Stochastic dominance-decision analysis with preferences unknown
Shauna Phillips
School of Economics
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Mid-sem exam
› 3 short(ish) answer questions
› All worth equal marks
› No calculations
› Next week we start the real options topic but it will NOT be in the mid sem exam.
› Question 1 : To analyse decision making under risk, agricultural economists sometimes need to incorporate a farmer’s beliefs about risky events. Suggest two different methods that economists might use to elicit the subjective probabilities that a farmer has about whether there might be a drought next year or not. Choose one and describe the steps in the method of elicitation. Do you think it is important to use these methods as opposed to just asking the farmer what they think the chance of drought next year is? Why or why not?
Guide to answer the questions well.
› Think what the question is asking.
› Think about the number of questions and/or subparts of questions and
answer them all.
› Don’t see some key words and write everything you know about them (i.e. no marks for redundant writing-focus on answering the question).
Theoretical limitations of EV analysis
› To be strictly valid, the distribution of outcomes must be normal, or the DM’s utility function over the payoffs must be quadratic with risk aversion.
› An alternative way to justify the use of E,V is as an approximation to the Taylor series expansion when derivatives of the utility function beyond the 2nd derivatives are small.
› Hardaker & Tanago (1973) note:
– many agricultural systems not normally distributed (often skewed and kurtotic) – quadratic utility implies increasing risk aversion for high magnitude payoffs
› Hence, another method that can be used to identify an efficient sub-set of strategies in the presence of risk is stochastic dominance analysis.
Stochastic efficiency analysis
› Alternative risky prospects/projects compared in terms of full distributions of outcomes (not just moments)
› Stochastic dominance method – pairwise comparisons of alternatives are made.
› Comparisons made at points on the distributions
› Degrees of stochastic dominance involve progressively stronger
assumptions about risk preferences.
› One strategy dominates another if it would be chosen by all DMs whose utility functions obey certain properties.
› First, second and third degree stochastic dominance – each separates the set of possible strategies/projects into efficient / inefficient sub-sets.
Stochastic efficiency analysis
› First, second and third degree stochastic dominance- each separates the set of possible strategies/projects into efficient / inefficient sub-sets.
› First degree stochastic dominance (FSD)
› Second degree stochastic dominance (SSD)
› Third degree stochastic dominance (TSD)
› FSD implies SSD which in turn implies TSD – more strategies can be ordered by TSD than SSD and then than FSD.
› Unlike E,V analysis, use of stochastic dominance not restricted by the underlying distribution ie doesn’t assume normality
› Does assume existence of a utility function of Bernoullian form, and criteria depend on assumptions about the form of the utility function (see below).
First degree stochastic dominance (FSD)
› 2 risky alternatives: A & B – each have uncertain payoffs of a and b
› Assume a monotonically increasing utility function such that u(a) < u(b) › Let the probability of payoff a for A is p, and for B is q.
› Then the utilities of these payoffs for A and B are: - U(A) = p(u(a)) + (1-p)(u(b))
- U(B) = q(u(a)) + (1-q)(u(b))
Here A will be preferable to B (stochastically dominates) if p < q ie. DMS prefer more to less, so prefer whichever of A and B has the lower probability of giving the lesser payoff.
› Extending this to cases with more than 2 payoffs – n possible payoffs of x (x1,.....xn) increasing in magnitude (so u(x1),...<.. u(xn))
› Probabilities for A (pi) and B (qi)
› Combiningthese,AwillbepreferredtoBifpj
with pi = qi for all i = j, k
› That is A is preferred to B if it has a lower probability of one less favourable payoff and a correspondingly increased probability of one more favourable payoff.
› FSD condition often written in terms of CDFs:
› A & B – have probability distributions of outcomes x defined by CDF: FA(x)
› Then A dominates B in FSD sense if: FA(x) ≤ FB(x) for all x with at least 1 strong inequality (FA(x) < FB(x) for at least 1 value of x). So, the CDF of preferred project must lie in part to the right and nowhere to the left to dominate. If the CDFs cross, neither dominate in FSD sense.
› Restriction on utility function: DM has positive marginal utility for the measure being considered (more is preferable to less) ie. U function is monotonically increasing in the payoff.
Stochastic dominance analysis – example (Source: Hardaker et al (2004))
Using past farming records for the area, and subjective judgements, the DM (or her advisor) can estimate the distributions of net returns from alternative rotations.
