Estimation of Structural Breaks in Large Panels with Cross-Sectional Dependence

Transcription

1 ISS X Department of Econometrics and Business Statistics Estimation of Structural Breaks in Large Panels with Cross-Sectional Dependence Jiti Gao, Guangming Pan and Yanrong Yang July 206 Working Paper 2/6

2 Estimation of Structural Breaks in Large Panels with Cross Sectional Dependence Jiti Gao and Guangming Pan and Yanrong Yang July 27, 206 Abstract This paper considers modelling and detecting structure breaks associated with crosssectional dependence for large dimensional panel data models, which are popular in many fields, including economics and finance. We propose a dynamic factor structure to measure the degree of cross-sectional dependence. The extent of such cross-sectional dependence is parameterized as an unknown parameter, which is defined by assuming that a small proportion of the total factor loadings are important. Compared with the usual parameterized style, this exponential description of extent covers the case of small proportion of the total sections being cross sectionally dependent. We establish a moment criterion to estimate the unknown based on the covariance of cross-sectional averages at different time lags. By taking into account the fact that the serial dependence of common factors is stronger than that of idiosyncratic components, the proposed criterion is able to capture weak cross sectional dependence that is reflected on relatively small values of the unknown parameter. Due to the involvement of some unknown parameter, both joint and marginal estimators are constructed. This paper then establishes that the joint estimators of a pair of unknown parameters converge in distribution to bivariate normal. In the case where the other unknown parameter is being assumed to be known, an asymptotic distribution for an estimator of the original unknown parameter is also established, which naturally coincides with the joint asymptotic distribution for the case where the other unknown parameter is assumed to be known. Simulation results show the finite sample effectiveness of the proposed method. Empirical applications to cross-country macro-variables and stock returns in SP500 market are also reported to show the practical relevance of the proposed estimation theory. Keywords: Cross-sectional averages; dynamic factor model; joint estimation; marginal estimation; strong factor loading. The corresponding author, Jiti Gao, Department of Econometrics and Business Statistics, Monash University, VIC 345, Australia jiti.gao@monash.edu. Guangming Pan, School of Physical and Mathematical Sciences, anyang Technological University, Singapore, GMPA@ntu.edu.sg. Yanrong Yang, Department of Econometrics and Business Statistics, Monash University, VIC 345, Australia yanrong.yang@monash.edu.

3 JEL Classification: C2, C32. Introduction The analysis of large dimensional panel data attracts ever-growing interest in some modern scientific fields, especially in economics and finance. Cross sectional dependence is common in large dimensional panel data analysis and the literature focuses on testing the existence of cross-sectional dependence. A survey on description and testing of cross-sectional dependence is given in Sarafidis and Wansbeek 202. Pesaran 2004 utilizes sample correlations to test cross-sectional dependence while Baltagi, Feng and Kao 202 extend the classical Lagrangian multiplier LM test to the large dimensional case. Chen, Gao and Li 202 and Hsiao, Pesaran and Pick 202 consider cross-sectional dependence tests for nonlinear econometric models. When more and more cross sections are grouped together, cross-sectional dependence appears to be quite natural and common. Cross-sectional independence is an extreme hypothesis. Rejecting such a hypothesis does not provide much information about the relationship between different cross sections under investigation. In view of this, measuring the degree of crosssectional dependence is more important than just testing for its presence. As we know, in comparison with cross sectional dependence tests, the literature contributes very little to accessing the extent of cross-sectional dependence. g 2006 uses spacings of cross-sectional correlations to exploit the ratio of correlated subsets over all sections. Bailey, Kapatanios and Pesaran 205 use a factor model to describe cross-sectional dependence and develop estimators that are based on a moment method. We will contribute to the descriptions and measures of the extent of cross-sectional dependence for large dimensional panel data with cross-section units and T time series. The first natural question is: how to describe the cross-sectional dependence? To deal with this issue, the panel data literature mainly discusses two different ways of modelling cross-sectional dependence: the spatial correlation and the factor structure approach see, for example, Sarafidis and Wansbeek 202. In this paper, we use the factor structure approach. The factor model is not only a powerful tool to characterize cross-sectional dependence for economic and financial data, but also efficient in dealing with statistical inference for high dimensional data from a dimension-reduction point of view. Some related work includes Fan, Fan and Lv 2008, Fan, Liao and Mincheva 20 and Pan and Yao With respect to factor structures, there are two common types of factor models: the static model and the dynamic model. The static model is defined as x it µ i + β if t + u it, i, 2,..., ; t, 2,..., T. In this model, {µ i, i, 2,..., } represent the means of sections, the components of the process {f t, t, 2,..., T } are called the common shocks or factors, and β i is a vector of factor loadings for unit i on the common factors f t for each i, 2,...,. The term static factor 2

