Weak and Strong Cross Section Dependence and Estimation of Large Panels

Transcription

1 Weak and Strong Cross Section Dependence and Estimation of Large Panels (with Alexander Chudik and Elisa Tosetti) Cambridge University and USC Econometric Society Meeting, July 2009, Canberra

2 Background Literature Related literature Weak vs strong cross section dependence Cross section dependence in dynamic panels Common factor models Estimation and inference Monte Carlo experiments

3 Related Literature Background Literature Growing literature on econometric methods for modelling and measuring cross section dependence in panel data model. Sources of cross dependence (CD): Omitted common e ects Spatial e ects Socio-economic networks Conditioning on variables speci c to the cross section units alone does not deliver cross section error independence. Neglecting cross section dependence can lead to spurious inference.

4 Background Literature How to take account of CD depends on the type of CD and the size of N (cross section dimension) relative to T (time series dimension). When T is large relative to N, SURE can be used. (Zellner, 1962). Currently, there are two main approaches to modelling CD in large panels: spatial processes and factor structures. Spatial processes were pioneered by Whittle (1954) and developed further in econometrics by Anselin (1988), Kelejian and Prucha (1999), and Lee (2002).

5 Background Literature Factor models were introduced by Hotelling (1933) and applied in economics rst by Stone (1947, JRSS). More recently, it has been applied extensively used in nance and economics (Chamberlain and Rothschild 1983; Connor and Korajczyk, 1993; Stock and Watson, 1998; Kapetanios and Pesaran, 2007), and in macroeconomics (Forni and Reichlin, 1998; Stock and Watson, 2002). The aim of this paper is to characterize the correlation pattern over the cross sectional dimension for a general class of processes, regardless whether they are represented by factor or spatial models or any other model featuring cross section dimension proposed in the literature.

6 Related de nitions provided by existing literature. Weak and Strong Cross Section Dependence Dynamic panels Common Factor Models Related de nitions provided by existing literature Forni and Lippi (2001, hereafter FL) introduce the notion of idiosyncratic process to characterize a weak form of dependence that involves both time series and cross sectional dimensions. Assume, for each N 2 N, the process z Nt = (z 1t,..., z Nt ) 0 is covariance stationary and the spectral measure of z Nt,F zn (ω), is absolutely continuous. FL (De nition 9) de ne the process fz it g as idiosyncratic if, for all weights w N satisfying lim N! kw N k = 0, we have lim N! Z 1 π wn 0 2π F zn (ω) w N dθ = 0. π

7 Related de nitions provided by existing literature. Weak and Strong Cross Section Dependence Dynamic panels Common Factor Models FL show that the boundedness of the largest eigenvalues of fz it g (at all frequencies) is necessary and su cient for the process to be idiosyncratic. Following FL, Anderson et al. (2009) de ne the concepts of weak and strong dependence for processes fz it g: De nition (Weak and strong dependence à la Anderson et al.) The double index processes fz it, i 2 N, t 2 Zg is weakly dependent if λ z N,1 (ω), the largest eigenvalue of F zn (ω), is uniformly bounded in ω and N. The process fz it g is strongly dependent if the rst m 1 eigenvalues (λ z N,1 (ω),..., λ z N,m (ω)) diverge to in nity for all frequencies as N!.

8 Related de nitions provided by existing literature. Weak and Strong Cross Section Dependence Dynamic panels Common Factor Models Both, FL and Anderson et al. assume that the underlying time series processes are stationary. This assumption might be quite restrictive and are not likely to hold in many applications, especially in nance. In this paper we consider a generalization where the asymptotic behavior of the weighted averages are considered at each point in time, which does not require any stationarity assumptions to be imposed on the time series properties of the underlying process.

9 Related de nitions provided by existing literature. Weak and Strong Cross Section Dependence Dynamic panels Common Factor Models Weak and Strong Cross Section Dependence Let z t = (z 1t,..., z Nt ) 0, with E (z t ji t 1 ) = 0, Var (z t ji t 1 ) = Σ t, where I t 1 is the information set at time t 1, and for each t where Σ t has diagonal elements 0 < σ ii,t K, for i = 1, 2,..., N. Let w t = (w 1t,..., w Nt ) 0 be a vector of weights satisfying the granularity conditions kw t k 2 = O N 2 1 w jt, = O N 1 2 for any j N kw t k 2 Obvious example is equal weights, w i = N 1. Consider z wt = w 0 tz t, E z 2 wt = w 0 t Σ t w t

10 Related de nitions provided by existing literature. Weak and Strong Cross Section Dependence Dynamic panels Common Factor Models De nition (Weak and strong cross section dependence.) The process fz it g is weakly cross sectionally dependent (CWD) at a point in time t if for all w t lim N! Var(w0 tz t ji t 1 ) = 0 The process fz it g is cross sectionally strongly dependent (CSD) at a point in time t if there exists w t such that Var(w 0 tz t ji t 1 ) K > 0 where K is a constant independent of N.

