Consistent Estimation of the Number of Dynamic Factors in a Large N and T Panel

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1 Consistent Estimation of the Number of Dynamic Factors in a Large N and T Panel Dante AMENGUAL Deartment of Economics, Princeton University, Princeton, NJ (amengual@rinceton.edu) Mark W. WATSON Woodrow Wilson School, Princeton University, Princeton, NJ (mwatson@rinceton.edu) Bai and Ng roosed a consistent estimator for the number of static factors in a large N and T aroximate factor model. This article shows how the Bai Ng estimator can be modified to consistently estimate the number of dynamic factors in a restricted dynamic factor model. The modification is straightforward: The standard Bai Ng estimator is alied to residuals obtained by rojecting the observed data onto lagged values of rincial-comonents estimates of the static factors. KEY WORDS: Aroximate factor model; Bai Ng estimator; Dynamic factor model. Downloaded by Princeton University at 12:52 25 June IRODUCTION Panel datasets with large time series dimension (T) and cross-sectional dimension (N) are being increasingly used in macroeconomics for both forecasting and structural analysis. Often, these data are analyzed in the context of an assumed latent factor structure of the form X t = F t + e t (1.1) for t = 1,...,T, where X t denotes an N 1 vector of observed variables, F t is an r 1 vector of latent factors, is a matrix of coefficients, and e t is a vector of errors. When the elements of e t have weak cross-sectional and serial correlation, the factors F t summarize the imortant cross-covariance roerties of the variables. A question of fundamental interest is the number of latent factors, r, that are required in (1.1). Significant rogress on addressing this roblem was made in Bai and Ng (2002) who roosed consistent estimators of r based on a enalized least squares objective function associated with the classic rincialcomonents estimator. However, in dynamic models, it is imortant to differentiate between the number of static factors (necessary to fit the covariance matrix of X) and the number of dynamic factors (necessary to fit the sectral density matrix of X). Whereas the Bai Ng estimator was develoed to estimate the number of static factors, this article shows that it can be easily modified to consistently estimate the number of dynamic factors. Dynamics can be incororated in the model by assuming that F t evolves as a VAR: F t = i F t i + ε t, (1.2) with innovations ε t that can be reresented as ε t = Gη t, where G is r q with full column rank and η t is a sequence of shocks with mean 0 and covariance matrix ηη = I q ; η t is the vector of dynamic factor shocks. Several articles show how (1.1) and (1.2) can be derived from a restricted version of a general dynamic factor model driven by q dynamic factors; in this case, F t contains linear combinations of current and lagged values of the dynamic factors and (1.1) (1.2) is analogous to the comanion form reresentation of the dynamic factor model. See Bai and Ng (2005, 2007), Forni, Hallin, Lii, and Reichlin (2005), Giannone, Reichlin, and Sala (2004), and Stock and Watson (2005, 2006). To see how the Bai Ng estimator might be used to estimate the number of dynamic factors, q, substitute (1.2) into (1.1) to obtain Y t = Ɣη t + e t, (1.3) where Y t = X t if t i and Ɣ = G. Thus, Y t can be reresented as a factor model with q factors that corresond to the common shocks η t.werey t observed data, q could be consistently estimated by alying the Bai Ng estimator to Y t. This is infeasible because Y t deends on unknown arameters and lags of the unobserved factors. This article studies the consistency roerties of the Bai Ng estimator alied to Ŷ t = X t ˆ i ˆF t i, where ˆ i is an estimator of i and ˆF t i is an estimator of F t i. The analysis roceeds in two stes. In the first ste, the Bai Ng estimator is shown to remain consistent if the estimation error Ŷ t Y t is sufficiently small (secifically T n (Ŷ it Y it ) 2 = O max(n, T)). The second ste shows that the rincialcomonents estimator of F and feasible estimators