Asymptotic Analysis of the Squared Estimation Error in Misspecified Factor Models

Transcription

1 Asymtotic Analysis of the Squared Estimation Error in Missecified Factor Models Alexei Onatski July 7, 2013 Abstract In this aer, we obtain asymtotic aroximations to the squared error of the least squares estimator of the common comonent in large aroximate factor models with ossibly missecified number of factors. The aroximations are derived under both strong and weak factors asymtotics assuming that the cross-sectional and temoral dimensions of the data are comarable. We develo consistent estimators of these aroximations and roose to use them for model comarison and for selection of the number of factors. We show that the estimators of the number of factors that minimize these loss estimators are asymtotically loss effi cient in the sense of Shibata (1980), Li (1987), and Shao (1997). 1 Introduction Emirical analyses of high-dimensional economic data often rely on aroximate factor models estimated by the rincial comonents method (see Stock and Watson (2011) for a recent survey of related literature). Many of these analyses intend to I would like to thank Serena Ng, Yuichi Kitamura, and Giovanni Urga for helful discussions and suggestions. I am grateful to J.M. Keynes Fellowshis Fund, University of Cambridge, for financial suort. 1

2 accurately estimate a low-dimensional common comonent of the data. For examle, the interest may lie in the art of multi-national data that can be attributed to a common business cycle, as in Forni and Reichlin (2001), or in the decomosition of sectoral outut growth rates into the common and idiosyncratic arts, as in Foerster et al (2011). Unfortunately, the estimation roblem is comlicated by the fact that the number of factors is tyically unknown and is likely to be missecified. This aer studies consequences of the missecification for the squared error of the estimated common comonent. Assuming that the cross-sectional and temoral dimensions of the data, n and T, are comarable, we derive asymtotic aroximations to the squared error loss through the order n 1 T 1. We consider both strong and weak factors asymtotics. Under the latter, the asymtotic loss turns out to be minimized not necessarily at the true number of factors. We develo an estimator of the loss which is consistent under both strong and weak factors asymtotics, and roose to use it for model comarison and for selection of the number of factors. We show that the estimator of the number of factors that minimizes the loss estimate is asymtotically loss effi cient in the sense of Shibata (1980), Li (1987), and Shao (1997). The majority of recently roosed estimators of the number of factors, including the oular Bai and Ng (2002) estimators, are asymtotically loss effi cient under the strong factors asymtotics, but not under the weak factors one. The basic framework of our analysis is standard. We consider an aroximate factor model X = ΛF + e, (1) where X is an n T matrix of data, Λ is an n r matrix of factor loadings, F is a T r matrix of factors and e is an n T matrix of idiosyncratic terms. Throughout the aer, we will treat Λ and F as unknown arameters. Equivalently, our results can be thought of as conditional on the unobserved realizations of random Λ and F. Suose that we estimate the first of the factors and the corresonding loadings by the least squares, and let us denote the estimates as ˆF 1: and ˆΛ 1:, resectively. As 2

3 is well known, ˆF 1: and ˆΛ 1: can equivalently be obtained by the rincial comonents (PC) method. That is, the columns of ˆF 1: / T are unit-length eigenvectors of X X, and ˆΛ 1: = X ˆF 1: /T. In the secial case where the idiosyncratic terms are i.i.d. N(0, 1), these are the maximum likelihood estimates subject to the normalization. Since we do not know the true value of r, may be smaller, equal or larger than r. We will say that the number of factors is missecified if r. We are interested in the effect of the missecification on the quality of the PC estimate ˆΛ 1: ˆF 1: of the common comonent ΛF of the data. This quality is measured by the average (over time and cross-section) squared error ] L = tr [(ˆΛ 1: ˆF 1: ΛF )(ˆΛ 1: ˆF 1: ΛF ) / (nt ). (2) Our interest in L is motivated by several reasons. First, accurate extraction of the common comonent is imortant in many alications. Second, in the secial case where the idiosyncratic terms are i.i.d. N (0, 1), L is roortional to the Kullback- Leibler distance between the true model (1) and the factor model with factors ˆF 1: and loadings ˆΛ 1:. Recall that the exected value of such a distance is usually aroximated by Akaike s (1973) information criterion (AIC). In Section 3, we show that the AIC aroximation does not hold in the large factor model setting. Finally, loss functions similar to L are widely used in the context of linear regression models. For examle, Mallows (1973) measure of adequacy for rediction of linear regression model Y = Z 1: β 1: + ε when the true model is Y = Zβ + u is given by (Ẑ1:ˆβ 1: Zβ) (Ẑ1:ˆβ 1: Zβ). The roblems of rediction, model selection, and model averaging with this loss function were extensively studied by Phillis (1979), Kunitomo and Yamamoto (1985), Shao (1997), and Hansen (2007), to name just a few studies. Since ΛF is unobserved, L can not be evaluated directly. In Section 2, we derive asymtotic aroximations for L that are easy to analyze and estimate. Subsection 2.1 considers the standard strong factors asymtotic regime (Bai and Ng (2008)). The strong factors asymtotics has been criticized by Boivin and Ng (2006), Heaton and Solo (2006), DeMol et al (2008), Onatski (2010, 2012), Kaetanios and 3

4 Marcellino (2010), and Chudik et al (2011) for not roviding accurate finite samle aroximations in alications where the factors are moderately or weakly influential. Therefore, in Subsection 2.2 we derive asymtotic aroximations for L using Onatski s (2012) weak factors assumtions. Using the derived asymtotic aroximations, Section 3 develos four different estimators of L. Under the strong factors asymtotics, all the roosed estimators are consistent for L after a shift by a constant that does not deend on. Moreover, the minimizers of these estimators are consistent for the true number of factors. Under the weak factors asymtotics, two of the roosed estimators rovide the asymtotic uer and lower bounds on the shifted loss. We show that the minimizers of these estimators bracket the actual loss minimizer with robability aroaching one as n and T go to infinity. The other two estimators are consistent for the shifted loss when there is either no cross-sectional or no temoral correlation in the idiosyncratic terms. In these secial cases, the number of factors that minimizes the corresonding estimator of the loss is consistent for the number of factors that minimizes the actual loss. The latter is not necessarily equal to the true number of factors. All the roosed loss estimators are simle functions of the eigenvalues of the samle covariance matrix. Monte Carlo exercises in Section 4 show that their quality is excellent when simulated factors are relatively strong. When the factors become weaker, the quality gradually deteriorates, but remains reasonably good in intermediate cases. In Section 5, we rovide an emirical examle of model comarison based on our loss estimators. We comare a two- and a three-factor model of excess stock returns, and find that estimating the third factor leads to a loss deterioration for the monthly data covering the eriod from 2001 to That is, a PC estimate of the threefactor model rovides a worse descrition of the undiversifiable risk ortion of the excess returns than a PC estimate of the two-factor model. Interestingly, this lossbased ordering is reversed when we use the data from 1989 to 2000, which suggests a decrease in the signal-to-noise ratio in the more recent excess returns data. Section 6 concludes. All roofs are given in the Aendix. 4

