Capturing Cross-Sectional Correlation with Time Series: with an Application to Unit Root Test

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1 Capturing Cross-Sectional Correlation with Time Series: with an Application to Unit Root Test Chor-yiu SIN Wang Yanan Institute of Studies in Economics (WISE) Xiamen University, Xiamen, Fujian, P.R.China, January 2, 2007 Abstract Traditional panel data analysis assumes cross-sectional uncorrelatedness. This is plausible when the cross-sectional units are households. Notably in the study of growth empirics or gravity models, panel data analysis is recently applied to cross-country data, in which cross-sectional dependence and unit root are common. Existing asymptotic theory either assumes that (i) T (the number of time-series units) goes to innity while N (the number of cross-sectional units) is xed; or (ii) the cross-sectional units are well-ordered or well-indexed with "economic distance". In this paper, we assume that N goes to innity while T is xed (as long as T is greater than q, where q is the number of restrictions under the null), which is more plausible in many panel data sets. On the other hand, no prior knowledge of the ordering or the indexation is assumed. Using the central limit theorem for stationary mixing random variables, we rst show the p N-consistency of an OLS estimator. We then construct a simple robust testing procedure that is insensitive to many possible cross-sectional correlations. The asymptotic critical values of this test can be simulated via Monte Carlo method. A number of Monte Carlo experiments suggest that our test has reasonable sizes in nite samples, even when N and T are as small as 50 and 2 respectively. It has non-trivial power when T is greater than or equal to 10. Key Words: Correlation-insensitive test; Cross-sectional correlation; Economic distance; Robust testing; Stationary mixing random elds; Unit root. JEL Classication C12; C21; C23 Acknowledgments: SIN thanks the comments from the participants at the Workshop on Sequential Analysis, Time Series and Related Topics held in Academia Sinica on December 27-28, 2004; and those at Departmental Seminar of National Taipei University on January 29, The usual disclaimers apply. 1

2 1 Introduction Throughout the paper, we consider the following linear regression model: y it = x it β + u it, (1.1) where i =1,...,N and t =1,...,T, T 2, x it is a kx1-vector while both y it and u it are scalars. By now there is a huge literature covering cases that the time-series dimension T goes to infinity. In this paper, we focus on cases that the cross-section dimension N goes to infinity. Given this assumption, it turns out that our analyses are much easier when we assume that T is fixed. We maintain these two assumptions in the balance of the paper. One major drawback in making inference on the parameter β is that as far as (asymptotic) distributions are concerned, it is hard to model and estimate the crosssectional correlations. More precisely, in one or the other estimators, one may need to model and estimate, for t =1,...,T, the following N(N 1)/2 terms: E[x it u it u jt x jt], (1.2) where i<j,andi, j =1,...,N. In a purely time-series context, the time-series correlations can be modelled using the natural ordering (that is, time) of the series. In a purely cross-section context, Conley (1999) models the cross-sectional correlations using a metric of economic distance. In virtue of the use of economic distance may be controversial, recently voluminous papers in the literature, in one way or the other, capture the cross-sectional correlations by further assuming that T also goes to infinity. See, for instance, Bai (2003). While the models suggested in many papers are found applicable to many data sets, it is unclear if they perform well when T is rather small. 2

3 In this paper, following the lines in Conley (1999), we first invoke some moment conditions and some mixing conditions to justify the asymptotic normal of our estimators. For making statistical inference on β, instead of using a metric of economic distance, we capture the cross-sectional correlation with the time-series units. In principle, T can be as small as 2. This contrasts tremendously with the existing literature in which T is assumed to go to infinity. After constructing a generic Wald test with an OLS estimator in Section 2, we show in Section 3 that a unit root test and a cointegration test can be cast as special cases of a Wald test. Some generalization and some extension can be found in Section 4. Unlike the conventional Wald test, our test does not distributed as a χ 2. The critical values need to be simulated by Monte Carlo method. Some values of certain special cases are tabulated in Section 5. We close this paper with some Monte Carlo experiments in Section 6 and conclusions and discussions in Section 7. 2 The OLS Estimator and the Wald Test y it = x it β + u it, (2.1) where i =1,...,N and t =1,...,T, T 2, x it is a kx1-vector while both y it and u it are scalars. Assumption (a). N and T is fixed. In an extension of this paper, we will let T at an appropriate rate of N. For sake of exposition, we first consider the case in which the time period is divided into two parts. The first part (with observations) is for estimating the parameter β (denoted as ˆβ) while the second part (with T 2 observations) is for estimating the variance-covariance matrix of ˆβ. Note T = + T 2. As one can see in the subsequent discussions, it is possible to have the estimation period and the testing period overlap to each other, though we still require that T 2. 3

