Department of Agricultural, Food, and Resource Economics Elton R. Smith Endowment

In the middle 1990s a graduate student taking a class of mine walked into my Washington State University office to give me a bit of unsolicited advice. He thought that it would be a good idea for me to read Stochastic Orders and their Applications by Moshe Shaked and J. George Shanthikumar. By golly he was right. Although agricultural economists have applied certain ways of ordering the distribution of a single random variable (say yield or price distributions) since the 1970s, these typically just allow us to infer what is better from a welfare perspective and not what implications there are for choices such as fertilizer use or acreage choice. Furthermore, putting order on multiple random variables is a far more vexing issue. Shaked and Shanthikumar (now available in an expanded second edition) brought coherence to a set of results on single variable and multiple variable distributions. I do not recall the student’s name for it was a difficult one for me to pronounce and I just remember that he dropped out of the program. But in the improbable chance that he is reading this, thank you very much.

Economic theory applications of single variable stochastic orders are made in:

• Hennessy, D.A. “The Ricardian Rent and the Allocation of Land Under Uncertainty: Comment.” European Review of Agricultural Economics, 24 (2 1997):313-317. Link
• Hennessy, D.A. “Capacity Choice in a Two-Stage Problem under Uncertainty.” Economics Letters, 65(November, 1999):177-182.Link
• Hennessy, D.A. “Crop Yield Skewness and the Normal Distribution.” Journal of Agricultural and Resource Economics, 34(April, 2009):34-52.Link

The third of these papers has pointed out that even if weather realizations are symmetrically distributed, decreasing returns to a weather input suggests that yields will likely be negatively skewed, i.e., with a thick and extended left tail. This point is relevant because the left tail determines yield insurance payouts. 

But this line of work needed strong formal analysts and empirical methodologists as collaborators. Long before the 2008 financial crisis, Harvey Lapan and I used the methods to provide a definition of systemic risk that was outside the multivariate normal straightjacket.

• Hennessy, D.A., and H.E. Lapan. “A Definition of ‘More Systematic Risk’ with Some Welfare Implications.” Economica, 70(August, 2003):493-507. Link

Empirical methods and data were used in 

• Roosen, J., and D.A. Hennessy. “Tests for the Role of Risk Aversion on Input Use.” American Journal of Agricultural Economics, 85(February, 2003):30-43. Link
• Roosen, J., and D.A. Hennessy. “Testing for the Monotone Likelihood Ratio Assumption.” Journal of Business and Economic Statistics, 22(June, 2004):358-366. Link

In both cases, Iowa nitrogen trials data were used. The first of these papers has had an impact because of its policy implications regarding freshwater quality and saltwater hypoxia in that it has suggested that institutions that provide more financial security, and so reduce risk aversion, may increase fertilizer use. The second paper was quite a way ahead of its time in that while theorists have been interested in the tested stochastic order for many years, it is only in recent years that other empiricists have turned to testing for it.

A separate methodology has also provided insights on opportunities for a better characterization of multiple variable distributions. In statistics a copula is function that links single variable distributions to multiple variable distributions. Although not really knowing it at the time, in,

• Babcock, B.A., and D.A. Hennessy. “Input Demand under Yield and Revenue Insurance.” American Journal of Agricultural Economics, 78(May 1996):416-427. Link
• Hennessy, D.A., B.A. Babcock, and D.J. Hayes. “Budgetary and Producer Welfare Effects of Revenue Insurance.” American Journal of Agricultural Economics, 79(August 1997):1024-1034. Link

 a methodology (Johnson and Tenenbein JASA 1981) that is essentially equivalent to the bivariate Gaussian copula was applied to model revenue insurance. The Gaussian copula formula, was advocated for use in modeling credit portfolio correlations by Li (Li, D.X. 2000 On Default Correlation: A Copula Function Approach". Journal of Fixed Income. 9 (4): 43-54). It subsequently became acclaimed and then infamous for its use in modeling systemic risks for portfolio and insurance products (Chris Arnade, The Real, and Simple, Equation That Killed Wall Street, Scientific American, January 30, 2013). The application to crop insurance products has fared much better, although even in that application there are issues. The controversial problem with the approach has to do with its characterizing of dependence structures among random variables in its left or right tails. There is reason to believe that this class of problem is relevant to crop insurance.

• Goodwin, B.K. 2015. Agricultural policy analysis: The good, the bad, and the ugly. American Journal of Agricultural Economics 97(2), 353-373. Link
• Goodwin, B.K., and A. Hungerford. 2015. Copula-based models of systemic risk in U.S., American Journal of Agricultural Economics 97(3), 879-896. Link
• Du, X., D.A. Hennessy, H. Feng, and G. Arora. “Land resilience and tail dependence among crop yield distributions.” Forthcoming at American Journal of Agricultural Economics. Link

The first two of the papers above present evidence on strengthened dependence in the tails and the failure of the gaussian copula model. The third provides an explanation for why this is to be expected, namely that in some regions climate and/or the dominant soil types provide little opportunity for resilience so that planted units tend to fail all together. A pseudocopula is estimated.

Archimedean copulas, which are flexible in use and admit tail dependence, have recently received much attention among empiricists. Harvey Lapan and I did not know about tail dependence concerns when we sought to popularize Archimedean copulas for the purpose of integrating applied theory models of multivariate risk analysis with stochastic ordering analysis of decision-making and methodologies for empirical analysis.

• Hennessy, D.A., and H.E. Lapan. “The Use of Archimedean Copulas to Model Portfolio Allocations.” Mathematical Finance, 12(April, 2002):143-154. Link

Since about 2012, copula methods have assumed widespread use in agricultural and natural resource economics. Examples in my work include:

• Du, X., and D.A. Hennessy. “Planting Real Option in Cash Rent Valuation.” Applied Economics, 44(6, 2012):765-776. Link
• Miao, R., D.A. Hennessy, and H. Feng. “The Effects of Crop Insurance and Sodsaver on Land Use Change.” Journal of Agricultural and Resource Economics, 41(2, 2016):247-265. Link