A note on the estimation of long-run relationships in panel equations with cross-section linkages

Di Iorio, Francesca; Fachin, Stefano Article A note on the estimation of long-run relationships in panel equations with cross-section linkages Economics: The Open-Access, Open-Assessment E-Journal Provided in Cooperation with: Kiel Institute for the World Economy – Leibniz Center for Research on Global Economic Challenges Suggested Citation: Di Iorio, Francesca; Fachin, Stefano (2012) : A note on the estimation of long-run relationships in panel equations with cross-section linkages, Economics: The Open-Access, Open- Assessment E-Journal, ISSN 1864-6042, Kiel Institute for the World Economy (IfW), Kiel, Vol. 6, Iss. 2012-20, pp. 1-18, https://doi.org/10.5018/economics-ejournal.ja.2012-20 This Version is available at: https://hdl.handle.net/10419/59035 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-nc/2.0/de/deed.en A Note on the Estimation of Long-Run Relationships in Panel Equations with Cross-Section Linkages Francesca Di Iorio University of Naples Federico II Stefano Fachin University of Rome "La Sapienza" Abstract The authors address the issue of estimation and inference in dependent nonstationary panels of small cross-section dimensions. The main conclusion is that the best results are obtained applying bootstrap inference to single-equation estimators, such as fully modified ordinary least squares and dynamic ordinary least squares. Seemingly unrelated regression estimators perform badly, or are even unfeasible, when the time dimension is not very large compared to the cross-section dimension. JEL C15, C23, C33 Keywords Panel cointegration; Fully modified ordinary least squares; Fully modified seemingly unrelated regression; Dynamic ordinary least squares; Dynamic seemingly unrelated regression Correspondence Francesca Di Iorio, Dip. TEOMESUS, University of Naples Federico II, via Leopoldo Rodinò 22 - 80138 Naples (Italy); [email protected]; Stefano Fachin, Dip. di Scienze Statistiche, University of Rome "La Sapienza", P.le A. Moro 5, 00185 Roma (Italy); [email protected] Citation Francesca Di Iorio and Stefano Fachin (2012). A Note on the Estimation of Long-Run Relationships in Panel Equations with Cross-Section Linkages. Economics: The Open-Access, Open-Assessment E-Journal, Vol. 6, 2012-20. http://dx.doi.org/10.5018/economics-ejournal.ja.2012-20 © Author(s) 2012. Licensed under a Creative Commons License - Attribution-NonCommercial 2.0 Germany Vol. 6, 2012-19 | June 6, 2012 | http://dx.doi.org/10.5018/economics-ejournal.ja.2012-20 conomics: The Open-Access, Open-Assessment E-Journal 1 Introduction Consider a panel of N units, with two non stationary variables, (say, Y and X) observed over T time periods. In each unit of the panel the two variables are known to be linked by a linear long-run equilibrium (cointegrating) relationship, so that the data generating process (DGP) is the following: yit =θi+βixit +uy it (1) xit =xit−1+ux it (2) where i=1,...,N,t=1,...,T, and ux it and uy it are stationary noises. The estimation of (1) when the covariance matrix of the noises is not diagonal, so that the units are dependent, is still a largely unsettled problem. Empirical applications, ignoring efficiency gains, are typically based on single-equation methods (see e.g., Kim et al. (2005), Herzer (2008), Westerlund 2008). This is not surprising, since system estimation with non-stationary variables is fraught with difficulties. Full information maximum likelihood (FIML) methods Groen and Kleibergen (2003) are feasible only when the number of time observations is much larger than the cross-section observations, thus precluding many of the non-stationary panels available in economics and finance. Seemingly unrelated regression (SUR) methods, namely Mark et al. (2005) dynamic SUR (DSUR) and Moon (1999) fully modified SUR (FM-SUR), which are respectively the sytem extensions of dynamic OLS (DOLS) and fully modified ordinary least squares (FM-OLS) are feasible with smaller T/N ratios. However, both require estimation the long-run covariance matrix of the system, a considerably more difficult task (Mark et al. (2005) describe it as ”thorny”) than obtaining the contemporaneuos covariance