4 model refers to the static relationship between x it and f t, but f t itself can be a dynamic process. The dynamic factor model is written as x it µ i + β ilf t + u it, i, 2,..., ; t, 2,..., T, where β i L is a vector of dynamic factor loadings of order s, i.e β i L β i0 + β i L + β i2 L β is L s. If s is finite, the model is called a dynamic factor model. If s can be infinity, it is called a generalized dynamic factor model. Under each case, f t CLε t, where {ε t, t Z} are independent and identically distributed i.i.d and CL is a coefficient matrix with time lags, i.e., {f t, t Z} is a dynamic process. Based on the concept of a factor model, the extent of crosssectional dependence in observed data x it can be reflected in the strength of factor loadings, and the cross-sectional dependence is caused by common factors f t. While it is a rare phenomenon to have cross-sectional independence for all sections, it is also unrealistic to assume that all sections are dependent. As cross-sectional dependence can be reflected in factor loadings, we impose some conditions on factor loadings in order to derive one part that contains cross-sectional dependent units and another part that includes cross-sectional independent sections. The simplest method is to assume that some factor loadings are bounded away from zero while others are around zero. In this paper, we assume that only [ α 0 ]0 α 0 of all factor loadings are individually important. This topic is quite related to the structural break literature developed in the econometrics literature. Bai 997 investigates multiple structural breaks in the mean for a single linear process, while Bai and Perron 998 estimate breaks in the coefficients of a linear model. Bai 200 considers common breaks in the mean for panel data and develops breakpoint analysis under large dimensional cases. By contrast, as far as we know, the literature that addresses the structural breaks for cross-sectional dependence is relatively limited. g 2006 is one of the few that contributes to breakpoint analysis of cross sectional dependence using the spacings. Instead of measuring the extent by α 0, we adopt the parameterization [ α 0 ]. The proportion of [ α 0 ] over the total is quite small which tends to 0 as 0 < α 0 <, while α 0 is comparable to because of the same order. In this sense, our model covers some sparse cases that only a small part of the sections are cross-sectionally dependent. With this description of the extent of cross-sectional dependence, the goal is directed to propose an approach to estimation of α 0. One advantage of our proposed statistic is based on an assumption for identification. There are several different ways of identifying factor models. Bailey, Kapatanios and Pesaran 205 assume that {x t x t, x 2t,..., x t : t, 2,..., T } is a high dimensional time series and decompose it into two parts: a common-factor part and an idiosyncratic part, both of which are weakly dependent stationary time series. The identification of their model lies on the assumption that cross-sectional dependence involved in the idiosyncratic part is weaker than that in the common-factor part. This assumption is common in the literature, e.g. Bai and g 2002, Fan, Fan and Lv 2008 and Fan, Liao and Mincheva 20. However, we consider an alternative factor model, similar to the idea of Lam and Yao 202, 3

5 which consists of two parts: the common-factor part driven by a lower-dimensional factor time series and the idiosyncratic part which is a stationary time series with relatively weaker serial dependence than the common-factor part. While the literature makes use of distinctive degrees of cross-sectional dependence in common components and idiosyncratic components respectively, we utilize distinctive extents of serial dependence in these two parts to attain identifications. From a point of replacing one condition by another in identification, our assumptions are quite weak. Moreover, one important advantage is that the new model identification condition leads to our proposed methodology for estimation of the exponents of cross-sectional dependence, which can eliminate the influence of idiosyncratic components in the estimation. The proposal of our estimation procedure is outlined as follows. An estimator for α 0 is proposed by calculating the covariance between x t and x t+τ for a larger range of α 0, i.e., 0 α 0, where x t i x it and τ > 0. Under the setting and structure of this paper, furthermore, we have weaker serial dependence in the idiosyncratic part than that in the common part. Then the leading term in cov x t, x t+τ will not contain the idiosyncratic part when τ tends to infinity. In other words, the idiosyncratic components do not bring any noise term to the proposed criterion. The main contribution of this paper is summarized as follows:. We construct two consistent estimators for α 0 by utilizing both joint estimation and marginal estimation respectively. As the parameter κ 0 involved in the proposed criterion is unknown, the joint estimation of α 0 and κ 0 is adopted. Otherwise, we use the marginal estimation for α We have been able to establish new asymptotic distributions for both the joint and the marginal estimators. The asymptotic marginal distribution coincides with that for the joint estimator for the case where κ 0 is assumed to be known. Finite sample performances of these two kinds of estimators are provided. 3. An additional contribution involves generalizing Theorem of Anderson 994. We establish a new central limit theorem for the sample covariance of a time series under the case where both the time lag and the sample size tend to infinity simultaneously. The rest of the paper is organized as follows. The model and the main assumptions are introduced in Section 2. Section 3 proposes both joint and marginal estimators that are based on the second moment criterion. Asymptotic properties for these estimators are established in Section 4. Section 5 reports the simulation results, which illustrate the effectiveness of the proposed methods. Section 6 provides empirical applications to cross-country macro-variables and stock returns in S&P 500 market respectively. Conclusions are included in Section 7. Justification for the assumptions and all the mathematical proofs are given in Appendices B and C. 4

6 2 The model Let x it be a double array of random variables indexed by i,..., and t,..., T, over space and time, respectively. The aim of this paper is to measure the extent of the crosssectional dependence of the data {x it : i,..., }. In panel data analysis, there are two common models to describe cross-sectional dependence: spatial models and factor models. In Bailey, Kapatanios and Pesaran 205, a static approximate factor model is used. As an extension, we consider a dynamic factor model: x it µ i + β i0f t + β if t + + β isf t s + u it µ i + β ilf t + u it, i, 2,..., ; t, 2,..., T, 2. where f t is the m vector of unobserved factors with m being fixed, β i L β i0 + β i L + β i2 L β is L s, in which β il β il, β il2,..., β ilm, l 0,,..., s are the associated vectors of unobserved factor loadings and L is the lag operator, here s is assumed to be fixed, and µ i, i, 2,..., are constants that represent the mean values for all sections, and {u it : i,..., ; t,..., T } are idiosyncratic components. Clearly, we can write 2. in the static form: x it µ i + β if t + u it, i, 2,..., ; t, 2,..., T, 2.2 where β i β i0 β i. and F t f t f t.. β is ms+ f t s ms+ The dimension of f t is called the number of dynamic factors and is denoted by m. Then the dimension of F t is equal to r ms +. In factor analysis, β if t is called the common components of x it. We first introduce the following assumptions. Assumption. The idiosyncratic component {u t u t, u 2t,..., u t : t, 2,..., T } follows a linear stationary process of the form: u it + φ ij + j0 s ξ js ν j,t s, 2.3 5

