## Abstract

We consider an empirical estimation of the environmental Kuznets curve (EKC) for carbon dioxide and sulphur, with a focus on confidence set estimation of the tipping point. Various econometric—parametric and nonparametric—methods are considered, reflecting the implications of persistence, endogeneity, the necessity of breaking down our panel regionally, trends and temporal instability, and the small number of countries within each panel. In particular, we propose a parametric inference method that corrects for potential weak-identification of the tipping point. Weak identification may occur if the true EKC is linear while a quadratic income term is nevertheless imposed into the estimated equation. Relevant literature to date confirms that non-linearity of the EKC is indeed not granted, which provides the motivation for our work. We also propose a non-parametric counterpart to the parametric confidence set, for sensitivity analysis. Viewed collectively, our results confirm an inverted U-shaped EKC in the OECD countries but generally not elsewhere, although a local-pollutant analysis suggests favorable exceptions beyond the OECD. Our measures of uncertainty confirm that it is difficult to identify economically plausible tipping points. Policy-relevant estimates of the tipping point can nevertheless be recovered from a local-pollutant long-run or non-parametric perspective.

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## Notes

For surveys, see

*e.g.*Carson (2010), Wagner (2008), Vollebergh et al. (2009), Brock and Taylor (2005), Cavlovic et al. (2000), Dinda (2004), Stern (2001; 2003; 2004; 2010), Yandle et al. (2004), Dasgupta et al. (2002), Levinson (2002), and the references therein. Other works are also discussed below.The range of published estimates is wide and covers values close to zero for the quadratic component, and controversial income elasticities.

A tipping point consistent with our definition may be hard to formulate from a general non-parametric perspective.

In Stern (2010, Table 4), for the case using Wagner’s carbon data, the fixed effects estimated turning point is $41,678 with a standard error of $4,043 without time effects and $15,837 with a standard error of $1,060 with time effects. This contrasts sharply with the between estimator where the turning point is $653,110 with a standard error of $2,084,513.

See Dufour (1997). Related results can also be found in the so called

*weak instruments*literature which is now considerable; see the surveys by Dufour (2003), Stock et al. (2002), and the viewpoint article by Stock (2010). Weak instruments and inference on ratios raise comparable local identification problems.Bolduc et al. (2010) find that the

*delta*and bootstrap method are spurious even in the simplest design they consider. Coverage rates collapsing to zero [which means that the probability of the estimated interval to include the unknown true value of the ratio is zero] are also documented for empirically relevant scenarios.For a parallel with the weak-instruments problem, refer to Stock (2010, pp. 86–87).

See, for example, Rühl and Giljum (2011).

Results when all regressors were instrumented are qualitatively similar so we do not report them for space considerations.

Instrumental variables (IV) methods [

*e.g.*Anderson and Hsiao (1982), Arellano and Bond (1991), and Blundell and Bond (1998)] may seem an attractive solution to treat endogeneity as well as persistence. Unfortunately, with small \(n\) which corresponds to the problem at hand, these IV methods, when applicable [see*e.g.*Bun and Kiviet (2006) for conditions on \(n\) relative to \(T\)], can be severely biased and highly imprecise; see*e.g.*Kiviet (1995), Judson and Owen (1999), Bruno (2005) and Bun and Carree (2005).Indeed, the above cited econometric literature provides many convincing simulation studies documenting this problem with standard Wald-type tests.

Results are not reported for space considerations but are available upon request.

The list of OECD countries includes countries that have been in the OECD for the majority of the time frame of this study, with the exceptions of Albania and South Korea. The latter two are included because, in our judgement, are anomalies with respect to their geographic peers and Albania is included because this group corresponded closest to its characteristics.

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This work was supported by the Canadian Network of Centres of Excellence [program on Mathematics of Information Technology and Complex Systems (MITACS)], the Social Sciences and Humanities Research Council of Canada, and the Fonds de recherche sur la société et la culture (Québec).

## Appendices

### Appendix 1: List of Countries

### 1.1 Countries Used for the \(\hbox {CO}_{2}\) Equation

**OECD.**
^{Footnote 16} (27 countries). Albania, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Hong Kong, Hungary, Iceland, Ireland, Italy, Japan, Malta, Netherlands, New Zealand, Norway, Portugal, South Korea, Spain, Sweden, Switzerland, Turkey, United Kingdom, United States.

**Asia.** (17 countries) Bangladesh, China, India, Indonesia, Kazakhstan, Kyrgyzstan, Malaysia, Mongolia, Pakistan, The Philippines, Singapore, Sri Lanka, Tajikistan, Thailand, Turkmenistan, Uzbekistan, Vietnam.

