Eigenstructure of nonstationary factor models

Working Paper 97-90 Statistics and Econometrics Series 29 December 1997 Estadística y Econometrfa Econometría Departamento de Estadfstica Universidad Carlos III de Madrid Calle Madrid, 126 28903 Getafe (Spain) Fax (341) 624-9849 EIGENSTRUCTURE OF NONSTATIONARY FACTOR MODELS ElGENSTRUCTURE Peña and Pilar Poncela" Daniel Pefta Abstract In this paper we present a generalized dynamic factor model for a vector of time series which seems to provide a general framework to incorporate all the common information included in a collection of variables. The common dynamic structure is explained through a set of common cornmon trends. Also, AIso, it factors, which may be stationary or nonstationary, as in the case of common may exist a specific structure for each variable. Identification of the nonstationary J(d) I(d) factors is made through the common cornmon eigenstructure of the generalized covariance matrices, properly I(d) factors, is the number of nonzero normalized. The number of common trends, or in general J(d) eigenvalues of the above matrices. It is also proved that these nonzero eigenvalues are strictly strictIy greater than zero almost sure. Randomness appears in the eigenvalues as well as the eigenvectors, but not on the subspace spanned by the eigenvectors. Key Words Cointegration and common cornmon factors, eigenvectors and eigenvalues, generalized covariance matrices, factor model, nonstationary J(d) I(d) factors, vector time series, Wiener processes "Departamento de Estadfstica Estadística y Econometrfa, Econometría, Universidad Carlos III de Madrid. E-mail: dpena@est-econ.uc3m.es and pilpon @est-econ.uc3m.es. AMS 1991 suject classifications: Primary 62M10; Secondary 62H25 Eigenstructure of Nonstationary Factor Models Daniel Pena Peña and Pilar Poncela Departamento de Estadistica Estadística y Econometria Econometría Universidad Carlos III nI de Madrid ABSTRACT: In this paper we present a generalized dynamic factor model for a vector of time series which seems to provide a general framework to incorporate all the common information included in a collection of variables. The common dynamic structure is explained through a set of common factors, which may be stationary or nonstationary, as in the case of common trends. Also, AIso, it may exist a specific structure for each variable. Identification of the nonstationary 1 I (d) factors is made through the common eigenstructure of the generalized covariance matrices, properly normalized. The number of common trends, or in general I(d) factors, is the number of non nonzero zero eigenvalues of the aboye matrices. It is also proved that these nonzero eigenvalues are strictly greater than above zero almost sure. Randomness appears in the eigenvalues as well as the eigenvectors, but not on the subspace spanned by the eigenvectors. AMS 1991 suject classifications: Primary 62MIOj 62MI0j Secondary 62H25. Key words and Phrases: Cointegration and common factors, eigenvectors and eigenvalues, generalized covariance matrices, factor model, nonstationary I(d) factors, vector time series, Wiener processes. The authors acknowledge support from the Catedra Cátedra BBV de Calidad and from DGICYT project PB96-0111. 1 Introduction Factor models are of great importance when dealing with reduction of dimensionality problems. When data is dynamic, this is specially important since for vector ARM ARMA A models, as well as for econometric models, the number of parameters to estimate grows rapidly with the number of observed variables. Dynamic factor models have been studied by Anderson (1963), Priestly et al (1974), Box and Tiao (1977), Brillinger (1981), Engle and Watson (1981), Shumway and Stoffer (1982), Watson and Engle (1983), Pefia Peña and Box (1987) and Velu et al (1986) among others. In the nonstationary case, estimating the nonstationary factors is equivalent to testing for cointegration, since as it was formally shown by Escribano and Pefia Peña (1994), both concepts are closely related. Engle and Granger (1987) presented a two step estimator