## 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.

## Appendix

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

By Assumptions B.2 and E, as *N _{g}, T*→∞

where the matrices Ω_{g}_{, 00}, Ω_{g}_{, 0}* _{g}* and Ω

_{g}_{,}

*have dimensions (*

_{gg}*r*

_{0}×

*r*

_{0}), (

*r*

_{0}×

*r*) and (

_{g}*r*

_{g}×

*r*), respectively, and are such that

_{g}Being positive semi-definite by construction, Ω will be positive definite if and only if it is non-singular. By Assumption E *π _{g}*>0 (

*g*=1,…,

*G*) and, by Assumption B.2, the matrices Ω

_{g}_{, 00}, Ω

_{g}_{,}

*and*

_{gg}*g*=1,…,

*G*).

Let

and

Note that Ω* _{SS}* 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.,

and (in general, the number within brackets will represent the iteration number).*k*_{1}=…=*k*=0) and estimate the model by computing the eigenvalues and eigenvectors of (_{G}*NT*)^{–1}*XX*′. The resulting estimated global factors and global loadings will be denoted byThe 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

subject to

Let

be a*T*×(

*T*–

*k*

_{0}) matrix such that

and

The solution of the above maximization problem is^{9}

where

is a (*T*–

*k*

_{0})×

*k*matrix with columns that are the orthogonal and normalized eigenvectors of the (

_{g}*T*–

*k*

_{0})×(

*T*–

*k*

_{0}) matrix

associated with its largest *k _{g}* eigenvalues. From (12), given

Compute

From Equation (13), the columns of are the*k*_{0}orthogonal and normalized eigenvectors of the latter matrix associated with its largest*k*_{0}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 thatUsing 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 M*_{1}*<∞ such that for all N _{g} and T*,

*Lemma A.2:** Under Assumption C.4, for every g*=1,…,*G and for any idempotent matrix P _{g}* (

*T*×

*T) of rank k*

_{0}

*+k*>0

_{g}*there exists some M*

_{2}<

*∞ such that for all N*,

_{g}and T*Lemma A.3:** Let*

*be any matrix*(

*T*×

*m) such that*

*and let*

*Under Assumptions A to D,*

*and*

*as N*→∞

_{g}*and T*→∞.

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

*For every g*=1,…,*G, let*^{10}

*and*

*Also let*

*and*

*where* Ω* _{g} are* (

*r*

_{0}+

*r*)×(

_{g}*r*

_{0}+

*r*0<

_{g}) matrices and*π*≤1 with

_{g}*Denote by*

*a solution of problem*

*subject to:*

*Under the assumption that* Ω* _{g} is symmetric positive definite* (

*g*=1,…,

*G),*

*for every**for all feasible solutions*and*if k*_{0}+k_{g}≥r_{0}+r_{g}and k_{g}≥r_{g}, then*if k*_{0}≥r_{0}and k_{g}≥r_{g}, then*and*

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

*be* ((*G*+1)×1) *vectors of non-negative integers. As N _{g}*→∞

*and T*→∞,

*for each g*:

*(i) There exists a ((r _{0}+r_{g})×(k_{0}+k_{g})) matrix*

*with rank *

*such that*

*(ii) for any*

*(T×(k*

_{0}+k_{g})) such that*(iii) If *

*and*

*then*

*and*

*(iv) If k _{0}≥r_{0} and k_{g}≥r_{g}, 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 withand

for any set of (orthogonal) matrices {*Q*_{00}, *Q*_{11},…,*Q _{GG}*}, with

*Q*(

_{gg}*r*×

_{g}*r*), such that

_{g}*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*r*

_{0}+

*r*(positive) eigenvalues of Ω

_{g}*.*

_{g}We will prove the Lemma by considering in turn all the possible cases:

**(1)** k_{0}≥r_{0}**and, for every** g, k_{g}≥r_{g}

(block diagonal and full rank *r*_{0}+*r _{g}*) and

is an optimal solution, for any matrices

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

**(2)** k_{0}<r_{0}**and, for every** g, k_{0}+k_{g}≥r_{0}+r_{g}

(full rank *r*_{0}+*r _{g}*) and

for every *g*.

**(3)** k_{0}≥r_{0}**and** k_{g}<r_{g}**for some** g

Without any loss of generality, let us suppose that *k _{g}*<

*r*for

_{g}*g*=1,…,

*G̅*and that

*k*≥

_{g}*r*for

_{g}is substituted for

For this benchmark case, the optimal solution is determined as in case (1). Thus, for every*g*, is full rank

*r*

_{0}+

*r*. Also, for every

_{g}*g*, attains its upper bound

*tr*(Ω

_{g}).

