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 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
Let
and
Note that ΩSS is positive definite.
We may write
where
and
The latter inverse exists because
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
subject to
Let
and
The solution of the above maximization problem is9
where
associated with its largest kg eigenvalues. From (12), given
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
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
Under Assumptions A to D,
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
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
(ii) for any
(iii) If
and
(iv) If k0≥r0 and kg≥rg, then
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
by (15), Assumptions A.1 and B.2 and because
Proof of Lemma A.4
First note that any optimal solution is not unique. If
and
for any set of (orthogonal) matrices {Q00, Q11,…,QGG}, with Qgg (rg×rg), such that
where
We will prove the Lemma by considering in turn all the possible cases:
(1) k0≥r0and, for every g, kg≥rg
(block diagonal and full rank r0+rg) and
is an optimal solution, for any matrices
(2) k0<r0and, for every g, k0+kg≥r0+rg
(full rank r0+rg) and
for every g.
(3) k0≥r0and kg<rgfor some g
Without any loss of generality, let us suppose that kg<rg for g=1,…,G̅ and that kg≥rg for
is substituted for
By construction, we have that
and
An optimal solution for k corresponds to any choice
Given an optimal solution for
and, correspondingly,
where
(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 the associated optimal solution
For
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
and
Hence, the principal components estimator also maximizes
subject to: (i)
for some matrices
In particular, for large T, there are two matrices
In general for any
and
Thus, by Assumption A.1,
First, note that if we partition
by (21) and taking into account that
As
where
Moreover, for
Now, let
with blocks
with Q00 (k0×k0), Qgg (kg×kg),
and
as Ng→∞ and T→∞. By Lemma A.4(i),
Moreover, if kg≥rg and k0+kg≥rg, by Lemma A.4(ii)
and
for any
From (24), we get that
implying that
Finally, by Lemma A.4(iii), if k0≥r0 and kg≥rg, then
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
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 Ng→N≤1 (g=1,…,G))
(because
Now consider k0≥r0 and kg≥rg for every g. Let us denote the limit of H(k) by
where Ωg,00, Ωg,0g, Ωg,gg and Ω are as defined in Appendix I,
and
VII. Proof of Theorem 2
Let
The upper bound results directly from the number of columns of
For the matrix
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
by Lemma A.5(i) and by Assumption B.2. The lower bound for
For sufficiently large Ng and T, by (26), if kg≥rg and k0+kg=r0+rg, matrix
Now let kg≥rg and k0+kg>r0+rg. By (26), for sufficiently large Ng and T, matrix
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 kg≥rg and k0+kg≥r0+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
Continue to admit that kg≥rg and k0+kg=r0+rg and let F̌
to complete the proof of Theorem 2 we will now show that
Let
subject to
where
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=[k0k1 … kg … kG]′
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
Lemma A.7:Suppose that the Assumptions A to E hold and let
k=[k0k1 … kg … kG]′
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
Lemma A.8:Suppose that the Assumptions A to E hold. Let
be (G+1)×1 vectors of non-negative integers. If kg≥rg,
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 k≤kmax such that kg≥rg and k0+kg≥r0+rg. This modification does not change the remaining steps of the proof, because in that case
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≤kmax 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+kg≥r0+rg
(IV) k0>r0 and kg=rg
(V) k0>r0, kg<rg and k0+kg≥r0+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 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
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
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, k≠r) and
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 δj>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 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+kg≥r0+rg, let
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 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
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
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 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
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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