The

Econometrics Journal Econometrics Journal (2013), volume 16, pp. 473–484. doi: 10.1111/j.1368-423X.2012.00392.x

Consistent co-trending rank selection when both stochastic and non-linear deterministic trends are present Z HENG -F ENG G UO † AND M OTOTSUGU S HINTANI ‡ †

International Monetary Fund, 1900 Pennsylvania Avenue NW, Washington, DC 20431, USA. E-mail: [email protected] ‡

Department of Economics, Vanderbilt University, 2301 Vanderbilt Place, Nashville, TN 37235, USA. E-mail: [email protected]

First version received: March 2011; final version accepted: September 2012 Summary This paper proposes a model-free co-trending rank selection procedure based on the eigenstructure of a multivariate version of the von Neumann ratio in the presence of both stochastic and non-linear deterministic trends. Our selection criteria can be easily implemented, and the consistency of the rank estimator is established under very general conditions. Simulation results suggest good finite sample properties of the new rank selection criteria. Keywords: Cointegrating rank, Smooth transition trend model, Trend breaks, von Neumann ratio.

1. INTRODUCTION For decades, one of the most important issues in the analysis of macroeconomic time series has been how to incorporate a trend. Two popular classes of trends are (a) stochastic trends such as the one suggested in Nelson and Plosser (1982), and (b) non-linear deterministic trends such as trend breaks considered in Perron (1989, 2006). Co-integration, introduced by Engle and Granger (1987), is an important concept in understanding the nature of co-movement among variables with stochastic trends. In co-integration analysis, the co-integrating rank, defined as the number of linearly independent co-integrating vectors, is often of main interest. Several modelfree consistent co-integrating rank selection procedures have been proposed in the literature. Analogous to co-integration, co-trend is a useful concept of co-movement in the presence of non-linear deterministic trends. The co-trend analyses of Bierens (2000), Hatanaka (2000) and Hatanaka and Yamada (2003) lie along this line of research. However, a consistent selection procedure of the co-trending rank, defined as a number of co-trending vectors, has not yet been developed. This paper proposes a model-free consistent co-trending rank selection procedure when both stochastic and non-linear deterministic trends are present in a multivariate system. Consistency here refers to the property that the probability of selecting the wrong co-trending rank approaches zero as sample size tends to infinity. Our procedure selects the co-trending rank by minimising the von Neumann criterion, similar to the one used by Shintani (2001) and Harris and Poskitt (2004) in their analyses of co-integration. We exploit the fact that the identification of the  C 2013 The Author(s). The Econometrics Journal  C 2013 Royal Economic Society. Published by John Wiley & Sons Ltd, 9600 Garsington

Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA, 02148, USA.

474

Z.-F. Guo and M. Shintani

co-trending rank is the same as the identification of three groups of eigenvalues of the generalised von Neumann ratio. Using the von Neumann criterion, we propose two types of co-trending rank selection procedures that are robust to model misspecifications. The simulation results also suggest that our co-trending rank selection procedures perform well in finite samples. Our analysis is closely related to that of Poskitt (2000) and Cheng and Phillips (2009), who propose consistent co-integrating rank procedures that do not require the estimation of a parametric vector error correction model used by Johansen (1991). In co-integration analysis, the co-integrating rank is the difference between the dimension of the system (number of variables) and the number of common stochastic trends.1 In the presence of both stochastic and nonlinear deterministic trends, however, the co-trending rank does not necessarily correspond to the difference between the dimension and the number of common non-linear deterministic trends. Because the number of common deterministic trends is also of interest, we introduce the notion of the weak co-trending rank as the difference between the dimension and the number of common deterministic trends. Our non-parametric procedure is designed to determine both the co-trending rank and weak co-trending rank. Our approach of separately identifying the co-trending rank and weak co-trending rank is potentially useful in many macroeconomic applications. For example, it can be used prior to the full estimation of a system of equations with specified non-linear deterministic trends. As discussed in Mills (2003), multiple shifts in the growth rate of gross domestic product (GDP) in the United Kingdom and Japan may be well described by a piecewise linear trend model or by a smooth transition trend model. Just as some types of technology shocks can have a permanent effect on only a specific group of industries, the timing and/or magnitude of shifts in growth may not be common to all the industries. Using an analogy to a vector error correction model, the prior knowledge of the weak co-trending rank determines the reduced rank structure of the coefficient matrix on non-linear trend functions. The co-trending rank further provides a restriction on the co-integrating vectors in modelling the stochastic trend components. A similar estimation strategy can also be applied to the analysis of international business cycles in the presence of country- or region-specific trends. In what follows, we denote a random sequence xT by Op (T λ ) if T −λ xT is bounded in probability, and by op (T λ ) if T −λ xT converges to zero in probability. The first difference of xt is denoted by xt .