5 2 Asymtotic aroximation for the loss 2.1 Strong factors asymtotics In what follows, µ i (M) denotes the i-th largest eigenvalue of a Hermitian matrix M. Further, A j and A j denote the j-th column and j-th row of a matrix A, resectively. We make the following assumtions. A1 There exists a diagonal matrix D n with elements d 1n d 2n... d rn > 0 along the diagonal, such that F F/T = I r and Λ Λ/n = D n. This assumtion is a convenient normalization. The only non-trivial constraint it imlies is the requirement that rank F = r and rank Λ = r. A2 As n, Λ Λ/n D, where D is a diagonal matrix with decreasing elements d 1 > d 2 >... > d r > 0 along the diagonal. Assumtion A2 is sometimes called the factor ervasiveness assumtion. It requires that the cumulative exlanatory ower of factors, measured by the diagonal elements of Λ Λ, increases roortionally to n. The assumtion is standard, but may be too strong in some alications. In Subsection 2.2, we consider an alternative assumtion that allows Λ Λ to remain bounded as n. Let n, T c denote the situation where both n and T diverge to infinity so that n/t c (0, ). This asymtotic regime is articularly useful for the analysis of data with comarable cross-sectional and temoral dimensions, such as many financial and macroeconomic datasets. It also does not reclude situations where n/t is small or large as long as n/t does not go to zero or to infinity. A3 As n, T c, (i) there exists ε > 0 such that Pr (tr[ee ]/(nt ) > ε) 1; (ii) for any j, k r, Λ jef k / nt = O (1); (iii) µ 1 (ee /T ) = O (1). Part (i) of A3 rules out uninteresting cases where the idiosyncratic terms e it are zero or very close to zero for most of i and t. Part (ii) of A3 is in the sirit of assumtions E (d,e) in Bai and Ng (2008). Validity of the central limit theorem for sequences 5

6 {Λ ij e it F tk ; i, t N} with j, k r is suffi cient but not necessary for A3 (ii). Part (iii) of A3 further bounds the amount of deendence in the idiosyncratic terms. Assumtion A3 (iii) is technically very convenient and has been reviously used by Moon and Weidner (2010). They rovide several examles of rimitive conditions imlying A3 (iii). In the Sulementary Aendix, we show that A3 (iii) holds for very wide classes of stationary rocesses {e t, t Z}. Proosition 1 Let P i:j be a T T matrix of rojection on the sace sanned by F i,..., F j, and let Q i:j be an n n matrix of rojection on the sace sanned by L i,..., L j. Under assumtions A1-A3, as n, T c, L = L (1) + o P (1/T ), where L (1) = { r j=+1 d jn + tr [ep 1: e + e Q 1: e] / (nt ) if r L (1) r + j=r+1 µ j (X X) / (nt ) if > r. (3) It is instructive to comare (3) to the loss in the case of known factors. This case is similar to the standard OLS regression with factor loadings laying the role of the regression coeffi cients. If the known factors satisfy A1, then a simle regression algebra shows that L known = { r j=+1 d jn + tr [ep 1: e ] / (nt ) if r L known r + tr [XP r+1: X ] / (nt ) if > r, where the suerscrit known is introduced to distinguish the case of known factors from that of latent factors. Comaring L known to L (1), we see that L (1) contains an extra term tr [e Q 1: e] / (nt ). The reason is that, in Proosition 1, not only loadings, but also factors are estimated. Hence, the exression for the loss becomes symmetric with resect to interchanging factors and factor loadings. More imortant, for > r, the term tr [XP r+1: X ] / (nt ) in L known is relaced by the term j=r+1 µ j (X X) / (nt ) in L (1). It is because when the over-secified factors are not known, they are chosen so as to exlain as much variation as ossible. In other words, the rojection P r+1: in tr [XP r+1: X ] / (nt ) is relaced by the rojection on the sace sanned by the r + 1,..., -th rincial eigenvectors of X X. 6

7 If we further assume homoscedasticity, E (e e) = nσ 2 I T, then we can write EL known = { r j=+1 d jn + σ 2 /T if r σ 2 /T if > r. (4) Hence, the exected loss, or risk, consists of the bias term r j=+1 d jn and the variance term σ 2 /T, with the bias term disaearing under correct or over-secification. In the case of latent factors, we have: Corollary 1 Suose that the elements of e are i.i.d. zero mean random variables with variance σ 2 and a finite fourth moment. Then, under assumtions A1-A2, as n, T c, L = L (1) + o P (1/T ), where EL (1) = { r j=+1 d jn + σ 2 (1/T + 1/n) if r σ 2 ( r) (1/ T + 1/ n) 2 + σ 2 r (1/T + 1/n) if > r. Comaring EL (1) to EL known, we see that the variance term of EL (1) is symmetric with resect to interchanging n and T. More imortant, in contrast to the case of known factors, the marginal effect on the variance term of EL (1) from adding -th factor deends on whether the model is under- or over-secified. It is σ 2 (1/T + 1/n) in the under-secified, but σ 2 (1/ T + 1/ n) 2 in the over-secified case. The unusual form of the term σ 2 (1/ T + 1/ n) 2 can be linked to the a.s. convergence µ 1 (ee /T ) σ 2 (1 + c) 2 as n, T c (Yin et al, 1988). Relacing c in σ 2 (1 + c) 2 by n/t, and dividing the obtained exression by n, we get σ 2 (1/ T + 1/ n) 2 (see the roof of Corollary 1 in the Aendix for more details on the link). 2.2 Weak factors asymtotics In this section we derive an asymtotic aroximation to L using alternative weak factor assumtions roosed and discussed in detail in Onatski (2012). A1w There exists a diagonal matrix n with elements δ 1n δ 2n... δ rn > 0 along the diagonal, such that F F/T = I r and Λ Λ = n. As n, n, 7