Using notation F to K these distributions are presented in the table below:
statistics
P(X<400) for FH(X) < P(X<400) for FF(X)
So P(X>400) will always be greater for FH(X) than for FF(X)
i.e. 1- FH(X) ≥ 1- FF(X)
So probability of drawing numbers higher than 400 will always be greater for FH(X) than for FF(X)
First degree stochastic dominance (FSD) (Hardaker et al 2004 cropping problem data)
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
FG(x) ≤ FF(x) for all x, so F is not in the efficient set
FH(x) ≤ FG(x) for all x, so G is not in the efficient set
CDFs for H and J cross – no FSD dominance – for low x, FJ(x) preferable, for high x, FH(x) preferable – need to appeal to SSD to make decision
0 100 200 300 400
500 600 700
Second degree stochastic dominance (SSD)
dU 0 d2U 0 dw dw2
SSD for normal distribution
0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
mean=0 σ = 1
-6 -4 -2-0.050 2 4 6
Different means and variances
Second degree stochastic dominance (SSD)
› SSD has more discriminatory power than FSD
› Efficient set under SSD is a subset of that under FSD
› Technically SSD should be calculated as the area under the CDF as above.
› Possible here to “eyeball” the CDFs and demonstrate the principle.
› Using crop data from Hardaker et al 2004 ch 7 for crops H, J & K
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
600 700 800 900
J dominates H
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
300 400 500
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
200 300 400
500 600 700
› minimum for H < minimum for J – comparing these 2: H can’t dominate J but J can dominate H in SSD sense. Comments to roughly compare areas below these CDFs:
Area below FJ(x) is smaller than area below FH(x) for all values of x For fractile range 0-0.8 FJ(x) lies to the right of FH(x)
This portion of area ( fractile range 0-0.8 ) > area between fractiles 0.8 – 1 in which FH(x) lies to the right of FJ(x)
Hence, J dominates H in SSD sense. Comparing J & K:
area below FK(x) is smaller for x values to about x=$430 or for fractiles 0-0.55 lies to the right of FJ(x)
fractiles above 0.55, area below FJ(x) is much smaller
› H is eliminated from the FSD set
Hard to determine SSD for this pair
› Third-degree stochastic dominance
› Same behavioural assumptions as for SSD plus the coefficient of absolute
risk aversion is decreasing with income or wealth.
› Method involves comparisons of new accumulations of areas – relatively less useful than FSD and SSD.
Application: Lucerne hay making in Spain (Hardaker & Tanago (1973))
FSD satisfied
FSD not satisfied – must appeal to SSD or TSD
Application: Lucerne hay making in Spain (Hardaker & Tanago (1973))
Application: Lucerne hay making in Spain (Hardaker & Tanago (1973))
Application: Lucerne hay making in Spain (Hardaker & Tanago (1973))
Application: genetically modified crops
› Nolan & Santos (2015) Genetic modification and yield risk: A stochastic dominance analysis of corn in the USA.
› GM technology involves inserting new DNA into the genome of an organism.
› GM plants- new DNA is transferred into plant cells, and seeds produced by these
plants will inherit the new DNA.
› Genomes contain genes that carry the instructions for making proteins (these give the plant its characteristics).
› In agriculture, genetic modification adds a specific stretch of DNA into the plant’s genome, which could change the way it grows or make it resistant to disease.
› GM is a new technology, and production risk may partially determine the adoption of new technologies
Nolan & Santos (2019)
› Quantify the changes in yield distribution associated with the development of genetically modified (GM) corn.
› The presence of the genes associated with the production of a protein found in the soil bacterium Bacillus thuringiensis which is toxic to lepidopterous insects allows for an almost complete protection against the most economically important pests of corn in the USA, the European corn borer and corn rootworm
› Superior level of control of these two pests when using GM traits, when compared with the use of pesticides, certainly helps explain their rapid adoption: by 2014, less than twenty years after they were first released, GM hybrids represented around 93% of the corn acreage in the United States, and have remained at that level since then.
Nolan & Santos (2019)
› Analyze a large dataset of experimental trials of corn hybrids run in the ten most important corn-producing states in the United States, over the period 1997-2009 (ie, the first 13 years since the commercial introduction of GM hybrids).
› Data extends beyond the Corn Belt, an area characterized by low level of yield risk for corn , so it’s possible to measure the effect of GM traits on the entire yield distribution where yield variability likely matters most.
› Stochastic dominance to rank the desirability of the conditional yield distributions generated by different technologies (here, GM hybrids versus conventional hybrids) under minimal assumptions both regarding decision makers’ objectives and the relation between input use and the moments of the yield distribution.
