6 Supervised Scale Measurement III: Linear Indices

Topics: Strategies for scale development without training data. Developing weights using qualitative expertise.

Required reading:

6.1 Seminar

In this class, we are going to do some analysis comparing the World Bank’s Human Capital Index to the UN’s Human Development Index. Both of these are measured at the country level. These are so obviously similar concepts, that the World Bank has a FAQ specifically addressing what is different about the Human Capital Index from the previously existing Human Development Index:

10. How does the Human Capital Index differ from UNDP’s Human Development Index?

“UNDP’s pioneering Human Development Index is a summary measure of average achievement along key dimensions of human development—a long and healthy life, being knowledgeable, and having a decent standard of living.

While both indices spotlight human capabilities as central to national development, the Human Capital Index also strengthens the economic case for investing in people. The two are highly complementary but differ in the way they are formulated.

The Human Capital Index links selected human capital outcomes with productivity and income levels. It is a forward-looking measure of how current health and education outcomes (including a new measure of learning-adjusted years of school) will shape productivity for the next generation of workers.”

The concept of development was defined by the authors of the Human Development Index as “a process of enlarging people’s choices”, so we can see that there is some conceptual distinction between the conceptualisation of HDI (human development facilitates individual human choices) and the conceptualisation of HCI (human capital facilitates economic productivity).

As discussed in the textbook, the Human Development Index is constructed as follows. HDI incorporates three sub-indices, a Life Expectancy Index \(I_{LE}\), an Education Index \(I_{E}\), and a Income Index \(I_{I}\). These sub-indices are defined in very minimalist ways, with just one or two indicators each (four overall). Each is defined such that the minimum score attainable is 0 and the maximum is 1. \(LE\) is life expectancy at birth. Mean years of schooling \(MYS\) is among those aged 25 and older while expected years of schooling \(EYS\) is a projection for those currently under 18. \(GNIpc\) is per capita gross national income in purchasing power parity US dollars:

\[I_{LE} = \frac{LE - 20}{85 - 20}\] \[I_{E} = \frac{1}{2} \cdot \frac{MYS}{15} + \frac{1}{2} \cdot \frac{EYS}{18}\] \[I_{I} = \frac{\log(GNIpc) - \log(100)}{\log(75000) - \log(100)}\]

As noted in the textbook, the benchmarks for what qualifies as full and what qualifies as zero on each of these sub-indices are quite important. These rescalings are designed to place all countries in the interval \(\left[0,1\right]\) and in the few cases where the rescaled values exceed that range, they are censored to the relevant extreme (this only occurs for the top end of the education and income indices). HDI then aggregates these three fractions:

\[HDI^3 = I_{LE} \cdot I_E \cdot I_I\]

This is a multiplicative aggregation: development is understood to arise out of jointly possessing life expectancy, education and income. These are complementary resources which do not substitute for one another.

One potential criticism of HDI is that there is not really a clear justification for why going from, for example, 68.75 to 85 years of life expectancy (\(I_{LE} = 0.75\) to \(I_{LE} = 1.00\)) is as consequential in terms of development as going from a GNI of $14331 to a GNI of $75000 (\(I_{I} = 0.75\) to \(I_{I} = 1.00\)). The Human Capital Index authors make a serious effort to justify these connections between marginal changes in their indices and marginal changes in productivity. They do not have direct measures of country-level productivity, so they cannot use the supervised strategy that we considered in our previous class, but they nonetheless are able to go beyond an equal weighting strategy in justifying how each indicator maps into the index. For full detail, see (World Bank 2018, p34–38).

The components of the HCI are combined into a single index by first converting them into contributions to productivity. Multiplying these contributions to productivity gives the overall HCI. The HCI summarizes how productive children born today will be as members of the future workforce, given the risks to education and health summarized in the components. The HCI is measured in units of productivity relative to a benchmark corresponding to complete education and full health.

In the case of survival, the relative productivity interpretation is stark: children who do not survive childhood never become productive adults. As a result, expected productivity as a future worker of a child born today is reduced by a factor equal to the survival rate, relative to the benchmark where all children survive.

In the case of education, the relative productivity interpretation is anchored in the large empirical literature measuring the returns to education at the individual level. A rough consensus from this literature is that an additional year of school raises earnings by about 8 percent. This evidence can be used to convert differences in learning-adjusted years of school across countries into differences in worker productivity. For example, compared with a benchmark where all children obtain a full 14 years of school by age 18, a child who obtains only 9 years of education can expect to be 40 percent less productive as an adult (a gap of 5 years of education, multiplied by 8 percent per year).