8 where is a diagonal matrix with decreasing elements δ 1 > δ 2 >... > δ r > 0 along the diagonal. By definition, δ jn equals the cross-sectional sum of the squared loadings of the j-th factor. Hence, δ jn can be thought of measuring the cumulative exlanatory ower, or strength, of factor j. The convergence δ jn δ j stays in contrast to assumtion A2, which imlies that δ jn = nd jn. As exlained in detail in Onatski (2012), the asymtotic regime described by A1w is meant to rovide an adequate aroximation to emirically relevant finite samle situations where a few of the largest eigenvalues of the samle covariance matrix are not overwhelmingly larger than the rest of the eigenvalues. A2w There exist n n and T T deterministic matrices A n and B T such that e = A n εb T, where (i) ε is an n T matrix with i.i.d. N (0, σ 2 ) entries; (ii) A n is such that tr (A n A n) = n and (A n A n) Λ = Λ; (iii) B T and (B T B T ) F = F. is such that tr (B T B T ) = T The idiosyncratic matrices of the form e = A n εb T were reviously considered in Bai and Ng (2006), Onatski (2010, 2012), and Ahn and Horenstein (2012). When A n and B T are not identity matrices, the idiosyncratic terms are both cross-sectionally and serially correlated. The assumtion restricts the covariance matrix of the vectorized e to be of the Kronecker roduct form σ 2 B T B T A n A n. This can be viewed as an aroximation to more realistic covariance structures. For a general discussion of the quality of aroximations with Kronecker roducts see Van Loan and Pitsianis (1993). As exlained in Onatski (2012,.247), A2w (ii),(iii) are simlifying technical assumtions. They allow Onatski (2012, Theorem 1) to obtain exlicit exressions for the bias of the PC estimator under the weak factors asymtotics. The analysis below will rely on these exlicit exressions. The Monte Carlo exercises in Section 4 show that the quality of the loss aroximation L (1) derived under A2w remains good if A2w (ii) and (iii) are relaxed. A theoretical investigation of this henomenon requires a substantial additional technical effort. We leave such an investigation for future research. 8

9 The last assumtion describes the asymtotic behavior of matrices A n and B T. Let G A (x) = 1 n n i=1 1 [µ i (A n A n) x] and G B (x) = 1 T T i=1 1 [µ i (B T B T ) x], where 1 [ ] denotes the indicator function. Hence, G A (x) and G B (x) are the emirical distribution functions of the eigenvalues of A n A n and B T B T, resectively. A3w There exist robability distributions G A and G B with bounded suorts [x A, x A ] and [x B, x B ], cumulative distribution functions G A (x) and G B (x), and densities d dx G A(x) and d dx G B(x) at every interior oint of suort x (x A, x A ) and x (x B, x B ), resectively, such that, as n, T c : (i) G A (x) G A (x) and G B (x) G B (x) for all x R, (ii) µ 1 (A n A n) x A and µ 1 (B T B T ) x B, and (iii) inf x (xa, x A ) d G dx A(x) > 0 and inf x (xb, x B ) d G dx B(x) > 0. Assumtion A3w holds for a broad range of matrices A n A n and B T B T. For examle, it is satisfied for large classes of widely used Toelitz matrices. Onatski (2012) shows that, under assumtions A1w-A3w, there is an asymtotic relationshi between the true strength of the j-th factor, δ j, and the j-th samle covariance eigenvalue. Precisely, as n, T c, µ j (XX /T ) σ 2 f ( δ j /σ 2), j r, (5) where function f ( ) deends only on G A and G B and can be evaluated numerically. In contrast, under the strong factor assumtions A1-A3, the r largest eigenvalues of XX /T, which are sometimes referred to as factor eigenvalues, diverge to infinity. Function f ( ) lays an imortant role in the analysis below. Its salient features are summarized in the following lemma. Lemma 1 Suose that assumtions A1w-A3w hold. Then, (i) there exists δ > 0, that deends on G A and G B, such that σ 2 f (δ/σ 2 ) = lim µ 1 (ee /T ) for any δ [ 0, δ ] ; (ii) As a function of z, f (z) is non-decreasing and continuous, and larger than z on z 0. Furthermore, it is differentiable on z < δ/σ 2 and on z > δ/σ 2, and is such that f(z)/z 1 as z ; (iii) the elasticity d ln f(z)/d ln z increases on z > δ/σ 2 and converges to one as z. 9

10 Proosition 2 Suose that assumtions A1w-A3w hold. Furthermore, suose that δ j δ for j = 1,..., r and let q be the largest {0, 1,..., r} such that δ > δ, where δ 0 =. Then, as n, T c, L = L (1) + o P (1/T ), where L (1) = { r j=1 δ jn/n + j=1 µ j (XX ) / (nt ) 2 L (1) q j=1 δ jnf (δ jn /σ 2 ) /n for q + j=q+1 µ j (XX ) / (nt ) for > q. Here f (z) denotes the derivative of f (z). For > r q, the increment to L (1) (6) due to over-secifying factors relative to 1 factors is aroximated by µ (XX ) / (nt ). As can be seen from (3), this coincides with the increment to L (1) due to the marginal increase in the over-secification under the strong factors asymtotics. For r, the weak and strong factors asymtotic aroximations to the loss are substantially different. Under the weak factors asymtotics, the increment L (1) L (1) 1 remains µ (XX ) / (nt ) for > q. In such cases, the -th factor is so weak that lim n δ n δ. For q, the increment becomes µ (XX ) / (nt ) 2δ n f (δ n /σ 2 ) /n. As formula (5) shows, this equals σ 2 f (δ /σ 2 ) /n 2δ f (δ /σ 2 ) /n+ o P (1/T ), which becomes negative for suffi ciently large n and T if and only if d ln f(z)/d ln z > 1/2 at z = δ /σ 2. (7) By Lemma 1 (iii), the elasticity d ln f(z)/d ln z increases with z. Thus, asymtotically, the loss is minimized for the largest such that (7) holds. Note that δ /σ 2 is large for relatively strong factors. Therefore, according to Lemma 1 (iii), d ln f (z) /dlnz 1 at z = δ /σ 2, so that (7) is satisfied. In other words, including very strong factors to the model leads to a decrease in the loss L. For very weak factors, d ln f (z) /dlnz = 0 at z = δ /σ 2 because f (δ/σ 2 ) is constant for δ δ by Lemma 1 (i), so that (7) is violated, and including such factors to the model leads to an increase in the loss. Inequality (7) tells us exactly how strong the factors should be so that including them to the model imroves the rediction of the common comonent. 10

