Accessible Requires Authentication Published by De Gruyter July 27, 2013

Determining the number of global and country-specific factors in the euro area

Francisco Dias, Maximiano Pinheiro and António Rua

Abstract

With the adoption of a single currency and a common monetary policy, the assessment of the existence of common macroeconomic factors driving the euro area as a whole is of key importance. In this paper, we try to contribute to this debate by admitting an approximate factor model with a common component comprising global and country-specific factors. Although it is arguable to allow for common factors shared by all the euro area member countries in the context of the European economic integration, it is also reasonable to admit that there may be country-specific factors. Hence, we end up with a two-level factor model allowing for both global and country-specific factors. We discuss the consistency of the principal components estimator within such framework and propose criteria for determining the number of global and group-specific factors. The consistency of the suggested criteria is established and the small sample properties are assessed through Monte Carlo simulations. Resorting to large datasets for the four major euro area countries, we find evidence of the existence of both global and country-specific factors.


Corresponding author: António Rua, Economics Research Department, Banco de Portugal, Av. Almirante Reis no 71, Lisboa, Portugal, e-mail:

Appendix

I. Proof that Ω is positive definite under Assumptions B.2 and E

By Assumptions B.2 and E, as Ng, T→∞

where the matrices Ωg, 00, Ωg, 0g and Ωg,gg have dimensions (r0×r0), (r0×rg) and (rg×rg), respectively, and are such that

Being positive semi-definite by construction, Ω will be positive definite if and only if it is non-singular. By Assumption E πg>0 (g=1,…,G) and, by Assumption B.2, the matrices Ωg, 00, Ωg,gg and

are all non-singular (g=1,…,G).

Let

and

Note that ΩSS is positive definite.

We may write

where

and

The latter inverse exists because

being the sum of positive definite matrices, is also positive definite.             □

II.  An algorithm to compute the principal components estimates (given k)8

  • First, let us consider only global factors (i.e., k1=…=kG=0) and estimate the model by computing the eigenvalues and eigenvectors of (NT)–1XX′. The resulting estimated global factors and global loadings will be denoted by

    and
    (in general, the number within brackets will represent the iteration number).

  • The first order conditions for the minimization of the overall MSIE subject to the identifying conditions (7)–(8) are

and, for g=1,…,G,

In iteration i (≥1), for each g, substitute

for
and
for
in (7), (8) and (14). The resulting expressions correspond to the first order conditions for the maximization with respect to
of

subject to

Let

be a T×(Tk0) matrix such that

and

The solution of the above maximization problem is9

where

is a (Tk0kg matrix with columns that are the orthogonal and normalized eigenvectors of the (Tk0)×(Tk0) matrix

associated with its largest kg eigenvalues. From (12), given

the corresponding estimate
is simply:

  • (c)

    Compute

    From Equation (13), the columns of
    are the k0 orthogonal and normalized eigenvectors of the latter matrix associated with its largest k0 eigenvalues and

(d) Steps (b) and (c) must be repeated until convergence is achieved.□

III. Proof that if is full rank, then

If

is positive definite, it is straightforward to show that

Using the latter equality,

because the trace of a positive semi-definite matrix is non-negative.□

IV. Lemmas

In order to prove the consistency of the principal components estimator, first it is convenient to present and prove several lemmas. Hereafter, rk(A) denotes the rank of matrix A.

Lemma A.1: Under Assumptions A, B and C, for every g=1,…,G there exists some M1<∞ such that for all Ng and T,

Lemma A.2: Under Assumption C.4, for every g=1,…,G and for any idempotent matrix Pg (T×T) of rank k0+kg>0 there exists some M2<∞ such that for all Ng and T,

Lemma A.3: Let

be any matrix (T×m) such that
and let

Under Assumptions A to D,

and
as Ng→∞ and T→∞.