of yield estimators Ŷ it that achieve this degree of accuracy. Together these results yield a feasible consistent estimator of the number of dynamic factors. The estimator studied in this article was roosed in Stock and Watson (2006) and alied to the roblem of estimating the number of dynamic factors in a large anel of U.S. macroeconomic time series. Stock and Watson (2006) did not study the consistency roerties of the estimator, and that is the urose of the resent article. Other estimators have also been roosed and used in alied work. Notably, Forni et al. (2000) suggested informal methods based on the relative size of eigenvalues from the estimated sectral density matrix for X; related methods have been roosed and alied in the emirical analysis of Forni, Lii, and Reichlin (2003), Giannone et al. (2004) and 2007 American Statistical Association Journal of Business & Economic Statistics January 2007, Vol. 25, No. 1 DOI /

2 92 Journal of Business & Economic Statistics, January 2007 Downloaded by Princeton University at 12:52 25 June 2012 elsewhere, and Hallin and Liška (2006) showed how a consistent estimator of q can be constructed from the estimated sectrum. Bai and Ng (2007) roosed an estimator for q based on the residual covariance matrix of the VAR in (1.3) estimated using the rincial-comonents estimator of F t and showed that the estimator is consistent. Section 3 studies the relative erformance of various consistent estimators using a simulation study. More generally, the lan of this article is as follows. Section 2 briefly summarizes the Bai Ng estimator, shows the estimator remains consistent when alied to data contaminated with a small amount of measurement error, and uses this result to show that the Bai Ng estimator alied to Ŷ is a consistent estimator of the number of dynamic factors. A Monte Carlo study is resented in Section 3 to gauge the erformance of the estimator, Section 4 contains some concluding remarks, and the Aendix includes the roofs to the results given in Section ASSUMPTIONS AND ASYMPTOTIC RESULTS 2.1 Review of Existing Work With a Small Extension We begin by reviewing results for the model (1.1) under a standard set of assumtions. Transosing (1.1) and stacking the T equations yields X = F + e, (2.1) where X is T N, F is T r, is N r, and e is T N. The tth rows of X, F, and e are X t, F t, and e t ;theith row of is λ i ;theith element of X t is X it and similarly for e it, so that X it = λ i F t + e it. Asymtotic roerties of various statistics generated by this model have been studied in Stock and Watson (2002), Bai and Ng (2002), Bai (2003), and Bai and Ng (2005, 2007) under a similar set of assumtions. The focus is on datasets in which both N and T are large, so that the asymtotics assume that N, T jointly equivalently that N = N(T) with lim T N(T) =. The minimum value of N and T lays an imortant role in the analysis and this value is denoted by s = min(n, T). The remaining assumtions concern moments and deendence roerties of the variables; for the uroses of this article, the following assumtions suffice: (A1) E(F t F t ) = I r. (A2) E(λ i λ i ) =, where is a diagonal matrix with elements σ ii >σ jj > 0fori < j. (When is deterministic, is interreted as the limiting emirical average.) (A3) T F t F t I r. (A4) N λ i λ i. (A5) () 1 T e 2 it σe 2 > 0. (A6) For some integer m 2 and for all integers j m, E trace(ee ) j =O( maxn, T j 1 ). (A7) E T Ts=1 ( N λ i F te is ) 2 = O( 2 ). (A8) E T N λ i λ ie 2 it = O(). (A9) E N T F t e it 2 = O(). Assumtions (A1) (A5) rule out exlosive or trending behavior in both the time series and cross-sectional dimensions; the articular values of E(F t F t ) and E(λ iλ i ) listed in (A1) and (A2) are normalizations (because F t = HH 1 F t for arbitrary H), and assumtion (A5) rules out degenerate cases in which the factors exlain all of the variance of the X it s. Assumtion (A6) limits the variability and deendence in the errors e it.forj = 1, it imlies that T E(e 2 it ) = O(); forj = 2, it imlies that Nj=1 ( T e it e jt ) 2 = T Tτ=1 ( N e it e iτ ) 2 = O ( maxn, T), and so forth for larger values