By construction, we have that

and

An optimal solution for *k* corresponds to any choice

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

*r*–

_{g}*k*columns of that have the smaller effect on i.e., make

_{g}^{12}

and, correspondingly,

where

is a (*r*×

_{g}*k*) matrix whose columns are columns of the identity matrix of order

_{g}*r*with Note that any solution that affects cannot be better than the candidate solution because it affects for all

_{g}*s*≠

*g*, thereby (in general) worsening the optimal value of the objective function without addressing the necessity to decrease the number of columns of from

*r*to

_{g}*k*. The same argument rules out changing or for any Hence, the candidate solution is optimal and

_{g}**(4)** k_{0}<r_{0}**and** k_{0}+k_{g}<r_{0}+r_{g}**for some** g

Again without any loss of generality, let us suppose that *k*_{0}+*k _{g}*<

*r*

_{0}+

*r*for

_{g}*k*

_{0}+

*k*<

_{g}*r*

_{0}+

*r*for The relevant “benchmark case” is now

_{g}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 (*r*+

_{g}*r*

_{0}–

*k*

_{0})×

*k*matrix whose columns are

_{g}*k*columns of the identity matrix of order (

_{g}*r*+

_{g}*r*

_{0}–

*k*

_{0}), 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*>*r*_{0}+*r _{g}* for every

*g*. Also let

*T*×(

*k*

_{0}+

*k*) matrix with

_{g}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

In general for any

, from (18) we getand

Thus, by Assumption A.1,

First, note that if we partition

into four blocks,by (21) and taking into account that

we getAs

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 haveNow, 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 *Q*_{00} (*k*_{0}×*k*_{0}), *Q _{gg}* (

*k*×

_{g}*k*),

_{g}and

as *N _{g}*→∞ and

*T*→∞. By Lemma A.4(i),

Moreover, if *k _{g}*≥

*r*and

_{g}*k*

_{0}+

*k*≥

_{g}*r*, by Lemma A.4(ii)

_{g}and

for any

such thatFrom (24), we get that

In addition, because (24) is valid for any*k*such that

*k*

_{0}+

*k*≥

_{g}*r*

_{0}+

*r*, it is also verified by any such that

_{g}implying that

Finally, by Lemma A.4(iii), if *k*_{0}≥*r*_{0} and* k _{g}*≥

*r*then

_{g},#### 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*N*→∞ and

_{g}*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(*N _{g}, T*) and taking into account Assumption E, we have, for every

*g*,

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

(because *N _{g}*→

*N*≤1 (

*g*=1,…,

*G*))

(because

)Now consider *k*_{0}≥*r*_{0} and* k _{g}*≥

*r*for every

_{g}*g*. Let us denote the limit of

*H*

^{(}

^{k}^{)}by

where Ω_{g}_{,00}, Ω_{g}_{,0}* _{g}*, Ω

_{g}_{,}

*and Ω are as defined in Appendix I,*

_{gg}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*N*and

_{g}*T*, but we will make this dependence implicit to simplify the notation. First we will show that for sufficiently large

*N*and

_{g}*T*

The upper bound results directly from the number of columns of

being*k*

_{0}+

*k*. As regards the lower bound, note that

_{g}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 *N _{g}* and

*T*, the rank of

*A*} such that ||

_{n}*A*–

_{n}*B*||→0, there exists such that rk(

*A*)≥rk(

_{n}*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*k*

_{0}+

*k*.

_{g}For sufficiently large *N _{g}* and

*T*, by (26), if

*k*≥

_{g}*r*and

_{g}*k*

_{0}+

*k*=

_{g}*r*

_{0}+

*r*, matrix

_{g}Now let *k _{g}*≥

*r*and

_{g}*k*

_{0}+

*k*>

_{g}*r*

_{0}+

*r*. By (26), for sufficiently large

_{g}*N*and

_{g}*T*, matrix

*k*

_{0}+

*k*, implying that is the identity matrix of order Let

_{g}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 *k _{g}*≥

*r*and

_{g}*k*

_{0}+

*k*≥

_{g}*r*

_{0}+

*r*. As regards the second term, for sufficiently large

_{g}*N*and

_{g}*T*, it is non-negative following the same argument as above for the case

*k*

_{0}+

*k*=

_{g}*r*

_{0}+

*r*, but with

_{g}*k*≥

_{g}*r*and

_{g}*k*

_{0}+

*k*≥

_{g}*r*

_{0}+

*r*, then

_{g}Continue to admit that *k _{g}*≥

*r*and

_{g}*k*

_{0}+

*k*=

_{g}*r*

_{0}+

*r*and let

_{g}*F̌*

_{g}(

*F̌*

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

Let

be an optimal solution of the problemsubject to

where

and*W*are matrices

_{g}*T*×(

*r*

_{0}+

*r*) and

_{g}*T*×(

*k*

_{0}+

*k*–

_{g}*r*

_{0}–

*r*), respectively. Let be any

_{g}*T*×(

*k*

_{0}+

*k*–

_{g}*r*

_{0}–

*r*) matrix such that and is a feasible solution of the above maximization problem. Thus, for all