7 where {ν is : i...,, 0,,... ; s 0,,...} is a double sequence of i.i.d. random variables with mean zero and unit variance, and Moreover, sup 0<j<+ where γ 2 i j satisfies φ ij < + and sup i + 0<j<+ s ξ js Eu it u j,t+τ γ τγ 2 i j, 2.5 γ 2 i j O 2.6 i,j and γ τ satisfies the condition 2.9 in Assumption 3 below. Assumption 2. For l 0,, 2,..., s and k, 2,..., m, β ilk v ilk, i, 2,..., [ α lk ] and i[ α lk ]+ β ilk O, 2.7 where [ α lk ] α lk is the largest integer part of α lk, 0 < αlk and v ilk i.i.d.µ v, σ 2 v has finite sixth moment, with µ v 0 and σ 2 v > 0. Moreover, {v ilk : i, 2,..., ; l 0,,..., s; k, 2,..., m} are assumed to be independent of the factors {f t : t, 2,..., T } and the idiosyncratic components {u it : i, 2,..., ; t, 2,..., T }. Assumption 3. The factors {f t, t, 2,..., T } are covariance stationary with the following representation: f t b j ζ t j, 2.8 j0 where {ζ t, t...,, 0,,...} is an i.i.d sequence of m-dimensional random vectors whose components are i.i.d with zero mean and unit variance, the fourth moments of {ζ t, < t < } are finite, and the coefficients {b j : j 0,, 2,...} satisfy j0 b j <. Furthermore, the unobserved factors {f t : t, 2,..., T } are independent of the idiosyncratic components {u t : t, 2,..., T }. Let γk, h Ef k,t f k,t+h and α 0 max l,k α lk. We assume u,f : where γ τ is defined in 2.5. γ τ [ 2α 0 2 ] s m l,l 2 0 k γk, τ l 2 + l o, 2.9 Let us briefly discuss how to verify 2.9 using a simple example. Consider the following model: x it µ i + β i f t + u it, i, 2,...,, t, 2,..., T, 2.0 6

8 where the factor loadings {β i : i, 2,..., } satisfy Assumption 2, the factor process {f t : t, 2,..., T } is AR, i.e., f t ρ f t + ε t for t, 2,..., T, and the idiosyncratic components u it can be decomposed into two independent parts: the serially correlated part and the cross section part, i.e., u it ζ t η i, with {ζ t : t, 2,..., T } being an AR, i.e., ζ t ρ 2 ζ t + ɛ t, t, 2,..., T. Moreover, {ε t : t, 2,..., T } and {ɛ t : t, 2..., T } are both white noises with zero mean and unit variance, and mutually independent. For model 2.0, it is easy to derive the values of γ τ, γ 2 i j and γ, τ defined in Assumption 2 and Assumption 3, i.e., γ τ ρ 2 τ, γτ ρ τ and Eη ρ 2 ρ 2 i η j γ 2 i j. Condition 2.9 is then u,f : ρ 2 τ ρ 2 o. It is equivalent to requiring that ρ [ 2α 0 2 ] ρ τ ρ 2 2 and ρ 2 are related by ρ 2 o[ 2α 0 τ ] ρ. We can then see that, if < α 2 0 <, τ can be taken as a constant. If 0 < α 0 < 2, τ should tend to + and ρ 2 should be smaller than ρ. Detailed justifications of Assumptions -3 are given in Appendix A. 3 The estimation method The aim of this paper is to estimate the exponent α 0 max l,k α lk, which describes the extent of cross-sectional dependence. As in Bailey, Kapatanios and Pesaran 205 BKP5, we consider the cross-sectional average x t / i x it and then derive an estimator for α 0 from the information of { x t : t, 2,..., T }. BKP5 use the variance of the cross-sectional average x t to estimate α 0 and carry out statistical inference for an estimator of α 0 before they show that V ar x t κ 0 [ 2α 0 2 ] + c + O α 0 2, 3. where κ 0 is a constant associated with the common components and c is a bias constant incurred by the idiosyncratic errors. From 3., we can see that, in order to estimate α 0, BKP5 assume that 2α 0 2 >, i.e. α 0 > /2. Otherwise, the second term will have a higher order than the first term. So the approach by BKP5 will fail in the case of 0 < α 0 < /2. In this paper, we propose a new estimator that is applicable to the full range of α 0, i.e., 0 α 0. Based on the assumption that the common factors possess serial dependence that is stronger than that of the idiosyncratic components, we construct a so called covariance criterion Cov x t, x t+τ, whose leading term does not include the idiosyncratic components for 0 α 0. In other words, the advantage of this covariance criterion over the variance criterion V ar x t lies on the fact that there is no interruption brought by the idiosyncratic components {u it : i, 2,..., ; t, 2,..., T } in Cov x t, x t+τ. ext, we illustrate how the covariance Cov x t, x t+τ implies the extent parameter α 0 in detail. ote that [ a ] a a 0 denotes the largest integer part. For simplicity, let [ b ] b 0 denote. Moreover, to simplify the notation, throughout the paper we also use the following [ b ] notation: [ ka ] : [ a ] k, [ a k ] : [ a ], a, k R. 3.2 k 7

9 But we would like to remind the reader that [ ka ] is actually not equal to [ a ] k. ext, we will propose an estimator for α 0 under two different scenarios: the joint estimator under the case of some other parameters being unknown while the marginal estimator for the case of some other parameters being known. 3. The joint estimator α, κ The joint estimator in this section is proposed when all the parameters involved are unknown. The marginal estimator proposed in the following section deals with the case where only α 0 is unknown. Without loss of generality, we assume that α lk α 0, l 0,, 2,..., s; k, 2,..., m. Let Assumption 2 hold. Let x nt be the cross sectional average of x it over i, 2,..., n with n. Similarly, β nlk : n n i β ilk. Then and E β { nlk µ v, n [ α 0 ] [ α 0 ] n v + K nlk n n > [ α 0 ], V ar β nlk { σ 2 v n, n [ α 0 ] [ α 0 ] n 2 σ 2 v, n > [ α 0 ], where K nlk n i[ α 0 ]+ β ilk. Then, we have Cov x nt, x n,t+τ s m l0 k + s l l 2 k E 2 β nlk + V ar β nlk Ef k,t l f k,t+τ l m E β nl ke β nl2 kef k,t l f k,t+τ l2 { κ 0 + On, n [ α 0 ] κ 0 [ 2α 0 ] n 2 + O [ α 0 ] n 2, n > [ α 0 ], 3.3 where s m κ 0 µ 2 v Ef k,t l f k,t+τ l2, 3.4 l,l 2 0 k in which µ v E[v ilk ]. Minimize the following quadratic form in terms of α and κ: [ α] α, κ 2 n ˆσ 3 n τ κ + Q n n[ α ]+ n 3 ˆσ n τ [ 2α ] 2, κ 3.5 n 2 8