of j. Assumtions (A7) (A9) limit the deendence across elements of, F, and e. Imortantly, all of these assumtions hold for sequences of iid random variables with the aroriate number of moments, and assumtions (A6) (A9) can be interreted as relaxing the iid assumtion to allow weak deendence. The Bai Ng estimators of r are based on enalized least squares objective functions. The enalty function deends on a deterministic function g(n, T) that satisfies g(n, T) 0 and s δ g(n, T) for δ = (m 1)/m, where m is given in assumtion (A6). The least squares objective function is conveniently written in terms of the eigenvalues of the XX moment matrix. Let ω i denote the ith largest eigenvalue of () 1 XX and consider the least squares roblem: min {λ k i }{F k t } () 1 N T (X it λ k i Fk t )2, where λ k i and F k t are arbitrary k 1 vectors. The usual rincial-comonents calculations imly that the average redicted sum of squares associated with the least squares solution is given by R(k, X) = k ω i. Letting ˆσ X 2 = () 1 T Xit 2 denote the average total sum of squares, the enalized average sum of squared residuals is PC(k, X) =ˆσ X 2 R(k, X) + kg(n, T), and the Bai Ng PC estimator is BN PC (X) = arg min PC(k, X). (2.2) 0 k r max Letting IPC(k, X) = lnˆσ X 2 R(k, X)+kg(N, T), thebai Ng IPC estimator is BN IPC (X) = arg min 0 k r max IPC(k, X), (2.3) where r max is a finite constant that satisfies r r max. Consistency of the Bai Ng estimator is given in the following lemma. Lemma 1 (Bai Ng). Under assumtions (A1) (A9), BN PC (X) r and BN IPC (X) r. As discussed in the last section, we will study consistency of the Bai Ng estimators alied to variables measured with error (Ŷ t in the notation of the last section). The following result shows that the Bai Ng estimators remain consistent in the resence of sufficiently small measurement error. Lemma 2. Suose (A1) (A9) are satisfied and X = X + b, where N T b 2 it = O (s 1 ). Then BN PC ( X) r and BN IPC ( X) r. Bai and Ng (2002) showed consistency of BN PC (X) and BN IPC (X) for δ = 1 using assumtions like those in (A1) (A9), but without (A6). However, there was an error in their roof. As shown in their errata, their roof is valid using a stronger condition on e. In articular, for e = RξH, where R and H are N N and T T matrices with bounded eigenvalues, ξ is required to be a T N matrix of indeendently distributed random variables with mean 0 and bounded seventh moments.

3 Amengual and Watson: Estimation of the Number of Dynamic Factors 93 Downloaded by Princeton University at 12:52 25 June Consistent Estimation of the Number of Dynamic Factors The results from Lemma 2 suggest that the estimators BN PC (Ŷ) and BN IPC (Ŷ) will be consistent for the number of dynamic factors if the error Ŷ Y is small. We consider two versions of Ŷ that are sufficiently accurate for this urose. Both rely on a first-stage estimate of F. Thus, let ˆF and ˆ denote the rincial-comonents estimators of F and constructed from (2.1) using a consistent estimator of r. Let ( ˆ 1, ˆ 2,..., ˆ ) denote the ordinary least squares (OLS) estimators from the regression of ˆF t onto ( ˆF t 1,..., ˆF t ).The first version of Ŷ is = X t ˆ ˆ i ˆF t i. (2.4) Ŷ A t The second version of Ŷ uses direct estimates of the regression of X t onto lags of ˆF t.let( ˆ OLS 1, ˆ OLS 2,..., ˆ OLS ) denote the OLS estimators from the regression of X t onto ( ˆF t 1,..., ˆF t ). The second version of Ŷ is Ŷt B = X t ˆ OLS i ˆF t i, (2.5) which does not imose the cross-equation constraint i = i. Consistency of the Bai Ng estimator for q is then readily shown if (1) the factor model (1.3) for Y satisfies the analogs of conditions (A1) (A9), and (2) the estimators ˆF, ˆ, ˆ, and ˆ OLS are sufficiently accurate. Thus, to begin, assume that (A1) (A9) hold with η relacing F and Ɣ relacing. Note that the normalization in (A1) (A2) can be achieved by aroriate choice of G. Stock and Watson (2002) and Bai (2003) discussed the accuracy of the estimators ˆF and ˆ under assumtions like those listed as (A1) (A9). As in Bai and Ng (2005, 2007), ˆ will be T 1/2 consistent under a standard set of assumtions for the VAR for F t : (A10) Let F t = (F t 1,...,F t ). Then 1. The stochastic rocess {F t } is stationary and ergodic. 