Lemma A.4: Consider the (G+1)×1 vectors of non-negative integers

For every g=1,…,G, let10

and

Also let

and

where Ωg are (r0+rg)×(r0+rg) matrices and 0<πg≤1 with

Denote by

a solution of problem

subject to:

Under the assumption that Ωg is symmetric positive definite (g=1,…,G),

  • for every

  • for all feasible solutions
    and

    if k0+kg≥r0+rg and kg≥rg, then

  • if k0≥r0 and kg≥rg, then

    and

Lemma A.5: Suppose that Assumptions A to E hold. Let

be ((G+1)×1) vectors of non-negative integers. As Ng→∞ and T→∞, for each g:

(i) There exists a ((r0+rg)×(k0+kg)) matrix

with rank

such that

(ii) for any

(T×(k0+kg)) such that

(iii) If

and
then

and

(iv) If k0≥r0 and kg≥rg, then

and

Lemma A.1 is a direct adaptation of Lemma 1 in Bai and Ng (2002). Only notation changes are required to adapt their proof. Lemma A.2 is a special case of Result 6 in Amengual and Watson (2005).11

Proof of Lemma A.3

Taking into account (5), we have

and

where

||(a)||→0 follows directly from Assumption B.2. Using the fact that |tr(A)|≤m||A|| for any m×m matrix A,

For (b), we have

by Lemma A.2, with

Now for (c),

by (15), Assumptions A.1 and B.2 and because

Thus ||(c)||→0 and

Proof of Lemma A.4

First note that any optimal solution is not unique. If

is optimal, then
is also optimal with

and

for any set of (orthogonal) matrices {Q00, Q11,…,QGG}, with Qgg (rg×rg), such that

(g=1,…,G). In any case,
and
share the same rank. Also note that, for any feasible solution,
and
will have all eigenvalues in the range [0;1]. Thus,

where

are the r0+rg (positive) eigenvalues of Ωg.

We will prove the Lemma by considering in turn all the possible cases:

(1) k0≥r0and, for every g, kg≥rg

with

(block diagonal and full rank r0+rg) and

is an optimal solution, for any matrices

and
such that
and
Indeed, it is a feasible solution and for every g

(2) k0<r0and, for every g, k0+kg≥r0+rg

with

(full rank r0+rg) and

is an optimal solution, for any matrix
such that
As in the previous case,

for every g.

(3) k0≥r0and kg<rgfor some g

Without any loss of generality, let us suppose that kg<rg for g=1,…, and that kgrg for

We will determine the rank of the optimal solution in this case by comparing it with the optimal solution in the relevant “benchmark case” for which

is substituted for

For this benchmark case, the optimal solution
is determined as in case (1). Thus, for every g,
is full rank r0+rg. Also, for every g,
attains its upper bound trg).

By construction, we have that

and

An optimal solution for k corresponds to any choice

that minimizes the latter difference while still complying with the problem constraints

Given an optimal solution for

a candidate for the optimal solution for k is the following: (i) for
and
ensuring that for these groups of variables
(ii) for
delete the rgkg columns of
that have the smaller effect on
i.e., make12

and, correspondingly,

where

is a (rg×kg) matrix whose columns are columns of the identity matrix of order rg with
Note that any solution that affects
cannot be better than the candidate solution because it affects
for all sg, thereby (in general) worsening the optimal value of the objective function without addressing the necessity to decrease the number of columns of
from rg to kg. The same argument rules out changing
or
for any
Hence, the candidate solution is optimal and

(4) k0<r0and k0+kg<r0+rgfor some g

Again without any loss of generality, let us suppose that k0+kg<r0+rg for

and that k0+kg<r0+rg for
The relevant “benchmark case” is now

and the associated optimal solution

is determined as in case (2).

For

and
and, consequently,
As for
adapting the argument presented for case (3), let
be a (rg+r0k0kg matrix whose columns are kg columns of the identity matrix of order (rg+r0k0), with
Chose those columns so that the difference

is as small as possible, when setting

Therefore, for

Proof of Lemma A.5

The principal components estimator was defined as the optimal solution of the problem of minimization of the overall MSIE subject to the restrictions (7)–(8). Let T>r0+rg for every g. Also let

be a T×(k0+kg) matrix with

and

Hence, the principal components estimator also maximizes

subject to: (i)