11 Related de nitions provided by existing literature. Weak and Strong Cross Section Dependence Dynamic panels Common Factor Models The process fz it g is CSD at time t 2 T if and only if lim N! 1 N λ 1 (Σ t ) = K > 0, i.e. λ 1 (Σ t ) increases to in nity at the rate N. If λ 1 (Σ t ) = O(N 1 ɛ ) for any ɛ > 0, then lim N! w0 tw t λ1 (Σ t ) = 0, and the underlying process will be CWD. Hence, the bounded eigenvalue condition is su cient but not necessary for CWD. Let fz it,a g and fz it,b g be CSD and CWD processes, respectively. Then z it,a and z it,b are weakly dependent on each other, in the sense that for all t and t 0, E ( z wt,a z wt 0,b)! 0, as N!.

12 Related de nitions provided by existing literature. Weak and Strong Cross Section Dependence Dynamic panels Common Factor Models Example of a CWD process with unbounded eigenvalue Suppose where with 1 < α < 0. u t = R α ε t, ε t s IID (0, I N ) N α R α = N α C 0 A N α 0 0 1

13 Related de nitions provided by existing literature. Weak and Strong Cross Section Dependence Dynamic panels Common Factor Models N α N α N α N α 1 + N 2α N 2α N 2α R α R 0 α = N α N 2α 1 + N 2α., C N 2α A N α N 2α N 2α 1 + N 2α kσk = kvar (u t )k = R α R 0 α = O N α+1 for 1 < α < 0. Thus the process u t is CWD, but the largest eigenvalue (in absolute value) of the variance matrix Σ is unbounded in N for α < 0.5. In particular, jλ max (Σ)j = O N 2α+1. Also see Kapetanios and Marcellino (2008).

14 Related de nitions provided by existing literature. Weak and Strong Cross Section Dependence Dynamic panels Common Factor Models Let z it,a and z it,b be independent processes and set z it = β a z it,a + β b z it,b If fz it,a g and fz it,b g are CSD, then fz it g is also CSD. If fz it,a g and fz it,b g are CWD, then fz it g is also CWD. If fz it,a g is CSD and fz it,b g is CWD, then fz it g is CSD. CWD and CSD can be de ned equally with respect to any information set, such as I M, for any xed M, or as M tends to in nity (if the underlying process is stationary).

15 Dynamic panels Related de nitions provided by existing literature. Weak and Strong Cross Section Dependence Dynamic panels Common Factor Models Suppose that for each N 2 N, cross section units collected into the vector z t = (z 1t, z 2t,..., z Nt ) 0 are generated from the following VAR model, z t = Φ t z t 1 + u t, (1) where Φ t is a N N dimensional matrix of unknown coe cients, which could be time-varying, and the vector u t of reduced-form errors has mean and variance E (u t ) = 0, E u t u 0 t = Σt. The initialization of the dynamic process could be from a nite past, t 2 T f M + 1,.., 0,..g Z, M being a xed positive integer; or we can let M!.

16 Related de nitions provided by existing literature. Weak and Strong Cross Section Dependence Dynamic panels Common Factor Models Object is to investigate the correlation pattern of fz it g across the cross sectional units. In our analysis, we set I t to contain only the starting values, z M, i.e. I t = I = fz M g. Suppose that M is xed, and for any t 2 T and any N 2 N, we have kφ t k < K <, kσ t k < K N 1 ɛ, where constants K and ɛ > 0 do not vary with N or t. Then the process z t given by the VAR model (1) is CWD at any point in time, conditional on information set I = fz M g. if kφ t k < 1 ɛ, then we can let M! and z t is again CWD at any point in time.