2. E(F t F t ) is nonsingular. 3. vec(f t η t ) is a martingale difference sequence with finite second moments. Finally, accuracy of ˆ OLS requires the additional assumtion: (A11) E N T F t e it 2 = O(). We then have the following result. Theorem 1. Consider the model (1.1) (1.3). Suose that (1.1) satisfies (A1) (A9), that the analogous assumtions are satisfied for (1.3), and that (A10) is satisfied. Then (a) BN PC (Ŷ A ) q and BN IPC (Ŷ A ) q. (b) In addition, suose that (A11) is satisfied. Then BN PC (Ŷ B ) q and BN IPC (Ŷ B ) q. The remaining ingredient in the testing roblem is, the number of lags in the VAR. It is straightforward to show that, under the usual VAR assumtions, can be estimated consistently by the Bayesian information criterion (BIC). In some models, innovations in a subset of the X t variables may deend on only a subset of the dynamic shocks η t.forexamle in Bernanke, Boivin, and Eliasz (2005) and Stock and Watson (2006), X t is artitioned into a set of slow moving variables and other variables, X t = (Xt slow Xt other ) where innovations in Xt slow deend on only a subset of the η t. It is straightforward to show that the size of this subset can be consistently estimated (N slow, T ) using BN IPC alied to the relevant subset of elements of Ŷ. 3. COMPARING THE ESTIMATORS USING SIMULATED DATA 3.1 Exerimental Design The exerimental design is taken from Bai and Ng (2007) where four data-generating rocesses (DGPs) are considered. DGPs. In the first design (DGP1), X it = λ i F t + e it and F t = F t 1 + Gη t, where F t is 5 1 and η t is 3 1, so that r = 5 and q = 3; {λ i }, {e it }, and {η t } are mutually indeendent, with {λ i } and {η t } iid standard normal random variables/vectors; is a diagonal matrix with elements (.2,.375,.55,.725,.90), and the columns of G are randomly chosen from the unit shere and are indeendent of the other random variables. To allow crosssectional deendence in the idiosyncratic errors, e t is N(0, ), where ij = ρ i j. Results are resented for ρ = 0 and ρ =.5. DGP2 is the same as DGP1, but with r = 3 and =.5 I 3. In the final two designs, X t is a moving average of factors f t that follow an autoregressive (AR) (DGP3) or moving average (MA) (DGP4) rocess. In DGP3, X it = (λ i0 + λ i1 L) f t + e it and f t = φf t 1 + η t, where f t is 2 1, so that r = 4 and q = 2. This model can be written as (1.1) and (1.2) with F t = (f t f t 1 ), λ i = (λ i0 λ i1 ), = φ 0 I 2 0, and G =I The factor loadings and errors are generated as in DGP1, and φ =.5 I 2. In DGP4, X it = (λ i0 +λ i1 L+λ i2 L 2 ) f t +e it and f t = (I 2 + L)η t, where f t is 2 1, so that r = 6 and q = 2. In this design, F t = (f t 2 ), but now F t follows a MA rocess, so that the VAR in (1.2) serves as an aroximation. The MA coefficient matrix is diagonal with elements.2 and.9. Estimators. The BN IPC estimators are imlemented using the enalty factor g(n, T) = ln(s )/A, where A = / (N + T). (This is the IPC2 enalty factor in Bai and Ng 2002.) r is estimated using BN IPC (X), where X is the standardized version of the data generated by DGP1 DGP4, and where r max = 10. q is estimated using BN IPC (Ŷ A ) and BN IPC (Ŷ B ) constructed using two lags of ˆF t for both secifications. Two alternative estimators, ˆq 3 and ˆq 4 from Bai and Ng (2007), were also constructed. These estimators use the eigenvalues of the residual covariance matrix of the VAR for ˆF t to estimate q. Secifically, let ˆε t = ˆF t ˆ i ˆF t i, where ˆF t is the ˆr 1 vector of factors estimated by rincial comonents using the normalization ˆ ˆ = Iˆr and ˆF ˆF = t f t 1 f diag( ˆσ ii ), ˆ ˆεˆε = T ˆε t ˆε t denote the estimated covariance matrix, and the ordered eigenvalues of ˆ ˆεˆε are denoted by c 1 c 2 cˆr.letd 1,k =c 2 k+1 / ˆr c 2 i 1/2 and D 2,k = ˆr i=k+1 c 2 i / ˆr c 2 i 1/2 ; the estimators are ˆq 3 = min k k : D 1,k < m/s 2/5, and ˆq 4 = min k k : D 2,k < m/s 2/5, where m is a ositive constant. Following Bai and Ng (2007), we imlement these estimators using m = 1.0.