2. THEORY We assume that an n-variate time series, yt = (y1t , . . . , ynt ) , is generated by yt = dt + st ,

t = 1, . . . , T ,

(2.1)

where dt = (d1t , . . . , dnt ) is a non-stochastic trend component, st = (s1t , . . . , snt ) is a stochastic component, respectively, defined below, and neither dt nor st is observable. We employ the following set of assumptions on dt and st .  ∞ 2 ASSUMPTION 2.1. (a) st = C(L)εt = ∞ j =0 Cj εt−j , C0 = In , j =0 j Cj  < ∞, s0 = 0,  where εt is iid with zero mean and covariance matrix εε > 0; (b) g = limT →∞ T −2 Tt=1 dt 1 See Stock and Watson (1988) and Bai and Ng (2004) on the number of common stochastic trends in a dynamic factor framework.

 C

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

Consistent co-trending rank selection

475

 and G = limT →∞ T −3 Tt=1 dt dt exist, all elements in the main diagonal of G are non-zero and  T −3 Tt=1 dt dt − G is O(T −1/2 ); (c) There exists an n × n orthogonal full rank matrix [B1 B2 B⊥ ], such that [B1 B2 ] G = 0, B1 C(1) = 0, B2 C(1) is of full row rank, B⊥ GB⊥ is of full rank, where B1 , B2 , B⊥ are n × r1 , n × r2 and n × (n − r1 − r2 ), respectively. Part (a) of the assumption implies that the stochastic component st contains at least one stochastic trend, unless C(1) = 0 so that all the elements of st are I (0) stationary random variables. Our procedure does not require the knowledge of a serial dependence structure of the stochastic component. Part (b) provides the class of deterministic trend functions allowed in our analysis. Clearly, a linear trend satisfies the condition. The class of trend functions, however, includes more general non-linear trends as long as they do not diverge faster than the linear trend in the sense that (b) holds. For example, piecewise linear trends, discussed in Perron (1989, 2006) and Mills (2003), and smooth transition trends, discussed in Bacon and Watts (1971), Lin and Ter¨asvirta (1994) and Leybourne et al. (1998), are allowed in our analysis. Quadratic trends or higher-order polynomial trends are excluded, but one may easily extend our co-trending rank selection criteria to incorporate other classes of trends by adjusting the speed of a penalty term.2 Part (c) describes the co-trending structure. First, B1 represents a set of r1 co-trending vectors that eliminate both deterministic and stochastic trends. We define the co-trending rank by the number of such vectors, r1 . Secondly, B2 represents a set of r2 vectors that eliminate deterministic trends but not stochastic trends. We define the weak co-trending rank by the total number of two types of vectors, r = r1 + r2 . Since the number of common non-linear deterministic trends corresponds to n − r, identifying the weak co-trending rank r is as important as identifying the co-trending rank r1 . Our two alternative definitions of co-trend are a natural consequence of the notion of a common feature introduced in Engle and Kozicki (1993). They define the common feature as ‘a feature that is present in each of a group of series . . . (and) there exists a non-zero linear combination of the series that does not have the feature (p. 370)’. When such a feature is a broad class of trends, namely a mixture of both stochastic and deterministic trends, the definition of cotrend requires a linear combination that eliminates both types of trends at the same time. We view this case as a stronger version of co-trending relationship represented by the co-trending rank r1 . In contrast, when such a feature is the dominant trend, namely the deterministic trend alone, a linear combination should eliminate the deterministic trend but not necessarily the stochastic trend. As long as the deterministic trend is eliminated, we consider the case as a weaker version of co-trending relationship represented by the weak co-trending rank r.3 It should be noted that the objective of the co-trend analysis here differs from that of the traditional co-integration analysis. In Assumption 2.1 (c), B1 not only represents a set of cotrending vectors but also co-integrating vectors since B1 C(1) = 0. However, there may be an additional n × r2∗ matrix B2∗ , such that B2∗ C(1) = 0 and csp(B2∗ ) ⊂ csp(B⊥ ), where csp(F ) is the column space of a matrix F . In such a case, the co-integrating rank is given by r ∗ = r1 + r2∗ . Because the focus of our analysis is the co-trending rank r1 or r, our proposed procedure below is not designed to find r2∗ . Using the terminology of Ogaki and Park (1997), our procedure cannot 2

