On measuring economic growth from outer space: a single country approach

Abstract

This article proposes a simple statistical approach to combine nighttime light data with official national income growth figures. The suggested procedure arises from a signal-plus-noise model for official growth along with a constant elasticity relation between observed night lights and income. The methodology implemented in this paper differs from the approach based on panel data for several countries at once that uses World Bank ratings of income data quality for the countries under study to produce an estimate of true economic growth. The new approach: (a) leads to a relatively simple and robust statistical method based only on time series data pertaining to the country under study and (b) does not require the use of quality ratings of official income statistics. For illustrative purposes, some empirical applications are made for Mexico, China and Chile. The results show that during the period of study there was underestimation of economic growth for both Mexico and Chile, while official figures of China over-estimated true economic growth.

Appendices

Appendix A: Obtaining the BLUE of D y

Let I be the (N − 1)-dimensional identity matrix, ${\varvec{\upeta}} = \left( {\eta_{2} , \ldots ,\eta_{N} } \right)^{\prime }$ and ${\varvec{\upvarepsilon}} = \left( {\varepsilon_{2} , \ldots , \varepsilon_{N} } \right)^{\prime }$, so that models (9) and (10) can be written as the system of linear equations

$$ \left( {\begin{array}{*{20}l} {D{\mathbf{z}}} \hfill \\ {D{\mathbf{x}}} \hfill \\ \end{array} } \right) = UD{\mathbf{y}}+\left( {\begin{array}{*{20}l} {\varvec{\upeta}} \hfill \\ {\varvec{\upvarepsilon}} \hfill \\ \end{array} } \right) $$

with $U = \left( {\begin{array}{*{20}c} I \\ {\beta I} \\ \end{array} } \right)$, $E\left( {\begin{array}{*{20}l} {\varvec{\upeta}} \hfill \\ {\varvec{\upvarepsilon}} \hfill \\ \end{array} } \right) = {\mathbf{0}}_{{2\left( {N - 1} \right)}}$ the zero vector of size 2(N − 1) and

$$ {\text{Va}}r\left( { \begin{array}{*{20}c} {\varvec{\upeta}} \\ {\varvec{\upvarepsilon}} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\sigma_{{}}^{2} I} & {0_{N - 1} } \\ {0_{N - 1} } & {\sigma_{\varepsilon }^{2} I} \\ \end{array} } \right) = \sigma_{\varepsilon }^{2} \left( {\begin{array}{*{20}c} {\alpha I} & {0_{N - 1} } \\ {0_{N - 1} } & I \\ \end{array} } \right) = \sigma_{\varepsilon }^{2}\Omega , $$

where $0_{N - 1}$ is the (N − 1)-dimensional zero matrix and $\alpha = \sigma_{{}}^{2} /\sigma_{\varepsilon }^{2}$.

Then, by assuming that the parameters $\beta$, $\sigma_{\varepsilon }^{2}$ and $\alpha$ are known, we apply GLS to obtain the best linear unbiased estimator (BLUE) of Dy, which is given by

$$ \widehat{{D{\mathbf{y}}}} = \left( {{\text{U}}^{{\prime }}\Omega ^{ - 1} U} \right)^{ - 1} U^{{\prime }}\Omega ^{ - 1} \left( {\begin{array}{*{20}c} {D{\mathbf{z}}} \\ {D{\mathbf{x}}} \\ \end{array} } \right) = \left( {\alpha^{ - 1} + \beta^{2} } \right)^{ - 1} \left( {\alpha^{ - 1} D{\mathbf{z}} + \beta D{\mathbf{x}}} \right), $$

that leads us to expression (11).