11 For secial cases of G A and G B, it is ossible to obtain exlicit formulae for f ( ) in (5). For examle, when G A and G B are degenerate robability distributions utting all mass at one (see Onatski (2006, Theorem 5)), 1 f ( δ/σ 2) = { (1 + c) 2 for 0 δ/σ 2 c (δ + σ 2 ) (δ + cσ 2 ) / (δσ 2 ) for δ/σ 2 > c Then, the asymtotic aroximation (6) given in Proosition 2 simlifies.. (8) Corollary 2 Suose that assumtion A1w holds, and let the elements of e be i.i.d. N (0, σ 2 ). Further, suose that δ j σ 2 c for j = 1,..., r, and let q be maximum {0, 1,..., r} such that δ > σ 2 c, where δ 0 =. Then, L = L (1) + o P (1/T ), where r L (1) j=+1 = δ jn/n + σ 2 (1/n + 1/T ) + 3 j=1 σ4 / (T δ jn ) for q L (1) q + ( q) σ (1/ 2 n + 1/ ) 2 T for > q. Note that, under the weak factors asymtotics, the minimum of L (1) is achieved not necessarily at = r, as is the case under the strong factors asymtotics. Instead, it is achieved at the maximum of {0, 1, 2,..., q} such that δ n /n > σ 2 (1/n + 1/T ) + 3σ 4 / (T δ n ). 3 Loss estimation In this section, we develo statistics ˆL that aroximate L. As mentioned in the introduction, although AIC is a natural candidate for ˆL, it fails in our setting. Let us exlore this in more detail. In the simlest secial case where the idiosyncratic terms are i.i.d. N (0, 1), the log-likelihood equals ln L (X Λ, F ) = nt 2 ln 2π 1 2 tr [ (X ΛF ) (X ΛF ) ], 1 Note that the constraints that A3w imoses on the densities of G A and G B in (x A, x A ) and (x B, x B ) are trivially satisfied even though G A and G B are not absolutely continuous with resect to Lebesgue measure because the intervals (x A, x A ) and (x B, x B ) are emty. 11

12 so that the Kullback-Leibler distance between the true model (1) and the model with arameters F, Λ is ( KL F, Λ; F, Λ ) = E ln L ( (X Λ, F L X Λ, F ) = 1 [ ( 2 tr ΛF Λ F ) ( ΛF Λ F ) ]. Hence, L equals 2KL (F, Λ; ˆF 1:, ˆΛ ) 1: / (nt ), which is the exact analog for the factor models of the loss used by Akaike (1973) to derive ( AIC. The loss L deends on only through 2E ln L X Λ, F ) / (nt ), evaluated at Λ = ˆΛ 1: and F = ˆF 1:. In our setting, the Akaike s (1973) idea is to aroximate this art of L by 2 ln L (X ˆΛ 1:, ˆF ) 1: / (nt ) and correct for the bias. The correction term, at least for the over-secified models, should be 2 times the arameter dimensionality divided by the samle size. Unfortunately, this simle rule does not hold here. For the sake of illustration, let there be no factors in the data (r = 0). Then, 2 ( nt E ln L X Λ, F ) Λ, F =ˆΛ1:, ˆF1: = ln 2π nt tr [ˆΛ1: ˆF 1: ˆF1: ˆΛ1: ] = ln 2π n µ j, where µ j is a shorthand notation for µ j (XX /T ). Furthermore, j=1 2 nt ln L ( X ˆΛ 1:, ˆF 1: ) = ln 2π + 1 n n µ j 1 n j=1 µ j. (9) j=1 Combining these two equalities, we obtain 2 nt [ ( E ln L X Λ, F ) ( ln L X ˆΛ 1:, Λ, ˆF ) ] 1: = 1 1 F =ˆΛ1:, ˆF1: n n µ j + 2 n j=1 µ j. ( (10) As shown in Onatski et al (2013, Lemma 12), n 1 1 n n j=1 j) µ d N (0, 2c) as n, T c. Hence, the term 1 1 n n j=1 µ j in the latter equality does not con- 12 j=1

13 tribute to the bias correction through the order 1/n. Further, by Yin et al s (1988) result, 2 j=1 µ a.s. j 2 (1 + c) 2. Relacing c by n/t, we see that 2 n j=1 µ j can be aroximated through the order 1/n by 2 ( n + T ) 2. In the factor model nt setting, the samle size is nt. Thus, had the Akaike s (1973) rule for the bias correction worked, we would have had ( n + T ) 2 as the arameter dimensionality. However, to the order n, the number of free arameters in the -factor model is (n + T ) ( n + T ) 2. Akaike s (1973) derivations of his bias correction rule is based on the quadratic aroximations to the log-likelihood and on the standard roerties of the maximum likelihood estimates. There are at least two reasons why this standard machinery does not work in the setting of large factor models. First, the number of arameters is increasing with the samle size. Second, arameters of an oversecified model are not identified (when the true loadings are identically zero, the corresonding factor may corresond to any oint on a shere of radius T in R T ). These roblems are related to the well-known incidental arameters roblem (Lancaster, 2000) and the non-standard inference in cases where some arameters are not identified under the null (Hansen, 1996). Although the AIC s bias correction rule does not work, equation (10) shows that the bias can be corrected simly by adding 2 j=1 µ j/n to 2 ln L (X ˆΛ 1:, ˆF ) 1: / (nt ). Even though under the general assumtions A1-A3 or A1w-A3w, the Kullback- Leibler interretation of L is lost, Proositions 1 and 2 suggest that, u to a quantity that does not deend on, L is still well aroximated by the right hand side of (9) after a bias correction which is a function of µ j. Secifically, at least for > r, adding 2 j=r+1 µ j/n to the right hand side of (9) will erfectly match L (1), after a shift by a quantity that does not deend on. Let ˆr be an estimator of the number of factors, such that lim ˆr = r under the strong factors asymtotics and lim ˆr = q under the weak factors asymtotics, where q is as defined in Proosition 2. One such ˆr is the estimator of the number of factors develoed in Onatski (2010). Below we study estimators of the loss L, shifted by a 13