The first condition in (b) also excludes non-convergent trends, such as (−1)t t. Co-trending rank is also related but not identical to co-breaking rank. See Hatanaka and Yamada (2003) and Hendry and Massmann (2007) on the relationship between the two concepts. 3

 C

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

476

Z.-F. Guo and M. Shintani

detect stochastic co-integration, namely the case where only the stochastic trend is eliminated by the co-integrating vector. In contrast, deterministic co-integration, the case in which both stochastic and deterministic trends are eliminated by the same co-integrating vector, can be detected by our procedure through the selection of r1 . Shintani (2001) and Harris and Poskitt (2004) utilise the multivariate version of the von Neumann ratio in their co-integration analysis.4 In the scalar case, the von Neumann ratio is the ratio of the sample second moment of the differences to that of the level of a time series. In the multivariate case, we can instead use the eigenvalues, λˆ 1 ≥ λˆ 2 ≥ · · · ≥ λˆ n ≥ 0, of a normalised −1 S00 , where second moment matrix S11 S11 = T −1

T 

yt yt ,

and

S00 = T −1

t=1

T 

yt yt .

t=2

Our co-trending rank selection method exploits the statistical properties of λˆ i s summarised in the following lemma. L EMMA 2.1. Suppose Assumption 2.1 holds. Then, as T → ∞, we have: (a) (λˆ 1 , . . . , λˆ r1 ) has a positive limit and is Op (1) but is not op (1) provided r1 > 0; (b) T (λˆ r1 +1 , . . . , λˆ r ) has a positive limit and is Op (1) but is not op (1), provided r − r1 > 0; and (c) T 2 (λˆ r+1 , . . . , λˆ n ) has a positive limit and is Op (1) but is not op (1). The proof of the lemma is omitted here because results can be derived using an argument similar to Hatanaka and Yamada (2003, Propositions 3.1–3.3). In the presence of both stochastic −1 and deterministic trends, the lemma shows that eigenvalues of S11 S00 can be classified into three groups depending on their rates of convergence, namely Op (1), Op (T −1 ) and Op (T −2 ). The number of eigenvalues in each group corresponds to the number of co-trending relationships (r1 ), the difference between weak co-trending and (strong) co-trending relationships (r2 = r − r1 ) and the number of common deterministic trends (n − r), respectively.5 Note that, the limit of each sequence is a set of random variables that only take positive values. Based on the results of Lemma 2.1, we construct the following two types of consistent co-trending rank selection procedures based on the von Neumann criterion, which consists of a partial sum of eigenvalues and a penalty term. The first is a ‘paired’ procedure that independently selects the co-trending rank r1 and the weak co-trending rank r by minimising each of VN1 (r1 ) = −

r1 

CT λˆ i + f (r1 ) , T i=1

rˆ1 = arg min VN1 (r1 ),

and

0≤r1
and

VN2 (r) = −

r 



C λˆ i + f (r) T2 , or T i=1

rˆ = arg min VN2 (r), 0≤r
where f (s), CT and CT are elements of the penalty function defined in detail below.

4

In contrast, Poskitt (2000) and Cheng and Phillips (2009) utilise canonical correlations. The von Neumann ratio can be computed using demeaned series instead of using raw series. Unlike co-trending rank tests that require the limiting distribution of the von Neumann ratio, our procedure relies only on the convergence rate of its eigenvalues. Therefore, all of our theoretical results hold for the demeaned version of the von Neumann ratio. 5

 C

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

Consistent co-trending rank selection

477

The second procedure is a ‘joint’ procedure that simultaneously determines both r1 and r by minimising VN(r1 , r) = −T α

r1 

λˆ i −

i=1

r  i=r1

C CT + f (r) T2 , or λˆ i + f (r1 ) T T +1

(ˆr1 , rˆ ) = arg min VN(r1 , r), 0≤r1 ≤r
where 0 < α < 1. Note that in VN(r1 , r), the sum of the first r1 eigenvalues is multiplied by a normaliser T α . The presence of the normaliser only for the first group but not for the second group is important in identifying the two types of the co-trending rank based on a single criterion. The main theoretical result is provided in the following proposition. PROPOSITION 2.1. (a) Suppose Assumption 2.1 holds, f (s) is an increasing function of s, CT → ∞, CT → ∞, CT /T → 0 and CT /T → 0 as T → ∞. Then the paired procedure using VN1 (r1 ) and VN2 (r) yields, lim P (ˆr1 = r1 , rˆ = r) = 1.