Appendix B: Unbiased estimation of $\sigma_{\varepsilon }^{2}$

Within the context of GLS, we can get an unbiased estimator of $\sigma_{\varepsilon }^{2}$ by defining the residual vectors of models (9) and (10) as

$$ {\hat{\boldsymbol{\upeta }}} = D{\mathbf{z}} - \widehat{{D{\mathbf{y}}}} \,\,\,{\text{and}}\,\,\, {\hat{\boldsymbol{\upvarepsilon }}} = D{\mathbf{x}} - \beta \widehat{{D{\mathbf{y}}}} $$

so that the stacked residual vector becomes

$$ \begin{aligned} \left( {\begin{array}{*{20}c} {{\hat{\boldsymbol{\upeta }}}} \\ {{\hat{\boldsymbol{\upvarepsilon }}}} \\ \end{array} } \right) & = \left( {\begin{array}{*{20}c} {D{\mathbf{z}}} \\ {D{\mathbf{x}}} \\ \end{array} } \right) - U\widehat{{{\text{D}}{\mathbf{y}}}} \\ & = \left[ {\left( {\begin{array}{*{20}c} I \\ {0_{N - 1} } \\ \end{array} \begin{array}{*{20}c} {0_{N - 1} } \\ I \\ \end{array} } \right){-}\left( {\alpha^{ - 1} + \beta^{2} } \right)^{ - 1} U\left( {\alpha^{ - 1} I \beta I} \right)} \right]\left( {\begin{array}{*{20}c} {D{\mathbf{z}}} \\ {D{\mathbf{x}}} \\ \end{array} } \right) \\ & = \left[ {\left( {\begin{array}{*{20}c} I \\ {0_{N - 1} } \\ \end{array} \begin{array}{*{20}c} {0_{N - 1} } \\ I \\ \end{array} } \right){-}\left( {\alpha^{ - 1} + \beta^{2} } \right)^{ - 1} \left( {\begin{array}{*{20}c} {\alpha^{ - 1} I} \\ {\beta \alpha^{ - 1} I} \\ \end{array} \begin{array}{*{20}c} {\beta I} \\ {\beta^{2} I} \\ \end{array} } \right)} \right]\left[ {UD{\mathbf{y}} + \left( {\begin{array}{*{20}c} { {\varvec{\upeta}}} \\ { {\varvec{\upvarepsilon}}} \\ \end{array} } \right)} \right] \\ \end{aligned} $$

Now, since

$$ \left( {\alpha^{ - 1} + \beta^{2} } \right)^{ - 1} \left( {\begin{array}{*{20}c} {\alpha^{ - 1} I} \\ {\beta \alpha^{ - 1} I} \\ \end{array} \begin{array}{*{20}c} {\quad \beta I} \\ {\quad \beta^{2} I} \\ \end{array} } \right)U = U $$

we get

$$ \left( {\begin{array}{*{20}c} {{\hat{\boldsymbol{\upeta }}}} \\ {{\hat{\boldsymbol{\upvarepsilon }}}} \\ \end{array} } \right) = \left[ {\left( {\begin{array}{*{20}c} {\text{I}} \\ {0_{{{\text{N}} - 1}} } \\ \end{array} \begin{array}{*{20}c} {\quad 0_{{{\text{N}} - 1}} } \\ {\quad {\text{I}}} \\ \end{array} } \right) - {\text{H}}} \right]\left( {\begin{array}{*{20}c} { {\varvec{\upeta}}} \\ { {\varvec{\upvarepsilon}}} \\ \end{array} } \right), $$

with

$$ H = \left( {\alpha^{ - 1} + \beta^{2} } \right)^{ - 1} \left( {\begin{array}{*{20}c} {\alpha^{ - 1} I} \\ {\beta \alpha^{ - 1} I} \\ \end{array} \begin{array}{*{20}c} {\quad \beta I} \\ {\quad \beta^{2} I} \\ \end{array} } \right) $$

a symmetric and idempotent matrix whose trace, ${\text{tr}}(H) = N - 1$, is the number of degrees of freedom for the residual sum of squares, given by