In the case of health, the relative productivity interpretation is based on the empirical literature measuring the economic returns to better health at the individual level. The key challenge in this literature is that there is no unique directly measured summary indicator of the various aspects of health that matter for productivity. This microeconometric literature often uses proxy indicators for health, such as adult height. This is because adult height can be measured directly and reflects the accumulation of shocks to health through childhood and adolescence. A rough consensus drawn from this literature is that an improvement in health associated with a 1-centimeter increase in adult height raises productivity by 3.4 percent.

HCI thus has three sub-indices (like HDI), which are survival \(I_{V}\), school \(I_S\), and health \(I_H\), which are aggregated multiplicatively: \[HCI = I_{V} \cdot I_{S} \cdot I_H\] Note the slight difference that \(HDI^3 = I_{LE} \cdot I_E \cdot I_I\) whereas \(HCI = I_{V} \cdot I_{S} \cdot I_H\), the former means that HDI is the cube root of the product of the indicators whereas HCI is just the product of the indicators, we will look at the consequences of this in the exercise below. The sub-indices are functions of the indicators that are motivated by the quoted discussion above:

\[I_{V} = \frac{1 - \text{under-5 mortality rate}}{1}\] \[I_{S} = exp\left( psi (\text{expected years of school} \cdot \frac{\text{harmonized test score}}{625} - 14) \right)\] \[I_H = exp\left( \frac{(\gamma_\text{ASR} \cdot (\text{adult survival rate} - 1) + \gamma_\text{stunting} \cdot (\text{not stunted rate} - 1))}{2}\right)\]

The components of the index are expressed here as contributions to productivity relative to the benchmark of complete high-quality education and full health. The parameter \(\psi = 0.08\) measures the returns to an additional year of school. The parameters \(\gamma_{ASR} = 0.65\) and \(\gamma_{\text{stunting}} = 0.35\) measure the improvements in productivity associated with an improvement in health, using adult survival and stunting as proxies for health. The benchmark of complete high-quality education corresponds to 14 years of school and a harmonized test score of 625. The benchmark of full health corresponds to 100 percent child and adult survival and a stunting rate of 0 percent.

hcihdi <- read.csv("week-6-hci-vs-hdi.csv")
  1. Create a subset of the data that only includes the six sub-indices of HDI and HCI, as defined above: \(I_L\), \(I_E\) and \(I_I\) for HDI, \(I_V\), \(I_S\), and \(I_H\) for HCI. Plot and calculate the pairwise correlations between all six of these sub-indices. Which are the strongest and the weakest pairwise correlations? Why do these patterns make sense? Code Hint:* You can use the pairs() command to create a base-R plot, or ggpairs() from the GGally package like last week.
  1. Make a table with the means and standard deviations of all six sub-indices. What do high/low means imply in this context? What do high/low standard deviations imply in this context? What do we learn from this table?
  1. Calculate the correlations of HDI with each of its three sub-indices. Calculate the correlation of HCI with each of its three sub-indices. One of the correlations will be notably high. In light of your response to 2, explain why you see this and what it means for understanding the quantity that is being measured by the relevant index.
  1. Does your finding in Q3 mean that there is something wrong with the index? If you assume, for sake of argument, that the index is really capturing variation in the relevant concept well, what is implied about variation across countries in the relevant concept?
  1. Overall, do you find HDI or HCI a more convincing measure of the quantities that they are respectively trying to measure. There is no right answer here, just give it some thought!
  1. As noted above, one way in which the two measures differ is that \(HDI^3 = I_{LE} \cdot I_E \cdot I_I\) whereas \(HCI = I_{V} \cdot I_{S} \cdot I_H\). Plot HDI and HCI against one another, then plot \(HDI^3\) against \(HCI\) (or plot \(HDI\) against \(HCI^{1/3}\)). Is one of these two measures generally “larger” than the other, when a fair comparison is made? Given the way they are designed, what does this imply about the extent to which countries are currently approaching the maximum possible level of “human development” and “human capital”?

References

World Bank. 2018. “The Human Capital Project.” International Bank for Reconstruction and Development / World Bank.