based on OL8 regressions. Phillips and Ouliaris (1988) proposed a method based in principal component analysis applied to the innovation sequence resulting after taking first differences of the series. Stock and \;Vatson (1988) developed a method to identify the number of common trends for VAR models. Residual based tests for cointegration are discused in Phillips and Ouliaris (1990). Related work on the topic is that of Tiao and Tsay (1989) and Gonzalo and Granger (1995). Johansen (1988, 1991) developed a maximum likelihood approach to estimate the linear space spanned by the cointegration vectors. Reinsel and Ahn (1992) have proposed a reduced rank model to deal with this problem and Ahn (1997) related it to the scalar component models of Tiao and Tsay (1989). Nonparametric cointegration analysis is considered in Bierens (1997). In order to model the dynamics of variables that exhibit cointegration, the cointegration relations that make the series stationary should be estimated and interpreted. Apart from problems that can arise due to arbitrary normalizations, when the number of series is moderate to large, the task becomes difficult. Then, the number of cointegration relations can be large, and since the basis from the cointegrating subspace that can be chosen is arbitrary, its interpretation can be very complicated. In this case, it could be better to estimate and interpret a small number of nonstationary factors that characterize the growing behaviour of pro pose a the series. This can be achieved using dynamic factor models. In this article, we propose method to identify the factor space by looking at the eigenvalues of the generalized (properly normalized) variance-covariance matrices of the observed series. It is shown that the nonzero eigenvalues converge to random quantities (functionals of Wiener processes) while the eigenvectors for those nonzero eigenvalues converge to a random basis of the vector space spanned by the factor loading matrix. The subspace of stationarity or cointegration is given by the eigenvectors of the zero eigenvalues. As it will be shown it is orthogonal to the subspace of nonstationarity. An important advantage of this approach is that no model is required. 1 Besides, it constitutes a simple extension to the one applied in the stationary case. This article is organized as follows. Section 2 presents the generalized dynamic factor model and study its properties. Section 3 presents the main result, which is the basis to separate the nonstationary factors from the stationary ones, and shows how this can be Peiia and Box (1987) for stationary carried out by a generalization of a method proposed by Peña factors and, finally, section 4 presents some sorne conclusions. 2 The Factor Model Let Yt be an m-dimensional vector of observable time series, generated by a set of not observable factors. We assume that each component of the vector of observed series, Yt, can be written as a linear combination of common factors and specific components; that is Yt m xl P It Jt mxr rx1 + nt mx1 + Et mx1 (1) where It Jt is the r-dimensional vector of common factors, P is the factor loading matrix, and nt is the vector of specific components and Et is white noise (0, (O, ¿f).Therefore, L:f).Therefore, the common dynamic structure comes through the common factors, Jt, ft, whereas the vector nt explains the dynamics specific to each time series.\iVe suppose that the vector of common factors follows a VARMA(p, q) model (2) where <I>(B) = 11 - <I>(l)B-,··· ,-<I>(p)BP, and 8(B) = 11 - 8(1)B-,··· ,-8(q)Bq, are r x r polinomial matrices and B is the backshift operator. The sequence of vectors at are normally L:a and are serially uncorrelated, distributed, have zero mean, a full rank covariance matrix ¿a that is (3) The vector of common factors, Jt, ft, can include stationary and nonstationary terms. We assume that the specific components, nt, if they exist, have stationary dynamic structure and follow an ARMA model, where <I>n and 8 n are mxm diagonal matrices given by cI>n(B) = 1I -cI>n(1)B-, ... ,-<I>n(p)BP, and 8 