$$ \left( {\begin{array}{*{20}c} {{\hat{\boldsymbol{\upeta }}}} \\ {{\hat{\boldsymbol{\upvarepsilon }}}} \\ \end{array} } \right) '\left( {\begin{array}{*{20}c} {\alpha^{ - 1} I} \\ {0_{N - 1} } \\ \end{array} \begin{array}{*{20}c} {\quad 0_{N - 1} } \\ {\quad I} \\ \end{array} } \right)^{ - 1} \left( {\begin{array}{*{20}c} {{\hat{\boldsymbol{\upeta }}}} \\ {{\hat{\boldsymbol{\upvarepsilon }}}} \\ \end{array} } \right) = \alpha^{ - 1} {\hat{\boldsymbol{\upeta }}}^{\prime } {\hat{\boldsymbol{\upeta }}} + {\hat{\boldsymbol{\upvarepsilon }}}^{\prime } {\hat{\boldsymbol{\upvarepsilon }}}. $$

The matrix H is called the “hat matrix” because it transforms the original data Dz and Dx into the estimated data $\widehat{{D{\mathbf{z}}}}$ and $\widehat{{D{\mathbf{x}}}}$ by means of the following relationship

$$ \left( {\begin{array}{*{20}c} {\widehat{{D{\mathbf{z}}}}} \\ {\widehat{{D{\mathbf{x}}}}} \\ \end{array} } \right) = U\widehat{{D{\mathbf{y}}}} = H\left( {\begin{array}{*{20}c} {D{\mathbf{z}}} \\ {D{\mathbf{x}}} \\ \end{array} } \right). $$

Thus, if $\alpha$ and $\beta$ were known, we obtain the following unbiased estimator

$$ {\hat{{\sigma }}}_{{{\varepsilon }}}^{2} = \frac{1}{N - 1}\left( {\alpha^{ - 1} {\hat{\boldsymbol{\upeta }}}^{\prime } {\hat{\boldsymbol{\upeta }}} + {\hat{\boldsymbol{\upvarepsilon }}}^{\prime } {\hat{\boldsymbol{\upvarepsilon }}} } \right). $$

In practice, we have to use the estimators $\hat{\alpha }^{ - 1}$ and $\hat{\beta }^{2}$, in place of the true parameters $\alpha^{ - 1}$ and $\beta^{2}$, so that two degrees of freedom are lost and the variance estimator becomes expression (13), where we should notice that ${\hat{\boldsymbol{\upvarepsilon }}} = \hat{\beta }(\widehat{{D{\mathbf{z}}}} - \widehat{{D{\mathbf{y}}}}).$

Appendix C: Original data employed in the empirical applications

Year	Mexico		China		Chile
	Ln(DN)	Ln(GDP)	Ln(DN)	Ln(GDP)	Ln(DN)	Ln(GDP)
1992	0.356540	29.3690	− 0.349342	29.1549	− 1.201073	31.0451
1993	0.548981	29.3883	− 0.266380	29.2859	− 0.896996	31.1126
1994	0.563912	29.4319	− 0.159449	29.4090	− 0.918416	31.1681
1995	0.745930	29.3677	0.021317	29.5125	− 0.657251	31.2691
1996	0.690892	29.4179	− 0.023675	29.6078	− 0.633634	31.3406
1997	0.593879	29.4834	− 0.017993	29.6967	− 0.671262	31.4046
1998	0.699476	29.5313	0.078083	29.7718	− 0.494210	31.4364
1999	0.746560	29.5693	0.048182	29.8451	− 0.513416	31.4288
2000	0.813918	29.6333	0.141074	29.9257	− 0.417595	31.4727
2001	0.802614	29.6317	0.180832	30.0055	− 0.380645	31.5059
2002	0.804416	29.6399	0.283795	30.0926	− 0.336200	31.5275
2003	0.652805	29.6533	0.191841	30.1879	− 0.491460	31.5659
2004	0.712251	29.6927	0.362953	30.2841	− 0.406507	31.6246
2005	0.679279	29.7242	0.319271	30.3830	− 0.477887	31.6787
2006	0.713111	29.7712	0.425282	30.4928	− 0.409212	31.7236
2007	0.769451	29.8027	0.521681	30.6150	− 0.376859	31.7693
2008	0.787806	29.8203	0.557354	30.7012	− 0.356730	31.8004