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A Spatial Econometric Approach to Designing and Rating Scalable Index Insurance in the Presence of Missing Data

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Abstract

Index-Based Livestock Insurance has emerged as a promising market-based solution for insuring livestock against drought-related mortality. The objective of this work is to develop an explicit spatial econometric framework to estimate insurable indexes that can be integrated within a general insurance pricing framework. We explore the problem of estimating spatial panel models when there are missing dependent variable observations and cross-sectional dependence, and implement an estimable procedure which employs an iterative method. We also develop an out-of-sample efficient cross-validation mixing method to optimise the degree of index aggregation in the context of spatial index models.

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Notes

  1. Alderman and Haque (2007); Woodard and Garcia (2008a, 2008b); Woodard (2008).

  2. See, for example, Anselin (1988); Elhorst (2003); Kelejian and Prucha (2007); LeSage and Pace (2009).

  3. For example, Woodard et al. (2012).

  4. Mude et al. (2012).

  5. LeSage and Pace (2004).

  6. See, for example, Wang and Lee (2013a, 2013b).

  7. Wang and Lee (2013a).

  8. Woodard and Sherrick (2012).

  9. Steinfeld et al. (2006); Herrero et al. (2013).

  10. Lybbert et al. (2004); Barrett et al. (2006).

  11. Jack and Suri (2011).

  12. Woodard and Garcia (2008a).

  13. The eMODIS data for East Africa are downloaded from FEWS-NET www.earlywarning.usgs.gov/fews/africa/web/imgbrowsc2.php?extent=eazd.

  14. Vrieling et al. (2014).

  15. Chen et al. (2004).

  16. See Anselin (1988).

  17. Elhorst (2003).

  18. A queen contiguity matrix is one that expresses contiguity whereby two locations are considered neighbours if they share a border or vertex (see LeSage and Pace, 2009). See LeSage and Pace (2009) as well for interpretation of the spatial filter and other spatial econometric basics.

  19. In typical applications testing is conducted to determine the likely form of the spatial dependence; however, to our knowledge, no such tests exist in the missing data case, so we proceed with the lag model for a variety of credibility reasons. Note that the fitted values from a spatial error type model will not be spatially smoothed necessarily, thus further motivating the lag approach. We comment on this further below in the contract design section.

  20. Although other methods exist as developed by Wang and Lee (2013a, 2013b), those works were not published and the code not available when this pilot was designed. We would not anticipate large differences from employing different methods. Nevertheless, further investigation of those alternative estimators is beyond the scope of this work and is left as an area of future investigation.

  21. LeSage and Pace (2004) also explain how the traditional approach of using the vectorised concentrated log-likelihood can be used to operationalise their approach by iteratively optimising the concentrated log-likelihood over the single spatial autoregressive parameter ρ, then constructing new estimates of β, then estimating a new value for σ (using only non-missing data), and then constructing a new conditional expectation of missing values that is conditional on the last iteration estimation of ρ, β and σ (using only non-missing values in the estimated error terms) to create a “repaired” dependent variable vector, and then iterating this process until convergence. Essentially, what we propose is similar except that our method does not require manipulation of the concentrated log-likelihood function. However, similar to how LeSage and Pace iteratively estimate using only non-missing data when recalculating the σ then constructing the “repaired” dependent variable vector, we simply recalculate the missing dependent variable values at each iteration using the equation for the fitted values conditional on the estimated errors, replacing those that correspond to missing values with zero. To our understanding, our proposed method and the iterative method articulated by Lesage and Pace are mathematically similar but implemented slightly differently computationally. Indeed, the logic behind both approaches is similar conceptually, although we go about it in an albeit more direct and simple manner from an implementation standpoint. It is also not apparent that the methods proposed by LeSage and Pace involve choosing starting values for missing observations in the initial estimation, whereas our approach explicitly does require the analyst to pick a set of starting values for the missing observations.

  22. For example, Elhorst (2003).

  23. We conducted Monte Carlo simulations to evaluate the performance of this technique and found results similar to those in LeSage and Pace (2004). We found that the model typically converged to a reasonable level in less than 15–20 iterations. We would caution that while our Monte Carlo results and those in this paper for the mortality models did not appear to have any issues converging, there could be cases in which this might not occur. Monte Carlo results and code are available from the authors on request.

  24. Chantarrat et al. (2012).

  25. We also investigated a model with division fixed effects for the intercept and district fixed effects for the NDVI terms and this model outperformed both competing models. For clarity and ease of exposition, we present these models though for illustration.

  26. We also evaluated within-division basis risk across individuals using a Bayesian random coefficients model (results not presented). The fraction of unexplained variance related to underpayments was approximately 25 per cent.

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Acknowledgements

This work was funded under International Livestock Research Institute Cooperative Work Agreement “ILRI-Cornell Collaborative Work Agreement for Special Joint Research Projects”. We would like to thank seminar participants at the 2014 International Agricultural Risk, Finance and Insurance Conference (Zurich) for helpful comments and suggestions. All errors are our own.

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Authors and Affiliations

  1. Charles H. Dyson School of Applied Economics and Management, Cornell University, 351B Warren Hall, NY, 14850, Ithaca, U.S.A.

    Joshua D Woodard

  2. Environment and Production Technology Division, International Food Policy Research Institute, Arusha, Tanzania

    Apurba Shee

  3. International Livestock Research Institute, Nairobi, Kenya

    Andrew Mude

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  1. Joshua D Woodard
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  2. Apurba Shee
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Woodard, J., Shee, A. & Mude, A. A Spatial Econometric Approach to Designing and Rating Scalable Index Insurance in the Presence of Missing Data. Geneva Pap Risk Insur Issues Pract 41, 259–279 (2016). https://doi.org/10.1057/gpp.2015.31

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