A method to evaluate the rank condition for CCE estimators

Abstract. We develop a binary classifier to evaluate whether the rank condition (RC) is satisfied or not for the Common Correlated Effects (CCE) estimator. The RC postulates that the number of unobserved factors, m, is not larger than the rank of the unobserved matrix of average factor loadings, ϱ. When this condition fails, the CCE estimator is inconsistent, in general. Despite its importance, to date this rank condition could not be verified. The difficulty lies in the fact that factor loadings are unobserved, such that ϱ cannot be directly determined. The key insight in this article is that ϱ can be consistently estimated with existing techniques through the matrix of cross-sectional averages of the data. Similarly, m can be estimated consistently from the data using existing methods. Thus, a binary classifier, constructed by comparing estimates of m and ϱ, correctly determines whether the RC is satisfied or not as (N,T)→∞. We illustrate the practical relevance of testing the RC by studying the effect of the Dodd-Frank Act on bank profitability. The RC classifier reveals that the rank condition fails for a subperiod of the sample, in which case the estimated effect of bank size on profitability appears to be biased upwards.


Introduction
In a seminal paper, Pesaran (2006) put forward the Common Correlated Effects (CCE) approach for √ NT-consistent estimation of panel data models with a multifactor error structure.The method aims to control for the unobserved common factors by augmenting the regression model with cross-sectional averages (CSA) of the observables.The CCE estimator has been applied to a large range of fields 1 , and it has also been extended to several theoretical settings. 2Such popularity of CCE can be attributed to the computational simplicity as well as the excellent finite-sample performance of the estimator in stylised setups.
Nevertheless, CCE comes at a cost.In particular, the CSA of the observables are valid proxies for the unobserved factors only if the number of factors, m, does not exceed the rank of the matrix of averaged factor loadings, .This so-called "rank condition" (RC) implies that there exist at least as many observables holding linearly independent information about the unobserved factors as there are factors.Westerlund and Urbain (2013) demonstrate that the CCE estimator is inconsistent when the RC fails and the factor loadings arecorrelated with the regressors.Furthermore, Karabiyik et al. (2019) and Juodis et al. (2021) show that even when the factor loadings are uncorrelated with the regressors, failure of the RC leads to a lower rate of consistency for the CCE estimator.
Despite the importance of the RC for the asymptotic properties of the CCE estimator, practitioners typically take this assumption for granted.The main reason is that the matrix of average factor loadings is unobserved and therefore its rank cannot be directly evaluated or estimated.
This article puts forward a binary classifier that evaluates the rank condition.The key insight is that the rank of the unobserved matrix of average factor loadings, , can be established from the matrix of CSA of the data.We shall show that can be estimated consistently using existing procedures developed for determining the true rank of an unknown matrix; see e.g., Camba-Mendez and Kapetanios (2009) and Al-Sadoon (2017) for an overview of this literature.Similarly, the number of factors, m, can be estimated from the data in a straightforward manner using existing methods, such as those developed by Onatski (2010), Ahn and Horenstein (2013), and Kapetanios (2010), among many others.Comparing consistent estimates of m and , m and , respectively, the rank condition is deemed to be satisfied when the classifier RC ≡ 1 − 1{ < m} = 1, where 1{•} is an indicator function that returns 1 when the argument inside the curly brackets holds true and 0 otherwise.RC is shown to be consistent, i.e., it correctly determines whether the rank condition is satisfied or not, with probability 1 as (N, T) → ∞.
When the RC is violated for the standard CCE approach, one can augment the model with additional CSA that contain new information about the factors.Several potential augmentations have been suggested (see e.g., Chudik and Pesaran, 2015;Juodis, 2022;Karabiyik et al., 2019).However, it is not always clear which set of additional CSA to choose, and whether the selected augmentation is sufficient to restore the RC. 3 To address these issues, we put forward a strategy that combines the classifier proposed in this article and the IC criterion of Karabiyik et al. (2019).The resulting procedure enables consistent CCE estimation of panel data models with a multifactor error structure, even in cases where the rank condition fails for the original CCE estimator.
We illustrate the practical relevance of our RC classifier and augmentation strategy by studying the effect of the Dodd-Frank Act of 2010 on bank profitability.In particular, based on a random sample of 450 banks, we analyze bank profitability conditional on several potential drivers, controlling for macro-risk factors and common shocks.To examine the impact of the Dodd-Frank Act, we estimate the model separately over two subperiods, namely 2006:Q1-2010:Q4 and 2011:Q1-2019:Q4.The RC classifier reveals that the rank condition fails for the first subperiod.By augmenting the standard set of CSA using external variables, our procedure is able to restore the rank condition.This proves to be important because the estimated effect of bank size on profitability is significantly lower when the RC is restored.
In what follows, we will use A † to denote the Moore-Penrose pseudo-inverse of the matrix A, rk(A) for its rank, |A| for the determinant and A = tr AA 1/2 for its Euclidean (Frobenius) matrix norm.
For an n × n matrix A, the λ 1 (A) ≥ λ 2 (A) ≥ • • • ≥ λ n (A) denote its n ordered eigenvalues.A vec(.) denotes the vectorization operation.Finally, a ( a ) is the floor (ceiling) function, which yields the largest (smallest) integer less than (greater than) or equal to a.

Model and assumptions
We study the following linear regression model with unobserved common factors where y i = [y i1 , . . ., y iT ] denotes a T × 1 vector of observations on the dependent variable for individual i, X i = [x i1 , . . ., x iT ] denotes a T × K matrix of covariates, where x it is K × 1, and β is a K × 1 vector of unknown parameters of interest with β < ∞.The error term is composite, such that F = [f 1 , . . ., f T ] denotes a T × m matrix of unobserved common factors, where f t is m × 1, and λ i denotes an m × 1 vector of factor loadings.The dimension m is fixed and finite.Finally, ε i = [ε i1 , . . ., ε iT ] is a T × 1 vector of purely idiosyncratic disturbances.Following Pesaran (2006), we assume that the covariates are also subject to a common factor structure, such that the data generating process (DGP) for X i is given by where i denotes an m × K matrix of factor loadings, and Replacing X i in Eq. ( 1) by the expression in Eq. ( 2), and stacking the observables into a T × (1 + K) matrix Z i = [y i , X i ] ≡ [z i1 , ..., z iT ] , yields where In what follows, it is important to note that C i can be written as C i = C i B, with Therefore, since B has full rank, the rank of C i is solely determined by the matrix of factor loadings C i .
The following assumptions are made throughout the article: Assumption 1. (Idiosyncratic errors) ε it and v it are mean zero, covariance-stationary, and independent across i, with E(ε 4 it ) < ∞ and E( v it 4 ) < ∞ for all i and t.Let U = [U 1 , . . ., U N ] be such that λ 1 (UU /M) = O p (1) and λ d c M (UU /M) ≥ c + o p (1) for some real c > 0 and d c ∈ [0, 1), where M = max{N, T} and M = min{N, T}.
Assumption 2. (Common factors) f t is covariance-stationary with E( f t 4 ) < ∞ and absolute summable autocovariances.In addition, rk(F) = m and T −1 F F → F as T → ∞, where F is positive definite.