n(B) = 1 I - 8 n(1)B-,··· ,-8n(q)BQ, and therefore each component follows an uniARMA(pi, qi), i = 1,2,··· ,m, being p=max(Pi) and q=max(qi) , i = 1,2,··· ,m. variate ARMA(Pi' 2 The sequence of vectors et are normally distributed, with zero mean and diagonal covariance matrix ｾ･Ｎ＠ We assume that the noises from the common factors and the specific components, are also uncorrelated for all lags, (4) and both noises are uncorrelated with the noise in model (1), Et, for all alllags lags (5) and (6) The model as stated is not identified, because for any r x r non singular matrix H the observed series Yt can be expressed in terms of a new set of factors, Yt = P* ft* + nt (7) (8) 1>*(B)ft = 8(B)*a; with p*' P* = (H- 1 )' P' P H-l, ft* = H ft, a; = Hat, H at, 1>*(B) = H1>H-l, 8*(B) = H8H-1, and ｾ＠ = ｈｾ｡ＧＮ＠ To solve the identification problem, we follow the work by Hannan (1969, 1971, 1976) and Kohn (1979) which has been more recently extended to nonstationary state space models by \Vall (1987), and look for parametrizations that are unique in their effect on first and second moments of the observed time series. Several identifying restrictions appear in the literature. Usually the factors noise covariI. Some Sorne ance matrix ｾ｡＠ is considered to be diagonal. Also P can be chosen such that P' P = 1. parameters of the processes followed by the factors may also be restricted: for example, if sorne stationary factors, the matrix 1> has already there is a common trend orthogonal to some \Ve assume ｾ｡＠ diagonal, it is also implied by the model sorne fixed paramenters. Note that if we some that E (Adj,r) = 0O Vt, T; for i # j. (9) This condition is not restrictive, since the factor model can be rotated for a better interpretation when needed (see for example Harvey (1989) for a brief discussion about it), and helps to make easier the derivation of the asymptotics. 3 3 Eigenstructure of nonstationary factor models When nt is white noise and the factors are stationary model (1) and (2) reduces to the factor model studied by Peña Pefia and Box (1987). These authors developed a method of identifying the number of common factors based in the common eigenstructure of the lagged covariances Nevertheless, in many cases real time series vectors are matrices of the vector of time series. Nevertheless, nonstationary. Suppose that the vector of time series is 1 J (d). In a general case, some sorne common factors will be stationary, while others will be nonstationary. In particular, a nonstationary aH the factor can be a common trend, in the sense of Stock and Watson (1988), driving all series. Assume that Yt is J(d), I(d), for d ::::: 1, we define the generalized sample covariance matrices Cy(k) properly normalized as (10) and we will see that these matrices play the role of proper sample covariance matrices for the stationary case. I(d) factors, !I,t h,t = (JI,t,,·· (Jt,t,," ,¡;})', ,I;})', r2 common zero Suppose that there are rl common J(d) Ui,t,"" ¡;})' I;})',, and that the specific components, nt = mean stationary factors, 12,t = Ui,t,···, (ni, ... ,nr)', if they exist, are zero mean stationary ones. Divide the vectors of comIf = (J{ t, ｉｾ＠ﾡ t) and ｡ｾ＠ = Ｈ｡ｾ＠ t, ｡ｾ＠ t), respectively, and the diagonal mon factors and noise as ¡f "ariance L:a = variance matrix for at as ¿a [Elo L:2'0O ].l. O " " ¿2 Assumption 1. Suppose that the equation for the common nonstationary factors is (d::::: 1) Ut Ut (11) w(B)al,t with E (al,t) = 0 O and var (al,t) = El = diag(ar," . ,a;l) > 0, O, h,-(d-l) = !I,-(d-2) h,-(d-2) = '" = !I,D h,o = 0, O, Ilwill = ｛ｴｲＨｷｾＩｰＯＲ＠ and I:>llwill ¿:>llwill < 00. oo. Define matrix w(1) = ｌｾｄ＠ｯ Wi with rank (w (1)) = rl' Then, the following foHowing result will help us to identify the nonstationary factors and to separate them from the stationary ones. Theorem 1. For the nonstationary factor model presented in sections 2 and 3 and as0,1,'" ,K, such that KIT -t O. Then: sumption 1, define Cy(k) as in (10) , for k = 0,1"" I(d) factors, r¡, rI, is the number of nonzero eigen(i) The number of common nonstationary J(d) values of lim Cy(k), k = 