17 Related de nitions provided by existing literature. Weak and Strong Cross Section Dependence Dynamic panels Common Factor Models Consider now VAR(1) model with time invariant coe cient matrix Φ t = Φ, and suppose that for each t 2 T, u t satis es E (u t ) = 0, E (u t u 0 t) = Σ,where Σ is a time invariant N N symmetric, nonnegative de nite matrix. In addition assume ρ (Φ) < 1 (i.e. fz it g is covariance stationary). Then process z t is weakly dependent in the sense of Anderson et al. (2009) if ρ (Σ) K < and kφk < 1. Notice that under the assumption that kφk < 1 and if, for at least one frequency ω 0, the matrix (I N e i ω 0 Φ) 1 (I N e i ω 0 Φ 0 ) 1 is non-singular, it is possible to show that weak dependence in the sense of Anderson et al. (2009) implies ρ(σ) K <.

18 Common Factor Models Related de nitions provided by existing literature. Weak and Strong Cross Section Dependence Dynamic panels Common Factor Models Consider the following in nite factor model for fz it g: where z it = γ i1 f 1t + γ i2 f 2t γ in f Nt + ε it, i = 1,..., N, f t = (f 1t,..., f Nt ) 0 is a covariance stationary process, with absolute summable autocovariances, distributed independently of ε it 0 for all i, t, t 0. Var (ε it ji t 1 ) = σ 2 i K <, and ε it, ε jt are independently distributed for all i 6= j and for all t. z it has conditional variance Var(z it ji t 1 ) = N γ 2 i` + σ2 i. `=1

19 Related de nitions provided by existing literature. Weak and Strong Cross Section Dependence Dynamic panels Common Factor Models Finiteness of Var(z it ji t 1 ) implies that f`t is said to be strong if N γ 2 i` K <, for i = 1,..., N. `=1 lim N! 1 N N E jγ i`j = K > 0. (2) i=1 f`t is said to be weak if the factor loadings are absolute summable N lim N! i=1 E jγ i`j = K <. (3)

20 Related de nitions provided by existing literature. Weak and Strong Cross Section Dependence Dynamic panels Common Factor Models The process fz it g is CWD at time t 2 T if f`t is weak for ` = 1,..., N. Suppose that the factor loadings are non-random. Then the process fz it g is CSD at time t 2 T if and only if there exists at least one strong factor. z it can rewritten as where u it = z it = u it + e it m N γ i`f`t ; e it = γ i`f`t + ε it `=1 `=m+1 and γ i` satisfy conditions (2) for ` = 1,..., m, and (3) for ` = m + 1,..., N, with m any nite number. Notice that u it is CSD and e it is CWD.

21 Related de nitions provided by existing literature. Weak and Strong Cross Section Dependence Dynamic panels Common Factor Models Spatial processes of the type z t = Rv t are examples of models with an in nite number of weak factors. Assumptions in the literature: Bai (2006): lim 1 N! N N i =1 γ2 i ` > 0, for ` = 1, 2,..., m. Onatsky (2006) and Paul (2007): let Σ et = σ 2 I N, then the `th factor is strong if N i =1 γ2 i ` > p cσ 2 ; the `th factor is weak if N i =1 γ2 i ` p cσ 2, with c such that T N c = o N 1/2. Onatsky (2006): principal components analysis of a weak factor structure yields inconsistent estimates of factors and their loadings.

22 Model CCE Estimation of Panel Data Models with In nite Factors Consider where y it = α 0 i d t + β 0 i x it + u it, (4) d t = (d 1t, d 2t,..., d nt ) 0 is a n 1 vector of observed common e ects x it is a k 1 vector of observed individual speci c regressors on the ith cross section unit at time t. The error term, u it, is given by the following general factor structure u it = m 1 `=1 γ i`f`t + m 2 `=1 λ i`g`t + e it,

23 Model CCE Estimation of Panel Data Models with In nite Factors x it can be correlated with any of the m 1 factors in f t : where x it = A 0 i d t + Γ 0 i f t + v it, (5) A 0 i and Γ 0 i are n k and m 1 k factor loading matrices with xed components, and v it is the individual component of x it, assumed to be distributed independently of the innovations u it and of the common factors.