10 where ˆσ n τ is a consistent estimator for Cov x nt, x n,t+τ of the form: ˆσ n τ T t xnt x n with x n T T t x nt and x 2 n T T t x n,t+τ. Then, the first order condition for κ is Q α, κ κ which is equivalent to [ α ] n ˆσ n3 n τ κ This derives κ κα : [ α ] We now introduce the additional expressions: 0, xn,t+τ x 2 n, + [ 2α ] n[ α ]+ n ˆσ n τ [ 2α ] κ 0. n 2 n n3ˆσ n τ + n[ α ]+ n[ 2α ]ˆσ n τ [ α ] n n [ 4α ] n[ α ]+ n [ α ] ˆq α n 3ˆσ n τ, ˆq 2 α n [ α ] α n 3 + n With this and 3.6, we can obtain Then n[ α ]+ [ α ] Q α, κ Q + κ 2 n 3 + κ 2 [ 4α ] n[ α ]+ [ 4α ] n, Q nˆσ n τ, n 3ˆσ nτ. 2 κ ˆq α + [ 2α ]ˆq 2 α. 3.7 α n n[ α ]+ n n 2κˆq α 2κ[ 2α ]ˆq 2 α Q + κ 2 α 2κ ˆq α + [ 2α ]ˆq 2 α ˆq Q α + [ 2α ]ˆq 2 α α Since Q does not depend on α, minimizing Q α, κ is equivalent to maximizing the term: ˆQ ˆq α + [ 2α ]ˆq 2 α 2 α. α In summary, the joint estimator α, κ can be obtained by α arg max α ˆQ ˆq α + [ 4 α ]ˆq 2 α α and κ. 3.9 α This joint estimation method estimates α 0 and κ 0 simultaneously. The above derivations show that it is easy to derive α and then κ. Of course, we can also use some other estimation methods to estimate κ 0 and then α 0. otice that we use the weight function wn n 3 in each summation part of the objective function Q α, κ of 3.5. The involvement of a weight function is due to technical necessity in deriving an asymptotic distribution for α, κ. 9

11 3.2 The marginal estimator α Although, for simplicity, the first [ α 0 ] sections are assumed to possess important factor loadings, the proposed marginal estimation procedure does not rely on the specification of the two categories in the sequence of the sections. From Assumption 2, we have β lk i β ilk [ α lk ] [ α lk ] i v ilk + [ α lk ] i[ α lk ]+ : [ α lk ] v lk + K lk, 3.0 [ α lk ] i v where v lk ilk and K [ α lk ] lk i[ α lk ]+ β ilk. A direct calculation then yields E β lk µ v [ αlk ] + O and β ilk V ar β lk 2 [ α lk ]σ 2 v [ α lk 2 ]σ 2 v. 3. Under model 2., it follows that for any t, 2,...,, x t E x t s,m x t+τ E x t+τ where x t i x it and ū t i u it. By 3.2, Assumptions and 2, we have l0,k s,m β lk f k,t l + ū t, l0,k β lk f k,t+τ l + ū t+τ, 3.2 Cov x t, x t+τ + s l,l 2 0 k s m l0 k s l l 2 k m E β β l k l2 kef k,t l f k,t+τ l2 + Eū t ū t+τ 3.3 E 2 β lk + V ar β lk Ef k,t l f k,t+τ l m E β l ke β l2 kef k,t l f k,t+τ l2 + Eū t ū t+τ. Substituting 3.0 into 3.3 ensures Cov x t, x t+τ + s l l 2 k + 2 m s m l0 k µ 2 v[ 2αlk 2 ] + O[ αlk 2 ] + O 2 Ef k,t l f k,t+τ l µ 2 v[ α l k+α l2 k 2 ] + O[ α l k 2 ] + O[ α l 2 k 2 ] + O 2 E u it u j,t+τ. i,j Ef k,t l f k,t+τ l2 0

12 Below we consider the case of α lk α 0, l 0,,..., s; k,..., m. Then Cov x t, x t+τ becomes Cov x t, x t+τ s m l,l 2 0 k + 2 µ 2 v[ 2α0 2 ] + O[ α0 2 ] + O 2 Ef k,t l f k,t+τ l2 E u it u j,t+τ. 3.4 i,j We then compare the orders of the two terms s l,l 2 0 m k µ2 v[ 2α 0 2 ]Ef k,t l f k,t+τ l2 and Eū t ū t+τ. From Assumption, we have Eū t ū t+τ 2 i,j where we have used Condition 2.6. By Condition 2.9, we have Eu it u j,t+τ 2 γ τ γ τγ 2 i j O, i,j Eū t ū t+τ s l,l 2 0 m k µ2 v[ 2α 0 2 ]Ef k,t l f k,t+τ l2 A simple manipulation of 3.4 and 3.5 yields 2 ln Cov x t, x t+τ lnκ α 0 4 ln, o. 3.5 which implies α 0 2 ln Cov x t, x t+τ lnκ ln +, 3.6 where κ 0 is defined in 3.4. Hence, for 0 α 0, α 0 can be estimated from 3.6 using a consistent estimator for Cov x t, x t+τ given by σ τ T x t x x t+τ x 2, 3.7 t where x T T t x t and x 2 T T t x t+τ. Thus, a consistent estimator for α 0 is given by α 2 log σ τ lnκ0 2 4 ln If {u t : t,..., T } are independent, the term Eū t ū t+τ will disappear in 3.4 for any τ. So under this case, we can take a finite lag τ. Furthermore, if not all α lk are equal to α 0, we can still get an expression similar to 3.6 but with a different value of κ 0, which can be estimated by the joint estimation method given in the previous section.