Assumption 3. (Factor loadings) C i is generated according to
where The setup described by the DGP in Eq. ( 3) together with Assumptions 1-4, is similar to that in Pesaran (2006) but deviates in the following respects.First, we focus on a model with homogeneous slope coefficients and without fixed effects.This is for ease of exposition only, as the results below also follow through under the assumption of independent random coefficients with a common mean, as in Pesaran (2006).See Section 3.5 for a discussion, and also Appendix C.2 for simulation evidence.Second, following Westerlund and Urbain (2013) and Karabiyik et al. (2019), Assumption 3 generalizes Pesaran (2006) by allowing λ i and i to be mutually correlated, although uncorrelated across crosssectional units i.Third, we introduce more explicit regularity conditions on the innovations, factors and their loadings compared to what is typically the case in the CCE literature.In particular, Assumption 1 places restrictions on the eigenvalues of the innovation covariance matrix as in Ahn and Horenstein (2013).They exclude the presence of factors in the innovations and bound the largest d c M eigenvalues away from zero.For the factors and loadings, the non central second moments are assumed to converge to a positive definite matrix.These regularity conditions are common in the factor literature (see e.g., Ahn and Horenstein, 2013;Bai and Ng, 2002) and allow us to consistently estimate m.Lastly, note that rk(F) = m of Assumption 2 implies that T ≥ m.

CCE and the rank condition
Since F enters into the data generating process of both y i and X i , and λ i and i are allowed to be mutually correlated, X i is endogenous.Therefore, standard panel data estimators, such as the two-way fixed effects estimator, fail to be consistent for the parameters of interest, β.The key idea of CCE is to replace F with CSA of the observables in Eq. (3).
In particular, taking sample averages over i in Eq. ( 3), we obtain where ) for all t = 1, . . ., T. As a result, the observed CSA converge to a linear combination of the m common factors at every t = 1, . . ., T: Suppose that C has full rank such that CC is invertible and bounded by Assumption 3. Pre-multiplying Eq. ( 7) by C and solving for f t yields Hence, as N → ∞, the common factor component at time t can be controlled for (or estimated) with the cross-section averages z t .The pooled CCE estimator for β is the least-squares estimator given by where The above idea of estimating factors with CSA crucially relies upon the assumption that C has full rank.This restriction, known as the "rank condition" (RC), corresponds to = m (10) where = rk(C).When < m, the RC fails and the CCE estimator is generally inconsistent.This is because the CSA do not contain enough information on f t , which implies that the factor estimator in Eq. ( 8) does not exist.
There are several cases where the RC may fail.To begin with, such failure occurs when m > K + 1, i.e., the number of factors is larger than the number of CSA, in which case ≤ min{m, K + 1} = K + 1 < m.In addition, although K + 1 ≥ m is a necessary condition for the RC to hold, it is by no means sufficient.For example, certain columns of Z can be asymptotically uninformative because the corresponding observables: (i) do not load on the common factors (e.g., some of the columns in i equal zero); (ii) have factor loadings that average out (e.g., = O p (N −1/2 )); or (iii) do not contain information on the common factors that is distinct from that already provided by other observables.In all these cases, the number of columns that are informative to estimate f t , as measured by , can be lower than m.5

Evaluating the rank condition
Despite the importance of the RC for the properties of the CCE estimator, this assumption is typically taken for granted.The main reason is that the population mean of the matrix of factor loadings, C, is unobserved and therefore its rank cannot be directly evaluated or estimated.The key insight of this article is that can be determined by estimating the rank of Z using existing techniques.Given a consistent estimate of , the RC is evaluated by direct comparison of that value with a consistent estimate for m.The latter can be determined from the observed data in a straightforward manner, based on e.g., Bai and Ng (2002), Alessi et al. (2010), Onatski (2010), and Ahn and Horenstein (2013).
The following two sections provide details for consistent estimation of and m as (N, T) → ∞.Section 3.3 puts forward a binary classifier that evaluates the rank condition correctly with probability 1 as (N, T) → ∞.Section 3.4 discusses a strategy for obtaining a consistent CCE estimator when the RC fails.