T →∞

(b) Suppose Assumption 2.1 holds, f (s) is an increasing function of s, CT /T α → ∞, CT → ∞, CT /T → 0 and CT /T → 0 as T → ∞. Then the joint procedure using VN(r1 , r) with 0 < α < 1 yields, lim P (ˆr1 = r1 , rˆ = r) = 1.

T →∞

REMARK 2.1. The proposition shows that both of the two co-trending rank selection procedures are consistent in selecting the co-trending rank without specifying a parametric model as long as the trend belongs to a certain class of non-linear functions. Consistency holds with or without stochastic trends in the stochastic component. The joint selection procedure requires slightly stronger assumptions on CT than the paired selection procedure. REMARK 2.2. Commonly employed CT in the literature of information criteria includes CT = ln(T ), 2 ln(ln(T )) and 2, which, respectively, lead to the Bayesian information criterion (BIC), Hannan–Quinn criterion (HQ) and Akaike information criterion (AIC). Part (a) of the proposition implies that the paired co-trending rank selection procedure is consistent when BIC and HQ type penalties are employed, but is inconsistent when an AIC type penalty is employed. In contrast, part (b) of the proposition implies that CT should diverge at a rate faster than T α for the joint co-trending rank selection procedure, thus none of CT = ln(T ), 2 ln(ln(T )) and 2 yield consistency. REMARK 2.3. By the definition of VN(r1 , r), co-trending ranks selected by the joint procedure always satisfy rˆ1 ≤ rˆ . For the paired procedure, selected co-trending ranks will satisfy rˆ1 ≤ rˆ if CT = T δ CT , where 0 ≤ δ < 1. This fact can be demonstrated by the following argument. The VN1 (s) > VN1 (ˆr1 ) for all s < rˆ1 . This result is equivalent selected co-trending rank rˆ1 implies  that ˆ1 to the partial sum of eigenvalues ri=s+1 λˆ i being greater than {f (ˆr1 ) − f (s)} CT T −1 (note that, λˆ i ≥ 0 and f (ˆr1 ) − f (s) > 0). To see if VN2 (s) > VN2 (ˆr1 ) for the corresponding s and rˆ1 , it  C

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

478

Z.-F. Guo and M. Shintani

 ˆ1 suffices to show that ri=s+1 λˆ i is greater than {f (ˆr1 ) − f (s)} CT T −2 . By substituting CT =  ˆ1 δ T CT , the latter becomes {f (ˆr1 ) − f (s)} CT T −1 × T −(1−δ) . Since T −(1−δ) < 1, ri=s+1 λˆ i > {f (ˆr1 ) − f (s)} CT T −1 > {f (ˆr1 ) − f (s)} CT T −1 × T −(1−δ) . The fact that VN2 (s) > VN2 (ˆr1 ), for all s < rˆ1 , implies rˆ1 ≤ rˆ . REMARK 2.4. For the estimation of a parametrically specified co-trend model, our nonparametric co-trending rank selection procedure offers useful prior information on its structure since it is robust to the misspecification of trend functions. For example, Hatanaka and Yamada (2003) considered a parametric co-trend model with its deterministic trend component given by dt = AB  dt−1 + MhT ,t , where A and B are n × r full column rank matrices, M is an n × m matrix satisfying m ≥ n − r and hT ,t = [1, I (t > [b1 T ]), . . . , I (t > [bm−1 T ])] with break fractions bj ’s satisfying 0 < b1 < · · · < bm−1 < 1.6 The selected weak co-trending rank r from our procedure pins down the dimension of A and B, and imposes a lower bound restriction on the dimension of M and hT ,t .7 In addition, suppose that the deterministic trend component is combined with a stochastic trend component generated from a parametric vector error correction model of order k, st = A∗ B ∗ st−1 +

k 

j st−j + εt ,

j =1

where A∗ and B ∗ are n × r ∗ full column rank matrices, and j s are n × n matrices. Then the selected (strong) co-trending rank r1 from our procedure imposes a restriction on the linear dependence structure of B and B ∗ . It should be noted that our procedure is not designed to select the co-integrating rank r ∗ because our primary focus is the deterministic co-trending relationship. However, in the absence of the deterministic trend components, the criterion function VN1 (r1 ) in the paired procedure can separately be used to select the co-integrating rank in a system of stochastic trends. In this case, rˆ ∗ from the minimisation of VN1 (r1 ) yields consistent co-integrating rank selection for r ∗ . In fact, the consistent co-integrating rank selection criterion considered in Harris and Poskitt (2004), C,T in their notation, is a special case of VN1 (r1 ) with the choice of CT = ln(T ) and f (s) = 2s(2n − s + 1). Their choice satisfies the required conditions for our procedure, since ln(T ) → ∞, ln(T )/T → 0 and df (s)/ds = 4(n − s) > 0.