14 quantity that does not deend on, of the following form { ( ) j=1 1 2ˆρj µj /n ˆL = ˆLˆr + j=ˆr+1 µ j/n for ˆr for > ˆr, (11) where ˆρ j are data deendent. Consider the following two secial cases of ˆL : L = ˆL with ˆρ j = 1, and (12) L = ˆL with ˆρ j = 1/(µ 2 j max{ ˆm ( ) µ ( ) j, m µ j }), (13) where and ˆm (x) = d dx ˆm (x) with ˆm (x) = (n ˆr) 1 n m (x) = d dx m (x) with m (x) = (T ˆr) 1 T where, for j > n, µ j is defined as zero. i=ˆr+1 (µ i x) 1, i=ˆr+1 (µ i x) 1, Proosition 3 (i) Let L = L or L = L. Then, under assumtions A1-A3, as n, T c, max L L (L r L r ) = o P (1) and (14) 0 <r L max L (L r L r ) = o P (1/T ). (15) r r max (ii) Under assumtions A1w-A3w, for any ɛ > 0, as n, T c, ( ) Pr[ min (L L 0 ) L ɛ/n] 1 and (16) 0 r max Pr[ max 0 r max ( (L L 0 ) L ) ɛ/n] 1. (17) Part (i) of Proosition 3 shows that both L and L can be thought of as asymtotic aroximations to a shifted version of L. As follows from Proosition 1, min 0 <r L is bounded away from zero with robability aroaching one. Hence, 14

15 the aroximation error in (14) is asymtotically negligible relative to the size of the loss. The ortion of the loss L that corresonds to r r max converges to zero. It can be shown (see the roof of Proosition 5) that, for r < r max, the rate of such convergence is 1/T, and the aroximation error in (15) is also negligible relative to the size of the loss. The reason why both L and L aroximate L well asymtotically is that, under the strong factors asymtotics, the difference L L converges to zero. Indeed, with robability aroaching one, µ j for any j ˆr, and µˆr+1 = O P (1). Therefore, ˆρ j with j ˆr defined in (13) converge in robability to one, which coincides with the value of ˆρ j in (12). Part (ii) of Proosition 3 shows that, under the weak factors asymtotics, L and L can be thought of as asymtotic lower and uer bounds on a shifted version of L. According to Proosition 2, an estimator ˆL that would aroximate L well under the weak factors asymtotics must have lim ˆρ j = d ln f (δ j /σ 2 ) /dln(δ j /σ 2 ). We were able to develo such ˆρ j only in the secial cases where either A = I n or B = I T, that is where there is either no cross-sectional or no temoral correlation in the idiosyncratic terms. Let ˆL (A=I) and resectively, where ˆL (B=I) be the estimators ˆL with ˆρ j = ˆρ (A=I) j and ˆρ j = ˆρ (B=I) j, ˆρ (A=I) j = (1 + ˆm(µ j )ˆσ 2 n/t ) ˆm(µ j )/(µ j ˆm (µ j )), and (18) ˆρ (B=I) j = (1 + m(µ j ) σ 2 T/n) m(µ j )/(µ j m (µ j )). (19) In the above exressions, ˆσ 2 = (n ˆr) 1 n i=ˆr+1 µ i, and σ 2 = (T ˆr) 1 T i=ˆr+1 µ i. Proosition 4 (i) Let L A1-A3, as n, T c, = ˆL (A=I) or L = max L L (L r L r ) = o P (1) and 0 <r L max L (L r L r ) = o P (1/T ). r r max ˆL (B=I). Then, under assumtions 15

16 (ii) Suose that assumtions A1w-A3w hold. If A = I n or B = I T, then, as n, T c, resectively, L (A=I) (A=I) max ˆL (L q ˆL 0 r max q ) L (B=I) (B=I) max ˆL (L q ˆL 0 r max q ) As can be seen from art (i) of Proosition 4, both = o P (1/T ) and = o P (1/T ). ˆL (A=I) and ˆL (B=I) aroximate a shifted loss L under the strong factors asymtotics. This result is similar to art (i) of Proosition 3. It holds because both ˆρ (A=I) j and ˆρ (B=I) j converge in robability to one under the strong factors asymtotics. For the weak factors asymtotics, aroximates a shifted loss L when A = I n, whereas ˆL (B=I) ˆL (A=I) aroximate a shifted loss L when B = I T. These aroximations imrove uon the asymtotic bounds L and L from art (ii) of Proosition 3. The aroximations to the shifted loss given in Proositions 3 and 4 can be used to assess changes in the loss that result from different secifications of the number of factors. Alternatively, they can be used to select an asymtotically loss effi cient number of factors. The concet of asymtotic loss effi ciency of model selection rocedures was studied in detail by Shibata (1980), Li (1987), and Shao (1997), among others. In the context of factor models and loss L, it can be described as follows. Let ˆ be an estimator of the number of factors. This estimator is called asymtotically loss effi cient if Lˆ min 0 rmax L 1. (20) Shao (1997) oints out that a suffi cient but not necessary condition for the asymtotic loss effi ciency is Pr (ˆ = ) 1, (21) where = arg min 0 rmax L. He calls this stronger roerty of ˆ consistency. Since the minimizer of the loss L does not necessarily coincide with the true number of factors r, we will call the roerty (21) otimal loss consistency instead. 16

17 Proosition 5 Let,, ˆ (A=I), and ˆ (B=I) be the minimizers of L, L, ˆL (B=I) on 0 r max, resectively. (A=I) ˆL, and (i) Suose that assumtions A1-A3 hold. Then any estimator consistent for r is otimal loss consistent. Furthermore, estimators ˆr,,, ˆ (A=I), and ˆ (B=I) are consistent for r, and thus, otimal loss consistent. (ii) Suose that assumtions A1w-A3w hold. Then, estimator ˆr is not, in general, otimal loss consistent. For any otimal loss consistent estimator ˆ, Pr ( ˆ ) 1. Moreover, L L L L + o P (1/T ) and L L L L + o P (1/T ). (iii) In the secial cases where assumtions A1w-A3w hold and A = I n or B = I T, estimators ˆ (A=I) or ˆ (B=I), resectively, are otimal loss consistent. By definition, the minimum of ˆL cannot be achieved for > ˆr. Under the weak factors asymtotics, ˆr q r. Therefore, arts (ii) and (iii) of Proosition 5 imly that, when factors are weak, the under-estimation of the number of factors r may lead to imrovements in the loss, even in large samles. The otimal loss consistent estimator will tend to be smaller than the reliminary estimator ˆr. Such an otimal estimator is asymtotically bracketed by the estimators and. Moreover, under the strong factors asymtotics, both and are otimal loss consistent and consistent for r. 4 Monte Carlo Exeriments In this section, we use Monte Carlo exeriments to assess the finite samle quality of the asymtotic loss aroximations L (1) and estimators L, L (A=I) (B=I), ˆL, and ˆL. Our simulation setting is similar to that used in many revious studies, including Bai and Ng (2002), Onatski (2010), and Ahn and Horenstein (2012). The data are generated from X it = C it + θe it, where the common comonent C it and the idiosyncratic comonent e it are indeendent and normalized to have variance one, and arameter θ measures the inverse of the signal-to-noise ratio. 17

