zhou, Y., 20. On the eigenvalues of specially low-rank perturbed matrices. Applied Mathematics and Computation, 27,

29 n (; #) (0; 0) (0:5; 0) (0; 0:5) (0; 0:5) (0:5; 0:5) 0:03 0:05 0:009 0:00 0:03 0:006 0:002 0:003 0:005 0:004 0:025 0:033 0:026 0:025 0:034 0:02 0:07 0:04 0:04 0:02 0:04 0:04 0:04 0:040 0:05 0:029 0:023 0:025 0:029 0:020 0:044 0:05 0:043 0:045 0:067 0:034 0:027 0:029 0:033 0:030 0:00 0:0 0:0 0:00 0:02 0:003 0:007 0:007 0:004 0:007 0:024 0:030 0:029 0:030 0:036 0:0 0:03 0:0 0:09 0:06 0:032 0:04 0:032 0:042 0:047 0:025 0:030 0:024 0:025 0:023 0:044 0:063 0:049 0:0 0:065 0:033 0:043 0:03 0:039 0:037 0:007 0:009 0:02 0:007 0:00 0:003 0:005 0:005 0:004 0:003 0:022 0:023 0:022 0:03 0:025 0:05 0:009 0:00 0:0 0:0 0:032 0:049 0:043 0:03 0:047 0:023 0:029 0:029 0:02 0:030 0:04 0:064 0:052 0:039 0:065 0:046 0:043 0:040 0:034 0:042 0:004 0:006 0:004 0:002 0:003 0:005 0:003 0:005 0:004 0:003 0:0 0:034 0:029 0:09 0:03 0:06 0:07 0:06 0:007 0:06 0:030 0:053 0:040 0:029 0:053 0:023 0:033 0:02 0:026 0:024 0:044 0:063 0:057 0:05 0:065 0:036 0:033 0:035 0:040 0:036 0:00 0:000 0:00 0:00 0:000 0:004 0:003 0:00 0:004 0:003 0:03 0:05 0:07 0:0 0:09 0:04 0:0 0:00 0:03 0:03 0:027 0:053 0:040 0:03 0:055 0:024 0:032 0:025 0:026 0:03 0:047 0:06 0:052 0:047 0:063 0:02 0:035 0:03 0:03 0:03 0:000 0:00 0:00 0:000 0:00 0:003 0:004 0:003 0:002 0:003 0:007 0:009 0:00 0:00 0:0 0:03 0:06 0:06 0:07 0:03 0:07 0:033 0:020 0:023 0:039 0:0 0:026 0:022 0:022 0:02 0:040 0:05 0:047 0:057 0:059 0:035 0:042 0:037 0:037 0:042 able. Empirical rejection frequencies for the null of no changes in the largest eigenvalue of. Data are generated according to (24). he empirical sizes have con dence interval [0:04; 0:06]. 2

30 r ln ln( ) n = 2= (; #) (0; 0) (0:5; 0) (0; 0:5) (0; 0:5) (0:5; 0:5) 0:035 0:04 0:33 0:2 0:235 0:293 0:427 0:533 0:039 0:06 0:7 0:53 0:3 0:374 0:460 0:54 0:040 0:0 0:20 0:4 0:344 0:409 0:56 0:627 0:032 0:020 0:27 0:203 0:356 0:440 0:52 0:66 0:006 0:029 0:2 0:24 0:440 0:54 0:623 0:727 0:000 0:035 0:2 0:20 0:46 0:55 0:644 0:764 0:02 0:00 0:2 0:073 0:0 0:5 0:302 0:3 0:026 0:06 0:7 0:099 0:245 0:244 0:337 0:395 0:034 0:02 0:54 0:2 0:247 0:247 0:339 0:397 0:026 0:02 0:57 0:6 0:247 0:24 0:364 0:430 0:00 0:00 0:36 0:45 0:35 0:35 0:390 0:47 0:000 0:04 0:26 0:52 0:296 0:3 0:40 0:493 0:027 0:00 0:2 0:04 0:2 0:22 0:37 0:424 0:02 0:020 0:3 0:2 0:246 0:29 0:377 0:469 0:037 0:06 0:49 0:45 0:257 0:305 0:37 0:42 0:033 0:02 0:3 0: 0:273 0:32 0:449 0:536 0:004 0:06 0:65 0: 0:365 0:4 0:43 0:570 0:000 0:02 0:64 0:204 0:377 0:447 0:6 0:594 0:034 0:006 0:22 0:0 0:9 0:232 0:335 0:40 0:026 0:05 0:3 0: 0:24 0:265 0:370 0:449 0:03 0:02 0:6 0:26 0:290 0:33 0:407 0:6 0:02 0:05 0:75 0:5 0:309 0:336 0:434 0:537 0:00 0:020 0:7 0: 0:34 0:426 0:493 0:574 0:000 0:05 0:49 0:20 0:33 0:466 0:52 0:603 0:06 0:005 0:2 0:052 0:67 0:40 0:259 0:272 0:025 0:02 0:03 0:075 0:97 0:6 0:2 0:309 0:033 0:00 0:32 0:04 0:205 0:93 0:2 0:32 0:025 0:00 0:40 0:00 0:209 0:92 0:29 0:347 0:003 0:009 0:2 0: 0:2 0:265 0:3 0:36 0:000 0:04 0:099 0:7 0:27 0:270 0:403 0:426 able 2a. Power of the test under the null of no changes in the largest eigenvalue of. Data are generated according to (24) and under the alternative hypothesis as in (27). 29