3. EXPERIMENTAL EVIDENCE We evaluate the finite sample performance of our non-parametric procedure by the frequencies of selecting the true co-trending rank (r1 , r) = (0, 0), (0, 1) or (1, 1) in a bivariate system (n = 2). The data-generating processes (DGPs) are similar to the ones used in Hatanaka and Yamada 6 Here, all the roots of det[I − (I − B  A)x] = 0 lie outside the unit circle and the rank of M  A is n − r, where A r r ⊥ ⊥ is an orthogonal complement matrix of A. 7 One referee points out the possibility of developing a new parametric co-trending rank selection procedure by generalising the parametric co-integrating rank selection of Aznar and Salvador (2002) with simple intercepts replaced by non-linear trend functions.

 C

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

Consistent co-trending rank selection

479

(2003) in their study of a parametric co-trending rank test. For the co-trend case, the deterministic component dt in yt = dt + st is generated from     1 1 0 dt = [−1, 0.5]dt−1 + h , 0 1 −0.5 T ,t where hT ,t = [1, I (t > [bT ])] with a break fraction b satisfying 0 < b < 1. To consider the case of (r1 , r) = (0, 1), the stochastic component st is generated from st = εt for DGP1 and   −1 st = [1, 0.5]st−1 + εt , 0.5 for DGP2, where εt is a bivariate iid random error term which follows N(0, 0.25 × I2 ).8 Likewise, to consider the case of (r1 , r) = (1, 1), stochastic components in DGP3 and in DGP4 are generated from     0.5 0 1 st = st−1 + εt and st = [−1, 0.5]st−1 + εt , 0 0.5 0 respectively. For the no co-trend case with (r1 , r) = (0, 0), the deterministic component dt is generated from   1 0 dt = h . 1 −0.5 T ,t DGP5, DGP6 and DGP7 combine this linearly independent (non-co-trended) deterministic trend For the dt with the stochastic components st used in DGP1, DGP4 and DGP3, respectively. √ paired procedure, we use a BIC-type penalty CT = ln(T ) along with CT = T ln(T ). Note that, the choice satisfies the condition CT = T δ CT with δ = 0.5. For the joint √ selection procedure, we set α = 0.5. Since ln(T ) cannot be used for CT , we use CT = CT = T ln(T ). Following Harris and Poskitt (2004), we use f (s) = 2s(2n − s + 1) for both procedures. Table 1 reports the frequencies of selecting the co-trending rank r1 and the weak co-trending rank r by the paired and joint co-trending rank selection procedures for sample sizes T = 50, 100 and 200 in 20,000 replications.9 Panel A reports the case where the true break fraction is given by b = 0.5. For the purpose of comparison, we also report the co-trending rank selected by the parametric sequential decision rule proposed by Hatanaka and Yamada (2003) based on the prior knowledge of a kinked trend and the break fraction b = 0.5.10 The paired and joint procedures perform similarly. While the selection frequencies are small in some cases with T = 50, they quickly approach one when the sample size becomes as large as T = 200. Overall, the parametric procedure performs better than the non-parametric procedure, except for the case of (r1 , r) = (1, 1). Panel B shows the case of b = 0.25. To see the effect of misspecification, 8 The performance of the co-trending rank selection procedure can depend on the choice of variance of the error term because it determines the contribution of st relative to dt . The value 0.25 is taken from the code used in Hatanaka and Yamada (2003), which is kindly provided by Hiroshi Yamada. 9 Initial values are set at d = s  0 −99 = (0, 0) and the first 100 observations of st are discarded. 10 We employ tests with the 5% level of significance in each step. Hatanaka and Yamada’s sequential procedure yields consistent rank selection except for the case of (r1 , r) = (0, 1), where the probability of selecting the true rank approaches 0.95 × 0.95.