13 3.3 Asymptotic Properties In this section, we will establish asymptotic distributions for the proposed joint estimator α, κ and the marginal estimator α, respectively. We assume that α lk α 0, l 0,,..., s and k, 2,..., m for simplicity. The notation a b denotes that a Ob and b Oa. For any i, j m and 0 h T, we define C ij h T h f i,t f j,t+h, c ij h σ ij h Ef i,t f j,t+h. 3.9 T h t The following theorem establishes an asymptotic distribution for the joint estimator α, κ. Theorem. In addition to Assumptions -3, we assume that i for some constant δ > 0, E ζ it 2+2δ < +, 3.20 where ζ it is the i-th component of ζ t and {ζ t :...,, 0,,...} is the sequence appeared in Assumption 3. ii The lag τ satisfies τ 0, as T, 3.2 δ/2δ+2 where δ is defined in iii The covariance matrix Γ of the random vector C ij h : i,..., m; j,..., m; h τ s,..., τ + s 3.22 is positive definite. iv As, T,, min[ α 0], γ τ 2α 0 max, /2 2α0 0, as 0 < α µ vσ τ µ v µ 0 vσ τ µ v 2 ; min[ α 0], /2 α 0 /2 0, as µ vσ τ µ v 2 < α 0 ; and as, T,, Then as, T,, 3.23 α 0/2 µ vσ τ µ v o [ 2 α α 0 /2 ] O P v κ 0, 3.25 /2 κ κ 0 O P v,

14 where v min[ α 0 ],, and κ v 2 α α 0 v κ κ 0 d 0 0, 4σ2 0 2σ0 2 2σ0 2 σ0 2, 3.27 where κ 0 is defined in 3.4, Σ τ EF t F t+τ and µ v µ v e ms+, in which e ms+ is an ms + vector with each element being, σ lim,t min[ α 0 ], lim 4µ,T [ α 0 ] vσ τ Σ v Σ τ µ v 3.28 min[ α 0 ], µ v µ vωµ v µ v and ωτ, τ ωτ, τ + s ωτ, τ s ωτ, τ ωτ +, τ ωτ +, τ + s ωτ +, τ s ωτ +, τ Ω , 3.29 ωτ s, τ ωτ s, τ + s ωτ s, τ s ωτ s, τ ωτ, τ ωτ, τ + s ωτ, τ s ωτ, τ with ωh, r Covf i tf j,t+h, f i2,tf j2,t+r : i, j, i 2, j 2 m. m 2 m 2 While Theorem may just establish a bivariate normal distribution with a singular covariance, it provides a joint distributional structure. We briefly show how to verify Conditions 3.23 and 3.24 based on the simple model 2.0. For model 2.0, µ vσ τ µ v µ 2 v ρ. Then and 3.24 are equivalent to the following three cases: i 0 < α 0, [ α 0 ρ ] <, 2 τ 2 ρ 2α 0 ρ o, o, ρ τ T α 0 α 0 o. 2 ii < α 2 0, [ α 0 ρ ] <, ρ o, α 0 T 2 α 0 o. 2 iii < α 2 0 <, [ α 0 ], ρ o. α 0 2 For each of these three cases, we provide a choice of T, and ρ 2. a 0 < α 0 2, [ α 0 ] <, [ ρ + δ 2τ α 0 ], T [ ρ + δ 4τ α 0 4τ ] + τ, ρ 2 ρ 2 [ 3 2 α 0 τ ]. b < α 2 0, [ α 0 ] <, [ ρ + δ 2τ α 0 ], T [ ρ + δ 4τ α 2τ 0 ] + τ. c < α 2 0, [ α 0 ], [ ρ α + δ 0 2 ], T [ ρ α + δ ] + τ, where 0 < δ 0 < δ, and δ > 0 is a constant. The following theorem establishes an asymptotic distribution for the marginal estimator α. Theorem 2. Under the conditions of Theorem, we have where σ 2 0 is defined in τ min[ α 0], 4 α α 0 κ 0 4σ 2 0 0,, ρ τ τα 0

15 Theorem establishes some asymptotic properties for the joint estimator α, κ. This result is consistent with that for the marginal estimator α derived in Theorem 2. From Theorem 2, one can see that α is a consistent estimator of α 0. Moreover, by a careful inspection on the proof of Theorem 2 one can see that Condition 3.23 is not needed to ensure the consistency of α under, T,. When the idiosyncratic components are independent, we can just use a finite lag τ instead of requiring τ. In this case, an asymptotic distribution for the estimator α is established in the following theorem. Theorem 3. In addition to Assumptions 2 and 3, suppose that τ is fixed and the following conditions i iii hold: i {u t : t,..., T } are independent with the mean of u t being 0 and its covariance matrix being Σ u, where 0 is an vector with zero components and the spectral norm Σ u is bounded. ii iii µ vσ τ µ v 0. min[ α 0], 2α 0 min[ α 0], /2 α 0 Then, as, T,, we have /2 0, as 0 < α 0 < 2 ; 3.3 /2 0, as 2 < α 0. min[ α 0], ln 2 α α 0 σ 2 0 /κ 2 0 d 0,, where κ 0 and σ 2 0 are defined in 3.4 and 3.28, respectively. Before we will give the proofs of Theorems 3 in Appendices B and C below, we have some brief discussion about Condition 3.3, which is actually equivalent to the following three cases: a 0 < α 0 2, [ α 0 ] <, 3 2 α 0 T 2 α 0 2 b 2 < α 0, [ α 0 ] < ; 2 T 2 o; o; c 2 < α 0, [ α 0 ], 2 α 0 o. Under these three cases, we can provide some choices for, T as follows: d 0 < α 0 < 2, [ α 0 ] < ; T τ + [ 2 3α 0 log]; f 2 < α 0, [ α 0 ] <, T τ + [ α 0 log]; g 2 < α 0, [ α 0 ], T τ + [ α 0 / log]. When τ, the term µ vσ τ µ v will tend to 0, because of Σ τ 0. So, as τ is very large, the value of lnµ vσ τ µ v may be negative in practice. Hence Theorem 2 provides an alternative 4