18 The common comonent is generated by rocess C it = r j=1 λ ijf tj / r, where r = 3, and λ ij and F tj are i.i.d. N (0, 1). Dividing by r insures that the variance of C it equals one. The idiosyncratic comonent e it is generated by rocess e it = ρe i,t 1 + v it + J j 0,j= J βv i j,t, where v it, i, t Z are i.i.d. N (0, σ 2 v) with σ 2 v = (1 ρ 2 )/(1 + 2Jβ 2 ), and J = min (n/20, 10). Following Ahn and Horenstein (2012), we consider (ρ, β) = (0, 0), (0.7, 0), (0, 0.5), or (0.5, 0.2). The signal-to-noise ratio θ 1 takes on five ossible values: 4, 2, 1, 1/2, or 1/4. The samle sizes are n = T = 50, 100, or 200, and r max is set to 8. To give the reader an idea on how the loss function in our MC exeriments looks like, Figure 1 shows two articular realizations of L for n = T = 100 and (ρ, β) = (0.7, 0). These realizations are suerimosed with the strong factors (dashed lines) and weak factors (dotted lines) asymtotic aroximations L (1), derived in Proositions 1 and 2. Function f( ) that aears in the weak factors asymtotic aroximation is comuted numerically using the MATLAB code develoed in Onatski (2012,.248). The left and right anels of the figure corresond to the lowest and the highest signal-to-noise ratio, resectively. In ractice, the relative strength of the signal is often assessed using the scree lot. Hence, we also rovide grahs (dots with abscissa 9) of the sorted eigenvalues of the samle covariance matrix, scaled to fit the icture. For the low signal-to-noise ratio (θ = 4), the first three eigenvalues do not clearly searate from the smaller eigenvalues, whereas for the large signal-to-noise ratio (θ = 1/4) the searation is obvious. We see that when θ = 1/4, so that the factors are relatively strong, L is minimized at the true number of factors = r = 3, and both aroximations to the loss are very close to the actual realization. When θ = 4, so that the factors are relatively weak, L is no longer necessarily minimized at = 3. For the articular realization shown at the icture, the loss is minimized at = 2, but the minimum is relatively large. Its value is 0.69, 18

19 Loss Loss 2 θ=4 2 θ=1/ Figure 1: Particular realizations of the loss L (solid lines) and the corresonding asymtotic aroximations L (1) (dotted lines weak factors aroximation, dashed lines strong factors aroximation). n = T = 100, (ρ, β) = (0.7, 0). which means, roughly, that 69% of the variation of the PC estimator of the common comonent that uses the otimal number of factors is due to the error (recall that the common comonent is normalized to have variance one). In this diffi cult case, the weak factors asymtotic aroximation is better than the strong factors one for r. Figure 2 shows the root mean squared errors (RMSE) of the strong factors (dashed lines) and weak factors (dotted lines) asymtotic aroximations of L, the mean being taken over 1000 MC relications. The figure corresonds to (ρ, β) = (0.7, 0) and n = T = 100. The other cases rovide qualitatively similar information and we do not reort them here. For relatively strong factors, the quality of the strong factors asymtotic aroximation is uniformly better than that of the weak factors aroximation. However, the scale of the difference between the qualities of the two aroximations is very small (both aroximations work very well). For relatively weak factors, the scale of the difference between the qualities of the aroximations increases, and the weak factors asymtotic aroximation become referable, ese- 19

20 RMSE RMSE θ= θ=1/ θ= θ=1/ θ= Figure 2: The root mean squared errors (over 1000 MC relications) of the strong (dashed lines) and weak (dotted lines) factors asymtotic aroximations of L. The dots with abscissa =9 show the MC average values of the sorted eigenvalues of the samle covariance matrix. (ρ, β) = (0.7, 0), n = T = 100. cially for r. We now turn to the analysis of the roosed loss estimators. Figure 3 shows RMSE of L (solid lines), L (dotted lines), and ˆL (A=I) 100 and (ρ, β) = (0.7, 0). Since n = T in our exeriments, ˆL (A=I) (A=I). In all our exeriments, estimator ˆL (dashed lines) for n = T = ˆL (B=I) coincides with erforms very similar to the simler estimator L. Therefore, in what follows, we focus on the analysis of L and L. Since, as exlained above, L and L estimate L only u to a shift, we shift the estimation error to zero at = q when comuting RMSE. For relatively strong factors (θ 2), q was equal to r = 3 in most of our exeriments. On Figure 3, this is reflected in the fact that RMSE is zero at = r = 3. From Figure 3, we see that estimator L dominates L for all values of θ. This situation remains the same for the other combinations of ρ and β, and the other samle sizes (not shown here). Therefore, in the rest of this section, we will restrict 20

21 RMSE RMSE 1.5 θ=4 θ= θ= θ=1/ θ=1/ Figure 3: The root mean squared errors (over 1000 MC relications) of the shifted versions of L (A=I).(solid lines), L (dotted lines), and ˆL (dashed lines). (ρ, β) = (0.7, 0), n = T = 100. attention to L. Comaring Figures 3 and 2, we see that the quality of L is comarable to the quality of the theoretical asymtotic aroximations to the loss, excet for relatively weak factors (θ 2) and < r, where the quality of L is substantially worse. It turns out that the main reason behind this quality deterioration is the inability of our reliminary estimator ˆr to accurately estimate q when factors are relatively weak. Figure 4 illustrates this finding. It shows the same realization of L as on the left anel of Figure 1 (solid line) suerimosed with a shifted version of L. For the corresonding data relication, q = 2, so the grahs coincide at = q = 2, by construction. However, our reliminary estimator ˆr = 1 < q. The dotted line shows what the value of L would have been, had ˆr been equal to q = 2. Clearly, the accurate estimation of q leads to much more accurate estimation of L for < r. Accurate estimation of q is diffi cult when factors are weak. The diffi culty is well 21