4 Assumption (b). Fort =1,...,T, N 1/2 N x it u it L ΓW k t, (2.2) where Γ is a positive definite matrix and W k t random vector. is a k dimensional standard normal Though not necessary, at some point we may assume that W k t s are independent across t. In a purely cross-sectional context, Conley (1999) shows that Assumption (b) holds in an even more general setting (see the proof of Proposition 2 there). Conley (1999) applies the CLT (central limit theorem) of the stationary mixing random field suggested in Bolthausen (1982). In an extension to, say a fluctuation test, we will modify Assumption (b) as: Assumption (b*). Fort =1,...,T, [rn] N 1/2 Assumption (c). Fort =1,...,T, x it u it L ΓWt k (r), r [0, 1]. N N 1 x it x it Ma.s., (2.3) where M is an kxk- invertible constant matrix. Note Assumption (c) of homogeneity can be relaxed a little at the expense of a more complicated estimation for the variance-covariance matrix. Given Assumption (c), for N sufficiently large, ( N x is x is ) 1 exists. We may consider the usual OLS estimator for β in this context (see, for instance, Hsiao, 2003): N ˆβ =( x is x N is ) 1 ( x is y is ). (2.4) It is not difficult to see that: (N 1 N x is x is) N( ˆβ β) = N 1/2 N x is u is L Γ Ws k. 4

5 Thus we have the following theorem for the limiting distribution of ˆβ. Theorem 2.1. Suppose Assumptions (a)-(c) hold. N( ˆβ β) L M 1 Γ( 1 Ws k ). (2.5) The difficult part in (2.5) is the estimation for Γ. In a time series context, there are numerous methods of estimation. See, for instance, de Long and Davidson (2000) and the references therein. Those references essentially assume some kind of mixing conditions in an ordered time series. In a cross-sectional context, Conley (1999) models the spatial correlation with a metric of economic distance; while Bai (2003) (and the reference therein) assumes that the time-series dimension (T in our context) goes to. In this paper, we do not model the economic distance on the one hand, and we allow T to be fixed on the other hand. First note that given the OLS estimator ˆβ, we may define the residual in a straightforward way: û it = y it x it ˆβ, where t =1,...,T.Fort = +1,..., + T 2, consider the following term: x it û it = x it (y it x it ˆβ) = x it (u it + x it β x it ˆβ) = x it u it x it x it ( ˆβ β) = x it u it x it x it ( N x is x N is ) 1 ( x is u is ). Therefore, in view of Assumptions (b)-(c), N N 1/2 x it û it N = N 1/2 N x it u it (N 1 x it x it )( N N 1 x is x is ) 1 (N 1/2 N x is u is ) 5