31 r ln ln( ) n = = (; #) (0; 0) (0:5; 0) (0; 0:5) (0; 0:5) (0:5; 0:5) 0:055 0:020 0:272 0:25 0:54 0:593 0:74 0:934 0:070 0:02 0:34 0:34 0:66 0:77 0:900 0:955 0:073 0:03 0:362 0:35 0:679 0:764 0:96 0:949 0:056 0:037 0:406 0:405 0:7 0:796 0:954 0:979 0:06 0:053 0:35 0:466 0:00 0:67 0:964 0:97 0:000 0:049 0:43 0:553 0:35 0:906 0:90 0:992 0:030 0:004 0: 0:36 0:30 0:43 0:652 0:762 0:04 0:022 0:206 0:6 0:420 0:447 0:704 0:795 0:043 0:09 0:253 0: 0:467 0:53 0:702 0:795 0:040 0:07 0:252 0:25 0:465 0:524 0:76 0:56 0:00 0:0 0:234 0:24 0:57 0:602 0: 0:3 0:000 0:024 0:224 0:22 0:572 0:60 0:25 0:902 0:045 0:02 0:23 0:2 0:4 0:50 0:753 0: 0:056 0:029 0:256 0:255 0:5 0:54 0:0 0:92 0:062 0:039 0:26 0:292 0:526 0:65 0:5 0:7 0:053 0:030 0:30 0:300 0:566 0:64 0:9 0:943 0:00 0:027 0:30 0:355 0:666 0:733 0:907 0:94 0:000 0:032 0:3 0:39 0:720 0:77 0:93 0:966 0:0 0:03 0:23 0:9 0:45 0:46 0:760 0:36 0:045 0:020 0:259 0:220 0: 0:579 0:75 0:79 0:063 0:02 0:295 0:274 0:55 0:63 0:46 0:95 0:039 0:027 0:299 0:30 0:59 0:674 0:66 0:920 0:04 0:037 0:332 0:357 0:6 0:764 0:903 0:954 0:000 0:037 0:290 0:37 0:7 0:77 0:932 0:97 0:023 0:00 0:64 0: 0:29 0:306 0:524 0:625 0:042 0:0 0: 0:9 0:356 0:349 0:56 0:65 0:04 0:03 0:29 0:6 0:40 0:406 0:592 0:67 0:043 0:009 0:222 0:6 0:403 0:49 0:665 0:74 0:004 0:03 0:207 0: 0:2 0:496 0:707 0:0 0:000 0:09 0:4 0:227 0: 0:499 0:74 0:27 able 2b. Power of the test for the null of no changes in the largest eigenvalue of. Data are generated according to (24) and under the alternative hypothesis as in (27). 30

