Estimating the Number of Common Factors in Serially Dependent Approximate Factor Models

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1 Estimating the Number of Common Factors in Serially Dependent Approximate Factor Models Ryan Greenaway-McGrevy y Bureau of Economic Analysis Chirok Han Korea University February 7, 202 Donggyu Sul University of Texas at Dallas Abstract A simple data-dependent ltering method is proposed before applying the Bai-Ng method to estimate the factor number. The asymptotic justi cation is provided and the nite sample performance is examined. Keywords: Factor Model, Prewhitening, LSDV lter, Factor Number Estimation, Cross Section Dependence JEL Classi cation: C33 Motivation and Empirical Example In practice, factor number estimation can be di cult in panels that exhibit persistency. Intuitively, when either the cross sectional units (N) or the time series observations (T ) are not su ciently large, weak dependence in the idiosyncratic component may be misinterpreted as dependence due to the factor structure, resulting in over- tting. Moreover, rst-di erencing (a typical treatment) can induce negative serial dependence, also leading to overestimation. This paper suggests that in these situations estimation can be sharpened by applying a datadependent lter to the panel before applying a selection criteria, and we provide the theoretical justi cation of the ltering procedure. First, however, we provide an empirical example illustrating the bene ts of data-dependent ltering using the Bai-Ng selection criteria. The views expressed herein are those of the authors and not necessarily those of the Bureau of Economic Analysis or the Department of Commerce. The authors thank Yoosoon Chang and H. Roger Moon for helpful comments. y Corresponding Author: r.greenaway-mcgrevy@bea.gov. Address: 44 L St. NW, BE-40, Washington, DC Phone: + (202) Fax: + (202)

2 For the approximate factor model X it = 0 if t +e it ; Bai and Ng (2002, BN hereafter) propose estimating the factor number r by minimizing IC(k) = ln V NT (k) + kg(n; T ); () with respect to k 2 f0; ; : : : ; k max g for some xed k max, where V NT (k) = min f k i g;fft kg (NT ) P N P T i= t= (X it h k0 i Ft k ) 2 and g() is a penalty function. Setting g (N; T ) = ln CNT 2 [(N + T ) =NT ], pt p i where C NT = min ; N, yields the IC p2 (k) criterion which is popular in practice and has the largest penalty on the tted factor number k, making it less sensitive to weak dependence in the idiosyncratic component than other IC(k) criteria. We estimate the factor number to industry-level employment growth. Annual wage and salary employment by North American Industry Classi cation System industry is obtained from table SA27 on the BEA website. We have 93 industries, and our sample spans Employment growth is annual log-di erences of total wage and salary employees. The panel is also standardized to remove the excessive heteroskedasticity. Table : Estimated factor number; Industry employment growth. Bai-Ng IC p2 (k) with k max = 5; N = 92; T = 9: sample level rst-di erence (FD) AR sub-sample robustness IC p2 (k) in levels always selects the maximum number of factors. This result does not change with other k max. With FD data, IC p2 (k) selects between 3 to 5 factors depending on the subsample. Note however that applying the IC p2 (k) criterion to the LSDV ltered panel yields a factor number estimate of one. This result holds for both the full sample and the subsamples. Next, we discuss this ltering procedure and provide an asymptotic justi cation for the lter. 2 Consistency of Filtering Procedure The whitening lter has been used in many areas of econometrics. The basic idea is to attenuate the temporal dependence in the data to make the transformed data closer to white noise. We employ an autoregressive ltering, and as such we must rst focus on two preliminary speci cation issues in order to ensure that the factor structure is preserved in the transformed data: (i) whether to perform an individual or a pooled ltering, and (ii) AR lag order. 2

3 To address the rst issue, consider the transformed data Z it = X it P p j= ijx it j ; X it = 0 if t + e it ; where the lter ij is permitted to be di erent for each i. Writing Z it as Z it = 0 P i F p t j= P ijf t j + e p it j= ije it j ; we see that only when ij = j (i.e., homogeneous for all i), the common factors of Z it are F t P p j= jf t j and the dimension of factors is preserved under the transformation. Without the homogeneity restriction in the ltering coe cients, the ltered common component 0 i(f t P p j= ijf t j ) cannot generally be expressed as a factor structure with the same dimension as F t. The second issue is the choice of the lag order p. Conveniently, an AR() tting (p = ) Z it = X it X it = X it + ( ) X it : (2) is su cient for consistent factor number estimation for many common panel processes, as we show below. Of course other orders p can also be used but we do not see any particular advantage in using more lags unless e it is more than once integrated. Hence we focus only on AR() ltering throughout the paper. Note that may not be a true AR() coe cient and X it X it may be dependent over t. We therefore consider an LSDV estimator, ^lsdv, obtained by regressing X it on X it and including individual intercepts. To show the validity of the AR() LSDV ltering, de- ne e it = 2 e;t e it and Ft = F F;T F t. Note e it and F t are divided by their variances rather than their standard deviations in the de nition of e it and F t, so (NT ) P N P T i= t= Ee2 it = 2 e;t and T P T t= F t Ft 0 = F F;T. The reason for this normalization is both to ensure that the variables e it and F t behave regularly when the original processes e it and F t are stationary, and to ensure that e it and F t are negligible when e it and F t are integrated. Now Z it of (2) can be rewritten as Z it = 0 i[f t + ( ) F F;T F t ] + [e it + ( ) 2 e;t e it ]; (3) so the factors of the transformed series Z it are F t + ( ) F F;T Ft and the idiosyncratic component is e it + ( ) 2 e;t e it. If is chosen such that ( ) F F;T and ( ) 2 e;t are bounded, then these transformed factors and idiosyncratic components are likely to satisfy Bai and Ng s (2002) regularity (called BN-regularity hereafter). For a rigorous treatment along this line, we make the following assumptions. Assumption For any constant b and b 2, f i g, ff t + b F t g and fe it + b 2 e it g are BN-regular. 3