Consistent estimation of
We make use of the fact that the rank of an unobserved matrix A can be determined through a √ N-consistent estimator of that matrix A N = A + O p (N −1/2 ); see e.g., Robin and Smith (2000) and Kleibergen and Paap (2006).Noting that that can be consistently estimated by applying a rank estimator to Z. 6Many popular rank estimators are based either on sequential testing procedures, as e.g., in Chen and Fang (2019), or on information criteria (IC), see e.g., Cragg and Donald (1997).Camba-Mendez and Kapetanios (2009) provide an overview and conclude that sequential testing procedures have an advantage over IC methods under several modeling scenarios.Therefore, in the remainder of this section, we closely follow the sequential testing procedure developed by Robin and Smith (2000).This is easy to implement and relies on relatively mild assumptions, in that it does not require the variancecovariance of the estimator of the unknown matrix FC to be full rank, or its rank to be known.
A major complication that arises in our setting, however, is that unlike Robin and Smith (2000), where the dimensions of the target matrix are fixed as the sample size grows, here FC and its estimator Z are of order T × n.7 Therefore, the number of rows increases with the time dimension such that Z To circumvent this issue, we introduce a narrow matrix of order n × T, such that Z is n × n and rk( FC) = rk(FC).That is, has the role of reducing the dimension of Z without altering the rank of the matrix it estimates.The following assumption is imposed: Assumption 5 places additional restrictions on the potential choices for , besides it being rank preserving.Assumption 5(i) requires that the entries of are sufficiently bounded.Assumption 5(ii) states that is asymptotically uncorrelated with U, the error term in Eq. (6).Assumption 5(iii) ensures that, by application of a suitable central limit theorem, Z remains a √ N-consistent estimator for FC and is asymptotically normally distributed with variance as (N, T) → ∞.This assumption is identical to Assumption 2.2 in Robin and Smith (2000), except that it is imposed on Z rather than Z itself.
In practice, there exist several options for that satisfy the rank preservation condition and Assumption 5.One option is to set = T −1/2 , with the entries of drawn from the standard normal distribution.The following theorem confirms that this choice is rank-preserving and satisfies the required consistency and boundedness conditions of Assumption 5.In turn, Assumption 5(iii) is easily seen to hold by application of a CLT.
Theorem 1.Let T > n and be a n × T random matrix with i.i.d.standard normal entries.(i) For a T × n matrix A, it holds that where FC = O p (1).
The proof of Theorem 1 is in Appendix B.
Remark 3.1.An alternative stochastic option would be = Z /T.However, this is ruled out because even though Z = Z Z/T is stochastically bounded and has the same rank as Z, it does not have an asymptotic normal distribution, i.e., it violates Assumption 5(iii).
The matrix can also be deterministic.Since n time periods contain the same information on the rank of C as do T observations, an obvious candidate is = [0 n×(T−n) , I n ], which considers only the last n rows of Z.One can also take averages over every n-th row in Z by setting = 1 T/n [ι T/n ⊗ I n ]I T/n n,T , where ι a is an a × 1 vector of ones.
Given the above, we propose estimating the rank of the n × n matrix Z by sequentially testing the null hypothesis H 0 : = * against the alternative H a : > * , using the following statistic: where λ 1 (A) ≥ • • • ≥ λ n (A) are the ordered eigenvalues of A ≡ ZZ .The procedure is implemented sequentially for * = 0, . . ., n − 1 and the estimated rank corresponds to the smallest value of * for which the null hypothesis is not rejected.Under the null, τ has a limiting distribution which is a weighted sum of independent χ 2 1 variables, with weights given by the (n − * ) 2 largest eigenvalues of (D * ⊗ R * ) (D * ⊗ R * ), where D * and R * denote the eigenvectors corresponding to the n − * smallest eigenvalues of Z Z and A, respectively.We shall assume in accordance with Robin and Smith (2000) The asymptotic variance of Z is unknown but can be estimated consistently by8 : As discussed in Robin and Smith (2000), the estimator for obtained from the test sequence is consistent when the employed significance level α N vanishes at an appropriate rate with N.This is because α N is the probability of over-estimating the true rank, Pr( > ), which must tend to zero for consistency.The authors show that α N = o(1) and −N −1 ln α N = o(1) are sufficient for consistency.This is summarized in the following proposition: The proposition follows from the arguments of Theorem 5.2 in Robin and Smith (2000), mutatis mutandis.We omit the proof to save space.Remark 3.2.Clearly, α N has to vanish sufficiently fast with N to limit the over-estimation frequency, but not too fast, as this results in under-estimation when N is small.We suggest to specify the nominal level as α N = αcN −1/γ .This way, for a given choice of α and γ , the small N significance level is controlled through c > 1, whereas the speed at which α N decreases with N is governed by γ > 0. For instance, choosing α = 5% and setting c = 20, γ = 1 fixes the nominal level to 5% for N = 20 and lets it decrease at rate N. Given that over-estimating may lead to false conclusions that the rank condition holds, we prefer a conservative estimator through a fast decrease with N (i.e., requiring strong evidence against the null before rejecting it in favor of a higher rank estimate).

Consistent estimation of m
Existing methods to estimate the number of factors from observed data rely on one of the following three approaches: looking at differences or ratios of adjacent eigenvalues (Ahn and Horenstein, 2013;Onatski, 2010), specifying threshold functions to separate bounded from unbounded eigenvalues (Alessi et al., 2010;Bai and Ng, 2002) or sequential tests to determine which eigenvalues are unbounded (Kapetanios, 2010;Trapani, 2018).Preliminary simulation evidence conducted for this article shows that the Growth Ratio (GR) by Ahn and Horenstein (2013) performs well in finite samples and outperforms other estimators. 9Therefore, in what follows we propose estimating m using the GR statistic.
In particular, let Z = [Z 1 , . . ., Z N ] denote a T × (K + 1)N matrix, where Z i (defined in Eq. ( 3)) collects all observables for individual i in a T × (K + 1) matrix.Also, let m max denote the maximum value of m considered in estimation, such that m max ≥ m.We define where V(j) = h k=j+1 λ k ZZ /NT with h = min{T, N(K + 1)}, and λ k ZZ /NT denotes the kth largest eigenvalue of ZZ /NT .
The GR statistic is easy to compute because it involves maximizing the "growth ratio" of two adjacent eigenvalues arranged in descending order.The main intuition is that the growth ratios of two adjacent eigenvalues of ZZ /NT are asymptotically bounded, except for the growth ratio involving the mth and (m + 1)th eigenvalues, which diverges to infinity.Remark 3.3.In exactly the same way as described above, the number of factors can also be estimated based on the T × T matrix YY /NT, where Y = [y 1 , . . ., y N ] is of dimension T × N.However, since both y i and X i share the same factors by assumption, it is natural to combine them together in order to increase the information set used to construct proxies for F. This strategy is in line with the rationale behind the CCE approach, which involves solving a system of equations, such that Eq. ( 3) includes LHS variables (observables) that are solely driven by a common factor component and purely idiosyncratic noise.Moreover, this strategy is consistent with Westerlund and Urbain (2015), who also estimate factors based on ZZ /NT.

A consistent classifier for the rank condition
Given consistent estimates and m of the rank of C and the number of factors, the rank condition is deemed to be violated when < m.We define the following classifier: where 1{•} is an indicator function that returns 1 when the argument inside the curly brackets holds true, and 0 otherwise.Hence, if RC = 1 the rank condition is considered to be satisfied, whereas RC = 0 indicates that ( 10) may be violated.The definition in ( 16) shows that we also take > m as a sign that (10) is satisfied.11Let RC ≡ 1 − 1{ < m} denote the indicator of the true state of the rank condition in the population.The following proposition summarizes the asymptotic properties of the classifier.
Proposition 2. Let Assumptions 1-5 hold true.Suppose also that is determined based on the sequential testing procedure outlined in Section 3.1, with α N = o(1), and −N −1 ln α N = o(1), and m is determined by Eq. ( 14) with m max ≥ m.Then, Pr That is, the probability that the classifier correctly identifies whether the rank condition is satisfied or not, converges to unity.The result follows directly from the consistency of as N → ∞ under Assumptions 1-5, given an appropriate rate of decay for α N , and the consistency of m as (N, T) → ∞ given appropriate specification of m max .