 C

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

480

Z.-F. Guo and M. Shintani Table 1. Two-dimensional co-trending rank selection. A. Kinked trend (b = 0.5) Non-parametric procedure Paired VN

DGP

(r1 , r)

T = 50

1

(0,1)

2 3 4

Parametric procedure

Joint VN

HY

100

200

T = 50

100

200

T = 50

100

200

46.7

65.6

83.9

47.3

65.9

84.1

85.6

87.9

88.8

(0,1) (1,1) (1,1)

56.4 95.8 100.0

75.0 100.0 100.0

90.0 100.0 100.0

58.1 90.0 100.0

76.1 100.0 100.0

90.4 100.0 100.0

88.1 10.6 18.1

88.8 45.7 76.2

89.4 98.5 100.0

5 6

(0,0) (0,0)

79.4 41.4

88.6 71.8

98.2 99.3

79.4 41.4

88.6 71.8

98.2 99.3

69.8 95.9

92.2 100.0

99.5 100.0

7

(0,0)

34.6

99.0

100.0 34.6 99.0 B. Kinked trend (b = 0.25)

100.0

99.3

100.0

100.0

Non-parametric procedure Paired VN

Parametric procedure

Joint VN

HY

DGP 1

(r1 , r) (0,1)

T = 50 49.4

100 67.8

200 85.6

T = 50 50.0

100 68.2

200 85.8

T = 50 86.6

100 88.7

200 89.3

2 3 4

(0,1) (1,1) (1,1)

59.6 94.8 100.0

76.9 100.0 100.0

90.5 100.0 100.0

61.4 88.1 99.9

78.1 100.0 100.0

91.3 100.0 100.0

88.7 10.1 16.8

89.6 44.0 72.4

89.8 98.2 100.0

5 6 7

(0,0) (0,0) (0,0)

68.5 45.9 28.0

73.9 63.9 95.8

91.7 94.5 100.0

68.5 45.9 28.0

73.9 63.9 95.8

91.7 94.5 100.0

17.8 13.6 7.2

20.1 24.8 22.4

35.0 71.5 94.1

C. Smooth transition trend (γ = 25, b = 0.25) Non-parametric procedure Paired VN

Parametric procedure

Joint VN

HY

DGP 1 2

(r1 , r) (0,1) (0,1)

T = 50 49.0 59.3

100 67.4 76.6

200 85.3 90.4

T = 50 49.6 61.0

100 67.8 77.7

200 85.6 91.1

T = 50 86.7 88.5

100 88.7 89.5

200 89.2 89.8

3 4

(1,1) (1,1)

94.6 100.0

100.0 100.0

100.0 100.0

87.6 99.8

100.0 100.0

100.0 100.0

10.1 17.1

44.4 74.1

98.3 100.0

5 6 7

(0,0) (0,0) (0,0)

67.2 43.1 16.3

71.0 58.2 85.3

88.9 90.9 100.0

67.2 43.1 16.3

71.0 58.2 85.3

88.9 90.9 100.0

19.7 16.9 8.9

25.6 36.0 28.1

44.9 84.7 96.9

Notes: The frequencies of selecting correct pairs of (r1 , r) are reported. Paired VN and Joint VN denote the paired and joint co-trending rank selection procedures based on the von Neumann ratio, respectively. HY denotes the parametric sequential decision rule of Hatanaka and Yamada (2003) using a kinked trend with prior knowledge of b = 0.5.

 C

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

Consistent co-trending rank selection

481

we use the same parametric procedure as before, setting a break point at b = 0.5. While the performance of the non-parametric method is similar to panel A, the performance of the parametric procedure deteriorates for the case of (r1 , r) = (0, 0). In addition to the kinked trend, we also consider the smooth transition trend represented by hT ,t = (1, G(γT , bT )) using the logistic transition function G(γT , bT ) = [1 + exp(−γT (t − bT ))]−1 , where γT = γ /T (γ > 0) is the (normalised) scaling parameter that controls the speed of transition, and bT becomes the transition mid-point. Panel C reports the frequencies of selecting the true rank when the deterministic trend is generated from a smooth transition trend with γ = 25 and b = 0.5. Again, in contrast to the non-parametric procedure, the parametric procedure does not perform well in some cases because the model is misspecified. In summary, simulation results suggest that our non-parametric procedure works well in a finite sample, and that it provides a useful alternative to the parametric procedure when the specification of trend functions is not known in advance.

4. CONCLUSION This paper has proposed a model-free co-trending rank selection procedure that is useful when both stochastic and non-linear deterministic trends are present in a multivariate system. The procedure selects two types of co-trending ranks by minimising two new criteria based on the generalised von Neumann ratio. Our approach is robust to misspecification of the model and consistent under very general conditions. Monte Carlo experiments suggest good finite sample performance by the proposed procedure.

ACKNOWLEDGMENTS We thank the co-editor, two anonymous referees, Walt Enders, Yanqin Fan, Junsoo Lee, Tong Li, Ron Masulis, Hiroshi Yamada and the seminar and conference participants at the University of Alabama-Tuscaloosa, University of Texas at Dallas, Kyoto University, Vanderbilt University, 20th Annual Meetings of the Midwest Econometrics Group, the 2010 NBER-NSF Time Series Conference and the 2012 Spring Meetings of Japanese Economic Association for helpful comments and discussion. The views expressed herein are those of the authors and should not be attributed to the IMF, its Executive Board or its management.