0,1,'" ,K, where limits are taken as T goes to infinity. (ii) Define the variability in {Yt}[=l as trace(Cy(O)). In the limit, when T -t 00, the amount 4 ce spann ed by the of variability is rando m (as in the finite sample case), but the subspa y factors is consta nt. eigenvectors corresponding to eigenvalues associated with nonsta tionar Proo f (i) Subst ituting Yt, expressed as in (1), in equation (10), we have 1 ¿ y)' T2d L (Yt-k - Y)(Yt - V)' - L,(ftL)ft- k - f)(ft - f)')P' + ｐＨｔｾ､＠ ｐＨｔｾ､＠ 1I L(ft- k + P(T 2d ¿(ft- + Ｈｾ､＠ ｦＩｅｾ＠ ¿L L,(ftL/ft- k - ｦＩｮｾ＠ f)')p' nt-k(f t - Inp' ""' Et-k ( Jt ""' nt-kEt' + (1 P 1 6 ft - Jf-)')' T2d 6 + T2d + 1 ""' 6 nt-knt' T2d + 1 ""' ' 2d 1 ""' 6 Et-kEt' 6 Et-kntT T2d 1 -k - f)(ft - f)')P' ｾＨｦｴＭｫ＠ P(T 2d ¿:(ft + op(l). fl,i' It is shown in Appendix 1 that all the terms f1 = liT L Jl,i' and /1 where vector l' = (/{, ＰｾＲＩ＠ but the ones associated with the I(d) factors are op(l). JI,t can be factors, !I,t From (12), and following the notati on in Tanaka (1996), the I(d) ) (d_1)}T . { (d_1)}T (d-1) (d) _ ",t _ (d) _ (d-1) 1 . I(d an IS lS t=l f1,t J1,t ,where fl,j Jl,j Ltj=l f1,t-1 + J1,t-1 fl,t f1,t - Jl,t fI,t - J1,t expressed as h,t = U1,t. For example, for process that can be defined recursively in a similar way with ｦｩｾｬ＠ｊ (1).. (1) T.IS the 1I (1) process J1,t {(l)} (1) ",t thlS (2) WIth thIS fl,t-l' Wüh fl,t = Ut + J1,t-1' t=1 lS f1,t t=l fl,j and J1,t Ltj=l J1,j f1,t = Ltj=1 I ( 2) factors J1,t 1 f(d-l) ",k-l ",k-1 J(d-1) - Jf Jf . notatlOn, 1,t-k - 1,t - Lti=O 1,t-i' so notatIO 1 T ¿: ｾ＠ ｔｾ､＠ t=k+1 From Chan and Wei (1988) and Tanaka (1996) T; also small relative to Ti Td!,/2 ｔｾ､＠ tfl,t t T ¿: fd' (fI,t - Id' fd (h,t (h,t (fI,t - Id T2d ＠ｾ t=k+1 T k-1 T -)' . (d-1»)( J1,t ("" 1 ""' - JI f1,t -!I fl,t-i 6 J1,t-i 6 T2d t=k+l t=k+1 i=O fd f1)(h,tt - Id (h,t-k - 11)(fI, 2 11,'' 2 l' '* W(l)E W(1)E:/ J"d:,t '* J"d;,t W(1)E:/ W(l)E t=l /,' L ｦｩｾＩ＠ｊ - 1 ) for finite i and i Op(T22dd-1) f{,t is Op(T )dr Fd-1(r Fd- 1(r)dr 2 )'W(l / d-11 (r)'dr (E:/ )'W(1 (r)FdFd-1 (r)F motio n and can be defined recursively by where Fd(T) is the d-fold integrated Brownian motion r1-dimension al Fd_ 1(S)ds, for d = 1,2, ... and FO(T) = W(T) where W(T) is the rI-dim Fd(T) = J; 5 Brownian motion. Then, by the continuous mapping theorem (Billingsley, 1968) ｔｾ､＠ ｾＨｦＱＬｴ＠ｬ T - f1)(fr,t fl)(fr,t - frY fr)' =} 1 ｗＨＱＩｾＯＲ＠ ｖ､｟ＱＨｔＩＧｾＯＲｗ＠ｬ (12) 1l Jo Fd(T)dT. Partitioning P as [P1l P2], where PI (P2) is the m x rI, rl, where Vd(T) = Fd(T) - Jo (m x r2) submatrix of the factor loading matrix associated to the nonstationay (stationary) factors and using again the continuous mapping theorem (Billingsley, 1968) 1 lim T2 L(Yt-k - Y)(Yt - y)' 1 lim P(T 2d [P P ] [ 1l 2 =} ｾＨｦｴＭｫ＠ - f)(ft f)(it - f)'p' 10Jo1l ｖ､｟ｬＨｔＩＧｾＯＲｗＱ＠ ｗＨＱＩｾＯＲ＠ OT2XT¡ OT2XTI PI ｗＨＱＩｾＯＲｬ＠ﾡ Vd- l (T) Vd- 1l (T)' dT) ＨｾＯＲＩＧｗＱ＠ ｐｾ＠ (13) O, as well as, for lag k, finite) have Note that all generalized covariance matrices (for lag 0, the same limiting distribution. 1l ､｟ＱＨｔＩｖＧｾｩＯＲ＠ｬＬ Let S = ｾｩＯＲ＠ Jo ization, leads to S = BAB' so ry , then its spectral decomposition for each real= AAA' where A = PI W(l)B and A has its TI eigenvalues different form zero. Therefore, the number of zero eigenvalues of r y is m - TI. Empirically, the number of common nonstationary factors can be found as the number of nonzero eigenval ues of Cy(k), since Cy(k) =} r y and the ordered eigenvalues are continuous functions of the coefficient matrix (Lemma 2 of Anderson et aI, al, 1983), applying the continuous mapping theorem, the ordered eigenvalues of Cy(k) converge weakly to those of rry. y• (ii) Define the variability in {Yt}f=l as tTace(Cy(O)). In the limit, when T -+ 00, the amount of variability is random (as in the finite sample case), but the subspace spanned by the eigenvectors corresponding to eigenvalues associated with nonstationary factors is constant. For matrices Cy(k), we just found their limiting distribution given by (13) and by the last paragraph of (i) T¡ TI trace( Cy(O)) =} ｾ＠ Ai i=1 where Ai, i = 1,'" ,TI are the diagonal elements of A, random quantities. The subspace