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Estimating Capabilities with Structural Equation Models: How Well are We Doing in a ‘Real’ World?

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Measuring capabilities is a major challenge for the operationalization of the capability approach. Structural equation models (SEM) are being increasingly used as one possible methodology for estimating capabilities, but a certain skepticism remains about their appropriateness. In this paper, we perform a unique simulation experiment for testing the validity of such estimators. Using an agent-based modeling tool, we simulate a ‘real’ life scenario with individuals of heterogeneous characteristics and behaviors, having different capability sets, and making different decisions. We then run a SEM (MIMIC) model on the data generated in this simulated world to estimate the individual capabilities. Thus our data generating process is completely disconnected with the econometric model used for estimation. Our results support the idea that SEM can coherently estimate the true capabilities. We find that the linear predictor from the structural part of the SEM provides better results than the ‘classical’ factor scores based on the full model.

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  1. Some examples of direct capability questions are ‘At present, how easy or difficult do you find it to enjoy the love, care and support your immediate family’, ‘I am able to express my political views’, ‘Is your current accommodation adequate or inadequate for your current needs’, ‘At work, have you recently felt that you were playing a useful part in things’, see e.g. Anand et al. (2009, 2011).

  2. See for instance Anderson and Gerbing (1988) and MacCallum and Austin (2000) for its use in the psychological literature and Judd and Milburn (1980) or Hurwitz and Peffley (1987) for an application to political attitudes.

  3. For a general introduction to agent-based models see Gilbert (2008) and Epstein and Axtell (1996).

  4. The extension of our study to a general SEM is relatively straightforward and should not alter the qualitative conclusions of our study. This can be envisaged in a follow-up work.

  5. For instance, the access to a certain service can depend on the location of the agent.

  6. The supply of education can be thought of as a combined package of quality and quantity. Though it is not complicated to distinguish between the two in our set-up and even introduce school ‘fees’, with all these aspects varying from one school to another, these extensions will not significantly alter the conclusions obtained in our simpler setting.

  7. It would be easy to include schooling fees in the model. However, for the purpose of our study it is not useful and would only make the model unnecessarily more complex.

  8. Using a t-test, we can reject the null hypothesis of equal means at less than the 0.1 % level.

  9. We standardize the two variables to have them on the same scale for comparison. This does not alter the analysis, because the latent factor has neither inherent scale nor level.

  10. We mean by fractional rank the rank measured on the interval between 0 (first individual) and 1 (last individual).


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Correspondence to Florian Chávez-Juárez.

Appendix: Additional Results

Appendix: Additional Results

1.1 Factor Scores Versus Linear Predictors

In this appendix we provide additional results and comparisons that underline the better performance of the linear predictors compared to the factor scores in our ABM simulation setting. Fig. 11 displays the cumulative distribution function of the absolute rank difference between the true and the estimated capabilities for both computation methods. The solid line corresponds exactly to the line displayed in Fig. 8 and the dashed line refers to the estimator \(\hat{f}\). From the graph it becomes evident that estimating the capabilities only using x performs much better than when also including y. While for the first case the difference is above 0.2 in less than 10 % of the cases, more than 30 % of cases show a difference of at least 0.2 when including the y. It seems that when including the information from y (outcome variables), these dominate the estimation and the predicted values proxy functionings rather than capabilities.

Fig. 11
figure 11

Comparison of both estimators for f

Fig. 12
figure 12

CDF of the true capabilities versus \(\hat{f}\)

Figure 12 is equivalent to Fig. 7 but displays the performance of \(\hat{f}\) rather than \(\tilde{f}\). Although the relatively bad performance of this second estimator is not obvious in this figure, the null hypothesis of equal distributions in the Wilcox test is rejected in all cases, suggesting that this second estimator does not coherently predict the true underlying capabilities in this baseline scenario.

1.2 Robustness Checks

As mentioned in Sect. 3.1, we have assumed a series of distributions in the baseline setting. It is important to make sure that the relatively high correlations are not due to too simplistic distributional assumptions. For this reason, we performed robustness checks in which we replace the uniform by non-symmetric distributions. Table 4 displays the baseline and the robustness check distributions:

Table 4 Baseline implementation of random variables

Figure 13 shows the baseline and skewed distributions that we simulate. To run the robustness check we simulated all possible combinations of uniform and skewed distributions, which sums up to 12 different settings.

Fig. 13
figure 13

Uniform and skewed distributions used for the robustness checks

Table 5 displays the results of the robustness checks. The first three columns indicate which distribution has been used in the skewed version. For each setting we performed 5 independent simulations with 2500 families each. The same random seed was used for each setting, allowing us to conclude that all differences are necessarily due to the distributions. In general, very little variation is observed in the rank correlations. In all cases the correlation between the true and the estimated capabilities is above 0.85.

Table 5 Robustness checks with skewed distributions

The right skewed income distribution slightly reduces the correlation, left skewed goal distribution increases it a little bit (because more students are constrained in this case!). Based on these findings, we argue that the MIMIC model is robust to changes in the distributions, which makes us confident for its use with real world data. These robustness checks suggest that the good performance of the estimator does not simply rely on basic distributional assumptions.

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Krishnakumar, J., Chávez-Juárez, F. Estimating Capabilities with Structural Equation Models: How Well are We Doing in a ‘Real’ World?. Soc Indic Res 129, 717–737 (2016).

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