22 Loss 2 θ= Figure 4: A realization of L (solid line) and L (dashed line). Dotted line corresonds to L based on the counterfactual ˆr = q. (ρ, β) = (0.7, 0), n = T = 100. illustrated by the relative osition of the eigenvalues shown on Figure 4. All methods of the number of factors estimation exlicitly or imlicitly look for a searation between r largest eigenvalues and the rest of the eigenvalues. For weak factors, the searation theoretically cannot occur in large samles for more than q eigenvalues, hence the focus on the estimation of q when the factors are weak. The eigenvalues reorted in Figure 4 do not show any visible searation, excet erhas between the first eigenvalue and the rest, which is catured by the fact that ˆr = 1 < q = 2. Under the weak factors asymtotics, only Onatski s (2010) estimator has been formally shown to be consistent for q (see Section 3 of Onatski (2012)). However, in rincile, other estimators may accurately estimate q in finite samles. Therefore, below, we comare the quality of the loss estimates based on various reliminary estimators ˆr. Our benchmark is Onatski s (2010) ED estimator, used above. We also consider Bai and Ng (2002) estimators based on their criteria BIC 3, P C j, and IC j with j = 1, 2, estimators ER and GR develoed by Ahn and Horenstein (2012), and Alessi et al s (2010) ABC estimator. Comaring quality of different loss estimates requires some care. Ideally, the measure of quality should deend on the use to which the estimate will be ut. 22

23 n, T θ ED ER GR ABC BIC PC1 PC2 IC1 IC / / / / / / Table 1: Values of rmax r max j=1 E MC ( Lj L j 1 (L j L j 1 ) ) 2 corresonding to different reliminary estimators of q. One might, for examle, use the estimate to comare the losses incurred by models with j 1 and j number of factors. In such a case, it is natural to measure the quality of L by the square root of E MC ( Lj L j 1 (L j L j 1 ) ) 2, where EMC denotes the oerator of taking mean over MC relications. Table 1 reorts values of 100 rmax r max j=1 E MC ( Lj L j 1 (L j L j 1 ) ) 2 for L based on different versions of ˆr, when (ρ, β) = (0.7, 0). Results corresonding to the other combinations of ρ and β are qualitatively similar and we do not reort them to save sace. Since the common comonent has variance one in all MC exeriments, the units of the quality measure reorted in Table 1 can be interreted, roughly, as ercents of variation of the common comonent. We see that for relatively strong factors all estimators fare very well. For weak factors, the quality substantially deteriorates, esecially for relatively small n and T. Overall, ED, ER, GR, and ABC show more robust erformance. However, none of these estimators clearly dominates. Another basis for comarison of different estimates of the loss is related to the 23

24 n, T θ ED ER GR ABC BIC PC1 PC2 IC1 IC / / / / / / Table 2: Value of E MC Lˆ /E MC L corresonding to different reliminary estimators of q quality of the corresonding loss effi cient estimates of the number of factors. Let ˆ = arg min 0 rmax L and = arg min 0 rmax L. Then, a natural measure of quality of L is E MC Lˆ /E MC L. Table 2 reorts this quality measure for L based on different versions of ˆr, when (ρ, β) = (0.7, 0). Results corresonding to the other combinations of ρ and β are similar, and are omitted to save sace. Loss estimates L based on ED, ER, GR, and ABC work reasonably well, and better than those based on the other reliminary estimators. However, the choice between ED, ER, GR, and ABC is not obvious. 5 Emirical illustration In this section we illustrate our loss estimation methodology by an analysis of excess return data. A fundamental assumtion of the Arbitrage Pricing Theory is that excess returns admit an aroximate factor structure. Statistical and fundamental 24

25 factor models (Connor, 1995) use factors estimated from the excess return data itself or constructed using additional information, such as the book-to-rice and market value, resectively. The oular Fama-French three factor model is an examle of a fundamental model. Many studies of statistical factor models, including Connor and Korajczyk (1993), Huang and Jo (1995), Bai and Ng (2002), and Onatski (2010) find only two factors in the excess returns data. If both the Fama-French model and the statistical factor model are correct, the number of factors in the two models must be the same. Assuming that there are indeed three factors in the data, as the Fama-French model ostulates, what is the loss from not estimating the third factor in the statistical factor model? This is a question that we can answer using our loss estimator. We use monthly excess return data constructed from the stock rice CRSP data and the historical data on the 3-month T-bill rate. Our data set consists of 284 stocks listed on NYSE selected as follows. First, we selected all stocks for which the rice data were available for the entire eriod from Jan2001 to Dec2012. For each of these stocks, we comuted the transaction volume (the roduct of the share rice and the share volume), and sorted the stocks according to the value of the cumulative transaction volume for the entire eriod. We selected the relatively more actively traded stocks that together constituted 90% of the entire transaction volume. Finally, we eliminated all remaining stocks with standard deviations above three times the median standard deviation. This left us with 284 stocks. Assuming that these data have three factors, and that the PC method does not break down for the third factor so that q = 3, we can estimate the loss function L by L, L, ˆL(A=I), and ˆL (B=I) with the reliminary estimator ˆr equal to the ostulated q = 3. Since the stock return data are oorly redictable, but have nontrivial idiosyncratic cross-sectional correlation, the assumtion B = I T is lausible, whereas A = I n is not. Further, since L does not erform well in our MC exercises, we restrict attention to estimators L (B=I) and ˆL. The left anel of Figure 5 reorts L (solid line) and ˆL (B=I) (dotted line) normalized to the units of the samle variance of the ooled excess return data, and shifted so that L (B=I) 0 = ˆL 0 = 0. Both estimates of the loss function are minimized 25

26 Shifted Loss Shifted Loss 0 Jan2001 Dec Jan1989 Dec Figure 5: Estimated loss of the PC estimator of a factor model of excess returns. L (B=I) solid lines, ˆL dotted lines. ˆr is set to 3. at = 2, desite our forcing ˆr = 3. Moreover, estimating three instead of two factors wies out all the benefit obtained from estimating two rather than one factor. In fact, according to L estimate, the marginal gain from estimating two rather than one factors is less than half the marginal loss from estimating three rather than two factors. Of course, the reason why estimating three factors is undesirable in these data, even after assuming that q = 3, is that the PC estimator of the third factor is too noisy to be useful. Interestingly, the entire exercise reeated for the Jan1989-Dec2000 data yields different results. As the right anel of Figure 5 shows, for that time eriod, estimating three factors would have been beneficial from the oint of view of minimizing the loss. Note that the ga between the first three and the fourth samle covariance eigenvalues (shown as dots with abscissas 9) in the Jan1989-Dec2000 eriod was much larger than that in the Jan2001-Dec2012 eriod. This can be interreted as lower signal-to-noise ratio in the more recent eriod, which hurts the recision of the PC estimator and makes estimating the third factor useless. 26