(ii) identical
for all g. Now define
such that
and
as T→∞. Given Assumption A.1, at least for sufficiently large T the columns of
span the T-dimensional space. Hence,
can be expressed as

for some matrices

In particular, for large T, there are two matrices

and
such that

In general for any

, from (18) we get

and

Thus, by Assumption A.1,

First, note that if we partition

into four blocks,

by (21) and taking into account that

we get

As

does not depend on g,
converges to a matrix
identical for all g. Also note that Assumption E, Lemma A.2 and (21) imply that

where

Moreover, for

, we also have

Now, let

be an optimal solution of maximizing
subject to:

with blocks

identical for all g. The optimal solution is not unique, because any block diagonal orthogonal transformation of
is feasible and attains the same optimal value of the objective function (see first part of the proof of Lemma A.4). Taking this into account, by (23), (20), (21) and (22), for every g there exists a matrix

with Q00 (k0×k0), Qgg (kg×kg),

and
such that

and

as Ng→∞ and T→∞. By Lemma A.4(i),

Moreover, if kgrg and k0+kgrg, by Lemma A.4(ii)

and

for any

such that

From (24), we get that

In addition, because (24) is valid for any k such that k0+kgr0+rg, it is also verified by any
such that

implying that

Finally, by Lemma A.4(iii), if k0r0 and kgrg, then

and
.   □

V. Proof of Theorem 1

The first part of Theorem 1 can be proved following step by step the proof of Bai and Ng’s Theorem 1, with the necessary adaptations of notation, and therefore the proof will not be repeated here. As regards the asymptotic rank of

let
be such that
as Ng→∞ and T→∞. From the definition of
Assumption B.2 and Lemma A.5(i),
and
  □

VI. Proof of Corollary 1.1

By Theorem 1,

Multiplying by min(N, T)/min(Ng, T) and taking into account Assumption E, we have, for every g,

Summing up the latter expressions for g=1,…,G, we obtain

(because NgN≤1 (g=1,…,G))

(because

)

Now consider k0r0 and kgrg for every g. Let us denote the limit of H(k) by

Taking into account Assumptions B.2 and E and Lemma A.5(i,iv),

where Ωg,00, Ωg,0g, Ωg,gg and Ω are as defined in Appendix I,

and
are as defined in Lemma A.5,

and

has full rank
because Π and Ω are non-singular matrices
(for the latter matrix, see Appendix I) and, by Lemma A.5(i,iv),
has full rank
   □

VII. Proof of Theorem 2

Let

The rank
depends on Ng and T, but we will make this dependence implicit to simplify the notation. First we will show that for sufficiently large Ng and T

The upper bound results directly from the number of columns of

being k0+kg. As regards the lower bound, note that

For the matrix

defined in Lemma A.5(i),

The first term on the right hand side converges to zero by Lemma A.3. As for the second term, note that

which goes to zero by Assumption B.2 and Lemma A.5(i). Thus

Therefore, for sufficiently large Ng and T, the rank of

is not smaller than the rank of
We use the fact that for any sequence of positive semi-definite matrices {An} such that ||AnB||→0, there exists
such that rk(An)≥rk(B) for all
But

by Lemma A.5(i) and by Assumption B.2. The lower bound for

in (26) follows directly from the lower bound on
and the fact that, by construction,
is the identity matrix of order k0+kg.

For sufficiently large Ng and T, by (26), if kgrg and k0+kg=r0+rg, matrix

will be positive definite. From Appendix III, we know that in this case

Now let kgrg and k0+kg>r0+rg. By (26), for sufficiently large Ng and T, matrix

may be singular. Let
be a
matrix that selects
linear independent columns of
That is, the columns of
are
columns of the identity matrix of order k0+kg, implying that
is the identity matrix of order
Let

Note that

The third term of the right hand side is zero because, by construction

By Lemma A.4(ii) and by Lemma A.5(iii), we also know that the first term converges to zero when kgrg and k0+kgr0+rg. As regards the second term, for sufficiently large Ng and T, it is non-negative following the same argument as above for the case k0+kg=r0+rg, but with

and
instead of
and
respectively. All in all, in this Part 2, we conclude that if kgrg and k0+kgr0+rg, then

Continue to admit that kgrg and k0+kg=r0+rg and let

Given that υg(

to complete the proof of Theorem 2 we will now show that

Let

be an optimal solution of the problem

subject to

where

and Wg are matrices T×(r0+rg) and T×(k0+kgr0rg), respectively. Let
be any T×(k0+kgr0rg) matrix such that
and
is a feasible solution of the above maximization problem. Thus, for all Ng and T,
implying that

But

by Lemmas A.5(ii) and A.5(iii), respectively. Therefore,

   □

VIII. More Lemmas

In addition to the lemmas presented in Appendix IV, to prove Theorem 3 and Corollary 3.1 we need the following three lemmas.