6 = N 1/2 N x it u it 1 (N 1/2 N x is u is )+o p (1) N = (N 1/2 x it u it 1 N 1/2 N x is u is )+o p (1) L Γ(Wt k 1 Ws k ). (2.6) By (2.5)-(2.6), we are about to construct a Wald test statistic for the null hypothesis H 0 : β = β 0. We need an additional assumption: Assumption (d). definite a.s. Tt=T1 +1 (W k t 1 T1 Ws k k )(Wt 1 T1 Ws k ) is positive Assumption (d) is non-trivial. Consider the simple case that = T 2 =1. If W1 k = W 2 k k a.s., the term (Wt 1 T1 Ws k ) is identically zero a.s. and Assumption (d) is not satisfied. In fact Assumption (d) tells us that our method does not apply to one single time point. We require that T 2. Assumption (d) assures that a.s., [ T t=t1 +1(N 1/2 N x it û it )(N 1/2 N x it û it ) ] 1 exists for N sufficiently large. Thus we may consider the following Wald test statistic: Ŵ = N( ˆβ β 0 ) ˆV 1 N( ˆβ β 0 ), (2.7) where ˆV 1 =(N 1 N x is x is)[ T t=t1 +1(N 1/2 N x it û it )(N 1/2 N x it û it ) ] 1 (N 1 N x is x is). Now we are able to state the major theorem in this section: Theorem 2.2. Suppose Assumptions (a)-(d) hold. Ŵ L T Ws k [ t= +1 (Wt k 1 W k s )(W k t 1 W k s )] 1 Ws k. (2.8) It should be noted despite the fact that a CLT is assumed for N 1/2 N x it u it (see Assumption (b)), unlike the conventional Wald test, the limiting distribution is not χ 2 k even when = 1. It is because unlike the usual estimator for the variancecovariance matrix of N( ˆβ β), ˆV around (2.7) does not converge in probability 6

7 to a constant matrix. Instead, as with a term in Abadir and Paruolo (1997), it converges in distribution to: M 1 Γ[ 1 T 2 1 T t= +1 (Wt k 1 W k s )(W k t 1 W k s )]Γ M 1. (2.9) As a result, the critical values from the limiting distribution in Theorem 2.2 needs to be simulated. Some critical values of certain special cases will be given in Section 5, after we discuss a general version of OLS and an extension to IV. For a general null hypothesis H 0 : Rβ = r 0,whereR is a qxk- matrix with full row rank of q and r 0 is a qx1- vector, q k. It is easy to see from (2.5) that under the null: N(R ˆβ r0 ) L RM 1 Γ( 1 Ws k T ). 1 RM 1 ΓW k s is normally distributed with mean zero and variance RM 1 ΓΓ M 1 R. Note there exists a qxq positive semi-definite matrix Λ such that ΛΛ = RM 1 ΓΓ M 1 R. Abusing the notation, we can write: N(R ˆβ r0 ) L Λ( 1 Ws q ), (2.10) where W q s is a q-dimensional standard normal random vector. Similar to (2.9), ˆV R R(N 1 (N 1 L RM 1 Γ[ 1 T 2 1 N x is x is) 1 [ T (N 1/2 N x it û it )(N 1/2 N x it û it ) ] t= +1 N x is x is) 1 R T (Wt k 1 Ws k )(Wt k 1 Ws k )]Γ M 1 R. t= +1 By the arguments similar to those for (2.10), ˆV R L Λ[ 1 T 2 1 T t= +1 (W q t 1 Ws q q )(W t 1 W q s )]Λ. (2.11) All in all, instead of Assumption (d), we impose the following condition which suffices for, a.s., the invertibility of ˆV R. 7

8 Assumption (e). Λ defined around (2.10) is positive definite. Tt=T1 +1(W q t 1 T1 Ws q )(W q t 1 T1 Ws q ) is positive definite a.s. Thus, for the general null hypothesis H 0 : Rβ = r 0, we may consider the following Wald test statistic: Ŵ R = N(R ˆβ r 0 ) 1 ˆV R N(R ˆβ r0 ). (2.12) The limiting distribution of the test statistic is presented in the next theorem. Theorem 2.3. Suppose Assumptions (a)-(c) and Assumption (e) hold. Ŵ R L Ws q [ T t= +1 (W q t 1 Ws q q )(W t 1 W q s )] 1 Ws q. (2.13) 3 UnitRootTestandTestforCointegration In this section, we investigate a unit root test for the time series {w it }. Following the lines in Fuller (1996), we assume that for each i, {w it } follows an AR(k). In other words, we have a linear regression model: w it = x it β + u it, (3.1) where x it =(w it 1, w it 1,..., w it k+1 ), t =1,...,T and i =1,...,N. In other words, the Augmented Dickey-Fuller test in this setting is simply testing β 1 =0orRβ =0,whereR is a 1xk- vector with the first element equals 1 and all other elements equal 0. If Assumption (a) holds, following the lines in Conley (1999), we may assume that some moment conditions and some mixing conditions hold for the cross-section series {x it u it }, t =1,...,T. In other words, Assumptions (b)-(c) hold. In addition, if we are willing to assume Assumption (e) holds, Theorem 2.3 applies. In fact, although the first part of Assumption (e) (the part about Λ) is hard to check, it may be easy to justify Assumption (d), which implies the second part of Assumption (e). First 8