matrix needed for the baseline SUR. Moon and Perron (2005) claim that SUR estimators are nevertheless superior to single-equation ones in a non-stationary set-up also. However, their simulation study considered a system of very small cross-section size (at most four units with one right-hand side variable, or two units with two variables) and large time dimension ( T=100,300 ), thus very different www.economics-ejournal.org 2 conomics: The Open-Access, Open-Assessment E-Journal from the typical non-stationary panel 1 . This prompts two main questions. First, with empirically relevant sample sizes how large are the efficiency gains (if any) actually delivered by SUR estimators relative to single-equation methods? Should these gains be small, then the widespread use of single-equation estimators would be largely legitimate. Our first goal is thus to compare the estimation performances of single-equation (FM-OLS and DOLS) and SUR system estimators (FM-SUR and DSUR) in panels with small to moderate cross-section dimension and moderate time dimension, characterised by short-run dependence across units. The results will lead to conclusions, hence, advice to practitioners, considerably different from Moon and Perron’s. The second question requires taking a completely different perspective. Efficiency improvements, such as those granted by SUR, are desired in order to have more accurate interval estimation and more reliable tests. Can we reach these targets applying some alternative inference procedure, such has the bootstrap, to standard single-equation estimators? The good simulation results reported for bootstrap inference on FM-OLS (Psaradakis (2001), Fachin (2004)) and unit root and cointegration tests (see inter alia, Park (2003), and, for panel extensions, Chang (2004), Fachin (2007), Fuertes 2008) suggest this point is worth investigating. We shall now first outline the set-up of the Monte Carlo experiment (Section 2) and discuss the results of the comparison between single-equation and SUR estimators (Section 3). In Section 4 we first recall the procedures for bootstrap inference on FM-OLS and then report their performances. Some conclusions are drawn in Section 5. 2 Monte Carlo Experiment: Design The key point here is that the aim of our simulation design cannot be that of obtaining fully general results, as there is a potentially infinite number of dependence structures among the units and variables of a panel. Rather, as mentioned above, we first of all wish to check if the results obtained by Moon and Perron (2005), 1 For instance, Coakley et al. (2006) describe as typical for macroeconomic panels sample sizes of 20 or 30 cross-section units with from 30 to 100 time observations, corresponding e.g., to about three decades of observations at annual or quarterly frequency for the OECD countries www.economics-ejournal.org 3 conomics: The Open-Access, Open-Assessment E-Journal hold for the sample sizes typical of non-stationary panels. In designing our experiment we will thus follow closely Moon and Perron (2005). The DGP is a simple generalisation of (1)-(2) to the case of K=2 explanatory variables: yit =θi+β1ix1it +β2ix2it +uy it ,i=1,...,N,t=1,...,T,(3) xkit =xkit−1+ux kit,k=1,2,i=1,...,N,t=1,...,T.(4) where ux kit,uy it are I(0) noises, so that both the x0s and y are I(1). In the nonstationary panels literature it is quite common (see, e.g., Pesaran 2006) to introduce some realism in the simulation design through parameters heterogenous across units. Here we will follow this practice, generating the regression coefficients respectively as θi∼U(2,4)and as βki ∼U(1,3),where k=1,2. The same set of coefficients has been used for all Monte Carlo replications. It should be remarked that, provided the error variances are suitably controlled to keep the signal-noise ratio constant, the use of heterogenous parameters instead of the homogenous ones used by Moon and Perron (2005) has no consequences on the performances of estimators which allow for heterogeneity 2 . Things are obviously different for pooled estimators, which are misspecified under heterogeneity. Since this class of estimators will not be examined in our experiment the point is irrelevant. The errors of equations (3) and (4) are drawn from a Multivariate Normal distribution with non-diagonal covariance matrix, so that there is short-run dependence across equations and units (the case of long-run dependence is ruled out, as FMSUR, which require the inversion of the long-run covariance matrix would then not be feasible). More precisely, letting ux t= [ux0 1tux0 2t...ux0 Nt ]0, where ux0 it = [ux 1it ux 2it ]0 and uy t= [uy 1tuy 2t...uy Nt ]0,we have uy t ux t(N+2N)×1∼iidN 0 0,R∆ ∆0Φ(N+2N)×(N+2N)!,(5) where R is a full N×N matrix governing the dependence across units in the uy0 it s, ∆ is a N×2N matrix governing the dependence between the ux and uy noises, and 2 The results of the simulazion with homogenous parameters, not included here for sake of brevity, is obviously available on request. www.economics-ejournal.org 4 conomics: The Open-Access, Open-Assessment E-Journal finally Φ is a 2N×2N matrix governing the dependence in the ux0s within and across units. Since Moon and Perron report the performances of both FM-OLS and FMSUR estimators to be negatively affected by the degree of endogeneity of the X ’s we decided to control this dimension of the experiment accurately, imposing an homogeneous endogeneity parameter δ and running two sets of experiments with δ=0.2 and δ=0.4. In both cases the ∆ matrix has a block form ensuring that there is constant correlation between the noise of any Xand that of the relevant Y equation, and no correlation across units: ∆N×2N=     δ δ 0 0 ... 0 0 0 0 δ δ ... 0 0 . . .. . .. . ..... . . 0 0 0 0 ... δ δ     .(6) We instead allow some heterogeneity across units in the dependence parameters, the elements of the Φ matrix. Without loss of generality, we assume x1it and x2it to be incorrelated. Letting φ(i j) lk =cov(ux li,ux k j) denote the covariance between the noise of the variable Xl in the ith unit and that of the variable Xk in the jth unit, we have: Φ2N×2N=              1 0 φ(12) 11 φ(12) 12 ... φ(1N) 11 φ(1N) 12 0 1 φ(12) 21 φ(12) 22 ... φ(1N) 21 φ(1N) 22 φ(21) 11 φ(21) 12 1 0 ... φ(2N) 11 φ(2N) 12 φ(21) 21 φ(21) 22 0 1 ... φ(2N) 21 φ(2N) 22 . . .. . .. . .. . ..... . .. . . φ(N1) 11 φ(N1) 12 φ(N2) 11 φ(N2) 12 ... 1 0 φ(N1) 21 φ(N1) 22 φ(N2) 21 φ(N2) 22 ... 0 1              (7) with φ(i j) lk ∼U(0.3,0.4) . The off-diagonal elements of R,ρi j , are also generated as U(0.3,0.4) , while ρii =1∀i. Again, the parameters thus generated have been kept fixed across the Monte Carlo repetitions. www.economics-ejournal.org 5 conomics: The Open-Access, Open-Assessment E-Journal The time and cross-section sample sizes have been chosen trying to strike a balance between empirical relevance (as mentioned above, most macroeconomic panels have N around 20 or 30 and T often much smaller than 100) and the requirements of the SUR estimator, which is feasible only with rather large T/N ratios. We thus fixed N=5,10 and T=50,100 . Finally, we set the number of Monte Carlo simulations (M) to 1000. 3 Simulation Results: Comparison of Single-Equation and SUR Estimators In Tables 1 and 2 we report some summary statistics describing the results obtained estimating (3) by single equation (FM-OLS and DOLS) and system (FM-SUR and DSUR) methods. To define the expressions for these estimators we need some notation. Following Moon (1999), for the simple bivariate case (1)-(2) first of all define ωt= (uy t,ux t)0 and assume that 1 √T∑[Tr] t=1ωt→B(r). The long-run covariance matrix of B(r) is Ω=∑∞ h=−∞E(ω0ω0 h), and the one-sided long-run covariance matrix Ξ=∑∞ h=0E(ω0ω0 h),both partitioned in the usual way as Ω=Ωyy Ω0 yx Ωyx Ωxx ,Ξ=Ξyy Ξ0 yx Ξyx Ξxx  where all blocks have dimension N×N. Further, let Xt=diag(x1t,...,xNt ),Yt= (y1t,...,yNt )0,Ωyy.x= (Ωyy −ΩyxΩ−1 xx Ωxy). Denoting by an hat a consistent estimate, then: ˆy+ t=yt−ˆ Ωyx ˆ Ω−1 xx ∆xt,e Y+ t= (˜y+ 1t,...,˜y+ Nt )0 ˜y+ it =yit −ˆ Ωii yx(ˆ Ωii xx)−1∆xit ,i=1,...,N. Also, b ξ= (b ξ0 1,...,b ξ0 n)0,b ξi=b Ξii xy −ˆ Ωii yx(ˆ Ωii xx)−1b Ξii xx ˆ πi=b Ξi. xx(( ˆ Ωii yx ˆ Ωii xx)−1 i.)0, and, finally, b ψi= (b Ξi. xy(ˆ Ω−1 yy.x)i.)0−(b ξi. xx(ˆ Ω−1 yy.xˆ Ωyx ˆ Ω−1 xx )i.)0 www.economics-ejournal.org 6 conomics: The Open-Access, Open-Assessment E-Journal with b ψ= ( b ψ0 1,..., b ψ0 N)0.The estimators are then defined as follows: ˆ βFM−OLS = T ∑ t=1 XtXt0!