32 n k = 2 [ln ( )] 3=2 (; #) (0; 0) (0:5; 0) (0; 0:5) (0; 0:5) (0:5; 0:5) 0:07 0:023 0:025 0:027 0:030 0:054 0:040 0:049 0:039 0:034 0:77 0:47 0:43 0:56 0:06 0:62 0: 0:0 0:06 0: 3 0:45 0:69 0:234 0:222 0:40 0:224 0:5 0:3 0:5 0:0 0:6 0:226 0: 0:467 0: :56 0:27 0:29 0:25 0:2 0:0 0:024 0:027 0:09 0:022 0:067 0:044 0:059 0:036 0:03 0:2 0:20 0:34 0:4 0:090 0:56 0:067 0:092 0:07 0: :549 0:69 0:242 0:254 0:64 0:26 0:57 0:93 0: 0:36 0: 0:22 0:497 0:466 0:22 0 0:5 0:44 0:56 0:07 0:24 0:04 0:02 0:026 0:09 0:03 0:057 0:04 0:05 0:043 0:037 0:202 0:25 0:54 0:4 0:090 0:6 0:07 0: 0:02 0: :536 0:52 0:253 0:247 0:7 0:247 0:45 0:77 0:5 0:03 0:909 0:23 0:4 0:477 0: :94 0:5 0:75 0:3 0:3 0:02 0:020 0:07 0:09 0:06 0:07 0:033 0:053 0:056 0:029 0:7 0:4 0:35 0:2 0:099 0:5 0:09 0:03 0:09 0:05 0 0:56 0:5 0:225 0:227 0:55 0:262 0:29 0:0 0:0 0:092 0:95 0:262 0:495 0:4 0:25 0 0:79 0:4 0:26 0:53 0:099 0:0 0:006 0:009 0:006 0:005 0:067 0:03 0:054 0:043 0:030 0:2 0:0 0:22 0:42 0:04 0:74 0:055 0:072 0:06 0:0 5 0:55 0:60 0:23 0:244 0:56 0:2 0:35 0:2 0:79 0:4 0:99 0:246 0:464 0:449 0: :55 0:06 0:26 0:33 0:07 0:07 0:00 0:003 0:002 0:00 0:074 0:034 0:039 0:037 0:003 0:94 0: 0:42 0:42 0:09 0:4 0:04 0:07 0:072 0: :5 0:67 0:225 0:225 0:62 0:249 0:2 0:7 0:3 0:079 0:93 0:259 0:475 0:444 0:26 0 0:6 0:24 0:46 0:24 0:07 able 2c. Power of the test for the null of no changes in the largest eigenvalue of. Data are generated according to (24), under the alternative speci ed as in (27). 3

33 n k h n + 2 [ln ( )]2 2 [ln ( )]5=2 3 p 0:07 0:07 0:07 0:059 0:054 0:037 0:037 0:62 0:77 0:2 0:0 0:9 0:62 0:42 0:64 0:43 3 0:45 0:4 0:304 0:609 0:224 0:7 0:249 0:64 0:6 0:34 0:904 0: :56 0:57 0:306 0:47 0:0 0:022 0:022 0:049 0:067 0:052 0:052 0:7 0:2 0:77 0:09 0:230 0:56 0:52 0:72 0: :549 0:534 0:362 0:702 0:26 0:95 0:29 0:704 0: 0:5 0:95 0:99 0 0:5 0:67 0:32 0:77 0:04 0:05 0:05 0:055 0:057 0:04 0:04 0:5 0:202 0:7 0:0 0:254 0:6 0:35 0:69 0: :536 0:546 0:35 0:63 0:247 0:4 0:273 0:69 0:909 0:94 0:930 0: :94 0:9 0:353 0:900 0:02 0:00 0:00 0:06 0:07 0:057 0:057 0:234 0:7 0:74 0:03 0:237 0:5 0:37 0:59 0: :56 0:56 0:375 0:709 0:262 0:202 0:292 0:7 0:95 0:936 0:942 : :79 0:95 0:356 0:905 0:0 0:009 0:009 0:047 0:067 0:064 0:640 0:29 0:2 0:4 0:094 0:246 0:74 0:30 0:5 0:45 5 0:55 0:596 0:422 0:762 0:2 0:97 0:296 0:757 0:99 0:936 0:957 : :55 0:76 0:337 0:92 0:07 0:004 0:004 0:033 0:074 0:630 0:630 0:25 0:94 0:7 0:07 0:276 0:4 0:46 0:4 0: :5 0:4 0:47 0:72 0:249 0:9 0:336 0:77 0:93 0:99 0:972 0: :6 0:75 0:369 0:935 able 2d. Power of the test for the null of no changes in the largest eigenvalue of. Data are generated as i.i.d., under the alternative speci ed in (27); the values of the time of change k are tabulated in the second row of the table. 32