4 94 Journal of Business & Economic Statistics, January Results Results are shown in Table 1 for each DGP and various values of N and T = 100. Panel A shows results with ρ = 0 (so that the e it errors are mutually uncorrelated), and anel B shows results with ρ =.5 (so that e it are correlated in the cross section). Looking first at anel A, five results stand out. First, the estimators are quite accurate for N as small as 50, at least for the simle designs considered. All of the estimators roduce the correct answer in more that 98% of the simulations when N = 50 and erform nearly as well when N = 40. Second, the constraint i = i used by BN IPC (Ŷ A ) but ignored by BN IPC (Ŷ B ) is useful: BN IPC (Ŷ A ) has a smaller root mean squared error than BN IPC (Ŷ B ) in all of the cases considered in the table. Third, for DGP1 and DGP2, BN IPC (Ŷ A ) achieves a higher roortion of correct values of q than the other estimators; for DGP3 and DGP4, ˆq 3 achieves the highest roortion of correct values. Fourth, in DGP1, while FF has rank 5, two of its eigenvalues are small and BN IPC (X) tends to underestimate the number of static factors when N is large. In site of Table 1. Simulation Results: cov(e it,e jt ) = ρ i j Downloaded by Princeton University at 12:52 25 June 2012 BN IPC (Ŷ A ) BN IPC (Ŷ B ) ˆq 3 ˆq 4 BN IPC (X) N T <q =q >q RMSE <q =q >q RMSE <q =q >q RMSE <q =q >q RMSE <r =r >r Panel A: ρ = 0 DGP1: X it = λ i F t + e it ; F t = F t 1 + Gη t ; r = 5, q = DGP2: X it = λ i F t + e it ; F t = F t 1 + Gη t ; r = 3, q = DGP3: X it = (λ i0 + λ i1 L) f t + e it ; f t = f t 1 + η t ; r = 4, q = DGP4: X it = (λ i0 + λ i1 L + λ i2 ) f t + e it ; f t = (I + L)η t ; r = 6, q = Panel B: ρ =.5 DGP1: X it = λ i F t + e it ; F t = F t 1 + Gη t ; r = 5, q = DGP2: X it = λ i F t + e it ; F t = F t 1 + Gη t ; r = 3, q = DGP3: X it = (λ i0 + λ i1 L) f t + e it ; f t = f t 1 + η t ; r = 4, q = DGP4: X it = (λ i0 + λ i1 L + λ i2 ) f t + e it ; f t = (I + L)η t ; r = 6, q = NOTE: The first two columns show the values of N and T used in the simulations. The next four columns summarize the results for the estimator BN IPC (Ŷ A ); the columns labeled <q, =q, and >q show the fraction of estimates that were less than, equal to, and greater than q; the column labeled RMSE is the root mean squared error of the estimates. The same entries are rovided for the other estimators of q. The final three columns summarize the results for the estimates of r. Results are based on 5,000 simulations.