9 we let F N t = σ{...,u N t 1,u N t }, whereu N t =(u 1t,...,u Nt ). It is plausible to assume that for all i =1,...,N, E[u N t F N t 1] =0. Thus, for all t, s T, t s, N Lim N E[(N 1/2 N x it u it )(N 1/2 x is u is )] = 0, (3.2) which justifies the independence between the W k t s and thus Assumption (d). With a similar setting, Levin and Lin (1992, 1993), Quah (1994) and Levin, Lin and Chu (2002) construct similar tests, with the assumption that both N and T go to infinity. While their approaches may not be applicable to cases in which T is rather small, more importantly, all the papers mentioned above assume that the data are identically and independently distributed across i. The approach adopted in this paper dispenses with the assumption of independence across i. Furthermore, it is our conjecture that with more elaborated analyses, the assumption of identical distribution in this paper can be relaxed, following the lines in Im, Pesaran and Shin (2003). It is interesting to note that unlike the pure time-series analysis, the rate of convergence of this unit root test is the same as that of tests of other parameters β 2,...,β k. In this paper we assume that T is fixed and only N goes to infinity. It is unclear if we obtain the some results on convergence, when T also goes to infinity. Next we consider a test for cointegration among a (k +1)x1- vector w it (w it0,w it1,...,w itk ). Following the lines in Phillips and Durlauf (1986), as well as a long series of subsequent papers, we first consider the following linear regression model: w it0 = x it β + u it, (3.3) where x it =(w it1,...,w itk ), t =1,...,T and i =1,...,N. 9

10 Presumably all the elements of w it are I(1). Based on (3.3), one form of testing for no cointegration can be cast as H 0 : β = 0. As a result, provided that Assumptions (a)-(d) hold, the limiting distribution of the Wald test defined in (2.7) (with β 0 = 0) can be found in Theorem 2.2. Having said that, as suggested in Phillips and Durlauf (1986), it is difficult to justify (3.2) and thus likely the Wt k s in Theorem 2.2 are independent. Our results here are similar to many in the literature. See, for instance, Pedroni (2004). However, as we argue in the discussions on our unit root test, those papers are deficient in two aspects, namely, assuming that T also goes to infinitely; and assuming that the data are independent across i. 4 Generalization and Extension In Section 2, we consider the case in which the time period is divided into two parts, the first part of which is used to estimate the parameter β (the OLS estimator is denoted as ˆβ), while the second part is used to estimate the variance-covariance matrix of ˆβ. In this section, we first generalize our test to a more flexible case. Then we consider an IV (instrumental variable) estimator. Define T {1,...,T}, wheret 2. Consider two subsets of T, and T 2, not necessarily disjoint. The numbers of elements in and T 2, denoted as # and #T 2 respectively, are non-zero. Instead of the OLS estimator in (2.4) above, we may consider an alternative one: ˆβ =( N x is x is) 1 ( N x is y is ). (4.1) s s We have the following theorem for the limiting distribution of ˆβ. Theorem 4.1. Suppose Assumptions (a)-(c) hold. N( ˆβ β) L M 1 Γ( 1 # s W k s ). (4.2) 10