_{g}*N*and

_{g}*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*=[*k*_{0}*k*_{1} … *k _{g}* …

*k*]′

_{G}*be a* (*G*+1)×1 *vector of non-negative integers. If* 1*≤k*_{0}*+k _{g}≤r*

_{0}

*+r*

_{g}, then there exists M_{4}

*<*∞

*such that for all N*

_{g}and T*where*

*is the matrix defined in Theorem 1.*

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

*k*=[*k*_{0}*k*_{1} … *k _{g}* …

*k*]′

_{G}*be a* (*G*+1)×1 *vector of non-negative integers. If k*_{0}*+k _{g}<r*

_{0}

*+r*>0

_{g}, then there exists τ_{g,k}*such that for all N*

_{g}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 k _{g}*≥

*r*

_{g},*k*

_{0}

*+k*≥

_{g}*r*

_{0}

*+r*

_{g}and*then there exists M*

_{5}<∞

*such that for all N*

_{g}and TLemmas 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 *k*≤*k*^{max} such that *k _{g}*≥

*r*and

_{g}*k*

_{0}

*+k*≥

_{g}*r*

_{0}

*+r*. This modification does not change the remaining steps of the proof, because in that case

_{g}*r*

_{0}

*+r*. 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)

_{g}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 *k*≠*r*, 0≤*k*≤*k*^{max} and *k*_{0}+*k _{g}*>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) *k*_{0}=*r*_{0} and *k _{g}*=

*r*

_{g}(II) *k*_{0}*+k _{g}*<

*r*

_{0}

*+r*

_{g}(III) *k _{g}*>

*r*and

_{g}*k*

_{0}

*+k*≥

_{g}*r*

_{0}

*+r*

_{g}(IV) *k*_{0}>*r*_{0} and *k _{g}*=

*r*

_{g}(V) *k*_{0}>*r*_{0}, *k _{g}*<

*r*and

_{g}*k*

_{0}

*+k*≥

_{g}*r*

_{0}

*+r*

_{g}Let us first admit that for all *g*, the comparison between (*k*_{0}, *k _{g}*) falls into one (and only one) of the previous cases. We will consider in turn those cases:

**Case (I):** If *k*≠*r*, 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 towhich has a positive limit by Lemma A.7. Hence, the left hand side of the inequality in (28) having a positive limit and *ψ*_{0}(*N _{g}, 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 *k*_{0}*+k _{g}*=

*r*

_{0}

*+r*.

_{g}**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 bothand

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, *k*≠*r*) and

*g*≤

*δ*

_{(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≤*g*≤*δ*_{(I)}, by Lemma A.8 we have

Consider any (II)≤(*j*)≤(V) for which *δ _{j}*>0 (it exists because

*k*≠

*r*). 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 *k*_{0}+*k _{g}*<

*r*

_{0}+

*r*, Lemmas A.1(iii), A.6 and A.7 imply that

_{g}^{13}

for some *ε _{g}*

_{,}

*>0. Thus,*

_{k}for some *δ _{g}*

_{,}

*>0.*

_{k}Now, when *k*_{0}+*k _{g}*≥

*r*

_{0}+

*r*, let

_{g}*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}Although Quah and Sargent (1993) already considered a moderate size panel with 60 variables.

^{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}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 ξ_{g,nt}with zero mean and uniformly bounded seventh moments and where Σ_{g}(N_{g}×N_{g}) and R_{g}(T×T) are arbitrary (possibly random) positive definite matrices with bounded eigenvalues.^{4}In the regular model with N variables, T time periods and k estimated factors, a set of k

is only defined up to an orthogonal transformation of matrix Q^{2}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^{(k)}. In other words, any estimator with Q^{(k)}orthogonal, also has the same optimal MSIE.^{5}The other situations were discarded to ease the presentation of the results.

^{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}All Matlab codes are available from the authors upon request.

^{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}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*×(*m*–*q*)) 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** ((*m*–*q*)×*n*) 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}Note that the block

does not depend on*g*.^{11}As

and Thus, Lemma 2 is proved by applying Result 6 of Amengual and Watson when their*P*is idempotent of rank_{g}*k*_{0}+*k*, there exists F_{g}(_{g}*T*×(*k*_{0}+*k*)) such that_{g}*m*→∞. Note that Assumption C.4 corresponds to their Assumption (A.6) with*m*→∞.^{12}Remark that, by (16) and (17),

and^{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