−1 T ∑ t=1 Xte Y+ t−Tb ξi! ˆ βDOLS = T−p ∑ t=p+1 XtX0 t!−1 T−p ∑ t=p+1 Xte Y+ t! ˆ βFM−SUR = T ∑ t=1 Xtˆ Ω−1 yy.xX0 t!−1 T ∑ t=1 Xtˆ Ω−1 yy.xb Y+ t−Tb ψ! ˆ βDSUR = T−p ∑ t=p+1 X−1 tˆ Ω−1 yy X0 t!−1 T−p ∑ t=p+1 Xtˆ Ω−1 yy Yt! where b β= (b β0 1,...,b β0 N) , and b βi= (θi,βi). Small sample point estimation performance of the estimators is usually evaluted by simulation on the basis of the mean over the M simulations of the relative bias, M−1∑M m=1(b βm−β)β−1. In a DGP such as (3)-(4), with N units and K explanatory variables, we evaluate overall point estimation performance by the average over units and parameters of the absolute value (so to avoid compensating errors in opposite directions) of the bias of the estimates of each parameter: bias = (KN)−1 K ∑ k=1 N ∑ i=1|M−1 M ∑ m=1 (b βkim −βki)β−1 ki |(8) Dispersion is analogously measured by the mean over the N units and K parameters of the relative Monte Carlo standard error, qM−1∑M m=1(b βkim −b βki)2β−1 ki : s.e.= (KN)−1 K ∑ k=1 N ∑ i=1" sM−1 M ∑ m=1 (b βkim −b βki)2!β−1 ki #(9) In our experiment we will also evaluate testing performances. Given the extremely different size performances of the single-equation and system estimators, we www.economics-ejournal.org 7 conomics: The Open-Access, Open-Assessment E-Journal will concentrate on Type I errors. We thus tested the hypothesis H0:βki =β(0) ki , k=1,...,K , i=1,...,N, where β(0) ki is the value of the slope parameter used in the Monte Carlo DGP. The first remark in order is that the all estimators are indeed more biased in DGP’s with an higher degree of endogeneity. The second, very important, remark is that the SUR procedure turned out to be practically unfeasible for T=50 and N=10 . The covariance matrix, although not exactly singular, was always so ill-conditioned that the estimators turned out highly numerically unstable even using a generalised Moore–Penrose inversion routine 3 . Hence, we do not report results for this ( T,N ) combination. Since these samples sizes are rather common in applied work on non-stationary panels (with indeed the time sample often actually smaller than this one) this is an important finding. Let us now go into some detail, considering point estimation first. All estimators are essentially unbiased even with the smaller time sample. However, from the first two columns of Table 1 we can appreciate that FM-OLS delivers a slightly better performance than DOLS, while the ranking of the two SUR estimators is not obvious (considering also that DSUR, contrary to FM-SUR, could be computed also for the T=50,N=10 combination). SUR estimators tend to be somehow more biased than the OLS ones. For instance, when T=50,N=5 and δ=0.2 (first raw of Tables 1 and 2) the average relative bias is essentially the same for FM-OLS, DOLS and FM-SUR (respectively, 0.32%, 0.32% and 0.33%) and higher for DSUR (0.45%); when T=100,N=10 and δ=0.4 (last raw of Tables 1 and 2) the bias of FM-SUR is larger than that of FM-OLS (0.51% against 0.16%), and DSUR (0.41%) which is even less biased than its single-equation counterpart (0.46%). 3 It should be remarked that the problem here, estimating the long-run covariance matrix, is considerably more difficult than the standard SUR problem examined by Foschi et al. (2003) www.economics-ejournal.org 8 conomics: The Open-Access, Open-Assessment E-Journal 5 Conclusions The Monte Carlo analysis conducted in this paper compares single equation (FM-OLS and DOLS) and system (FM-SUR and DSUR) estimators of long-run relationships in panels under more realistic time series and cross-section dimensions than previous studies. The Monte Carlo results unambiguously suggest that single-equation FM-OLS alongside a block-bootstrap method provides more accurate estimation and inference. These conclusions, in stark contrast to Moon and Perron (2005), should not come as a surprise. As remarked by Mark et al. (2005), the properties of SUR estimators depend critically upon the quality of the estimate of the covariance matrix. 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Please go to: http://dx.doi.org/10.5018/economics-ejournal.ja.2012-20 The Editor © Author(s) 2012. Licensed under a Creative Commons License - Attribution-NonCommercial 2.0 Germany