16 form for the asymptotic distribution of α α 0 instead of α α 0, and the case of τ being fixed is discussed in Theorem 3. We now evaluate the finite sample performance of the proposed estimation methods and the resulting theory in Sections 4 and 5 below. 4 Simulation 4. Data Generating Process First, we consider the following two-factor static model x it µ + β i f t + β i2 f 2t + u it, i, 2,..., ; t, 2,..., T. 4. The factors are generated by f jt ρ j f j,t + ρ 2 j ζ jt, j, 2; t 49, 48,..., 0,,..., T, 4.2 with f j, 50 0 for j, 2 and ζ jt i.i.d 0,. The idiosyncratic components are generated by in which η i i.i.d 0,, u it ε t η i for i, 2,, and t, 2,..., T, 4.3 ε t ρ 2 ε t + ɛ t, t, 2,..., T 4.4 and ɛ t i.i.d 0,, and {ɛ t : t, 2,..., T } are independent of {ζ jt : t, 2,..., T ; j, 2}. The factor loadings are generated as β ir v ir, for i, 2,..., M; r, 2; β ir ρ i M, for i M +, M + 2,..., ; r, 2, 4.5 where v ir i.i.d U0.5,.5, M [ α 0 ] and ρ 0.5. Moreover, we set µ and ρ j 0.5 for j, Data Generating Process 2 Second, we consider a dynamic model as follows. x it µ + β i,0 f t + β i,02 f 2t + β i, f,t + β i,2 f 2,t + u it, i,..., ; t,..., T, 4.6 where µ, and the factor loadings are generated as β i,jk v i,jk, for i, 2,..., M k ; β i,jk ρ i M k, for i M k +, M k + 2,...,, 5

17 for j 0,, k, 2, M k [ α 0 ] and ρ 0.5, in which v i,jk i.i.d U0.5,.5. The generating procedures for f t and u it are the same as those in Data Generating Process. The factor loadings are generated as 4.5 in the first static model. For the two data generating processes, we consider values of α 0 0.2, 0.4, 0.6, 0.8,, 00, 200, 500, 000 and T 00, 200, 500. All the experiments are based on 500 replications. For each replication, the values of α, ρ, ρ 2 are given as above. These parameters are fixed across all replications. The values of v ji, j, 2 are drawn randomly for each replication. The bias and root mean square error RMSE results for the marginal estimator α and joint estimator α are summarized in Tables -4, and show that the proposed estimation methods work well numerically. 5 Empirical applications In this section, we show how to obtain an estimate for the exponent of cross sectional dependence, α 0, for each of the following panel data sets: quarterly cross-country data used in global modelling and daily stock returns on the constitutes of Standard and Poor 500 index. 5. Cross-country dependence of macro-variables We provide an estimate for α 0 for each of the datasets: Real GDP growth RGDP, Consumer price index CPI, ominal equity price index OMEQ, Exchange rate of country i at time t expressed in US dollars FXdol, ominal price of oil in US dollars POILdolL, and ominal short-term and long-term interest rate per annum Rshort and Rlong computed over 33 countries. The observed cross-country time series, y it, over the full sample period, are standardized as x it y it ȳ i /s i, where ȳ i is the sample mean and s i is the corresponding standard deviation for each of the time series. Table 5 reports the corresponding results. For the standardized data x it, we regress it on the cross-section mean x t i x it, i.e., x it δ i x t + u it for i, 2,...,, where δ i, i, 2,...,, are regression coefficients. With the availability of the OLS estimate δ i for δ i, we have the estimated versions, û it, of the form: û it x it ˆδ i x t. Since our proposed estimation methods rely on the different extent of serial dependence of the factors and idiosyncratic components, we provide some autocorrelation graphs of { x t i x it : t, 2,..., T } and {ū t i u it : t, 2,..., T } for each group of the real dataset under investigation see Figures 4. From these graphs, it is easy to see that CPI, OMEQ, FXdol and POILdolL have distinctive serial dependence in the factor part x t and idiosyncratic part ū t. All the observed real data x it are serially dependent. Figures -4 near here The datasets are downloaded from 6

18 Due to existence of serial dependence in the idiosyncratic components, we use the proposed second moment criterion. The marginal estimator α and the joint estimator α for these real data are provided in Table 5. We use τ 0 for two estimators. We can see from Table 5 that the values of α and α are different from the those provided by Bailey, Kapatanios and Pesaran 205. Some estimated values are not. This phenomenon implies that a factor structure might be a good approximation for modeling global dependencies, and the value of α 0 typically assumed in the empirical factor literature might be exaggerating the importance of the common factors for modelling cross-sectional dependence at the expense of other forms of dependencies that originate from trade or financial inter-linkage that are more local or regional rather than global in nature. Furthermore, note that our model is different from that given by Bailey, Kapatanios and Pesaran 205 BKP5 and difference mainly lies on that our model only imposes serial dependence on factor processes and assumes that the idiosyncratic errors are independent. Different models may bring in different exponents. Table 5 near here 5.2 Cross-sectional exponent of stock-returns One of the important considerations in the analysis of financial markets is the extent to which asset returns are interconnected. The classical model is the capital asset pricing model of Sharp 964 and the arbitrage pricing theory of Ross 976. Both theories have factor representations with at least one strong common factor and an idiosyncratic component that could be weakly cross-sectional correlated see Chamberlain 983. The strength of the factors in these asset pricing models is measured by the exponent of the cross-sectional dependence, α 0. When α 0, as it is typically assumed in the literature, all individual stock returns are significantly affected by the factors, but there is no reason to believe that this will be the case for all assets and at all times. The disconnection between some asset returns and the market factors could occur particularly at times of stock market booms and busts where some asset returns could be driven by some non-fundamentals. Therefore, it would be of interest to investigate possible time variations in the exponent α 0 for stock returns. We base our empirical analysis on daily returns of 96 stocks in the Standard & Poor 500 S&P500 market during the period of January, 20-December, 202. The observations r it are standardized as x it r it r i /s i, where r i is the sample mean of the returns over all the sample and s i is the corresponding standard deviations. For the standardized data x it, we regress it on the cross-section mean x t i x it, i.e., x it δ i x t + u it for i, 2,...,, where δ i, i, 2,...,, are the regression coefficients. Based on the OLS estimates: δ i for δ i, we define û it x it ˆδ i x t. The autocorrelation functions ACFs of the cross-sectional averages x t i x it and ū t i u it are presented in Figure 5. Figure 5 near here 7