**Sub-Saharan Africa**. (16 countries) Angola, Benin, Botswana, Cameroon, Congo, Cote d’Ivoire, Gabon, Ghana, Kenya, Namibia, Nigeria, Senegal, South Africa, Togo, Zambia, Zimbabwe.

**The Middle East & North Africa.** (16 countries) Algeria, Bahrain, Egypt, Eritrea, Iran, Jordan, Kuwait, Lebanon, Morocco, Oman, Saudi Arabia, Sudan, Syria, Tunisia, United Arab Emirates, Yemen.

**South America.** (11 countries) Argentina, Bolivia, Brazil, Chile, Colombia, Ecuador, Guatemala, Paraguay, Peru, Uruguay, Venezuela.

**Central America & The Caribbean.** (10 countries). Costa Rica, Dominican Republic, El Salvador, Haiti, Honduras, Jamaica, Mexico, Nicaragua, Panama, Trinidad & Tobago.

**Other.** (17 countries) Armenia, Azerbaijan, Belarus, Bulgaria, Croatia, Czech Republic, Georgia, Latvia, Lithuania, Macedonia, Moldova, Poland, Romania, Russia, Slovakia, Slovenia, Ukraine.

### 1.2 Countries Used for the \(\hbox {SO}_{2}\) Equation

**OECD.** (27 countries). Albania, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Hong Kong, Hungary, Iceland, Ireland, Italy, Japan, Malta, Netherlands, New Zealand, Norway, Portugal, South Korea, Spain, Sweden, Switzerland, Turkey, United Kingdom, United States.

**Asia.** (12 countries). Bangladesh, China, India, Indonesia, Malaysia, Mongolia, Pakistan, Philippines, Singapore, Sri Lanka, Thailand, Vietnam.

**Sub-Saharan Africa.** (11 countries). Botswana, Cameroon, Cote d’Ivoire, Gabon, Ghana, Kenya, Senegal, South Africa, Togo, Zambia, Zimbabwe.

**The Middle East & North Africa.** (13 countries) Algeria, Bahrain, Egypt, Iran, Jordan, Kuwait, Morocco, Oman, Saudi Arabia, Sudan, Syria, Tunisia, United Arab Emirates.

**South America.** (10 countries). Argentina, Bolivia, Brazil, Chile, Colombia, Guatemala, Paraguay, Peru, Uruguay, Venezuela.

**Central America & The Caribbean.** (7 countries). Costa Rica, Dominican Republic, El Salvador, Honduras, Mexico, Panama, Trinidad & Tobago.

**Other.** (2 countries). Bulgaria, Romania.

### Appendix 2: The Fieller Solution

The Fieller method requires solving inequality (3.5) for \(d_{0}\), which may be re-expressed as

Except for a set of measure zero, \(A\ne 0\). Similarly, except for a set of measure zero, \(\Delta =B^{2}-AC\ne 0\). Real roots equal to \(\frac{-B\pm \sqrt{\Delta }}{A}\) exist if and only if \(\Delta >0\). Let \(d_{01}\) refer to the smaller root and \(d_{02}\) to the larger root, then

Bolduc et al. (2010) further show that: (i) if \(\Delta <0\), then \( A<0\) and \(FCS\left( d;\alpha \right) =\mathbb {R}\); (ii) \(FCS\left( d;\alpha \right) \) contains the point estimate \(\hat{g}_{1}/\hat{ g}_{2}\) and thus cannot be empty, and (iii) asymptotically, Fieller’s solution and the *Delta* method give similar results when the former leads to an interval, *i.e.* when the denominator is far from zero. Taking the exponential of the limits of \(FCS\left( d;\alpha \right) \) provides a confidence set for \(\exp (d)\).

### Appendix 3: B-splines

The method from Ma and Racine (2013) uses a B-spline function for \(f(.)\), which is a linear combination of B-splines of degree \(m\) defined as follows

where \(b_{c}\) are denoted “control points”, \(k_{0},\ldots ,k_{N+1}\) are known as a knot sequence [an individual term in this sequence is known as a knot],

which is referred to as the ‘intercept’, and

The unknown function \(f(.)\) is estimated by least squares as

Explicitly, this requires the estimation of the control points \(b_{c}\). If covariates are considered endogenous and instruments provided, 2SLS is also possible. Underlying best fit parameters are selected by cross-validation; see Ma et al. (2012) for further details. Further description of this R-package is available at: http://cran.r-project.org/web/packages/crs/crs.pdf.

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Bernard, JT., Gavin, M., Khalaf, L. *et al.* Environmental Kuznets Curve: Tipping Points, Uncertainty and Weak Identification.
*Environ Resource Econ* **60**, 285–315 (2015). https://doi.org/10.1007/s10640-014-9767-y

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DOI: https://doi.org/10.1007/s10640-014-9767-y