Lemma A.6:Suppose that the Assumptions A to E hold and let

k=[k0k1kgkG]′

be a (G+1)×1 vector of non-negative integers. If 1≤k0+kg≤r0+rg, then there exists M4<such that for all Ng and T

where

is the matrix defined in Theorem 1.

Lemma A.7:Suppose that the Assumptions A to E hold and let

k=[k0k1kgkG]′

be a (G+1)×1 vector of non-negative integers. If k0+kg<r0+rg, then there exists τg,k>0 such that for all Ng and T

where

is the matrix defined in Theorem 1.

Lemma A.8:Suppose that the Assumptions A to E hold. Let

be (G+1)×1 vectors of non-negative integers. If kgrg,

k0+kgr0+rg and
then there exists M5<∞ such that for all Ng and T

Lemmas A.6, A.7 and A8 are direct adaptations of Lemmas 2, 3 and 4 in Bai and Ng (2002), respectively. The proofs of the latter propositions can be easily adapted step by step to prove Lemmas A.6, A.7 and A.8, with the necessary notation changes. However, two remarks are needed in relation to the proof of Lemma A.8. The first remark regards the adaptation of the first expression in Bai and Ng’s proof of their Lemma 4 (page 217):

In our model, for the group g of variables, the maximum refers to the maximum for all kkmax such that kgrg and k0+kgr0+rg. This modification does not change the remaining steps of the proof, because in that case

has rank r0+rg. The second remark refers to the last part of the proof of Bai and Ng’s Lemma 4 (bottom of page 218 and top of page 219), which is not correct. More precisely, as acknowledged by Bai and Ng (2005), the proof that (for our model and in our notation)

is invalid. Under our Assumption C.4, we can use Lemma A.2 (Appendix IV) to complete the proof.   □

IX. Proof of Theorem 3

We need to prove that for all k such that kr, 0≤kkmax and k0+kg>0 (g=1,…,G)

as N→∞ and T→∞. Given k, for every variables group g we will have one of the following (mutually exclusive) cases:

(I) k0=r0 and kg=rg

(II) k0+kg<r0+rg

(III) kg>rg and k0+kgr0+rg

(IV) k0>r0 and kg=rg

(V) k0>r0, kg<rg and k0+kgr0+rg

Let us first admit that for all g, the comparison between (k0, kg) falls into one (and only one) of the previous cases. We will consider in turn those cases:

Case (I): If kr, this case cannot happen.

Case (II): The left hand side of the inequality in (28) may be rewritten as

Lemma A.6 implies that the first and third terms converge to zero. As regards the second term, note that

because
and
asymptotically span the same space. Thus the second term is asymptotically identical to

which has a positive limit by Lemma A.7. Hence, the left hand side of the inequality in (28) having a positive limit and ψ0(Ng, N, T) tending to zero for g=0, 1,…,G are sufficient conditions to ensure that the probability converges to zero.

Case (III): Multiplying both sides of the inequality in (28) by min(N, T) we get

By Lemma A.8, the left hand side of the inequality is bounded. The probability goes to zero because conditions (ii) and (iii) ensure that the right hand side of the inequality diverges to –∞. Note that condition (iii) is required whenever k0+kg=r0+rg.

Case (IV): Similar to case (III).

Case (V): The left hand side of the inequality in (28) may be rewritten as

where

Thus, it is sufficient for (28) to prove that both

and

The proofs of (29) and of (30) are similar to those presented for cases (IV) and (II), respectively.