24 Model CCE Estimation of Panel Data Models with In nite Factors CCE estimation of panel data models with in nite factors Pesaran (2006) introduced CCE estimators in a panel model where m 1 is xed and generally small, m 2 = 0, and γ 0 i f t represents a strong factor structure. Contrary to what Bai (2006, page 2) suggests, CCE estimators are valid regardless of the number of strong common factors m 1 (which must be nite if the variables to be explained to have nite variances). Kapetanios, Pesaran, and Yagamata (2006, revised 2009) extended Pesaran (2006) by allowing common factors to follow a unit root process, and Pesaran and Tosetti (2009) consider panels where the errors follow spatial MA or AR processes.

25 Model CCE Estimation of Panel Data Models with In nite Factors This paper provides further extensions of Pesaran (2006) to in nite factor structures. Assume: m 1 does not change with N f`t can be weak, strong or semi-weak (i.e. neither weak nor strong) factor m 2 rises with N such that m 2 /N! { 0, and g`t is a weak factor Then CCE estimators proposed by Pesaran (2006) continues to be consistent and asymptotically normal.

26 Monte Carlo Design Monte Carlo Design Estimators Considered in the MC experiments in Detail MC Findings y it = α i d 1t + β i1 x i1t + β i2 x i2t + u it, (6) for i = 1, 2,..., N and t = 1, 2,..., T. We assume heterogeneous slopes, and set β ij = β j + η ij, with η ij IIDN (1, 0.04),for i = 1, 2,..., N and j = 1, 2. The errors, u it, are generated as u it = 3`=1 γ i`f`t + m 2 `=1 λ i`g`t + ε it, where ε it N(0, σ 2 i ), σ2 i IIDU (0.5, 1.5), for i = 1, 2,..., N (the MC results will be robust to serial correlation in ε it ) and unobserved common factors are generated as AR(1) independent processes with autoregressive parameter 0.5 and unit variance.

27 Monte Carlo Design Estimators Considered in the MC experiments in Detail MC Findings The rst three factors are assumed to be strong, and their loadings are generated as γ i` IIDU(0, 1), for i = 1,..., N, ` = 1, 2, 3. Regressors x ijt are assumed to be correlated with strong unobserved common factors and generated as follows: where x ijt = a ij1 d 1t + a ij2 d 2t + 3`=1 γ ij`f`t + v ijt, j = 1, 2, γ ij` IIDU(0, 1), for i = 1,..., N, ` = 1, 2, 3; j = 1, 2. v ijt = ρ υij v ijt 1 + ϑ ijt, i = 1, 2,..., N; t = 49,..., 0, 1,.., T, ϑ ijt IIDN(0, 1 ρ 2 ϑ ij ), v ij, 50 = 0, ρ ϑij IIDU(0.05, 0.95) for j = 1, 2.

28 Monte Carlo Design Estimators Considered in the MC experiments in Detail MC Findings The observed common e ects are generated as d 1t = 1; d 2t = 0.5d 2t 1 + v dt, t = 49,..., 0, 1,.., T, v dt IIDN(0, ), d 2, 50 = 0, When generating v ijt and the common factors f`t, g`t and d 2t the rst 50 observations have been discarded to reduce the e ect on estimates of initial values. The factor loadings of the observed common e ects do not change across replications and are generated as α i IIDN(1, 1), i = 1, 2,..., N, (a i11, a i21, a i12, a i22 ) IIDN(0.5τ 4, 0.5I 4 ), where τ 4 = (1, 1, 1, 1) 0 and I 4 is a 4 4 identity matrix.

29 Monte Carlo Design Estimators Considered in the MC experiments in Detail MC Findings Experiment A: m 2 factors fg`t g are weak with their loadings given by λ i` = η i` 2 N i=1 η i`, η i` IIDU(0, 1), for ` = 1,..., m 2, and i = 1,..., N. Notice that for each `, N i=1 jλ i`j = O(1) and for each i, m 2 `=1 λ i`2 = O(m 2 /N 2 ). Therefore, as N!, the R 2 of the individual relations are only a ected by the strong factors, even if m!. Experiment B: Semi-strong (weak) factors fg`t g: λ i` = η i` q, η i` IIDU(0, 1), for ` = 1,..., m 2, and i = 1, 2,..., N 3 N i=1 η 2 i` In this case for each `, N i=1 jλ i`j = O(N 1/2 ), and for each i, m 2 `=1 λ2 i` = O(m 2/N).