11 Correspondingly, we define the residual as: û it = y it x it ˆβ, where t = 1,...,T. Correspondingly we replace Assumption (e) with the following: Assumption (e ). Λ defined around (2.10) is positive definite. t T 2 (W q t 1 # s Ws q q )(Wt 1 # s Ws q ) is positive definite a.s. A Wald test statistic for the null hypothesis H 0 : Rβ = r 0 is defined as: Ŵ R = N(R ˆβ r0 ) 1 ˆV R N(R ˆβ r0 ), (4.3) where ˆV R = R(N 1 N s x is x is ) 1 [ t T 2 (N 1/2 N x it û it )(N 1/2 N x it û it ) ] (N 1 N s x is x is ) 1 R. The limiting distribution of Ŵ is presented in the next theorem. Theorem 4.2. Suppose Assumptions (a)-(c) and Assumption (e ) hold. Ŵ L Ws q [ s (W q t 1 Ws q )(W q t 1 t T 2 # s # s W q s )] 1 s W q s. (4.4) Next we consider a Wald test derived from an IV estimator. Instead of the OLS estimator in (4.1) above, suppose we have an instrument z it,whichisalsoa kx1-vector. Define the following IV estimator: β =( N z is x is ) 1 ( N z is y is ). (4.5) s s Correspondingly, we replace Assumptions (b) and (c) with the followings: Assumption (b ). Fort =1,...,T, N 1/2 N where Γ is a positive definite matrix and W k t random vector. Assumption (c ). Fort =1,...,T, z it u it L ΓW k t, (4.6) is a k dimensional standard normal N N 1 z it x it Ma.s., (4.7) 11

12 where M is an kxk- invertible constant matrix. We have the following theorem for the limiting distribution of β. Theorem 4.3. Suppose Assumptions (a), and Assumptions (b )-(c ) hold. Define the residual as: N( β β) L M 1 Γ( 1 # s W k s ). (4.8) ũ it = y it x it β, where t =1,...,T. A Wald test statistic for the null hypothesis H 0 : Rβ = r 0 is defined correspondingly as: W R = N(R β r 0 ) ṼR 1 N(R β r0 ), (4.9) where ṼR = R(N 1 s N x is z is ) 1 [ t T 2 (N 1/2 N z it ũ it )(N 1/2 N z it ũ it ) ] (N 1 s N z is x is) 1 R. The limiting distribution of WR is presented in the next theorem. Theorem 4.4. Suppose Assumption (a), Assumptions (b )-(c ), and Assumption (e ) hold. W R L Ws q [ s (W q t 1 Ws q )(W q t 1 t T 2 # s # s W q s )] 1 s W q s. (4.10) It should be noted that the limiting distribution in Theorem 4.4 is the same as that in Theorem 4.2, which is free from the nuisance parameters Γ and M. Nevertheless, it depends on the dependence between the W q t s as well as the way we choose and T 2. In the next section, assuming that the W q t s are independent across t, q =1, 2, we tabulate the critical values of the following two cases: (i) The disjoint case. More precisely, = {1,..., } and T 2 = { +1,..., + T 2 }; (ii) The fully overlapping case. More precisely, = T 2 = T. 12

13 5 Simulating the Critical Values TABLE 5.1 Quantiles of the Limiting Distribution in (5) or (8), k = 1. α th simulated quantiles T rv DISJ OV ER z DISJ OV ER z DISJ OV ER z DISJ OV ER z DISJ OV ER z DISJ OV ER z DISJ OV ER z DISJ OV ER z DISJ OV ER z DISJ OV ER z DISJ OV ER z DISJ OV ER z DISJ OV ER z DISJ OV ER z χ

14 6 Monte Carlo experiments TABLE 6.1(a) Rejection Percentage under H 0 : β 1 =0,ρ =0. Size T Test 10% 5% 1% 2 DISJ OV ER WHITE DISJ OV ER WHITE DISJ OV ER WHITE TABLE 6.1(b) Rejection Percentage under H 0 : β 1 =0,ρ =0.5. Size T Test 10% 5% 1% 2 DISJ OV ER WHITE DISJ OV ER WHITE DISJ OV ER WHITE

15 TABLE 6.1(c) Rejection Percentage under H 0 : β 1 =0,ρ =0.9. Size T Test 10% 5% 1% 2 DISJ OV ER WHITE DISJ OV ER WHITE DISJ OV ER WHITE TABLE 6.2(a) Rejection Percentage under H a : β 1 =0.1, ρ =0. Size T Test 10% 5% 1% 2 DISJ OV ER WHITE DISJ OV ER WHITE DISJ OV ER WHITE