REFERENCES Aznar, A. and M. Salvador (2002). Selecting the rank of the cointegration space and the form of the intercept using an information criterion. Econometric Theory 18, 926–47. Bacon, D. W. and D. G. Watts (1971). Estimating the transition between two intersecting straight lines. Biometrika 58, 525–34. Bai, J. and S. Ng (2004). A PANIC attack on unit roots and cointegration. Econometrica 72, 1127–77. Bierens, H. J. (2000). Nonparametric nonlinear cotrending analysis, with an application to interest and inflation in the United States. Journal of Business and Economic Statistics 18, 323–37.

 C

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

482

Z.-F. Guo and M. Shintani

Cheng, X. and P. C. B. Phillips (2009). Semiparametric cointegrating rank selection. Econometrics Journal 12, S83–104. Engle, R. F. and C. W. J. Granger (1987). Co-integration and error correction: representation, estimation, and testing. Econometrica 55, 251–76. Engle, R. F. and S. Kozicki (1993). Testing for common features. Journal of Business and Economic Statistics 11, 369–80. Harris, D. and D. S. Poskitt (2004). Determination of cointegrating rank in partially non-stationary processes via a generalised von-Neumann criterion. Econometrics Journal 7, 191–217. Hatanaka, M. (2000). How to determine the number of relations among deterministic trends. Japanese Economic Review 51, 349–74. Hatanaka, M. and H. Yamada (2003). Co-trending: A Statistical System Analysis of Economic Trends. Tokyo: Springer. Hendry, D. F. and M. Massmann (2007). Co-breaking: recent advances and a synopsis of the literature. Journal of Business and Economic Statistics 25, 33–51. Johansen, S. (1991). Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59, 1551–80. Leybourne, S., P. Newbold and D. Vougas (1998). Unit roots and smooth transitions. Journal of Time Series Analysis 19, 83–97. Lin, C.-F. J. and T. Ter¨asvirta (1994). Testing the constancy of regression parameters against continuous structural change. Journal of Econometrics 62, 211–28. Mills, T. C. (2003). Modelling Trends and Cycles in Economic Time Series. Basingstoke: Palgrave Macmillan. Nelson, C. R. and C. I. Plosser (1982). Trends and random walks in macroeconomic time series: some evidence and implications. Journal of Monetary Economics 10, 139–62. Ogaki, M. and J. Y. Park (1997). A cointegration approach to estimating preference parameters. Journal of Econometrics 82, 107–34. Perron, P. (1989). The great crash, the oil price shock and the unit root hypothesis. Econometrica 57, 1361– 401. Perron, P. (2006). Dealing with structural breaks. In K. Patterson and T. C. Mills (Eds.), Palgrave Handbook of Econometrics, Volume 1: Econometric Theory, 278–352. Basingstoke: Palgrave Macmillan. Poskitt, D. S. (2000). Strongly consistent determination of cointegrating rank via canonical correlations. Journal of Business and Economic Statistics 18, 77–90. Shintani, M. (2001). A simple cointegrating rank test without vector autoregression. Journal of Econometrics 105, 337–62. Stock, J. H. and M. W. Watson (1988). Testing for common trends. Journal of the American Statistical Association 83, 1097–107.

APPENDIX Proof of Proposition 2.1: (a) Consistency of the paired procedure: Let r1 be the true co-trending rank and r1 be an integer value satisfying 0 ≤ r1 < n. To prove the consistency, we only need to show that the probability of VN(r1 ) > VN(r1 ) approaches one if r1 is not equal to the true co-trending rank r1 . When r1 < r1 , we have VN1 (r1 ) − VN1 (r1 ) =

r1 

λˆ i + (f (r1 ) − f (r1 ))CT T −1 ,

i=r1 +1  C

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

483

Consistent co-trending rank selection

where λˆ i , for i = r1 + 1, . . . , r1 , is positive and Op (1) but is not op (1) from Lemma 2.1. Since CT T −1 → 0 as T → ∞, the first positive term dominates and the probability of VN1 (r1 ) > VN1 (r1 ) approaches one. When r1 > r1 , we have 

VN1 (r1 )

− VN1 (r1 ) = −

r1 

λˆ i + (f (r1 ) − f (r1 ))CT T −1 ,

i=r1 +1

where λˆ i , for i = r1 + 1, . . . r1 , is Op (T −1 ) but is not op (T −1 ) from Lemma 2.1. By multiplying both sides by T , we have 