5 Amengual and Watson: Estimation of the Number of Dynamic Factors 95 Downloaded by Princeton University at 12:52 25 June 2012 this, BN IPC (Ŷ a ) and BN IPC (Ŷ b ) accurately estimate the number of dynamic factors. Finally, comaring the results from DGP3 and DGP4, the AR aroximation for DGP4 does not aear to lead to a serious deterioration of erformance in any of the estimators. Panel B shows that the erformance of the BN IPC deteriorates when there is cross-sectional correlation in the errors: BN IPC (X) tends to overestimate r, the number of static factors, and, while not as severe, this uward bias is also evident in BN IPC (Ŷ a ) and BN IPC (Ŷ b ). ˆq 3 and ˆq 4 suffer only a small deterioration in accuracy. Both BN IPC (Ŷ) and ˆq rovide accurate estimates of the number of dynamic factors when N = SUMMARY AND CONCLUDING REMARKS This article has roosed a modification of the Bai Ng (2002) estimator and shown that the modification rovides a consistent estimator for the number of dynamic factors in an aroximate dynamic factor model. The modification uses a result (Lemma 2) that shows that the Bai Ng estimator remains consistent even when the data are contaminated with a suitably small amount of error. This result may rove useful in other settings, for examle, in models in which the equation for X it has the form X it = λ i F t + β i Z it + e it, where Z it are observed regressors and β i must be estimated. We leave these calculations for future work. ACKNOWLEDGMES This research is an outgrowth of joint work with Jim Stock, who we thank for his comments and suggestions. Thanks also to Jushan Bai, Serena Ng, two referees, and the associate editor for their useful comments. This work was funded in art by NSF grant SBR APPENDIX: PROOFS This aendix summarizes key details of roofs to the results given in the text. A comlete set of roofs is given in the detailed aendix (D-Aendix hereafter) available at htt:// mwatson. Proof of Lemma 1 This is a version of theorem 1 and corollary 1 in Bai and Ng (2002) under slightly different assumtions. See D-Aendix for a detailed roof using the assumtions listed reviously. Proof of Lemma 2 Let ω k denote the kth ordered eigenvalue of () 1 X X.As shown in D-Aendix, Lemma 2 is imlied by (1) ω k ω k = o (1) for k r and (2) ω k ω k = O (s δ ) for k > r. Toverify (1) and (2), let µ denote the largest eigenvalue of () 1 bb. Then ω k + µ 2(ω k µ) 1/2 ω k ω k + µ + 2(ω k µ) 1/2 (A.1) follows from Horn and Johnson (1991, thm ). By the assumtion of the lemma, trace(bb ) = O (s 1 ), so that µ = O (s 1 ). For k r, ω k σ kk (D-Aendix R11), so that ω k ω k = o (1) for k = 1,...,r follows from (A.1), and this shows (1). For k > r, ω k = O (s δ ) (D-Aendix R28); thus, (A.1) imlies ω k ω k = O (s 1 ) + O (s (1+δ)/2 ), and this shows (2). Proof of Theorem 1 Let = ( 1, 2,..., ), =, and F t = (F t 1,..., F t ), so that F t = F t +Gη t and Y t = X t F t.letπ i denote the ith row of and γ i denote the ith row of Ɣ. Then X it = η t γ i +F t π i +e it. The following results are versions of theorem 1 in Bai and Ng (2002) (see D-Aendix): ˆF t J F t 2 = O (s 1 ), J a nonsingu- where J is an r r matrix that satisfies J lar matrix, ˆF t J F t 2 = O (s 1 ), J a non- where J is a (r) (r) matrix that satisfies J singular matrix, The following lemma is useful. ˆλ i J 1 λ i 2 = O (s 1 ). (A.2) (A.3) (A.4) Lemma 3. Let ˆπ i denote an estimator of π i and b it = ˆF t ˆπ i F t π i.if N ˆπ i J 1 π i 2 = O (s 1 ), then T 1 T N b 2 it = O (s 1 ). Proof. Write ˆF t = J F t + ( ˆF t J F t ) and ˆπ i = J 1 π i + ( ˆπ i J 1 π i), so that b it = F t J ( ˆπ i J 1 π i)+( ˆF t J F t ) J 1 π i + ( ˆF t J F t ) ( ˆπ i J 1 π i). Thus, + + N b 2 it F t 2 J 2 ˆF t J F t 2 ˆF t J F t 2 J 1 2 ˆπ ii J 1 π i 2 π i 2 ˆπ i J 1 π i 2, and the result follows from T F t 2 = O (1), (A10), J 2 J 2 <, N ˆπ i J 1 π i 2 = O (s 1 ) (assumtion of the lemma), and T ˆF t J F t 2 = O (s 1 ) from (A.3).