Turning now to “mixed cases,” given k, let δ(j) (≥0) be the number of groups of variables g that fall into case (j) (j=I, II, III, IV, V). We have δ(I)<G (because, by construction, kr) and

Without loss of generality, admit that groups with gδ(I)fall into case (I) and that groups with
fall into case (j).

If δ(I)=0, to prove (28) then it is sufficient to show that for j=II, III, IV, V

The proofs are similar to those for the corresponding “pure cases” presented in Part 1.

Finally, when 0<δ(I)<G, note that for any 1≤gδ(I), by Lemma A.8 we have

Consider any (II)≤(j)≤(V) for which δj>0 (it exists because kr). Substitute

for (31). Owing to (32), the additional terms do not change substantially the proof and the same arguments apply.   □

X. Proof of Corollary 3.1

For k0+kg<r0+rg, Lemmas A.1(iii), A.6 and A.7 imply that13

for some εg,k>0. Thus,

for some δg,k>0.

Now, when k0+kgr0+rg, let

be any (G+1)×1 vector of non-negative integers. If
and
Lemmas A.1(iii) and A.8 imply that

and thus that

Therefore, the proofs in Appendix IX remain valid after substituting

for

   □

  1. 1

    Although Quah and Sargent (1993) already considered a moderate size panel with 60 variables.

  2. 2

    The outlier adjustment corresponds to replacing the observations of the transfomed series with absolute deviations larger than six times the interquartile range by the median value of the preceding five observations [see, for example Stock and Watson (2005)].

  3. 3

    Bai and Ng (2005) acknowledged that the assumptions in their 2002 paper were not sufficient to prove their Lemma 4 and Theorem 2. They considered two alternatives to complete the set of assumptions, one being similar to our Assumption C.4. The other alternative would consist in assuming that (for our model and in our notation)

    , where
    are T×N matrices of independent variables ξg,nt with zero mean and uniformly bounded seventh moments and where Σg (Ng×Ng) and Rg (T×T) are arbitrary (possibly random) positive definite matrices with bounded eigenvalues.

  4. 4

    In the regular model with N variables, T time periods and k estimated factors, a set of k2 restrictions is required to achieve exact identification. In practice, typically only k(k+1)/2 restrictions are explicitly considered, meaning that the principal component estimator

    is only defined up to an orthogonal transformation of matrix Q(k). In other words, any estimator
    with Q(k) orthogonal, also has the same optimal MSIE.

  5. 5

    The other situations were discarded to ease the presentation of the results.

  6. 6

    We also considered another variant where the number of series is 30 and the other one to 90 (so that the total number of series remains unchanged at 120). As expected, there is a significant deterioration of the performance of the criteria for the group less represented in the data set with the remaining results almost unchanged.

  7. 7

    All Matlab codes are available from the authors upon request.

  8. 8

    A similar algorithm can be envisaged if the following alternative (partial) identification restrictions (on the loadings, instead of on the factors) are considered:

  9. 9

    Consider the problem

    subject to Z′Z=1 and to B′Z=0, where Z, A and B are (m×n), (m×m) and (m×q) matrices, respectively, with n<m, A symmetric positive semi-definite and B such that B′B=I. Let B be the (m×(mq)) orthogonal complement of B, with
    and
    . The restriction
    implies that the columns of the optimal solution Z* are linear combinations of the columns of
    and therefore the solution of the above optimization problem may be expressed as
    where W* ((mqn) is the solution of
    subject to
    By Theorem 11.6 in Magnus and Neudecker (1988, p. 205), the columns of W* are the normalized and orthogonal eigenvectors of
    associated with the n largest eigenvalues of the latter matrix.

  10. 10

    Note that the block

    does not depend on g.

  11. 11

    As Pg is idempotent of rank k0+kg, there exists Fg (T×(k0+kg)) such that

    and
    Thus, Lemma 2 is proved by applying Result 6 of Amengual and Watson when their m→∞. Note that Assumption C.4 corresponds to their Assumption (A.6) with m→∞.

  12. 12

    Remark that, by (16) and (17),

    and

  13. 13

    The argument is similar to the one presented for case (II) in Part 1 of the proof of Theorem 3.

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Published Online: 2013-07-27
Published in Print: 2013-12-01

©2013 by Walter de Gruyter Berlin Boston