Comparing two distributions: B first order stochastically Dominates A.
Two distributions for B and A:
As it is clear from which presents the probability density functions of these two technologies (fA and fB), a mean-variance comparison of these alternatives does not allow us to decide which one is preferable: although B has a higher mean, this change comes accompanied by a higher variance.
GM corn and yield risk: Data and empirical results
› Data from experimental field trials of corn hybrids, independently run by the State Agricultural Extension Services of universities in the ten most important corn-producing states in the United States, in the first 13 years since the commercial introduction of GM corn in 1997.
› 163,941 observations of 14,614 hybrids, at 339 locations including information about the genetic make-up of each hybrid, detail on agronomic practices (yield, seeding rate, nitrogen application, cultivation type, previous crop, whether the trial is early or late, and irrigation) and environmental conditions (soil type, rainfall and temperature)
› Agronomic & environmental variables potentially influence yield and its variability.
› Data from experimental field trials avoids the problems of using data at county or state level, which underestimate farm level risk.
Table 1. Summary statistics.
, (2019) Genetic modification and yield risk: A stochastic dominance analysis of corn in the USA. PLOS ONE 14(10): e0222156. https://doi.org/10.1371/journal.pone.0222156
https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0222156
Estimated yield function
› Estimate a (linear) production function with hybrid fixed effects, which can be written as:
›𝑦 =𝛼+𝑋′+𝑢it 𝑖𝑡 𝑖 𝑖𝑡
› where yit is the yield of hybrid i in year t,
› Xit is the set of covariates
› αi is the unobserved (fixed) effect of the underlying germplasm
› uit is the idiosyncratic error term.
› These estimates used to identify the effect of GM traits on the entire distribution of conditional yield.
3. Stochastic dominance: CB vs. conventional hybrids, 2001-2005.
, (2019) Genetic modification and yield risk: A stochastic dominance analysis of corn in the USA. PLOS ONE 14(10): e0222156. https://doi.org/10.1371/journal.pone.0222156
https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0222156
Fig 4. Stochastic dominance: Triple stacked vs. conventional hybrids, 2005-2009.
, (2019) Genetic modification and yield risk: A stochastic dominance analysis of corn in the USA. PLOS ONE 14(10): e0222156. https://doi.org/10.1371/journal.pone.0222156
https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0222156
Analysis of higher orders of stochastic dominance
• Analysis of higher orders of stochastic dominance based on the empirical likelihood ratio tests presented from Davidson & Duclos (2003).
• The approach involves comparing stochastic dominance curves at different order (first, second and third) and estimating the crossing points (or critical values) of the dominance curves at which there is a reversal of the ranking of the curves (if any).
• Additionally, test whether any difference between the corresponding distributions is statistically significant.
Table 2. Stochastic dominance results
, (2019) Genetic modification and yield risk: A stochastic dominance analysis of corn in the USA. PLOS ONE 14(10): e0222156. https://doi.org/10.1371/journal.pone.0222156
https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0222156
Hybrids with trait combinations found in the large majority of the varieties being trialed second order stochastically dominate conventional hybrids.
Finally, our results show that double stacked hybrids containing the corn borer and herbicide tolerance traits (CBHT), which represent approximately 20% of the GM hybrids under trial, third order dominate conventional hybrids.
Hence, it seems that under relatively weak assumptions regarding producers’ preferences (producers are risk averse), the most commonly trialed GM hybrids would be preferred to conventional hybrids.
› Davidson.R.&Duclos(2003)StatisticalInferenceforStochasticDominanceandfortheMeasurementof Poverty and Inequality, Econometrica 68 (6) https://doi.org/10.1111/1468-0262.00167
› Hardaker, Huirne and Anderson (2004) Coping with Risk in Agriculture, CAB International.
– Chapter 7
› Hardaker,J.B, & A.G. Tanago (1973) Assessment of the Output of a stochastic decision model. Australian Journal of Agricultural Economics, Volume 17, Issue 3
› Nolan,E.&Santos,P.(2019)Geneticmodificationandyieldrisk:Astochasticdominanceanalysisofcorn in the USA. Plos One 14(10). https://doi.org/10.1371/journal.pone.0222156
› Watson, A.S & J.H Duloy (1964). The effect of wheat acreage shifts upon the mean and variance of total yields in NSW. Australian Journal of Agricultural Economics, December 1964
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