spanned by the columns of PI is the rank of Pr, Fr, of dimension rl, rI, and it can be called the subspace of nonstationarity since it is associated to the J(d) I(d) factors. The null space of PI 6 Pto It can be called caHed the subspace of stationarity or is orthogonal to the one spanned by Pt. cointegration. cointegrationo The next result establish that the limit random matrix has TI rl eigenvalues strictly greater than zero almost sure. sureo Theorem 2. For the model of sections 2 and 3 and assumption 1, r y has TI rl eigenvalues greater than zero almost sure and m - TI rl equals zero. zeroo Proof For the factor model of sections 2 and 3, under assumption 1, it was proved in theorem 1, equation (12) that ST = ｔｾ､＠ I)h,t fl)(h,t - !d fd = } I)h,t - !I)(h,t ｗＨＱＩｾｩＯＲ＠ｉ ｖ､｟ｬＨｔＩＧｾｩＯＲｗＱＮ＠ｉｯ The eigenvalues of the limiting sequence ST are all aH greater than zero, since this is always a positive definite symmetric matrix. matrixo This is easily seen if we apply the next equality given po 49) , for s = 1,2,· 1,2, o.. oo ,rI, ,TI, which proves that all aH the principal minors have in Bellman (1960, p. definiteo Let xi, i = 1,2, ... ooo s be a determinant greater than zero and therefore ST is postive definite. RI, ooo ,fLT)' set of T-dimensional vectors, T 2: s, given by Xi = ft It - Rl, where f{ If = ULl' fL2' IL2'··· ,ILT)' is the T x 1 vector of sample values of the i-th nonstationary factor, R = liT ¿:f=l 2:f=l and l' I' = (1,· (1, o.. oo ,1) is a T x 1 vector of ones. oneso Then . . I(xt, XJ )kj=1,2, .. ,s 1¿ = s!12: {is} {is} XlIl21 XlI222 x 2I222 .T oTi12 x2 ｸｾ＠ Ss Il tI X I2 22 R xl25Is x2 x t2Iss fL IL 2 x!?t5Is where (Xi, xj) = "L,I=l ｸｾｻ＠ is element i, j of the matrix in the left hand side whose determinant we are calculating and the sum in the equality is over all aH sets of integers {is}, with ooo ::; To 1 ::; i l ::; ii22 ::; ... ::; is ::; T. Not only the limiting sequence is positive definite, but also in the limit it cannot be zero. zeroo l Vd- l (T)V First, it will be shown that M = Iol Jo Vd-l (T)Vd-l sureo Denote by d- l (T)'dT is nonsingular almost sure. Vj(w, V¡(w, T) the j-th component of the process V9(T), for 9 = 0,1, ... 000', d -1. If M were singular, oooCr Cr 1)' =1= 0 O such that e'Me c'Mc = 0. 00 Therefore ＢｌＬｪｾｬ＠ｉ CjVLl(T) = 0, for 0::; O::; T ::; l. lo then :3 ec = (Cl, ... t O, j Ci =1= 0, such that VLl(T) = ＲＺｪｾｬＬＣｩｃｖｌＨｔＩＮ＠﾿ｯ Since ec =1= 0, But for each realization of l Jo Fd_l(T)dT this means that the proccess Vd-l(T) = Fd-l(T) - Iol 7 or 10l I; where K = 1/Ci -¿J!=l,#i ｾｪｬＬＣｩ＠ Cj J; FLl FLl (r)dr - Jo Ftl FJ-l (r)dr. But with probability 1 Ftl FJ-l (r) 2d I FLl'· .. ' Fti, ｆｪｾｩＬ＠ ... , ｆ［ｾｬＧ＠ cannot lie in the span of Fi-l'· since var (Fd-I(r)) = r - /((2d1)((d - 1)!)2) 1)1)2) xX Ir¡ IT! is diagonal. It can also be checked that var (Vd-l(r)) is a full rank diagonal matrix, then with probability 1 VLI Vd-I (r) cannot be a linear combination of the VLI remaining components Vi-I (r), j sure and if ｾｬ＠ > 0, O, ｬｾｩＯＲ＠ Ｎｬ｜ＱＨｾｩＯＲＩＬ＠ 1l ry = ｐｉＧｬＡＨＱＩｾｩＯＲ＠ Jo ｖ､｟ｬＨｲＩＧｾｩＯＲＱＡｐｻ＠ 10 1 =1= i, j = 1, ... , TI. Therefore M is nonsingular almost =1= 0 O almost sure. Since 'l!(1) and PI have rank Tl, TI, is nonsingular almost sure. Then, for any m x 1 vector A =1= 0 O and by the Portmanteau theorem (see theorem 2.1 in Billingsley, 1968), which constitutes the desired result: with probabillity 1, matrix r y is positive definite. Similar results are found if we use generalized sample second moments matrices, Ay(k) = T;d -¿ ｾ＠ ｙｴｾＭｫＧ＠ instead of generalized covariance matrices. In this case (14) ,y. ",here TI matrix. A similar result is found for the eigenvalues of Ty. where PI is an m x Tl Lemma 1. For the model of sections 2 and 3 and assumption 1, IY fY has Tl TI eigenvalues TI equals zero. greater than zero almost sure and m - Tl Proof is given in Appendix 2. 3.1 Nonstationary 1(1) factors From a practical point of view and due to its broad applicability, special attention is paid to the 1(1) case. Suppose now that the vector of observed time series is 1(1). Then, the nonstationary factors are also al so 1(1). In particular, they can be common trends in the sense of Stock and Watson (1988). Assumption 2. The equation for the common nonstationary factors is fr,t h,t Ut h,t-l fr,t-I + Ut Ut 'l!