4 Assumption 2 The common factors F t and idiosyncratic errors e it satisfy T TP E[F t t= F 0 t] = O() and NT NP i= t= TP E[e it e it ] = O(): Theorem (Consistency of LSDV Filtering) Given Assumptions and 2, if 2 e;t = o(t ) and F F;T = o(t ), then ^k(^ lsdv )! p r where r is the true factor number and ^k estimated factor from LSDV ltering. ^lsdv is the Assumptions and 2 are satis ed by a variety of processes in e it and F t, including I () and square summable I (0), heterogenous, as well as local asymptotic processes. While rst di erencing works well when the process is closer to unit root or even integrated, the LSDV ltering typically performs better than rst di erencing if the process does not exhibit strong serial correlation. In practice the strength of the dependence is unknown, so it will be useful to provide a method which combines the two ltering methods and which is at least as good as the two ltering methods separately. A simple way to enhance the small sample performance is to choose the minimum factor number estimate from the rst di erencing and the LSDV ltering, i.e., o ^k min = min n^k(); ^k(^lsdv ) : (4) This minimum rule is justi ed by the fact that serial correlation usually causes overestimation rather than underestimation of the factor number. The method can also be applied to the restricted dynamic models based on F t = P p j= jf t j + G t, where t is q and G is r q with full column rank (Amengual and Watson, 2007; Bai and Ng, 2007). For this model, ltering the data using the methods suggested above preserves the dimensions of both the static and dynamic factors because F t F t = px j (F t j F t j ) + G( t t ); j= where the transformed static factors F t F t are still r and the transformed primitive shocks t t are still q. 3 Monte Carlo Studies We consider the following data generating process: X it = P r j= jif jt + e it ; F jt = F jt + v jt for j = ; : : : ; r; JP e it = i e it + " it ; " it = u i k;t + u it for J = bn =3 c: k= J;k6=0 4

5 where bc denotes the largest integer not exceeding the argument. Note that e it can exhibit cross sectional and time series dependence through and i, respectively. We draw u it N(0; s 2 i ); v jt N (0; ), and ji N 0; r =2. We set r = 2. We consider the IC p2 (k) criterion only because it uses the largest penalty, so the probability of overestimation is the smallest amongst BN s IC criteria. We consider the following two cases. Case : Moderate heterogenous idiosyncratic serial dependence We generate the moderate heterogeneity by setting i U [ 0:; 0:9], s i U [0:5; :5] ; = 0:5; = 0:; and 2 v = : Table 2 shows the results with 2,000 replications. Evidently the AR lter outperforms the FD lter by a wide margin either N or T is small. The minimum rule improves on the AR lter slightly, while IC p2 (k) in levels performs poorly. Case 2: Extremely heterogenous autoregressive parameters In this set of simulations we draw i from iid U[ 0:; 0:] for i = ; : : : ; 2 N and iid U[0:7; 0:9] for i = 2N + ; : : : ; N. Other settings are as in Case. As in Case above, in this framework there is a marked disparity between the degree of serial dependence in the AR parameter. But within each subgroup the degree of heterogeneity is low. Table 3 exhibits the results. Notably the AR lter performs much better than the FD or LEV methods. As in case, this result is attributable to the FD lter inducing large negative correlation in many of the cross sections. The e ect is more noticeable in this DGP because more cross sections have an AR() coe cient close to zero. Table 2: Finite Sample Performances with Moderate Heterogeneity i U [ 0:; 0:9] ; s i U [0:5; :5] ; = 0:5; = 0:; 2 v = LEV FD AR MIN N T < = > < = > < = > < = > Note: <, =, >: under, correct, and over estimation, respectively. 5