What if the rank condition is violated?
When RC = 0, the standard CCE estimator is generally inconsistent unless the regressors are uncorrelated with the unobserved factor loadings.One may seek to restore the RC by augmenting the model with additional CSA (see Appendix A for several options).This brings about two important issues.
The first one is how to choose relevant additional CSA from a set of candidate expansions, as not all candidates are necessarily informative about F. The second one is whether the selected additional CSA are also able to restore the RC.
To tackle the first question, Karabiyik et al. (2019) have proposed an IC selection procedure.To illustrate, let Z + be the matrix of available expansions where (say) Z (1) + = Z w 1 contains a matrix of CSA arising from a new weighting variable w 1 , and similarly Z (3) + = Z w 2 for a weight w 2 (see Appendix A for details).The appropriate set of expansion CSA can be selected from Z + by minimizing where where cols(•) denotes the number of columns of the matrix within brackets and = { 1 , 2 , ...} gathers the indices of the considered expansions from Z + .As such, for (say) = { 1 , 3 }, we have Z (e) , Z w 2 ], and M ( ) A .A desirable property of the IC selection procedure is that it identifies the CSA that bring in new information about the factors in Z i given what is already present in Z. Candidates that are uninformative, or informative on factors that do not feature in Z i , will be excluded (asymptotically).However, the IC does not by itself signal whether the additional CSA are sufficient to restore the RC.For example, if the IC does not select additional CSA besides Z, this could be either because the rank condition is satisfied with Z, or because no further informative CSA are available in the proposal set Z + .To overcome this problem, we propose combining the IC with our RC classifier, as outlined in Algorithm 1.
Remark 3.4.An alternative strategy would be to combine our classifier with the regularization approach proposed by Juodis (2022).The latter makes use of the Singular Value Decomposition in order to remove the asymptotically redundant singular values of appropriately normalized CSA.We leave this possibility for future research.

Algorithm 1: CCE A algorithm
(1) Estimate the model parameters using the standard CCE approach and calculate IC 0 = IC(∅) (no expansions).Proceed to step 2; (2) Evaluate the rank condition for Z.If RC = 1, no further steps are required.If RC = 0, proceed to step 3; (3) Employ the IC in Eq. 19 to select from Z + = {Z The key benefit of Algorithm 1 is that it either returns a consistent CCE estimator or signals to the researcher that alternative CSA need to be sought.If Z or the potential expansions Z + contain sufficient informative CSA to satisfy or restore the RC, they will (asymptotically) be identified and the CCE estimates emerging from Algorithm 1 are consistent.In this case, the algorithm will end in step (5) (or step (2)).If on the other hand Z does not satisfy the RC and Z + does not contain the right (or insufficient) expansions to restore it, then this will be signaled by landing in step (6).The researcher then knows that alternative CSA candidates need to be sought and fed into the algorithm before consistent CCE estimates can be obtained.
Remark 3.5.It is also possible to evaluate the RC for all potential augmentations until RC = 1.However, this strategy bares the risk of selecting CSA that load on different factors than those in Z i , and so they are irrelevant for approximating the factor space.This is because such CSA will increase the rank of the augmented loading matrix, despite being irrelevant, and they will therefore be incorrectly favored by the classifier.A preliminary pass-through by the IC selection, as in Algorithm 1, eliminates such irrelevant options.
Remark 3.6.We refer to Appendix D for a discussion and results pertaining to inference with the augmented CCE estimator after the application of Algorithm 1.

Heterogeneous slopes
The preceding results and theorems apply similarly to heterogeneous random slope models as in Assumption 4 of Pesaran (2006), where β i = β + ν i , ν i ∼ IID(0 K×1 , ν ), with ν nonnegative definite and ν g is independent of f t , ε is , v jl , υ h for all t, i, s, j, l, g, h.Note that in this setting, the first column of However, since ν i is an i.i.d.random variable with finite variance, expectation zero and independent of the other model components, it straightforwardly follows that Assumptions 1-5 listed above directly follow through.In particular, the critical eigenvalue bounds and √ N-consistency of Z for FC remain unaffected.The rank condition can thus be evaluated with the same procedures as outlined above.This is also confirmed by the Monte Carlo simulations reported in Appendix C.

Monte Carlo simulation
In this section, we investigate the small sample performance of the rank condition classifier proposed in Section 3 using Monte Carlo simulations.

Design
Data are generated from Eq. ( 3), broadly following Westerlund and Urbain (2013).We set m = 2, K = 1, β = 3 and sample the time series in F, ε i and V i assuming independent autoregressive processes with a common AR coefficient ρ = 0.8 and normally distributed mean zero innovations with variance (1 − ρ 2 ) for the factors and (1 − ρ 2 )/2 for the idiosyncratic errors.For the factor loadings λ i and i , we specify the following three scenarios: Thus, in Experiment 1 the RC is satisfied for the simple CSA Z ( = m = 2).In Experiment 2, the basic CSA contain some information for estimating the factors ( = 1), yet not sufficient to satisfy the RC.Since the loadings in y i and X i are (perfectly) correlated, the standard CCE estimator is not consistent.In Experiment 3 the standard CSA contain no information at all about the factors ( = 0 < m), in which case consistent CCE estimation is also not possible with Z.
We evaluate the RC in each MC iteration, using Algorithm 1 of Section 3.4.The number of factors (m) is estimated by the GR statistic of Ahn and Horenstein (2013), setting m max = 7.The rank of the loading matrix ( ) is estimated as in Section 3.1 with a random dimension reduction = T −1/2 , containing i.i.d.standard-normal entries, and the nominal significance level given by α N = cαN −1/γ , with c = 20, γ = 1 and α = 5%.
Additional CSA are constructed using the following weighting schemes: which results in CSA calculated over the first (Z w,1 ) and second (Z w,2 ) group of N/2 cross-sectional units.This choice of weights presumes the existence of an exogenous grouping of the cross-sectional units, as in Experiment 2. It is not an appropriate RC-restoring expansion for experiments 1 and 3, as no such grouping exists for these experiments.We also consider candidate CSA originating from the T × 2 matrix of external variables i , where the columns of (e)  i are generated as AR(1) processes with autoregressive coefficient ρ = 0.8 and mean zero normally distributed innovations with variance (1 − ρ 2 )/2, while As the Z (e) i load on the same factors as those in Z i , the matrix i is an informative, RCrestoring, expansion in experiments 2 and 3. We also accommodate in our simulations the fact that in practice not all external variables will load on the same factors as those in Z i .These irrelevant candidates are generated from where the factors G, loadings C i is informative about G but not F, and Z (g) is therefore not an appropriate expansion in any of the considered experiments.The total set of candidate expansions that is fed into Algorithm 1 is thus a mixture of both relevant and uninformative candidates, and is given by In accordance with Algorithm 1, the augmented estimator CCE A selects expansions from Z + using the Information Criterion by Karabiyik et al. (2019) given in Eq. ( 19).The RC is re-evaluated when expansions are selected.We generate 10000 datasets for each combination of N = (20, 50, 100, 200, 500, 1000) and T = (20, 50, 100, 200), and calculate the under/over-estimation frequencies for and m, and the classification accuracy of RC, i.e., the % of MC draws where the RC is correctly evaluated.When the RC is not satisfied for the standard CCE estimator (experiments 2 and 3), we also consider the CCE A estimator and compute the 'RC satisfied rate' as the % of MC draws where Algorithm 1 selects expansions that restore the rank condition.

