Energy consumption and GDP: a panel data analysis with multi-level cross-sectional dependence

Abstract

A fractionally integrated panel data model with a multi-level cross-sectional dependence is proposed. Such dependence is driven by a factor structure that captures comovements between blocks of variables through top-level factors, and within these blocks by non-pervasive factors. The model can include stationary and non-stationary variables, which makes it flexible enough to analyze relevant dynamics that are frequently found in macroeconomic and financial panels. The estimation methodology is based on fractionally differenced block-by-block cross-sectional averages. Monte Carlo simulations suggest that the procedure performs well in typical samples sizes. This methodology is applied to study the long-run relationship between energy consumption and economic growth. The main results suggest that estimates in some empirical studies may have some positive biases caused by neglecting the presence non-pervasive cross-sectional dependence and long-range dependence processes.

Introduction

Panel data models are used in economics and finance to analyze complex systems and phenomena controlling individual heterogeneity and identify effects that are not detectable in simple pure cross section or time series frameworks. In macroeconomics and finance, where panel data sets typically present large cross section $\left(N\right)$ and time series dimension $\left(T\right),$ cross section units are exposed to the influence of factors creating complex interdependencies often in the form of cross-sectional dependence, which can noticeably complicate the statistical inference.

Recent literature on panel modeling has focused on the treatment of cross-sectional dependence. A particular branch studies cases of “full” dependence among cross-sectional units (or “strong dependence” as referred by Pesaran and Tosetti (2011)). A common approach is based on the idea that a small number of common factors may drive the cross-sectional dependence, so that their estimates can be used as additional regressors to augment the model. Pesaran (2006) proposes the Common Correlated Effects (CCE) estimates that use cross-sectional averages of the observables as good proxies for the unobservable common factors, while Bai (2009) suggests estimating the factor structure with principal components. Westerlund and Urbain (2015) formally compare both approaches.

The CCE approach has attracted attention because the number of unobserved common factors is not required. For estimation and inferential theory of CCE methods under different frameworks see, among many others, Pesaran and Tosetti (2011), Harding and Lamarche (2011) for I(0) cases, and Kapetanios et al. (2011), and Banerjee and Carrion-i Silvestre (2017) for non-stationary cases.

Until very recently, only a few papers study panel data that may be composed by economic or financial variables that exhibit long-range dependence of non-integer order. Robinson and Velasco (2015) propose four estimation techniques for a panel data with fixed effects. Ergemen and Velasco (2017), and Ergemen (2017) consider type-II fractionally integrated panel data models that allow for cross-sectional dependence driven by common factors that are pervasive to all cross section units.

In this paper, I propose a type-II fractionally integrated panel data model, which is divided in $R$ blocks of data that can be defined with respect to geographic locations, economic indexes, or by statistical criteria. The main difference with respect to Ergemen and Velasco (2017)’approach (see Ergemen et al. (2016) for an application of their method) is that the cross-sectional dependence assumed in this paper is characterized by two orthogonal levels to ensure that their effects do not leak among them and into other blocks. Pervasive (top-level) factors drive the cross-sectional dependence between blocks, while, block-specific factors characterize dependence within blocks. Then, the model proposed can be useful in situations where some covariations are not sufficiently pervasive to affect all individuals in a panel. This type of factor structure has recently been studied in the literature of factor models, see e.g. Wang (2010), Breitung and Eickmeier (2016), and Choi et al. (2018) and, in panel data models, see Ando and Bai (2016), Ando and Bai (2017), and Kapetanios et al. (2020).

The model proposed can also be useful for a wide range of empirical applications where panel data incorporates variables that exhibit long-range dependence on non-integer orders. The model considers long-range dependence on both levels of common factors covering stationary and nonstationary cases as in Ergemen and Rodríguez-Caballero (2016), so common factors are not restricted to the I(1) case of Kapetanios et al. (2011).

The estimation methodology is executed as follows. Cross-sectional averages are separately computed in each block of data and used as proxies for the unobservable factor space controlling the cross-sectional dependence. Then, the CCE estimators are obtained once the factor structure is filtered out. I refer to this methodology as the block-by-block CCE procedure. The estimation strategy is based on the methodology proposed by Ergemen and Velasco (2017) who use (fractionally) differenced cross-sectional averages to augment the model.

Although the model allows for fractional cointegrating relationship through model’s residual memory parameters, it is does not require to estimate the slope parameters in practice. These residuals memory estimates are based on the conditional-sum-of-square (CSS) criterion proposed by Hualde and Robinson (2011), and Nielsen (2015). Monte Carlo simulations show that this methodology works well even in relatively small panels.

The model proposed is used to study the relationship between economic growth and energy consumption. I argue that there are possible country-specific, regional, and global covariations that may completely characterize the interdependence between countries. This nexus has been extensively examined using a wide range of econometric methods. VAR, VEC, and Panel models have been primordially used, see, among many othrs, Osman et al. (2016), Rodríguez-Caballero and Ventosa-Santaulària (2017), and Leiva and Liu (2019). Ozturk (2010) and Tiba and Omri (2017) provide broad and extensive literature surveys.

I use a panel of 69 countries divided in four regions according to the geographic location to analyze the economic nexus. I assume that the cross sectional dependence is driven by a multi-level factor structure. In general, it is found that the elasticity of energy consumption with respect to economic growth is smaller than those usually reported in the literature as documented by Damette and Seghir (2013), Jalil (2014), and Liddle and Lung (2015), among many others. This positive bias could be a consequence of neglecting a multi-level cross-sectional dependence between countries, which may result in erroneous energy policies.

The paper is organized as follows. Section 2 introduces the model, its assumptions, and the estimation procedure. Section 3 presents the finite-sample properties based on some Monte Carlo designs. Section 4 presents the empirical application. Section 5 concludes the paper.