# **Appendix C. Statistical framework of Henderson et al. (2012)**

We introduce the statistical framework to relate lights to GDP growth in Henderson, Storeygard, and Weil (2012) (HSW hereafter). We specify how we apply the modified version to our analysis.

Let _y*_ be the growth (or log difference) in true real GDP, _y_ the officially reported version, and _l_ the growth of observed light. HSW assumes the following relationships between reported GDP growth, light growth, and true GDP growth: GDP growth: ∗ ; (A.1) yy ii,tt ii,tt∗ ii,tt yy = yy + εε , (A.2) ll where _i_ and _t_ index region and year, respectively. ii,tt _ε_<sup>_y_</sup> in Eq. (A.1) is the classical measurement error in the official ii,tt ii,tt ll = ββyy + εε 2 2 2 GDP growth data. _β_ is the elasticity of lights to true GDP. The variance of _y_<sup>∗</sup> , _ε_<sup>_y_</sup> and _ε_<sup>_l_</sup> are denoted ∗ , ∗ and , respectively. The error terms of the above two equations are assumed to be uncorrelated, i.e., ) = 0. yy yy ll σσ σσ σσ Then we use lights growth to predict GDP growth with the following regression: YYc (εε<sup>yy</sup> ,εε<sup>ll</sup> , (A.3) where _ψ_ is the elasticity of light growth to the true real GDP growth. In practice, Eq. (A.3) is estimated by using ii,tt ii,tt ii,tt yy = ψψll + μμ levels of lights and GDP, augmented with region and year fixed effects. Here the unit of analysis is the city. We further modify Eq. (A.3) to allow for region-specific slopes of lights so that the relationship between nighttime lights and real GDP may differ for regions at different stages of development. We classify China into eight economic regions as mentioned earlier. We thus run the following regression: , (A.4) where _i_ , _r,_ and _t_ index city, region, and year respectively. To improve precision, the fitted values from the above ii,rr,tt rr ii,rr,tt ii,rr tt ii,rr,tt llYY( G PP ) = αα+ ψψ llYY( NNYYNNhttllYYNNhttYY ) + μμ + δδ + εε regression are combined with the official GDP growth figures to arrive at a composite estimate which will have a lower error than either one separately. That is ∗ , (A.5) where is the light-predicted growth from Eq.(A.4), and λ is the optimal weight that minimizes the variance of ii,tt ii,tt ii,tt yy� = λλyy + (1 −λλ)yy� measurement error in the above estimate relative to the true value of economic growth, i.e., ii,tt yy� ). (A.6) From Eq. (A.6), we solve for the optimal weight yy<sup>2</sup> as: ccY (yy�<sup>∗</sup> −yy<sup>∗</sup> ) = λλ<sup>2</sup> σσ + (1 −λλ)<sup>2</sup> ccY (yy�−yy<sup>∗</sup> 2 2 = 2 <u>y2</u> λλ2<sup>∗</sup> 2 = <u>φ ρ</u> , (A.7) where is the correlation between λλ<sup>∗</sup> _l_ and σσyy<sup>2</sup> _y_ (ββ. In our context, we also allow for region-specific<sup>2</sup> σσyy∗σσll+σσσσ lly∗)+σσll σσyy∗ 1−ρρφ−<sup>22</sup> when calculating the optimal . ρρ ρρ Based on Eqs. (A.1) and (A.2), we can obtain three sample moments from the observed data: + 2 λλ<sup>∗</sup> 2 2 2 2 ∗ , ∗ and ∗ . However, we still need one more equation to solve for the above 2 ccY (yy) = σσyy<sup>2</sup> <u>y</u> unknown parameters. To achieve this, we specify a value for σσyy ccY (ll) = ββ σσyy + σσll YYc (ll, yy) = ββσσyy 2 ~~,~~ i.e., the ratio of signal to the total variance σσ y∗ in measured GDP growth. If we divide the countries into two groups based on the levels of measurement error in φφ= σσyy∗+σσyy2 GDP following HSW, by assuming a specific value for _φ_ for the countries with high quality of GDP measurement (developed countries), e.g., 0.9, we can derive the corresponding value of _φ_ for the other group (developing countries like China), i.e., 0.594 in this case. Substituting 0.594 and the region-specific ρ into Eq. (A.7), we get the optimal weight _λ_ . For our sample, the variation of _λ_ is quite small among the eight regions, ranging from 0.576 to 0.594.

0.594. Under this framework, GDP manipulation naturally follows as the difference between the officially reported growth and the predicted true growth in Eq. (A.5): ). rr ii,rr,tt ii,rr,tt (1 −λλ )(yy −yy� 59