T (VN1 (r1 )

r1 

− VN1 (r1 )) = −T

λˆ i + (f (r1 ) − f (r1 ))CT .

i=r1 +1

Since CT → ∞ as T → ∞ and f (r1 ) − f (r1 ) > 0 when r1 > r1 , the second term on the right-hand side dominates and is positive. Thus, the probability of VN1 (r1 ) > VN1 (r1 ) approaches one and the probability of selecting r1 = r1 approaches zero when the co-trending rank is selected by minimising VN1 (r1 ). Analogously, one can establish the consistency of the weak co-trending rank selection by minimising VN2 (r). (b) Consistency of the joint procedure: Let r be the true weak co-trending rank and r  be an integer value satisfying r1 < r  < n. To prove the consistency, we first consider the case of r1 = r1 , for any given pair of r  and r. When r1 < r1 , we have VN(r1  , r  ) − VN(r1 , r) = T α

r1 

λˆ i +

i=r1 +1

⎧ ⎨ ⎩



−

r 

r 

λˆ i +

i=r1 +1

λˆ i

i=r1 +1

⎫ ⎬ ⎭

+O

CT T

,

where λˆ i , for i = r1 + 1, . . . , r1 , is positive and Op (1) but is not op (1) from Lemma 2.1. Since the second term in curly brackets is bounded in probability and CT T −1 → 0 as T → ∞, the first positive term dominates and the probability of VN(r1 , r  ) > VN(r1 , r) approaches one. When r1 > r1 , we have 

r1 

VN(r1 , r  ) − VN(r1 , r) = −T α

λˆ i +

i=r1 +1

⎧ ⎨ ⎩

+ (f (r1 ) − f (r1 ))



−

r 

λˆ i +

i=r1 +1

CT +O T

r 

λˆ i

i=r1 +1



CT T2

⎫ ⎬ ⎭

,

where λˆ i , for i = r1 + 1, . . . r1 , is Op (T −1 ) but is not op (T −1 ) from Lemma 2.1. By multiplying both sides by T 1−α , we have 

T 1−α (VN(r1 , r  ) − VN(r1 , r)) = −T

r1  i=r1 +1

λˆ i +

⎧ ⎨ ⎩

+ (f (r1 ) − f (r1 ))



r 

− T 1−α

λˆ i + T 1−α

i=r1 +1

CT +O Tα



CT T 1+α

r  i=r1 +1

λˆ i

⎫ ⎬ ⎭

.

The first term on the right-hand side is Op (1) and the second term in curly brackets is Op (T −α ). Since CT /T α → ∞ and CT T −1 → 0 as T → ∞ and f (r1 ) − f (r1 ) > 0 when r1 > r1 , the third term on the right-hand side dominates and is positive. Thus, the probability of VN(r1 , r  ) > VN(r1 , r) approaches one.  C

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

484

Z.-F. Guo and M. Shintani Next, we consider the case of r1 = r1 , for the various pairs of r  and r. When r  < r, VN(r1 , r  ) − VN(r1 , r) =

r 



C λˆ i + (f (r  ) − f (r)) T2 , T

i=r  +1

where λˆ i , for i = r  + 1, . . . , r, is positive and Op (T −1 ) but is not op (T −1 ) from Lemma 2.1. By multiplying both sides by T , we have T (VN(r1 , r  ) − VN(r1 , r)) = T

r 



i=r  +1

C λˆ i + (f (r  ) − f (r)) T . T

CT T −1

Since → 0 as T → ∞, the first term on the right-hand side dominates and is positive. Hence, the probability of VN(r1 , r  ) > VN(r1 , r) approaches one. When r  > r, 

VN(r1 , r  ) − VN(r1 , r) = −

r 



C λˆ i + (f (r  ) − f (r)) T2 , T i=r+1

where λˆ i , for i = r + 1, . . . r  , is Op (T −2 ) but is not op (T −2 ) from Lemma 2.1. By multiplying both sides by T 2 , we have 

T 2 (VN(r1 , r  ) − VN(r1 , r)) = −T 2

r 

λˆ i + (f (r  ) − f (r))CT .

i=r+1

CT



→ ∞ as T → ∞ and f (r ) − f (r) > 0 when r  > r, the second term on the right-hand side Since dominates and is positive. Hence, the probability of VN(r1 , r  ) > VN(r1 , r) approaches one. Combining all the cases yields the consistency of the joint selection procedure. 

 C

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