i i i i i# # i# # i# #)} where i# 2 i i 2 2} and ττ̂ = EE(ττ= ~~�~~ |(vv −ττ )) = μμ). In implementing this formula, we replace + {σσ φφ(μμ /σσ )}/{ΦΦ(μμ /σσ μμ i= {−σσi uu by its estimate from the (ττ −vv )}/{σσττ + σσvv SF model in Step 2. The overall gap is then predicted by summing the predicted values of i . ττ<sup>2</sup> vv<sup>2</sup> ττ<sup>2</sup> vv<sup>2</sup> σσ<sup>#</sup> σσ σσ /(σσ + σσ ττ −vv Almost all the SF models are specified in log (or rates of change) form. In these formulations, inefficiency ii ηη and ττ (shortfall from the potential outcome) can be interpreted as the percentage (when multiplied by 100) shortfall in the outcome variable, especially when inefficiency is small. However, the exact percentage shortfall is 1 − exp( − inefficiency) , where exp( − inefficiency) is the measure of efficiency. Since we have two components of inefficiency, we can obtain the corresponding estimates of the two efficiency components. Thus, persistent efficiency is ) and time-varying efficiency is exp( i ). Then overall technical efficiency (OTE) is the product of the persistent and time-varying efficiency. Note that ii the overall technical inefficiency and OTE are closely related. For small values of inefficiency, the overall exp (−ηη −ττ technical inefficiency ≈ 1- OTE. For reporting purposes, for each indicator one can use either the mean of overall inefficiency for each country, which is , or the average of OTE for each country. Either the average inefficiency or efficiency scores can be used to rank countries for each indicator. Note that the ranking shows the scope ii ii ii for improvement for each country and indicator. The inefficiency scores show potential for improvement uu� = ηη + ττ̅ for the indicator in question and not the value of the indicator, which might confuse some readers. We clarify this issue using a graph, which is used for illustration purposes. Since the model we estimate for each indicator is i 0 i′ ′ i i , the country average is 0 ′ ′ �= 0 since i . If we denote 0 + ′ ′ � ii ii ii � , yy = ββ + xx. Now we use the information on ββ+ zz γγ+ αα −ηη −ττ + vv and in ii ii ii ii ii ii ıı yȳ = ββ + xx̄ ββ+ zz γγ+ αα −ηη −ττ̄ assuming vv EE(vv ) = 0 ∀ii ββ Figure 1 and measure the difference between them as the potential for improvement. ii ii ii ii ıı ii ıı ıı ii ıı ıı ii xx̄ ββ+ zz γγ+ αα bbyy mm yy�= mm −uu� where uu� �= ηη + ττ yy mm To be more specific, let us assume that is educational quality, measured by the PISA reading performance, and the points A and B in figure 1 represent � for two countries, say Vietnam and the ıı Republic of Korea. The points C and D represent their respective yy _m_ values, that is, their own potential. ıı Clearly both actual and potential values for Korea are higher than those of Vietnam. However, Vietnam yy is much closer to its frontier value (distance AC < BD) and therefore has smaller scope for improvement. Thus, if we rank the countries in terms of their improvement potential (efficiency), Vietnam will be ranked higher than Korea in terms of educational quality. This does not mean, however, that educational quality in Korea is lower than in Vietnam. Our ranking is in terms of efficiency, not in terms of the raw indicator. We discuss this in detail for various indicators in the results section.

# **5. Data** **_5.1 The indicators (_** i **_)_** yy

Following the illustration presented in the conceptual framework, we draw on the theoretical and empirical literatures on economic growth, poverty reduction, and resilience to identify some variables that can be used as indicators of social and economic development (Foa 2013; Loayza, Fajnzylber, and Calderón 2005; World Bank 2013).

Initially, we built a database for 146 indicators covering 207 countries for the period 1950-2014. However, for practical reasons, we narrowed the number of indicators down to 10 indicators and the period of study 11