Estimating and m
Results for the performance of the estimators for and m are presented in Table 1 in A/B format, with A and B the percentage of MC iterations where or m are, respectively, under-and over-estimated.The left panel contains results for estimating the rank of the loading matrix and reveals that both the over-and under-estimation frequencies tend to zero as N → ∞.This is consistent with the main result of the paper that can be estimated consistently from Z.As expected, we find that the rank estimator is somewhat sensitive to the size of the cross-section dimension, which needs to be sufficiently large (i.e., N of at least 50) to achieve an accuracy of 75%.In contrast, its performance is largely invariant to the size of T, which supports the projection strategy to guarantee computability of the estimator and large N consistency when also T → ∞.Note that the rank estimator is conservative in the sense that the true rank is more likely to be under-estimated than over-estimated.This is a consequence of our chosen significance level α N = 20αN −1 , of which its fast decay in N implies that strong evidence against the null = * is required before it is rejected in favor of a higher rank > * .Yet, the observed underestimation frequency is reasonable and vanishes sufficiently fast with N. 2/0 0/0 0/0 = 0, m = 2 50 0/2 0/2 0/2 0/2 7/9 0/0 0/0 0/0 100 0/1 0/1 0/1 0/2 6/7 0/0 0/0 0/0 200 0/1 0/1 0/1 0/1 5/6 0/0 0/0 0/0 500 0/0 0/0 0/0 0/0 5/5 0/0 0/0 0/0 1000 0/0 0/0 0/0 0/0 5/5 0/0 0/0 0/0 Notes: (i) Based on 10, 000 MC iterations.(ii) Reported in the left panel is the percentage of under/over-estimation of the true rank by the rank estimator applied to Z, with = T −1/2 , drawn from the standard-normal distribution α N = 20αN −1 and α = 5%.
(iii) The right panel is the percentage of under/over estimation of the true number of factors m = 2 by the GR estimator with m max = 7.  16) correctly identifies whether the RC is satisfied or not.(iii) The RC classifier uses the GR estimator of Ahn and Horenstein (2013) with m max = 7 to estimate m, and the Robin and Smith (2000) rank estimator with a standard-normal projection matrix and significance level α N = 20αN −1 to estimate .(iv) The left panel evaluates the rank condition for the standard CCE estimator that uses the matrix of CSA Z to control for the unobserved factors.The right panel evaluates the rank condition for the CCE A estimator, which is the outcome of Algorithm 1 presented in Section 3.4.That is, if RC = 1 for Z, then only Z is employed in the estimation.If on the other hand Z yields RC = 0, then expansion CSA are selected from Z + using the IC in (19).
The right panel of Table 1 reports results for estimating the number of factors m = 2 with the GR estimator of Ahn and Horenstein (2013).The estimator performs very well despite the high serial dependence in the generated data, in which case many of its competitors in the literature tend to behave more poorly.The finite sample performance of the GR approach appears to be primarily driven by the time series dimension T. Yet, its small-sample performance is more than adequate as the approach displays low error frequencies even when T = 20, and identifies m without error when T > 50.

Experiment 1: rank condition satisfied
In Experiment 1, the RC is satisfied for the CCE estimator that uses the standard set of CSA in Z.The classification accuracy reported in Table 2 shows that the RC classifier is reasonably accurate in detecting that the rank condition is indeed satisfied.Even for smaller samples, the RC is correctly confirmed for at least 70% of the MC iterations, the only exception being the smallest N = 20 setting where the lowest rate is 64%.As the sample size grows, the accuracy improves and we find that it tends to 1 as both (N, T) → ∞, as required.The results also show that the main determinant for finite sample performance is the cross-section dimension N, rather than T.This is as expected from the results in Table 1, which show that (i) is more prone to finite sample error than m.The latter is practically error-less when T ≥ 50; and (ii) converges at a slower rate and only as N grows.Hence, is the main driver of the finite sample performance of RC.Therefore, in line with the properties of the CCE estimator itself, it will mainly be N that needs to be sufficiently large to be able to correctly assess the rank condition in practice.Finally, note that the samples where we incorrectly obtained RC = 0 for the CCE estimator prompted the application of the augmentation strategy outlined in Algorithm 1 of Section 3.4.As shown in Table C1 in the appendix, an expansion was only selected in the smallest samples and in at most 2% of the MC iterations.Hence, the rank evaluation results for the augmented CCE A estimator reported on the right panel of Table 2 are almost identical to those for the CCE estimator.This also confirms the effectiveness of the IC selector.

Experiment 2: rank condition violated for basic weights
In Experiment 2, the RC is violated when using the standard set of CSA Z, and since factor loadings are (perfectly) correlated, the CCE estimator is inconsistent for β in this setting. 12The left panel of Table 3 shows that the RC-classifier strongly signals that the RC is violated for the CCE estimator.The proportion of samples where the classifier wrongly concludes that the RC holds quickly diminishes as When the RC is found to be violated, Algorithm 1 is applied by letting the IC search among the proposal expansions for additional CSA.In this experiment, this leads to the selection of at least one of the valid augmentations (Z w,1 , Z w,2 , Z (e) ) in the majority of combinations of N and T (see Table C1 in Appendix C).Accordingly, the rank condition was successfully restored in around 90% of the MC iterations, even in the smallest samples T = N = 20.The proportion of samples where the RC is restored is given in the lower panel of Table 3 for the CCE A estimator, and can be seen to converge to 1 as (N, T) → ∞.Hence, Algorithm 1 leads to a consistent CCE A estimator as (N, T) → ∞, when provided with appropriate rank-increasing CSA. 13 Note that the algorithm performs well in finite samples.The cases where the RC is not satisfied for CCE A are mostly due to the miss-classification as RC = 1 in the 'CCE' panel, which vanishes as the sample size grows.2 and 3. CCE A,sub refers to using Algorithm 1, with the set of potential augmentations given by Z In practice, selecting expansion CSA with the IC does not guarantee that the rank condition is also satisfied, leaving the researcher unsure about the state of the RC.Hence, Algorithm 1 incorporates a reevaluation with the classifier after expansions have been chosen.The top right panel of Table 3 reveals that this re-evaluation is able to confirm with good accuracy that the rank condition is satisfied in those cases where the right expansions have been selected.The overall classification accuracy is over 70% in the smallest samples and gradually converges to 1 as (N, T) → ∞.Finite T fluctuations in the table are induced by the estimator of m (which we have demonstrated to be sensitive to T), and the variability of the averages selected by the IC estimator for finite T settings.See to that end the selection probabilities reported in Table C1 of the Appendix.