6 Downloaded by Princeton University at 12:52 25 June Journal of Business & Economic Statistics, January 2007 Part (a) of Theorem 1. The feasible OLS estimator of is T 1 ˆ = ˆF t ˆF t ˆF t ˆF t 1 ˆF t ( ˆF t J F t ) J 1 π i. + ˆF t η t γ i + ˆF t e it. Using F t = F t + Gη t, ˆF t can be written as ˆF t = J J 1 ˆF t + J Gη t + ( ˆF t J F t ) Thus, ˆ J J 1 = J G J J 1 T 1 η t ˆF t + T 1 J J 1 ( ˆF t J F t ). ( ˆF t J F t ) ˆF t ( ˆF t J F t ) ˆF t ˆF t ˆF t 1. Straightforward calculations (see D-Aendix) show that each of the terms T η t ˆF t, T 1 T ( ˆF t J F t ) ˆF t, and T ( ˆF t J F t ) ˆF t are O (s 1/2 ) and that T 1 T ˆF t ˆF t JE(F t F t )J, which is nonsingular. Thus, ˆ J J 1 = O (s 1/2 ). To comlete the roof, let ˆπ i = ˆ ˆλ i and write ˆλ i = J 1 λ i + (ˆλ i J 1 λ i) and ˆ = J J 1 + ( ˆ J J 1 ), so that ˆπ i J 1 π i = J 1 J (ˆλ i J 1 λ i) + ( ˆ J J 1 ) J 1 λ i + ( ˆ J J 1 ) (ˆλ i J 1 λ i). Thus, N ˆπ i J 1 π i 2 = O (s 1 ) follows from ˆ J J 1 = O (s 1/2 ) and N ˆλ i J 1 λ i 2 = O (s 1 ). Part (a) then follows from Lemma 3. Part (b) of Theorem 1. The feasible OLS estimator of π i is 1 ˆπ OLS i = ˆF t ˆF t ˆF t X it. Using X it = F t π i + η t γ i + e it = ˆF t J 1 π i ( ˆF t J F t ) J 1 π i + η t γ i + e it, 1 ˆπ i OLS J 1 π i = ˆF t ˆF t Straightforward calculations (see D-Aendix) show that each of the terms N T ˆF t ( ˆF t J F t ) J 1 π i 2, N T ˆF t η t γ i 2, and N T ˆF t η t γ i 2 are O (s 1 ) and that T 1 T ˆF t ˆF t JE(F t F t )J, which is nonsingular. Thus, N 1 N ˆπ i OLS J 1 π i 2 = O (s 1 ) and the result follows from Lemma 3. Received October Revised July REFERENCES Bai, J. (2003), Inferential Theory for Factor Models of Large Dimensions, Econometrica, 71, Bai, J., and Ng, S. (2002), Determining the Number of Factors in Aroximate Factor Models, Econometrica, 70, (2005), Confidence Intervals for Diffusion Index Forecasts and Inference for Factor-Augmented Regressions, unublished manuscrit, University of Michigan. (2006), Determining the Number of Factors in Aroximate Factor Models, Errata, available at htt:// ngse/aers/ correctionecta2.df. (2007), Determining the Number of Primitive Shocks in Factor Models, Journal of Business & Economic Statistics, 25, Bernanke, B. S., Boivin, J., and Eliasz, P. (2005), Measuring the Effects of Monetary Policy: A Factor-Augmented Vector Autoregressive (FAVAR) Aroach, Quarterly Journal of Economics, 120, Forni, M., Hallin, M., Lii, M., and Reichlin, L. (2000), The Generalized Factor Model: Identification and Estimation, The Review of Economics and Statistics, 82, (2005), The Generalized Dynamic Factor Model: One-Sided Estimation and Forecasting, Journal of the American Statistical Association, 100, Forni, M., Lii, M., and Reichlin, L. (2003), Oening the Black Box: Structural Factor Models versus Structural VARs, CEPR Discussion Paer Giannone, D., Reichlin, L., and Sala, L. (2004), Monetary Policy in Real Time, in NBER Macroeconomics Annual 2004, No. 19, eds. M. Gertler and K. Rogoff, MIT Press, Hallin, M., and Liška, R. (2006), The Generalized Dynamic Factor Model: Determining the Number of Factors, Journal of the American Statistical Association, to aear. Horn, R. A., and Johnson, C. R. (1991), Toics in Matrix Analysis, Cambridge, U.K.: Cambridge University Press. Stock, J. H., and Watson, M. W. (2002), Forecasting Using Princial Comonents From a Large Number of Predictors, Journal of the American Statistical Association, 97, (2005), Imlications of Dynamic Factor Models for VAR Analysis, NBER Working Paer w (2005), Macroeconomic Forecasting Using Many Predictors, in Handbook of Economic Forecasting, eds. G. Elliott, C. Granger, and A. Timmerman, North Holland,