34 n (; #) (0; 0) (0:5; 0) (0; 0:5) (0; 0:5) (0:5; 0:5) 0:047 0:000 0:000 0:000 0:000 0:052 0:000 0:000 0:000 0:000 0:045 0:000 0:000 0:000 0:000 0:043 0:000 0:000 0:000 0: :056 0:036 0:03 0:03 0:035 0:044 0:024 0:023 0:02 0:029 0:062 0:044 0:04 0:040 0:04 0 0:042 0:040 0:03 0:040 0:040 0:005 0:000 0:000 0:000 0:000 0:009 0:000 0:000 0:000 0:000 0:032 0:000 0:000 0:000 0:000 0:02 0:000 0:000 0:000 0: :0 0:030 0:032 0:03 0:02 0:023 0:03 0:06 0:020 0:04 0:065 0:042 0:04 0:039 0: :037 0:032 0:030 0:03 0:032 0:002 0:000 0:000 0:000 0:000 0:00 0:000 0:000 0:000 0:000 0:02 0:000 0:000 0:000 0:000 0:09 0:000 0:000 0:000 0: :047 0:03 0:024 0:025 0:026 0:020 0:00 0:009 0:00 0:00 0:059 0:03 0:03 0:03 0:06 0 0:029 0:032 0:032 0:02 0:023 0:00 0:000 0:000 0:000 0:000 0:002 0:000 0:000 0:000 0:000 0:06 0:000 0:000 0:000 0:000 0:009 0:000 0:000 0:000 0: :037 0:034 0:030 0:029 0:029 0:02 0:0 0:02 0:03 0:06 0:065 0:04 0:04 0:040 0: :040 0:039 0:036 0:035 0:036 0:000 0:000 0:000 0:000 0:000 0:000 0:000 0:000 0:000 0:000 0:00 0:000 0:000 0:000 0:000 0:003 0:000 0:000 0:000 0: :040 0:037 0:033 0:03 0:030 0:022 0:024 0:029 0:032 0:030 0:07 0:056 0:040 0:04 0: :039 0:032 0:030 0:040 0:040 able 3a. Empirical rejection frequencies for the null of no change in. Data are generated according to (24). he empirical sizes reported here have con dence interval [0:04; 0:06]. 33

35 n (; #) (0; 0) (0:5; 0) (0; 0:5) (0; 0:5) (0:5; 0:5) 0:490 0:224 0:20 0:276 0:4 0:46 0:22 0:3 0: 0:2 0:573 0:320 0:356 0:32 0:344 0:55 0:290 0:2 0:3 0:36 3 0:62 0:2 0:55 0:476 0:52 0:605 0:47 0:3 0:524 0:47 0:724 0:6 0:576 0:555 0:59 0 0:737 0: 0:554 0:53 0:523 0:33 0:000 0:000 0:002 0:000 0:2 0:000 0:000 0:000 0:000 0:40 0:65 0:97 0:20 0: 0:33 0: 0:23 0:54 0:65 4 0:567 0:444 0:462 0:4 0:406 0:547 0:39 0:42 0:4 0:427 0:24 0:77 0:794 0:69 0: :45 0:2 0:76 0:02 0:739 0:092 0:000 0:000 0:000 0:000 0:06 0:000 0:000 0:000 0:000 0:407 0:203 0:222 0:24 0:20 0:33 0:26 0:93 0: 0:24 5 0:624 0:9 0:52 0:570 0:552 0:579 0:476 0:0 0:4 0:4 0:5 0:9 0:723 0:74 0: :67 0:06 0:6 0:799 0:73 0:03 0:000 0:000 0:000 0:000 0:07 0:000 0:000 0:000 0:000 0:3 0:27 0:34 0:20 0:36 0:235 0:62 0:56 0:94 0: :66 0:70 0:4 0:536 0:499 0:593 0:56 0:52 0:49 0:7 0:99 0:734 0:05 0: 0:55 0 0:92 0:62 0:797 0:93 0:2 0:000 0:000 0:000 0:000 0:000 0:000 0:000 0:000 0:000 0:000 0:293 0:09 0:03 0:06 0:04 0:6 0: 0:076 0:077 0: 7 0:67 0:60 0:606 0:63 0:577 0:534 0:533 0:62 0:59 0:562 0:92 :000 0:903 0:926 0: :903 :000 0:945 0: 0:92 able 3b. Power of the test for the null of no changes in. Data are generated according to (24) and under the alternative hypothesis speci ed in (27) with =