Experiment 3: rank condition violated
In Experiment 3, the loading matrix C for the standard CSA has rank zero.Intuitively, the effect of the factors is averaged out in Z such that the CSA are uninformative for estimating the factor space.The top panel of Table 4 reveals that our RC evaluation method is highly accurate in this setting even for very small N.This is due to the large discrepancy between m = 2 and = 0. Since we have specified a conservative estimator for , such a large over-estimation error almost never occurred (recall the bottom panel of Table 1).
Given the strong signal by the classifier that the RC is violated, Algorithm 1 in the 'CCE A ' panel has led to a search for expansion CSA in nearly all MC samples.We find that the sole rank-restoring expansion Z (e) was selected with high probability, as indicated by the high proportion of samples for which the RC has been restored (see the bottom panel of Table 4).This again confirms the performance of the IC approach to select only relevant candidates (i.e., loading on the same factors) for expansion.Note, however, that compared to Experiment 2, the classifier appears less capable to confirm that the rank condition is restored when N is very small.Accuracy is only 40% when N = 20.Closer analysis reveals that this is caused by a relatively large under-estimation rate (60%) of the true rank in N = 20 samples when the correct expansion was selected.A possible cause is that the expanded matrix has a potential rank (number of columns = 4) which is twice the true rank (2).This suggests a relatively high level of estimation noise, and a cross-section dimension of N = 20 appears too small to estimate the rank accurately in such cases.Yet, the performance of the estimator improves quickly with N, and classification accuracy recovers to 85% or higher for N = 100.We consider also the empirically relevant scenario where the proposal set Z + does not contain sufficient informative CSA to restore the rank condition.To that end, we report in the CCE A,sub panel of Table 4 the outcomes of Algorithm 1 when the set of proposal expansions is Z +,sub = {Z w,1 , Z w,2 , Z (g) } instead of Z + .Hence, Z +,sub contains insufficient valid expansions to restore the RC, and Algorithm 1 should signal that the RC remains violated even when expansions have been selected from it.It is furthermore important that the IC does not select Z (g) , the CSA that load on factors other than those in Z i , as it would lead to false conclusions that the RC is satisfied by the classifier (see Remark 3.5).This makes the setting particularly challenging.However, the results summarized in the CCE A,sub panel of Table 4 show that the classifier confirms with high accuracy that the RC fails even after expansions were chosen.

Experiments with heterogeneous slopes
To confirm the validity of the RC classifier in settings with heterogeneous slopes, we also run Experiments 1-3 with the slopes in Eq. ( 1) generated as β i = β + ν i , where ν i ∼ N (0, 1).We generally find that slope heterogeneity has little effect on the performance of the classifier, or Algorithm 1, save that the rank estimator incurs a slightly higher error rate due to the additional noise (max +5%).Conclusions are thus practically identical to the common slope results reported above, and we refer to Appendix C.2 for the results in the interest of space.This confirms the consistency of the procedures posited in Section 3.5.

Application: the impact of the Dodd-Frank Act on the profitability of U.S. banks
Studies on the profitability of banking institutions are vital for obtaining better understanding of the causes of financial crises, economic recessions, and growth.Profits constitute the first line of defense against losses from credit impairment, since retained earnings are an important source of capital.When it comes to large banks, high profitability may also signal excessive market power through stronger brand image or implicit regulatory protection; this is the so-called "too-big-to-fail" (TBTF) hypothesis, which postulates that large financial institutions may be so widely interconnected to the rest of the economy that their failure would generate a disastrous domino effect for the whole economy.To the extent that governments effectively subsidize downsize risk for financial institutions with TBTF status, large banks face artificially lower costs of capital, and thus reap more profits.
The present illustration contributes to this literature by examining the impact of the well-known "Dodd-Frank Act" (DFA) on profitability in the U.S. banking sector.The DFA is a U.S. federal law enacted in 2010 that has instituted a new failure-resolution regime, which seeks to ensure that losses resulting from bad decisions by managers are absorbed by equity and debt holders, thus potentially reducing moral hazard.Existing empirical evidence on the extent to which the DFA has alleviated the TBTF is relatively sparse and not in agreement.For example, while Baily et al. (2020) conclude on a positive influence of the DFA towards resolving moral hazard, other studies point in the opposite direction (see e.g., Bordo and Duca, 2018).In what follows, we apply the CCE estimator and rank test methodology developed in the present paper to shed further light on this important topic.

Data and model specification
We make use of a panel data set consisting of 450 U.S. banking institutions over the period 2006:Q1-2019:Q4. 14We analyze the impact of major drivers of bank profitability, with emphasis on bank size.
where i = 1, . . ., N, t = 1, . . ., T, = τ 1 1{t < 2011 : Q1} + τ 2 1{t ≥ 2011 : Q1}.Essentially, the model is estimated for two sub-periods, namely 2006:Q1-2010:Q4 and 2011:Q1-2019:Q4.The first sub-period belongs to the Basel I-II period, whereas the second corresponds to the DFA and coincides with the introduction of the Basel III internationally. 15he variables of the model are defined as follows: ROA it is the return on assets (annualized net income expressed as a percentage of average total assets on a consolidated basis); SIZE it denotes the natural logarithm of bank total assets; CAR it stands for "capital adequacy ratio" (ratio of Tier 1 capital over average total assets minus ineligible intangibles).Higher values of this ratio imply higher levels of capitalization; LIQUIDITY it is proxied by the loan-to-deposit ratio.Higher values imply a lower level of liquidity; QUALITY it is computed as the total amount of loan loss provisions expressed as a percentage of assets; and RISK it denotes the ratio of non performing loans to total loans.Higher values of RISK indicate that banks ex-ante took higher lending risk and therefore they have accumulated ex-post more bad loans.
The error term u it is composite.In particular, η i captures bank-specific effects, such as ownership and location.The m × 1 vector f t denotes unobserved economy-wide factors that influence bank profits, albeit with heterogeneous intensities λ i .Last, ε it is an idiosyncratic error.
The above set of explanatory variables originate from bank accounts (balance sheets and/or profit and loss accounts) and are tied to management decisions.As such, they are viewed as "internal".Bank profitability is also driven by "external" factors that lie beyond the control of management, such as business cycle effects, monetary shocks, and financial innovation.These are absorbed in our model by the common factor component specified in the error term, λ i f t .Although in some cases external drivers can be measured and included directly in the model, often the details of measurement may be difficult and/or contentious. 16e note that internal and external drivers of bank profitability are likely to be mutually correlated.For example, asset quality may depend on the position of the business cycle, since contractionary phases are typically associated with a higher level of default risk.Therefore, standard panel data approaches that fail to control for external drivers are likely to face an endogeneity problem and, hence, to yield inconsistent parameter estimates.The CCE approach allows for consistent estimation, provided that the rank condition is satisfied such that the external drivers are adequately controlled for.
For notational convenience, let Z i denote the T × 6 matrix with the observables where y i = y i1 , . . ., y iT is a T × 1 vector such that y it ≡ ROA it , and similarly for the remaining variables, where x