(B)al,t 8 (15) with E (al,t) = 0O and var (al,t) = ｾＱ＠ｬ = diag(a?,··. ,a;1) > 0, O, and ¿:illwill 2:il\Will < 00, IIwill = ｛ｴｔＨｗｾＩｰＯＲＮ＠ Ilwill Lernrna Lemma 2. The model given in sections 2 and 3 with nonstationary factors as in assump- tion 2 (model M2) is equivalent (in the sense that it gives equal first and second moments of the observed series and of the auxilliary process that defines the short mn run dynamics) to a model (M1) (MI) with the same number of common trends, Tl, TI, and Tl TI more stationary common factors. Proof is given in Appendix 3. Theorm 1 applied to the 1(1) case tells us that the generalized covariance matrices are T22 and converge to now divided by T where \1(r) = W(r) - Jo1l¥(r)dr JollV(r)dr is the demeaned BTownian BTownían motion, motíon, and 111(r) l¥(r) is the TlTIdimensional standard Brownian motion. S = ｾｩＯＲＨｊｏｬ＠ ｖＨｲＩＧ､ｾｩＯＲＱ＠Ｌ is a nondiagonal matrix and that all Note that matrix 5 generalized covariance matrices (for lag 0, O, as well as, for lag k, finite) have the same limiting distribution: they all tend to a symmetric random matrix of rank TI' Remarks: (1) Similar results are found if we use generalized sample second moments matrices, Ay (k) d=l d=1 rk rh ¿: 2: ｙｴｾＭｫＧ＠ instead of generalized covariance matrices. In this case, also with (16) (2) These convergence results can also be found in Phillips and Durlauf (1986) and Chan and \iVei (1988) and apply to processes that satisfy more general assumptions of the innovations, that what is needed here. In particular, these results can be generalized to the case where the innovations present heterogeneity. Also AIso normallity is no needed. (3) The expected value of ｾｩＯＲ＠ E ＨｾＱＯＲ＠ f ｗＨｲＩｬ､ｾＱＯＲＧ＠ J ｗＨｲＩＧ､ｾｩＯＲ＠ = ｾＱＯＲ＠ is f E ＨｗｲＩｬ､ｾＱＯＲＧ＠ = ｾ､ｩ｡ｧＨ＠ 2 since E (f wiwjdr) = 1/2 if i = j and 0O otherwise. Lemma 1 and theorem 2 also apply to the case of 1(1) factors. Therefore TI eigenvalues strictly grater than zero and m - TI equals zero. almost sure Tl 9 ,+) -;', -;'" ... .. a1 aT1 al r1 ry and "(y have 4 Conclusions Several authors (Engle and Granger, 1987, Phillips and Durlauf, 1986, Stock, 1987) have proposed estimating a cointegration vector by using the fact that if and ergodic then the sample variance Zt = bb'' Yt is stationary whereas if Zt is nonstationary 1(1) J(l) then where the constant ec depends on covariances of the differenced stationary process. Therefore, oo. This leads to finding cointegrating if Zt is nonstationary its sample variance will go to 00. vectors by minimizing the sample variance of Zt. The usual procedure is to assume a normalization of b such that the coefficient of the first component of Yt is unity, which implies finding b by regressing the first component on all the others. If instead of looking at the cointegration relationships we look at the orthogonal factor a/Yt space it is clear that a reasonable procedure for finding a vector a such that Zt = a'Yt is nonstationay is by maximizing the variance of Zt which leads to a principal component analysis of the covariance matrix of the series. This approach was initially followed by Stock and 'Vatson \Vatson (1988) who proposed to base their cointegration test on the linear combinations generated by the principal components of the covariance matrix of the series, although afterwards these authors abandoned this approach in favor of a regression procedure (Stock and 'Vatson, \Vatson, 1993). Their approach differs from ours in the following aspects. First, principal components are introduced in an intuitive way, whereas in our model the formal justification is the stability of the factor space in the eigenvalues of alllag covariance matrices. Second, our approach is general and can be applied to factors with different orders of rl factors, h,t, fr,t, that are I(dr) I(d¡) and T2, r2, h,t, integration. For example, suppose that there are TI that are 1(d J(d2 ), with d dI1 > d2 , plus