6 Table 3: Finite Sample Peformance with Extreme Heterogeneity i U [ 0:; 0:] for i N=2 and i U [0:7; 0:9] for i > N=2 LEV FD AR MIN N T < = > < = > < = > < = > We also considered other parameters settings and DGPs, as well as other estimation methods such as the Hallin and Liska (2007) cross validation approach. In the interests of brevity we do not report the results here, but are avaliable from the authors upon request. 4 Conclusion We demonstrate that even a moderate degree of serial correlation in the idiosyncratic errors (relative to the given sample size) can cause the Bai-Ng criteria to overestimate the true factor number. To overcome this problem, we suggest ltering the panel. We theoretically analyze how the ltering method can work for general processes with serial correlation and verify the applicability of the method by simulations. 5 Appendix The following three lemmas are helpful to prove Theorem. Lemma Under Assumption, if (i) ( ) F F;T converges to a nite limit, (ii) ( ) 2 e;t = O () and (iii) (^ ) 2 X;T! p 0, then ^k(^)! p r. Lemma 2 If T 2 X;T = O () ; then under Assumption 2, ( lsdv) 2 X;T = O () : Lemma 3 Suppose that (i) var(xit 2 ) M4 X;T, and (ii) P k= cov(x2 it ; X2 it+k ) M 4 X;T for all i and t for some M <. If T 2 X;T = o(), then (^ lsdv lsdv ) 2 X;T = o p(). 6

7 Proof of Lemma Let ^a = (^ ) 2 X;T ; ^h(k) = h NT (k; ^Z it ) and h(k) = h NT (k; Z it ). The goal is to show that (i) ^h(k) does not shrink to zero for k < r, and (ii) ^h(k) = O(C 2 NT ) for k > r. (See Bai and Ng, 2002, proof of Theorem 2.) Note that ^Z it = Z it + ^axit, where Xit := X it= 2 X;T. where (i) When k < r : It su ces to show that ^h(k) h(k)! p 0. But ^h(k) h(k) = ^ r ^k, ^j = max F 2< T j NT NX ^Z ip 0 F ^Zi i= max F 2< T j NT NX ZiP 0 F Z i : Note that j max f max gj max jf gj, so j^ j j! p 0 since ^a! p 0 and all the averages are stochastically bounded. (ii) For the case with k > r : i= All conditions satisfy BN-reqular. But Lemma 4 in BN should be proved with new assumptions. Particularly we have to show that max F 2< T k NT NX i= t= (u i ^ae i; ) 0 M F (u i ^ae i; ) = O p (C 2 NT ); which corresponds to () of Bai and Ng (2006). Since both u it and and e it satisfy the assumptions of Bai and Ng (2002, 2006) and ^a! p 0; the left term becomes O p (C 2 NT ): Proof of Lemma 2 De ne ^ lsdv = ^ 0 ^ and lsdv =E ^ 0 E^ = 0;T ;T : Since ( lsdv ) 2 X = 0;T ( 0;T ;T ) 2 X;T and 0;T is nite, it su ces to show that ( 0;T ;T ) 2 X;T = O() or P N P T NT i= t= E X ~ it X ~ it = O() where the notation stands for the within-group transformation: It is easy to show since T E X ~ it t= ~ X it = T t= EX it X it T 2 t= s= EX it X is ; and the rst and second terms are bounded by Assumption 2. Proof of Lemma 3 3 is boiled down to the boundary condition of T 2 t= Under Assumption 2 and conditions in Lemma 3, the proof of Lemma T var X it Xit 2 X + T 2 t= s=t+ cov X it X it ; X is X is : Since var(x it X it ) =var(x2 it )=4 X = O( 2 X ) = O(), so the rst term is O(T ), and the second term is also O(T ): By using this boundary condition, Lemma 3 can be proved. The detailed proof is omitted. 7

8 Proof of Theorem The rst di erenced process X it clearly gives a consistent estimate. For X it ^lsdv X it, we note that the assumptions that T 2 e;t = o() and T F;T = o() imply that T 2 X;T = o(). Then it is straightforward to see that conditions for Lemma 2 and 3 are satis ed under the regularity Assumptions -2. The result follows from Lemmas 2 and 3. References [] Amengual, D. and M. W. Watson, 2007, Consistent estimation of the number of dynamic factors in a large N and T panel, Journal of Business and Economic Statistics 25, [2] Bai, J. and S. Ng, 2002, Determining the number of factors in approximate factor models, Econometrica 70, [3] Bai, J. and S. Ng, 2006, Determining the number of factors in approximate factor models, Errata, available at sn2294/papers/correctionecta2.pdf. [4] Bai, J. and S. Ng, 2007, Determining the number of primitive shocks in factor models, Journal of Business and Economic Statistics 26, [5] Hallin, M. and R. Liska, 2007, Determining the number of factors in the generalized dynamic factor model, Journal of the American Statistical Association 02,