Evaluating the RC
Before looking at the CCE estimation results, it is important to examine whether the RC is satisfied.
The number of factors m is estimated from the T × (K + 1)N matrix Z = [Z 1 , . . ., Z N ], using the Growth Ratio statistic of Ahn and Horenstein (2013).The rank of the matrices of CSA that we consider, Notes: 'Basel I-II' refers to the 2006:Q1-2010:Q4 period, while 'Dodd-Frank Act' to the 2011:Q1-2019:Q4 period.m is the number of factors estimated from the T × (K + 1)N matrix Z, using the GR statistic of Ahn and Horenstein (2013). is the rank estimator of Robin and Smith (2000), with α N = 20αN −1 and = I T .RC is the rank condition classifier defined in Eq. ( 16).
to be defined shortly, is determined based on the sequential testing procedure of Robin and Smith (2000).Since T is small in both sub-samples, there is no need to reduce the row-dimensionality using a projection matrix.
We start with the standard CCE estimator based on the unweighted CSA Z = 1 N N i=1 Z i .Table 5 reports results for evaluating the RC.The first and second columns correspond to the standard CCE estimator applied to the periods 2006:Q1-2010:Q4 (Basel I-II) and 2011:Q1-2019:Q4 (Dodd-Frank Act).For the period under Basel I-II, m = 3.The standard CSA Z appear unable to proxy these factors as the RC is found to be violated, RC = 0.For the period under the Dodd-Frank Act, we obtain m = 2 and the rank condition now appears to hold for the standard CCE estimator.
Given that the RC is violated for the standard CCE approach in the first sub-period of the sample, we consider a set of potential expansion of CSA, given by

Z
(1) 6) , x (7) ] is a T × 2 matrix, where x (6) and x (7) denote the simple CSA of two external variables, namely the return to equity (ROE), and the tier 1 risk-based capital ratio.ROE is defined as annualized net income expressed as a percent of average total equity on a consolidated basis.The risk-based capital ratio is defined as the tier 1 (core) capital expressed as a percent of risk-weighted assets.As these variables present alternative measures of profitability (y i ) and capitalization (x (2) i ), respectively, they are expected to be driven by the same common factors as those entering into the regression model.Z (2) + and Z (3) + denote T × (K + 1) matrices of weighted CSA, computed from Z i in Eq. ( 23).Z (2) + is calculated using as aggregation weight the initial value of the bank-specific debt ratio (defined as total liabilities over total assets).This variable has been employed in the literature as a measure of interconnectedness of banks (Fernandez, 2011).Thus, banks with similar levels of debt ratio may be hit by common shocks in an alike manner and therefore they take a similar weight in the computation of the CSA of Z i .Z + uses the size of each bank in the beginning of the sample as averaging weight.This implies that banks of similar size get a similar weight in the computation of Z (3) + is a T × 2(K + 1) matrix of CSA, obtained using two weights that are constructed by grouping banks according to their size.In particular, we take CSA over the bottom and the top quintile of banks.
Table 6 reports IC results for each of the suggested additional CSA.Under Basel I-II, where the RC was found to be violated, the IC selects Z (1) + as a relevant expansion.The other expansions (Z do not provide new information about the factor space.In the DFA period none of the expansions are selected, since the RC was already satisfied.
Given the IC results, we consider the augmented CCE A estimator with CSA Z A = [Z, Z + ].Whether this augmented set of CSA is also sufficient to restore the rank condition needs to be verified with the RC classifier.Results are reported in the right panel of Table 5.As we can see, the augmentation has restored the rank condition ( RC = 1) for the first sub-period.As expected, the RC remains satisfied in the second sub-period should we also augment the CCE estimator with Z (1) + .

CCE and CCE A estimation results
Table 7 reports CCE and CCE A estimates for the two sub-periods 2006:Q1-2010:Q4 and 2011:Q1-2019:Q4.The RC evaluation results imply that in the first sub-period the CCE A estimator is consistent, whereas CCE is not.Such discrepancy is mainly noticeable in the estimated coefficients of SIZE and LIQUIDITY.In both cases, the inconsistent CCE appears to overestimate the impact of these variables on bank profitability.For the period 2011:Q1-2019:Q4, RC holds for both CCE and CCE A .Hence, there is no need to augment the model with additional CSA.Notably, the estimated coefficients obtained by the two estimators are not statistically different.
Turning to a comparison of the results across the two sub-periods, SIZE appears to be substantially less important in terms of driving profitability of banks under the DFA period. 17More specifically, the difference between β (τ 1 ) 1 = 0.647 and β (τ 2 ) 1 = 0.267 equals 0.38 and is statistically significant at the 10% level of significance, with a p-value that is roughly equal to 0.061 (one-tailed test). 18That is, if large banks exercised market power and implicitly relied on regulatory protection based on a "too-big-to-fail" presumption, such type of behavior seems to be less prevalent after the introduction of the Dodd-Frank Act.Further, note that if we use the standard CCE estimator in both sub-periods, the difference between β (τ 1 ) 1 and β (τ 2 ) 1 amounts to 0.959 − 0.267 = 0.692.Hence, the impact of the DFA is estimated to be twice as large as that obtained based on our approach.This further highlights the importance of evaluating the rank condition for CCE-type estimators. 17In practice, we note that β ( ) 1 captures not only the effect of market power and implicit regulatory protection via TBTF, but also the effect of economies of scale.Such scale effects will be positive (negative) if there exist economies (diseconomies) of scale.However, to the extent that the degree of returns to scale in the banking sector has remained unaltered during the sampling period of the analysis, the difference in the coefficient of SIZE it between the two sub-periods, i.e., β 1 , will measure the impact of the Dodd-Frank Act on TBTF, conditional on the remaining covariates. 18The t-statistic is t = (0.647 − 0.267)/ 0.196 2 + 0.149 2 = 1.54.Note that since the CCE A and CCE estimates are based on different samples, it is natural to assume that their covariance equals zero.