27 6 Conclusion In this aer, we study the effect of missecification of the number of factors in aroximate factor models on the quadratic loss from the estimation of the common comonent. We derive asymtotic aroximations for the quadratic loss through the terms of order O (1/T ) O (1/n) under both weak and strong factors asymtotics. We develo several estimators of the loss, all of which are consistent under the strong factors asymtotics. The consistency under the weak factors asymtotics requires either no cross-sectional or no temoral correlation in the idiosyncratic terms. The estimators of the number of factors that minimize the roosed estimators are shown to be asymtotically loss effi cient. When the idiosyncratic terms exhibit both cross-sectional and temoral correlation and factors are weak, we derive uer and lower bounds on the loss. The minimizers of these bounds bracket the loss-minimizing number of factors with asymtotic robability one. Many imortant issues are not considered in this aer. As exlained in Bai and Ng (2008,.95), static factor models are suffi ciently flexible to accommodate dynamic factor models with loadings reresented by lag olynomials of fixed finite order. However, the generalized dynamic factor models introduced by Forni et al (2000) cannot be reresented in the form (1), and their study is left for future research. The quadratic loss from the estimation of the common comonent is not the only interesting loss that can be considered in the factor model context. A large number of alications of factor models are related to diffusion index forecasts (Stock and Watson, 2006). From the oint of view of these alications, a natural and interesting loss to consider would be the squared forecast error. Finally, this aer does not derive the standard errors of the roosed loss estimators. Finding these standard errors and roosing methods of their estimation remains an imortant task for future studies. 27

28 7 Aendix For any matrix A, let A denote the sectral norm of A, that is A equals the maximum singular value of A. Let e (j,k) = Λ jef k / d jn nt. Sulementary Aendix contains roofs of the following two auxiliary lemmas. Lemma 2 Under assumtions A1-A3, for j r, as n, T c, µ j (X X)/ (nt ) = d jn + 2 d jn / (nt )e (j,j) + F je ef j / ( nt 2) +Λ jee Λ j / ( d jn n 2 T ) + o P (1/T ). Let P j = F j ( F j F j ) 1 F j = F j F j/t be the rojection on the sace sanned by F j, and let P 0 be the rojection on the subsace of R T orthogonal to all columns of F. Similarly, let Q j = Λ j ( Λ j Λ j ) 1 Λ j be the rojection on the sace sanned by Λ j, and let Q 0 be the rojection on the subsace of R n orthogonal to all columns of Λ. Further, let ˆQ j = ˆΛ j (ˆΛ j ˆΛ j ) 1 ˆΛ j and let ˆP j = ˆF j ( ˆF j ˆF j ) 1 ˆF j. Lemma 3 Let k and j be integers such that 0 < k, j r. Then, under assumtions A1-A3, as n, T c, { ] (i) tr [P k ˆPj = { ] (ii) tr [Q k ˆQj = Proof of Proosition 1. 1 Λ jee Λ j / ( d 2 jnt n 2) + o P (1/T ) if k = j o P (1/T ) if k j, and 1 F je ef j / (d jn nt 2 ) + o P (1/T ) if k = j o P (1/T ) if k j. Oening brackets in (2) and using the definition of ˆΛ 1: and ˆF 1: and assumtion A1, we obtain L = tr[ˆλ ˆΛ 1: 1: ]/n + tr [Λ Λ] /n 2 tr[ˆλ 1: ˆF 1: F Λ ]/ (nt ) r r = µ j (X X/ (nt )) + d jn 2 (Λ k ˆΛ j )(F k ˆF j )/ (nt ). (22) j=1 j=1 28 k=1 j=1

29 To simlify (22), consider the identity (Λ k ˆΛ j )(F 2 = Λ k 2 ˆΛ j 2 tr[q k ˆQj ] F k 2 ˆF j 2 tr[p k ˆPj ] k ˆF j ) = d kn nt 2 µ j (X X/T ) tr[q k ˆQj ] tr[p k ˆPj ]. (23) For j r and j k, by Lemmas 2 and 3, µ j (X X/T ) tr[q k ˆQj ] tr[p k ˆPj ] = o P (1/T ). Therefore (Λ k ˆΛ j )(F k ˆF j )/ (nt ) = o P (1/T ). (24) This equality holds also for j > r. Indeed, according to a singular value analog of Weyl s eigenvalue inequalities (see Theorem of Horn and Johnson (1991)), for any n T matrices A and B, µ 1/2 ( i+s 1 (A + B) (A + B) ) µ 1/2 i (AA ) + µ s 1/2 (BB ), (25) where 1 i, s min {n, T }. Setting A = F Λ / nt, B = e / nt, i = r + 1, and s = j r, and noting that µ r+1 (F Λ ΛF /(nt )) = 0, we get µ j (X X)/ (nt ) µ j r (e e)/ (nt ). (26) Similarly, setting A = F Λ / nt, B = X / nt, i = r + 1, and s = j, we get µ j (X X)/ (nt ) µ j+r (e e)/ (nt ). (27) Further, for j > r, we have 0 tr[p k ˆPj ] tr[p k ˆPj ] + tr[p k (I T ˆP j ˆP k )] = 1 tr[p k ˆPk ]. Hence, by Lemma 3, tr[p k ˆPj ] = O P (1/T ). Similarly, tr[q k ˆQj ] = O P (1/T ). The latter two equalities together with (23) and the fact that, by (26), µ j (X X/T ) µ 1 (e e/t ) = O P (1) imly (24). Using (24) together with (22), we obtain L = µ j (X X/ (nt ))+ j=1 r j=1 min{,r} d jn 2 (Λ k ˆΛ k )(F k ˆF k )/ (nt )+o P (1/T ). (28) k=1 Now, by (23), (Λ k ˆΛ k )(F k ˆF k ) = ±(d kn nt 2 µ k (X X/T ) tr[q k ˆQk ] tr[p k ˆPk ]) 1/2. On the other hand, 29