16 TABLE 6.2(b) Rejection Percentage under H a : β 1 =0.5, ρ =0. Size T Test 10% 5% 1% 2 DISJ OV ER WHITE DISJ OV ER WHITE DISJ OV ER WHITE TABLE 6.2(c) Rejection Percentage under H a : β 1 =0.9, ρ =0. Size T Test 10% 5% 1% 2 DISJ OV ER WHITE DISJ OV ER WHITE DISJ OV ER WHITE

17 7 Conclusions and Discussions In this paper, we propose a Wald test for the parameters in a linear regression model, in which the correlations between N cross-sectional units (where N goes to infinity) are hard to model, and are hard to estimate. We need T time-series unit, where T is fixed and in principle it may be as small as 2. This is in contrast to cases in the literature that either (i) the correlations between the cross-sectional units do not exist, see, for instance, Anderson (1978), Anderson and Hsiao (1981), Holz-Eakon, Newey and Rosen (1988), and Moon and Phillips (1999); (ii) the cross-sectional correlations are modelled with the geographical distance (see, for instance, Kelejian and Prucha, 1999 and the references in the field of spatial statistics therein) or the economic distance proposed by Conley (1999); or (iii) T also goes to infinity, see, for instance, Kao (1999), Bai and Ng (2002), and Bai (2003). In Section 3, we also consider a unit root test and a test for cointegration when we have data in both the time-series dimension and the cross-sectional dimension. Both topics are overwhelming in the field of economics over the past ten years. See, for instance, Levin and Lin (1992, 1993), Quah (1994), Levin, Lin and Chu (2002), Im, Pesaran and Shin (2003), and Pedroni (2004), all of which assume that both N and T go to infinity. As in the aforementioned papers, in Section 3 of this paper, these two tests are found to be special cases of the Wald test in the linear regression model developed in Section 2. On the other hand, we extend our OLS estimation to IV (instrumental variable) estimation in Section 4. The method proposed in this paper has an interesting analogy with the classical z test for the population mean. Consider a special case of Model (1.1), in which k =1,x it =1and = T 2 = T = {1,...,T}: y it = β + u it. (7.1) If we sum all the terms in (7.1) against i and multiply them by N 1/2, we will 17

18 get: v Nt = N Nβ + N 1/2 u it, (7.2) where v Nt N 1/2 N y it. Given Assumption (b), for t =1,...,T, v Nt Nβ = N 1/2 N u it L ΓW 1 t, (7.3) where Γ 2 = Lim N E(N 1/2 N u it ) 2. Refer to the Wald test statistic in (2.7). First, we re-write the null hypothesis as H 0 : Nβ = Nβ 0. Second, it is easy to see that: 1 T N ˆβ = v Ns v N, T ˆV = 1 T (v T 2 Nt v N ) 2. t=1 And if we further assume the W 1 t in (7.3) be identically distributed, by Theorem 4.1 (for the case when k =1),asN, T ( vn Nβ 0 ) T ( v N Nβ 0 ) Tt=1 = (v Nt v N ) 2 T 1 Tt=1 (v Nt v N ) 2 /(T 1) L T T 1 t T 1, (7.4) where t T 1 denotes a random variable which is t distributed with T 1 degrees of freedom. In spite of the analogy, hypotheses with the linear regression model in (1.1) is much more general than a classical z test. In the further study, we will extend Model (1.1) to cases with more heterogeneity. In particular, we will allow some fixed effects, random effects, random coefficients (see, for instance, Hsiao, 2003), and common factors (see, for instance, Bai, 2003). In a classical z test, ast, the limiting distribution is standard normal. It is interesting to derive the limiting distribution of our Wald test statistic, when apart from N,# and/or #T 2 also do. Furthermore, it has been shown 18

19 that under certain assumptions, the classical z test is optimal in testing for the population mean. It is also interesting to see the optimality results, especially those refer to the choice of and T 2. 19

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