30 Monte Carlo Design Estimators Considered in the MC experiments in Detail MC Findings We report bias, RMSE, size and power for six estimators: The FE estimator with standard variance. The CCEMG and CCEP estimators (Pesaran, 2006, equation 53 and 65, respectively). CCE estimators are based on regressions augmented by cross section averages of dependent variable and regressors, which is fully satisfactory to capture CS dependence for the purpose estimation of slope coe cients, regardless the number of factors. The MGPC and PPC estimators proposed by Kapetanios and Pesaran (2007), where common factors are estimated by Principal Components (PC). The iterative PC estimator proposed by Bai (2006, Theorem 3).

31 Monte Carlo Design Estimators Considered in the MC experiments in Detail MC Findings Estimators Considered in the MC experiments in Detail CCE approach The CCEMG and CCEP estimators are computed by running individual or pooled regressions augmented with the cross section averages z t = (y t, xt) 0 0 to compute ^b i and ^b CCEP. The CCEMG is given by ^b CCEMG = N 1 ^b i = (X 0 i MX i ) 1 X 0 i My i, M = I T H and H = (D, Z). The CCEP estimator is ^b CCEP =! 1 N Xi 0 MX i i=1 N ^b i (7) i=1 N i=1 H H H X 0 i My i (8)

32 Monte Carlo Design Estimators Considered in the MC experiments in Detail MC Findings Variance matrices are computed as dvar ^b CCEMG = 1 N (N 1) N ^b i ^b CCEMG ^b i 0 ^b CCEMG i=1 (9) dvar ^b CCEP = 1 N ^S 1^R ^S 1 (10) where ^S = N i=1 ^R = 1 N X 0 N 1 i MX i ^b i i=1 T 1 X 0 i MX i N T ^b CCEMG ^b i ^b CCEMG 0 X 0 i MX i T

33 PC augmentation approach Monte Carlo Design Estimators Considered in the MC experiments in Detail MC Findings The MGPC and PPC estimators are similar to MGCCE and PCCE except that z t = (y t, x 0 t) 0 is replaced by ^F computed as the T (m + n) matrix of observations on ^f t, the vector of (m + n) principal components extracted from z it = (y it, x 0 it )0.

34 Monte Carlo Design Estimators Considered in the MC experiments in Detail MC Findings Bai s PC iterative approach ^b PC, ^F is the solution to the following set of non-linear equations: 1 NT ^b PC = N i=1! 1 N N X i M ˆF X i X i M ˆF y i i=1 i=1 y i X i^b PC y i X i^b PC 0 ^F = ^F^V where M ˆF = I T ^F ^F^F 0 1 ^F 0, and ^V is a diagonal matrix with the m largest eigenvalues of the matrix 0 y i X i^b PC y i X i^b PC arranged in decreasing 1 NT N i=1 order.

35 Monte Carlo Design Estimators Considered in the MC experiments in Detail MC Findings The variance estimator of ^b PC is dvar ^b PC = 1 NT D 0 1 D Z D0 1 where ˆσ 2 i = 1 T T t=1 ˆε2 it, Z i = M ˆF X i D 0 = 1 NT D Z = 1 N 1 N N k=1 T z it zit 0 t=1 N ˆσ 2 i i=1 1 T! T z it zit 0 t=1 ^λ0 i ^L^L 0 1 /N ^λ0 k M ˆF X k, and ^L = ^λ1,..., ^λ N 0 is the matrix of estimated factor loadings.

36 Monte Carlo Design Estimators Considered in the MC experiments in Detail MC Findings When T /N! ρ > 0, ^b PC is biased and we estimate the bias as bias = 1 N 2 D 0 1 N X i i=1 where ^V i = 1 N N j=1 ^λ 0 i T ^V i 0 ^F ^L^L 0 1 /N ^λ0 jx j. ^L^L 0 N! 1 ^λi ˆσ 2 i

37 Monte Carlo Design Estimators Considered in the MC experiments in Detail MC Findings For extracting factors we use the selection procedure proposed by Bai and Ng (2002). Each experiment was replicated 2, 000 times for all pairs of N and T = 20, 30, 50, 100, 200. For each N we shall consider m = 0, N/5, 3N/5, N. Bai estimator was computed only for selected experiments (N and T = 20, 100, and R = 1000) due to higher computational demands.