19 From Figure 5, we can see that the serial dependences of the common factor components are stronger than those of the idiosyncratic components. We use the estimates α and α to characterize the serial dependences of the common factors and the idiosyncratic components. The estimates α and α are calculated with the choice of τ 0. Table 6 reports the estimates with several different sample sizes. As comparison, the estimates from BKP5 are also reported. From the table, we can see that their estimation method does not work when α is smaller than /2. The results also show that the cross-sectional exponent of stock returns in S&P500 are smaller than. This indicates the support of using different levels of loadings for the common factor model as assumed in Assumption 2, rather than using the same level of loadings in such scenarios.. Table 6 near here Furthermore, Figure 6 provides the marginal estimate α and the joint estimate α for the first 30 days of all the period. It shows that the estimated values for α 0 with the two methods are quite similar. On the other hand, since a 30-day period is short, meanwhile, it is reasonable that the estimates didn t change very much. Figure 6 near here 6 Conclusions and discussion In this paper, we have examined the issue of how to estimate the extent of cross-sectional dependence for large dimensional panel data. The extent of cross-sectional dependence is parameterized as α 0, by assuming that only [ α 0 ] sections are relatively strongly dependent. Compared to the estimation method proposed by BKP5, we have proposed using a dynamic factor model to characterize the extent of inter-connections in large panel data and developed a new moment method to estimate α 0. In detail, based on the assumption that stronger serial dependence exists in the factor process than that for the idiosyncratic errors, we have recommended the use of the covariance function between the cross-sectional average values of the observed data at different lags to estimate α 0. One main advantage of this new approach is that it can deal with the case of 0 α 0 /2. Due to some unknown parameters involved in the panel data model, in addition to the proposed marginal estimator, we have also construct a joint estimation method for α 0 and the related unknown parameters. The asymptotic properties of all the estimators have all been established. The simulation results and an empirical application to two datasets have shown that our estimation methods work well numerically. Future research includes discussions about how to estimate factors and factor loadings in factor models, and determine the number of factors for the case of 0 < α 0 <. Existing 8

20 methods available for factor models, such as Bai and g 2002, Bai 2003, Onatski 2009, for the case of α 0, may not be applicable, and should be extended to deal with the case of 0 < α 0 <. Such issues are all left for future work. 7 Acknowledgements The first and the third authors acknowledge the Australian Research Council Discovery Grants Program for its support under Grant numbers: DP & DP Thanks from the second author also go to the Ministry of Education in Singapore for its financial support under Grant # ARC 4/. References T.W. Anderson 994. The statistical analysis of time series. London: Wiley 2th edtion. J. S. Bai 997. Estimating multiple breaks one at a time. Econometric Theory 33, J. S. Bai, P. Perron 998. Estimating and testing linear models with multiple structural changes. Econometrica 66, J. S. Bai, S. g Determine the number of factors in approximate factor models. Econometrica 70, J. S. Bai Inferential theory for factor models of large dimensions. Econometrica 7, J. S. Bai 200. Common breaks in means and variances for panel data. Journal of Econometrics 57, Bailey, G. Kapetanios, M.H. Pesaran 205. Exponent of cross-sectional dependence: estimation and inference. Available at Journal of Applied Econometrics DOI: 0.002/jae B.H. Baltagi, Q. Feng, C. Kao 202. A large multiplier test for cross-sectional dependence in a fixed effects panel data model. Journal of Econometrics 70, J.M. Bardet, P. Doukhan, J.R. León A functional limit theorem for η-weakly dependent processes and its applications. Statistical Inference for Stochastic Processes 3, G. Chamberlain 983. Funds, factors and diversification in arbitrage pricing theory. Econometrica 55, J. Chen, J. Gao, D. G. Li 202. A new diagnostic test for cross-section uncorrelatedness in nonparametric panel data models. Econometric Theory 28, J. Q. Fan, Y. Y. Fan, J. C. Lv High dimensional covariance matrix estimation using a factor model. Journal of Econometrics 47, J. Fan, Y. Liao, M. Mincheva 20. High dimensional covariance matrix estimation in approximate factor models. Annals of Statistics 39, E. J. Hannan 970. Multiple time series. John Wiley and Sons, Inc.. 9

21 C. Hsiao, M. H. Pesaran, A. Pick 202. Diagnostic tests of cross-section independence for limited dependent variable panel data models. Oxford Bulletion of Econometrics and Statistics 74, C. Lam, Q. W. Yao 202. Factor modeling for high-dimensional time series: inference for the number of factors. Annals of Statistics 402, S. g Testing cross-section correlation in panel data using spacings. Journal of Business and Economic Statistics 24, A. Onatski Testing hypothesis about the number of factors in large factor models. Econometrica 775, J. Pan, Q. Yao Modelling multiple time series via common factors. Biometrika 952, M. H. Pesaran General diagnostic test for cross section dependence in panels. Working Paper at University of Cambridge & USC. S. Ross 976. The arbitrage theory of capital asset pricing. Journal of Economic Theory 3, J.P. Romano, M. Wolf A more general central limit theorem for m-dependent random variables with unbounded m. Statistics and Probability Letters 47, V. Sarafidis, T. Wansbeek 202. Cross-sectional dependence in panel data analysis. Econometric Reviews 35, W. Sharpe 964. Capital asset prices: a theory of market equilibrium under conditions of risk. Journal of Finance 93, Figure : ACF of RGDP and CPI 20

22 Figure 2: ACF of OMEQ and FXdol Figure 3: ACF of Rshort and Rlong This material includes three appendices, i.e. Appendices A C. Appendix A presents justification of Assumptions 3 in the main paper. Appendix B provides the proofs of Theorems and 2 in the main paper. Some lemmas used in the proofs of Theorems and 2 are given in Appendix C. The proof of Theorem 3 in the main paper is omitted since it is similar to that of Theorem 2. Throughout this material, we use C to denote a constant which may be different from line to line and to denote the spectral norm or the Euclidean norm of a vector. In addition, the notation a n b n means that a n O P b n and b n O P a n. 8 Appendix A: Justifications of Assumptions In this section, we provide some comments on Assumptions 3 in the main paper. The three assumptions are mild and can be satisfied in many cases. ext, we will discuss them in detail.. Justification of Assumption : The weak stationarity assumption on the idiosyncratic components {u t : t, 2,..., T } is a commonly used condition in time series analysis. Rather than independence assumption, weak cross-sectional correlation and serial correlation are imposed via γ 2 i j and γ τ, respectively. The levels of weakness are described by 2.6 and 2.8. ote 2