36 n (; #) (0; 0) (0:5; 0) (0; 0:5) (0; 0:5) (0:5; 0:5) 0:736 0:600 0:59 0:567 0:66 0:72 0:555 0:599 0:602 0:59 0:55 0:77 0:756 0:2 0:734 0:33 0:03 0:799 0:77 0:3 3 0:944 0:906 0:92 0:36 0:99 0:939 0: 0:47 0:935 0:92 0:995 :000 :000 :000 : :996 :000 :000 :000 :000 0:270 0:36 0:22 0:46 0:5 0:227 0:204 0:23 0:2 0:99 0:72 0:74 0:632 0:6 0:65 0:675 0:60 0:66 0:639 0: :932 0:99 0:73 0: 0:76 0:937 0:936 0:93 0:952 0:90 :000 :000 :000 :000 :000 0 :000 :000 :000 :000 :000 0:204 0:099 0:072 0:070 0:0 0:35 0: 0:0 0:22 0:62 0:729 0:70 0:69 0:732 0:677 0:66 0:67 0:696 0:64 0: :973 :000 0:99 0:33 0:9 0:964 :000 0:972 0:959 0:932 :000 :000 :000 :000 :000 0 :000 :000 :000 :000 :000 0:077 0:000 0:000 0:000 0:000 0:032 0:000 0:000 0:000 0:000 0:72 0:654 0:67 0:6 0:700 0:59 0:623 0:64 0:673 0:6 6 0:977 :000 0:99 0:973 :000 0:97 :000 :000 :000 :000 :000 :000 :000 :000 :000 0 :000 :000 :000 :000 :000 0:000 0:000 0:000 0:000 0:000 0:000 0:000 0:000 0:000 0:000 0:652 0:53 0:603 0:652 0:669 0:525 0:72 0:6 0:624 0:73 7 0:97 :000 :000 0:997 :000 0:972 :000 :000 :000 :000 :000 :000 :000 :000 :000 0 :000 :000 :000 :000 :000 able 3c. Power of the test for the null of no changes in. Data are generated according to (24) and under the alternative hypothesis as in (27) with = 2. 35

37 n (; #) (0; 0) (0:5; 0) (0; 0:5) (0; 0:5) (0:5; 0:5) 0:077 0:000 0:000 0:000 0:000 0: 0:000 0:000 0:000 0:000 0:6 0:065 0:03 0:099 0:06 0:25 0:072 0:064 0:074 0: :9 0:20 0:97 0:23 0:99 0:25 0:03 0:2 0: 0:09 0:539 0:592 0:60 0:59 0: :063 0:073 0:076 0:079 0:052 0:052 0:000 0:000 0:000 0:000 0:095 0:000 0:000 0:000 0:000 0:03 0:00 0:003 0:000 0:002 0:090 0:000 0:000 0:000 0: :57 0:9 0:30 0:277 0:32 0:09 0:063 0:053 0:040 0:069 0:7 0:599 0:600 0:522 0:35 0 0:02 0:09 0:03 0:079 0:067 0:023 0:000 0:000 0:000 0:000 0:074 0:000 0:000 0:000 0:000 0:005 0:000 0:000 0:000 0:000 0:076 0:000 0:000 0:000 0: :06 0: 0:279 0:275 0:74 0:02 0:09 0:067 0:079 0:092 0:373 0:39 0:40 0:402 0:3 0 0:066 0:09 0:079 0:097 0:02 0:007 0:000 0:000 0:000 0:000 0:02 0:000 0:000 0:000 0:000 0:047 0:000 0:000 0:000 0:000 0:059 0:000 0:000 0:000 0: :09 0:024 0:036 0:02 0:000 0:04 0:000 0:000 0:000 0:000 0:243 0:333 0:394 0:390 0:46 0 0:067 0:03 0:027 0:025 0:044 0:000 0:000 0:000 0:000 0:000 0:002 0:000 0:000 0:000 0:000 0:029 0:000 0:000 0:000 0:000 0:060 0:000 0:000 0:000 0: :07 0:000 0:032 0:02 0:000 0:065 0:000 0:000 0:000 0:000 0:47 0:390 0:334 0:33 0:63 0 0:079 0:02 0:00 0:002 0:00 able 3d. Power of the test for the null of no changes in. Data are generated according to (24) and under the alternative hypothesis speci ed in (26), with k = h n + and = I n. 36