Conclusion
It is well known that the so-called Rank Condition is crucial for the statistical properties of the CCE approach developed by Pesaran (2006).However, to date this rank condition could not be verified as it relates to the rank of the unobserved matrix of factor loadings.Therefore, in practice the rank condition is typically assumed to hold true.In this article, we have outlined a procedure to evaluate whether the rank condition holds in the model of interest given a chosen set of cross-sectional averages.If the rank condition is found to be violated, the procedure can be applied in an augmentation strategy, which combines our proposed classifier with an Information Criterion, to determine the set of CSA that restores the rank condition.Therefore, our approach is generally applicable for checking whether the chosen cross-section averages are sufficient to satisfy the rank condition, or whether additional variables should be explored.
Consider the row-matrix representation of a product of two matrices: It where 1 and 2 are n 1 × T and (n − n 1 ) × T respectively with rank( 1 ) = n 1 with probability 1.Thus, the product between and M can be written as Based on exactly the same arguments as in Case 1, it can be shown that rank( 1 M 1 ) = n 1 with probability 1.However, since 1 M 1 is a submatrix of M, rank( M) ≥ n 1 .Therefore, we have Hence, rank( M) = n 1 = rank(M) with probability 1.This completes part (i) of the theorem.
By the independence of and U we have for every k = 1, . . ., n that E( T t=1 φ kt u t ) = 0 1×n and Pesaran (2006) under Assumptions 1 and 4. Hence, T −1/2 U = O p (N −1/2 ) as (N, T) → ∞.Consider next T −1/2 F. By the independence of and F we have E ( F) = 0 n×m , and since F = [f 1 , . . ., f T ] we can write the k-th row of F as T t=1 φ kt f t .Therefore, 1) for every t (Assumption 2).Hence, we have Thus, combining the results above yields as (N, T) → ∞, where also Hence, the proof of part (ii) of the theorem is complete.Notes: Reported are percentages out of 10000 Monte Carlo iterations that the CSA stated in the column has been selected as an expansion by the IC given in Eq. ( 19).Since multiple expansions can be selected on each sample size, the percentages do not necessarily sum to 100.The bottom panel displays selection frequencies in Experiment 3 when Z (e) is not a selectable option.That is, the set of proposal expansions is Z +,sub = {Z w,1 , Z w,2 , Z (g) }. #true RC=1 conclusions+#false RC=0 conclusions ), and the rightmost panel gives the 'Specificity' , or the rate of correctly obtaining RC = 0 when the RC is indeed violated/not restored ( #true RC=0 conclusions #true RC=0 conclusions+#false RC=1 conclusions ).Note that when there are no RC = 0 cases and also no RC = 0 conclusions, then Specificity = 1, and similarly for the Sensitivity.The inverse of Sensitivity and Specificity give respectively the false positive (false RC holds conclusions) and false negative rates (false RC violated conclusions).(ii) The RC classifier employs the GR estimator with m max = 7 to estimate m, and the rank estimator employs the random projection with α N = 20αN −1 .(iii) The bottom panel gives outcomes for Algorithm 1 when the rank-restoring expansion CSA Z (e) is not among the set of proposal expansions such that it is impossible to restore the RC.(iv) Note that the classifier Sensitivity/Specificity are not separately reported when evaluating the RC based on Z, because they are identical to the 'Classification Accuracy' reported in the main text.That is, when the RC is satisfied for Z (experiment 1), then Specificity = 1 and Sensitivity equals the classification accuracy reported in table 2. Conversely, when RC is violated for Z, then Sensitivity = 1 and Specificity is the reported accuracy.Notes: Reported are percentages out of 10000 Monte Carlo iterations that the CSA stated in the column has been selected as an expansion by the IC given in Eq. ( 19).Since multiple expansions can be selected on each sample size, the percentages do not necessarily sum to 100.The bottom panel displays selection frequencies in Experiment 3 when Z (e) is not a selectable option.That is, the set of proposal expansions is Z +,sub = {Z w,1 , Z w,2 , Z (g) }.Notes: see notes Table C2.3/1 1/0 0/0 = 0, m = 2 50 0/2 0/2 0/2 0/2 8/10 0/0 0/0 0/0 100 0/1 0/1 0/1 0/1 7/8 0/0 0/0 0/0 200 0/1 0/1 0/1 0/1 6/6 0/0 0/0 0/0 500 0/0 0/0 0/0 0/0 6/6 0/0 0/0 0/0 1000 0/0 0/0 0/0 0/0 5/5 0/0 0/0 0/0 Notes: (i) Based on 10000 MC iterations.(ii) Reported in the left panel is the percentage of under/over-estimation of the true rank by the rank estimator applied to Z, with = T −1/2 , drawn from the standard-normal distribution α N = 20αN −1 and α = 5%.

C.2.1. Evaluating the rank condition
(iii) The right panel is the percentage of under/over estimation of the true number of factors m = 2 by the GR estimator with m max = 7.  16) correctly identifies whether the RC is satisfied or not.(iii) The RC classifier uses the GR estimator of Ahn and Horenstein (2013) with m max = 7 to estimate m, and the Robin and Smith (2000) rank estimator with a standard-normal projection matrix and significance level α N = 20αN −1 to estimate .(iv) The left panel evaluates the rank condition for the standard CCE estimator that uses the matrix of CSA Z to control for the unobserved factors.The right panel evaluates the rank condition for the CCE A estimator, which is the outcome of Algorithm 1 presented in Section 3.4.That is, if RC = 1 for Z, then only Z is employed in the estimation.If on the other hand Z yields RC = 0, then expansion CSA are selected from Z + using the IC in (19).  2 and 3. CCE A,sub refers to using Algorithm 1, with the set of potential augmentations given by Z +,sub = {Z w,1 , Z w,2 , Z (g) } instead of Z + .

Table 1 .
Under/over-estimation frequency of the estimators for and m. m
Notes: (i) Based on 10, 000 MC iterations.(ii) Reported is the Classification Accuracy (CA), which is the proportion of MC samples in which the classifier RC defined in Eq. (

Table 3 .
Evaluating the rank condition: Experiment 2.See notes to Table2.The 'RC satisfied rate' is the % of MC samples in which the algorithm behind CCE A selects CSA augmentations that restore the rank condition.

Table 5 .
US bank profitability: Evaluating the rank condition.

Table 6 .
US bank profitability: IC for additional CSA.

Table 7 .
US bank profitability: CCE and CCE A estimation results

Table C
Notes: (i) Reported in the left panel is the fraction of MC samples where the rank condition ( = m) is satisfied (restored) after application of Algorithm 1.The middle panel displays the 'Sensitivity'of the RC classifier, or the rate of correctly obtaining RC = 1 for the cases where the RC is satisfied/restored ( #true RC=1 conclusions

Table C .
3. Estimation results for β in Experiment 1. Reported are estimation bias for β and root mean square error (rmse).
Note: Reported are estimation bias for β and root mean square error (rmse).Note: Reported are estimation bias for β and root mean square error (rmse).CCE A,sub denotes the outcome of Algorithm 1 with the set of potential augmentations given by Z +,sub = {Z w,1 , Z w,2 , Z (g) } instead of Z + .

Table C .
8. Under/over-estimation frequency of the estimators for and m (heterogeneous slopes).

Table C .
9. Evaluating the rank condition: Experiment 1 (heterogeneous slopes).Based on 10000 MC iterations.(ii) Reported is the Classification Accuracy (CA), which is the proportion of MC samples in which the classifier RC defined in Eq. (

Table C .
10. Evaluating the rank condition: Experiment 2 (heterogeneous slopes).See notes to Table2.The 'RC satisfied rate' is the % of MC samples in which the algorithm behind CCE A selects CSA augmentations that restore the rank condition.