23 Figure 4: ACF of POILdolL Figure 5: ACF of averages of 96 stock returns that when {u it } is independent across i, t, we have γ τ 0 and γ 2 i j 0 which satisfy Conditions Justification of Assumption 2: The degree of cross-sectional dependence in {x t : t, 2,..., } crucially depends on the nature of the factor loadings. This assumption groups the factor loadings into two categories: a strong category with effects that are bounded away from zero, and a weak category with transitory effects that tend to zero. From this point, the first [ α 0 ] sections are dependent while the rest are independent. Here α 0 maxα lk : l 0,, 2,..., s; k, 2,..., m. To simplify the proof of Theorem 2, we require the factor loadings to have the finite sixth moments. However, we believe that the finite second moment condition may just be sufficient by performing the truncation technique in the proof of Lemma Justification of Assumption 3: The common factors {f t : t, 2,..., T } are also weak stationary time series. The important condition 2.9, which connects {f t : t, 2,..., T } and {u it : i, 2,..., ; t, 2,..., T }, requires stronger serial dependence existed in the factors than that in the idiosyncratic components. This requirement assures the leading term position for the common factor part rather than the idiosyncratic part. 22

24 Figure 6: 30-day joint and marginal estimators for 96 stocks of S&P Appendix B: Proofs of Theorems and 2 This section provides the proofs of Theorems and 2. The proofs will use Lemmas and 2, which are given Appendix C below. For easy of presentation, we first prove Theorem 2 which is for the marginal estimator. 9. Proof of Theorem 2 Proof. Based on model 2.2 in the main paper, we have x t x it µ + β F t + ū t, where β / i β i, µ / i µ i and ū t i u it. Then we have x T x t µ + β F T + ū, x 2 t t where F T T T t F t, F T +τ T T t F t+τ, ū T Then the auto-covariance estimator σ τ can be written as σ τ i T x t+τ µ + β F T +τ + ū 2, T t ūt and ū 2 T T t ūt+τ. T β F t F T + ū t ū β F t+τ F T +τ + ū t+τ ū 2 t T t where C c + c 2 + c 3 with β F t F T F t+τ F T +τ β + C, B. c T ū t ū ū t+τ ū 2, t c 2 T t c 3 T t β F t F T ū t+τ ū 2, β F t+τ F T +τ ū t ū. 23

25 Table : DGP: Bias and RMSE for the marginal estimator α with τ [ T /4]. α /T Bias RMSE Bias RMSE Bias RMSE Bias RMSE /T Bias RMSE Bias RMSE Bias RMSE Bias RMSE /T Bias RMSE Bias RMSE Bias RMSE Bias RMSE Denote S τ T F t F T F t+τ F T +τ. t From 3.0 in the main paper, we can obtain β S τ β [ 2α 0 2 ] v S τ v + R, B.2 where R [ α 0 2 ] v S τ K ρ + [ α 0 2 ]K ρs τ v + 2 K ρs τ K ρ. B.3 Here we would like to remind the reader that D becomes an identity matrix since we assume that 24

26 Table 2: DGP: Bias and RMSE for the joint estimator α with τ [ T /4]. α /T Bias RMSE Bias RMSE Bias RMSE Bias RMSE /T Bias RMSE Bias RMSE Bias RMSE Bias RMSE /T Bias RMSE Bias RMSE Bias RMSE Bias RMSE α lk α 0 for simplicity. Therefore, from B. and B.2, we have ln σ τ 2 ln β S τ β 2 + ln + C β S τ β 2 4α 0 ln + ln v S τ v 2 + ln + R 2 + ln [ 2α0 2 ] v S + τ v It follows from 3.6 in the main paper and B.4 that C 2. β S τ β 4 α α 0 ln + lnκ 2 0 ln v S τ v 2 R 2 C 2 ln + [ 2α0 2 ] v S + ln + τ v β S τ β. B.4 B.5 From Lemma in Appendix C, which provides the central limit theorem for v S τ v, and condition 3.24 in the main paper, we conclude that, as, T, v S τ v µ vσ τ µ v i.p. 0. B.6 25

27 Table 3: DGP2: Bias and RMSE for the marginal estimator α with τ [ T /4]. α /T Bias RMSE Bias RMSE Bias RMSE Bias RMSE /T Bias RMSE Bias RMSE Bias RMSE Bias RMSE /T Bias RMSE Bias RMSE Bias RMSE Bias RMSE Evidently, K ρ C. Moreover, by Assumption 2, s m s m E v 2 E E l0 k v 2 lk l0 k [ α 0 ] [ α 0 ] i,j v ilk v jlk C B.7 and by Assumption 3, we have E S τ T E F t F t+τ t T t E s f t+τ jf t j C. So v O P and S τ O P. These derivations, together with B.3, ensure j0 R O P [ α 0 2 ]. B.8 We conclude from B.8 and B.6 that R [ 2α0 2 ] v S O P τ v 26 [ α 0 ]µ vσ τ µ v. B.9

28 Table 4: DGP2: Bias and RMSE for the joint estimator α with τ [ T /4]. α /T Bias RMSE Bias RMSE Bias RMSE Bias RMSE /T Bias RMSE Bias RMSE Bias RMSE Bias RMSE /T Bias RMSE Bias RMSE Bias RMSE Bias RMSE Therefore where r ln + R 2 [ 2α0 2 ] v S r + o P r O P τ v [ α 0 ]µ vσ τ µ v 2R R 2, + [ 2α 0 2 ] v Sτ v [ 2α 0 and we have used the simple fact that 2 ] v Sτ v It follows that min [ α 0 ], ln +, B.0 ln + x x lim 0. B. x 0 x R 2 [ 2α0 2 ] v S OP τ v [ α0/2 o ]µ P. B.2 vσ τ µ v Meanwhile, based on the decomposition of C 3 i c i, we evaluate the orders of the following c terms: i β Sτ β for i, 2, 3. 27