38 monthly data weekly data mean std. dev skew kurt AR() ARCH(7) mean std. dev skew kurt AR() ARCH(7) m m m m m m m m m m y y y y y y y y able 4. Descriptive statistics for monthly and weekly data. We report the mean, standard deviation, skewness and kurtosis, the AR() coe cient and the p-value of the ARCH(7) test for the returns -,, and denote rejection of the null of no ARCH e ects at 0%, 5% and % levels, respectively. 37

39 level 0% 5% % (; p) cv 2:044 3:076 3:575 (63; ) cv 3 3:97 3:4222 3:9344 cv 3:743 3:9690 4:273 cv 2:947 3:9 3:6379 (732; ) cv 3 3:3266 3:553 3:79 cv 3:7779 4:005 4:5603 cv 6:97 6:4075 6:7747 (63; ) cv 3 6:6 6:646 7:0053 cv 6:3066 6:5202 6:9209 (732; ) cv 3 6:62 6:040 7:49 cv 2:7952 3:0336 3:595 (66; ) cv 4 3:26 3:523 3:974 (66; 0) cv 0 5:074 5:372 5:796 able 5. Critical values for the applications in Sections 5. and 5.2. cv N refers to the critical value to be used when N hypotheses are being tested for, in order to have a procedure-wise level of 0%, 5% and %, respectively. Panels with (; p) = (63; ) and (732; ) contain critical values for unidimensional tests (monthly and weekly frequencies, respectively) to test for changes in eigenvalues or to verify the stability of the diagonal elements of one at a time. Panels with (; p) = (63; ) and (732; ) contain critical values for tests with hypotheses under the null designed for tests for the stability of one eigenvector. Finally, panels with (; p) = (66; ) and (66; 0) contain critical values for the application in Section 5.2: the former are for changes in the eigenvalues, both individual and family-wise tests unidimensional tests (monthly and weekly frequencies respectively), the latter are for testing the stability of the whole matrix. 3

40 H 0 : ii constant H 0 : i constant H 0 : i constant i monthly weekly monthly weekly monthly weekly m 2:699 2:36 3m 2:7656 3:7004 :692 3:756 [st week, 2/7] x 3:942 7:07 [3rd week, 03/] 6m 2:7394 3:770 9m 2:3924 2: :553 2:5 x 2 4:39 7:2960 [3rd week, 04/] 2m :53 3:266 5m :499 2: :432 [0/] 2:7495 x 3 4:2340 7:2526 [2nd week, 03/] m :6467 2:9063 2m :065 3:092 24m :927 3:274 30m 2:07 3:369 3y 2:05 3:5926 4y :934 5y :964 6y :369 7y :7677 y :960 9y 2:046 0y 2:967 4:00 [3rd week, 09/7] 4:70 [st week, 2/7] 4:2779 [st week, 2/7] 4:2595 [st week, 2/7] 4:3342 [st week, 2/7] 4:3549 [st week, 2/7] 4:436 [last week, 0/] able 6. ests for changes in the variances of the term structure; in the volatilities of each principal component; and in the eigenvalue-normed eigenvectors. Rejection at 0%, 5% and % levels are denoted with, and respectively. Where present, numbers in square brackets are the estimated breakdates, de ned as arg max () : 39

41 monthly data weekly data st subsample 2nd subsample st subsample 2nd subsample able 7. Proportion of the total variance explained by principal components (, 2 and 3 refer to the level, slope and curvature respectively) for each subsample. he samples are split based on the results in able 6. When considering monthly data, the sample was split at January ; when using weekly data, at the rst week of December 7. Figure. erm structure of the US interest rates. Maturities correspond to m, 3m, 6m, 9m, 2m, 5m, m, 2m, 24m, 30m, 3y, 4y, 5y, 6y, 7y, y, 9y, 0y over the period April 997-November