38 Monte Carlo Design Estimators Considered in the MC experiments in Detail MC Findings Table 1: Small sample properties of FE estimator, N = 100. m 2 nt Bias (100) RMSE (100) Experiment A: m 1 = 3 strong factors and m 2 weak factors Experiment B: m 1 = 3 strong factors and m 2 semi-weak factors Size (100) Power (100) Experiment A: m 1 = 3 strong factors and m 2 weak factors Experiment B: m 1 = 3 strong factors and m 2 semi-weak factors

39 Monte Carlo Design Estimators Considered in the MC experiments in Detail MC Findings Table 2: Small sample properties of CCEMG estimator, N = 100. m 2 nt Bias (100) RMSE (100) Experiment A: m 1 = 3 strong factors and m 2 weak factors Experiment B: m 1 = 3 strong factors and m 2 semi-weak factors Size (100) Power (100) Experiment A: m 1 = 3 strong factors and m 2 weak factors Experiment B: m 1 = 3 strong factors and m 2 semi-weak factors

40 Monte Carlo Design Estimators Considered in the MC experiments in Detail MC Findings Table 3: Small sample properties of CCEP estimator, N = 100. m 2 nt Bias (100) RMSE (100) Experiment A: m 1 = 3 strong factors and m 2 weak factors Experiment B: m 1 = 3 strong factors and m 2 semi-weak factors Size (100) Power (100) Experiment A: m 1 = 3 strong factors and m 2 weak factors Experiment B: m 1 = 3 strong factors and m 2 semi-weak factors

41 Monte Carlo Design Estimators Considered in the MC experiments in Detail MC Findings Table 4: Small sample properties of MGPC estimator, N = 100. m 2 nt Bias (100) RMSE (100) Experiment A: m 1 = 3 strong factors and m 2 weak factors Experiment B: m 1 = 3 strong factors and m 2 semi-weak factors Size (100) Power (100) Experiment A: m 1 = 3 strong factors and m 2 weak factors Experiment B: m 1 = 3 strong factors and m 2 semi-weak factors

42 Monte Carlo Design Estimators Considered in the MC experiments in Detail MC Findings Table 5: Small sample properties of PPC estimator, N = 100. m 2 nt Bias (100) RMSE (100) Experiment A: m 1 = 3 strong factors and m 2 weak factors Experiment B: m 1 = 3 strong factors and m 2 semi-weak factors Size (100) Power (100) Experiment A: m 1 = 3 strong factors and m 2 weak factors Experiment B: m 1 = 3 strong factors and m 2 semi-weak factors

43 Monte Carlo Design Estimators Considered in the MC experiments in Detail MC Findings Table 6: Small sample properties of Bai estimator, N = 100. Bias (x100) RMSE (x100) Size (x100) Power (x100) m 2 N / T Experiment A: m 1 = 3 strong factors and m 2 weak factors Experiment B: m 1 = 3 strong factors and m 2 semi-weak factors

44 Monte Carlo Design Estimators Considered in the MC experiments in Detail MC Findings Figure 1: Power Curves for the CCEP t-tests in experiments with N = 100, T = 100, m 1 = 3 strong factors, and varying number m 2 of weak factors (left chart) and semi-weak factors (right chart) 3 strong factors and varying number of weak factors 100% 80% 60% 40% 20% 0% strong factors and varying number of semi weak factors 100% 80% 60% 40% 20% 0%

45 This paper: Considers the statistical analysis of large panel data sets where even after conditioning on common observed e ects the cross section units might remain dependently distributed. Introduces the concepts of time-speci c weak and strong cross section dependence, and investigates their relationship with the notions of weak and strong common factors. Investigates Common Correlated E ects (CCE) estimators of panel data model with an in nite factor error structures.

46 The CCE approach has been applied successfully to the analysis of PPP by Imbs et al. (2005 QJE) Trade ows in EU by Serlenga and Shin (2007, JAE) US house prices by Holly et al. (2009, JoE, forthcoming) Health expenditure in the US by Tosetti and Moscone (2009, Health Economics) Labour productivity in UK industrial sector by Vecchi and Byrne (2008, Applied Economic Letters)

47 Aggregation bias in PPP by Robertson, Kumar,and Dutkowsky (2009, Journal of Development Economics) Fiscal Spillovers and Trade Relations in Europe, by Barrell, Holland, Liadze and Pomerantz (NIESR, 2007) Import growth and globalisation, by Barrell, Liadze and Pomerantz (NIESR, 2008) Savings in OECD, by Mark Holmes (2006, Journal of Economics and Finance) Taxes across U.S. States, by Bob Chirinko and Daniel J. Wilson (2008, CESifo).