stationary factors. We can apply the method defining generalized covariance matrices as in (10), divided by T 2d l . After we find the TI rl I(d 1I ) P¡fI,t. We can now apply nonstationary factors, we define the auxilliary process Zt = Yt - Pdl,t. the same procedure to Zt defining generalized covariance matrices for Zt and normalizing them by T 2d2 to obtain the T2 r2 common 1(d J(d 2 ) factors. Third, the method can be generalized to nonstationary fractional factors. If instead of defining the d-fold integrated Brownian motion recursively, we use the definition valid for 10 real d such as d> -1/2, convergence results could also be found. 11 Appendix 1 In this appendix, it is shown that for nt, Jt ft and! and ! defined as in sections 2 and 3, (a) ｔｾ､＠ L: ｮｴＭｫｾ＠ -4 Omxm, rk rh L: ｅｴＭｫｾ＠ -4 Omxm and rk rh L: ｅｴＭｫｮｾ＠ -4o mxm , -,p 1 -,p 1 (b) TU L:(ft-k - f)nt f)n t -=-+ Orxm and TU L:(ft-k - f)E f)Ett -=-+ Orxm and 1 , P (c) T2d L:(fI,t-k - JdJ2,t fdf2,t -=-+ Orlxr2. In each case and in what follows foHows O 0 is a matrix of appropriate dimensions or an scalar. (a) Let nt be an m x 1 vector of specific components. By the stationary assumption, exists and is finite, rk rh L: ｮｴＭｫｾ＠ -4 O. Also AIso ｔｾ､＠ E (EtEt-k) = Omxm and ｔｾ､＠ ｌＺｅｴＭｫｮｾ＠ ｅｴＭｫｮｾ＠ -4 E (Et-knt) = Omxm· L: Omxm' ft be an Tr x 1 vector of common factors with TI rl common nonstationary factors (b) Let Jt and T2 r2 common, zero mean, stationary factors, such that T r = r1 + r2, then ｾ＠ L: ｮｴＭｫｾ＠ ｌＺｅｴＭｫｾ＠ ｅｴＭｫｾ＠ L: -4 -4 ｅＨｮｴＭｫｾＩＮ＠ Since ｅＨｮｴＭｫｾＩ＠ _1 "'(f "'(I T2d L t-k - = _1 ' " J-) f-) , nntt T2d L ｦＱＩｮｾ＠ [ (fI,t-k f 2,t-k n 't (bl) (b1) First, it will be shown that for the stationary factors ｔｾ､＠ 1 L: ｨＬｴＭｫｮｾ＠ -4 0O which is easily seen since both processes nt and ht are stationary, then ｾ＠ L ｨＬｴＭｫｮｾ＠ ｾ＠ E(h,t-knD, finite, therefore ｔｾ､＠ l: ｨＬｴＭｫｮｾ＠ ｾ＠ 0, O, for d ｾ＠ 1. (b2) Now, No"" for the term associated with the nonstationary common factors, rh L(fI,t-kｦｬＩｮｾ＠ ｾ＠ O. Denote by JL, fL, (nD the i-th component of vector J1,t, f1,t, (nt). Element (i,j) of the previous matrix is defined as ai,j = ｔｾ､＠ l:U{,t-k - fDnl, for i = 1, ... rl, j = 1, ... , m. It will be shown that ai,j ｾ＠ 0, O, for all aH i = 1, ... T1 r1 and j = 1, ... , m. 1 T12d T 2d L(f{,t-k - i{)n{ f{)n{ ｾ＠ ｔｾ､＠ L(f{,t-k - fD 1 ］ＭＺｾ＠ max Td-1/2 l<t<T - - Jl . ｬｾｴ＠ 1 1n In t ITd+1/2 In{1 L r. 1,t-k T t=k+1 From Tanaka (1996) we know that Tdl1/2 L:i=k+l J{,t-k f{,t-k is Op(l), and since n{ is a stationary process and d ｾ＠ 1, ｔ､ｾＱＯＲ＠ ｾｴ＠ｬ In{ I -4 O. Therefore ai,j -4 o.O. So, ],2 L:(fI,t-k - ｦ､ｮｾ＠ﾡＩ -4 o. 1 - , p And from (bl) (b1) and (b2), T2d L(ft-k - f)n t -=-+ O. The proof for L:Ut-k - ｦＩｅｾ＠ -4 Orxm goes exactly like the one before since sin ce a stationary process. rh Et is also (c) Now, No"" for the term involving stationary and nonstationary common factors, I 1" ", , -,p T2d L(fI,t-k - JdJ fdf 2,t -=-+ 0 O The proof goes like (b2), with h,t instead of nt, since it is the limit of the sum of the product of a lagged k, k = 0,1, ... K, nonstationary factor by an stationary one. 12 Appendix 2 PROOF OF LEMMA 1. The proof goes like in Theorem 2, but now So all aH we have to prove is that J Fd- I (r)F {r)Fd- 1 Fl(r) the 1 (r)'dr is nonsingular. Denote by Fl{r) j-th component of the process Fg(r), Fg{r), for 9 = 0,1, ... We will prove by induction that P(I J Fd- I (r)F O) = 0, since otherwise 3FLIIFLI lies in the span of that P{I {r)Fd-1 d_1 (r)'drl = 0) FLI' ... , Fti, ｆｪｾｩＬ＠ ... , ｆ［ｾｉ＠ and this is not possible. For d = 1, Fd-I(S) = W(s), where Hr (s) is the TI-dimensional standard Brownian motion, with all HI aH its components independent among them. Therefore, P(I J1V(r)W(r),drl = 0) O) = 0, since otherwise 3W i lW i lies in sin ce aH all the componets of the span of W\ ... , W i - \ Wi+\ ... , WTl and this is not possible since 1V(8) are independent among each other. Now suppose that it is true for d - 1, this means that P(I J Fd-l(r)Fd-l(r)'drl = 0) O) = 0, or equivalently with probability zero FLI lies in \Ve will see that P(I J Fd(r)Fd(r),drl Fd(r)Fd(r)'drl = 0) O) = 0, O, the span of FLI' ... , ｆｴｌｪｾｩＬﾷ［ｉＮ＠ because if not 3FjIFj lies in the span of FJ, ... , ｆｴ｜ｾＫ＠ ... ,F2, that is Fj can be expressed ... ,F2 or it exists CYI, 0ó1, ... ... ,OóT1-I ,CYT1-I not all of them as a linear combination of FJ, ... Ｌｆ､ＭｉｾＫ＠ simultaneousHy zero such that simultaneouslly Tl-I Fj = L CYjF1. Oó F1. j j=1 Differentiating the above aboye equation, Tl-I FLI = ¿OójFLI L:CYjFLI j=1 which occurs with probability zero. Therefore "/y ,,/y is nonsingular almost sure. Appendix 3 2. Since the specific components and the common stationary factors not derived by the dynamic structure of the nonstationary ones are not involved in this proof, let existo To distinguish both models, us suppose, just for ease of exposition, that they do not exist. the factors and system matrices will be denoted by in M2 and without it in model MI. So, let us suppose model M2 with Tl common 1(1) factors with dynamic structure Ml. PROOF OF LEMMA expressed as in assumption 1 and a model with TI common trends plus TI common stationary factors. We will see when they give the same first and second moments of the observed series. 13 First, let us show that they give the same limiting distribution of the generalized covariance matrices. For model M2 and by theorem 1 (A3.1) For model M1 MI and by theorm 1 (A3.2) Pr These distributions are the same if and only if ｐｉｾＨＱＩｩＺＯＲ＠ = ｐｉｾＯＲＮ＠ If ｾｉ＠ = ｾｉＧ＠Ｌ then = PI ｾＨＱＩＬ＠ but notice that many other possibilities are still open. Now for the short run, let us show that the dynamics generated by the structure of the /(1) factors can be expressed as rl common stationary factors, h,t in the equivalent model = ｐｉｾＨｌＩ｡ｬＬｴＫｦ＠ｅ where MI. Define the auxilliary process in model M2, Xt = ｙｴＭｐｉｾＨｬＩｪＬ＠ M1. ｾＨｌＩ＠ = (1 - ｌＩＭｉＨｾ＠ - ｾＨＱＩＬ＠ ｾｪ＠ = - E:j+1 E:j+1 ｾｩＮ＠ The mean of the auxilliary process is 0 O and second moments are given by 00 00 00 j=O i=j+1 i=j+1 00 00 00 j=O i=j+l+k i=j+1 Define an auxilliary process, Zt, related to model M1 MI as Zt = Yt - Pdl,t = P2!2,t + Et, ft, with h,t = cI>(B)a2,t, a set of generic r2 stationary common factors. Let us see who are the h,t factors. E(zt)=O and second moments are given by 00 2.:= ｐＲ｣ｉ＾ｩｾ＠ + ｾｦ＠ i=O 00 E Ｈｚｴｾ｟ｫＩ＠ = 2.:= ｐＲ｣ｉ＾ｩｾ｟ｫＮ＠ i=k Both auxilliary process are the same, if we take r¡ rl common stationary factors in MI M1 and 00 L i=k ｐＲ｣ｉ＾ｩｾ｟ｫ＠ 00 = L 00 P1( Pl( j=O L i=j+l+k which \\'hich is satisfied, for example, for P2 = PI and cI>i 14 00 ｾｩＩｴｬＨ＠ = ｾｩ＠ = - L ｾ､ｐｻ＠ i=j+l :E:j+l ｾｩＮ＠ References ARN, S. K. (1997) Inference of vector autoregressive models with cointegration and scalar components. Journal 01 of the American Statistical Association 92 350-67. ANDERSON, S. A., BRONS, H. K., JENSEN, S. T. (1983) Distribution of eigenvalues in of Statistics 11 392-415. Multivariate Statistical Analysis The Annals 01 ANDERSON, T. W. (1963) The use of factor analysis in the statistical analysis of multiple time series Psychometrika 28, 1-25. BELLMAN, R. (1960) Introduction to Matrix Analysis McGraw-Hill. BlERENS, BIERENS, H. J. (1997) Nonparametric cointegration analysis Journal 01 of Econometrics 77 379-404. 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Cl CI Madrid, 126. 28903 Getafe (Madrid), Spain. e-mail: pilpon@est-econ.uc3m.es 17 WORKING PAPERS 1997 Business Economics Series 97-18 (01) Margarita Samartín Samartin "Optimal aIlocation allocation of interest rate risk" 97-23 (02) Felipe Aparicio and Javier Estrada "Empirical distributions of stock returns: european securities markets, 1990-95" 97-24 (03) Javier Estrada "Random walks and the temporal dimension of risk" 97-29 (04) Samartfn Margarita Samartín "A model for financial intermediation and public intervention" 97-30 (05) Clara-Eugenia García Garcfa "Competing trough marketing adoption: a comparative study of insurance companies in Belgium and Spain" 97-31 (06) Gómez and Fernando Zapatero Juan-Pedro G6mez "The role of institutional investors in international trading: an explanation of the home bias puzzle" 97-32 (07) Isabel Gutiérrez, Gutierrez, Manuel Núñez Nufiez Niekel and Luis R. 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