[SOLVED] 程序代写 AREC3005- see Cerroni. (2020))

Shauna Phillips
School of Economics
Quantifying uncertainty(I)

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Risk and farm decision making
› Wide range od ag. decisions: technology adoption, crop selection and diversification, contract farming, paying for insurance, and responding to environmental policies.
› Much empirical work has been done on risk perceptions due their importance in farmer behaviour (especially policy response).
› Probabilities may be unknown and farmers may not have enough information or experience to form a unique and well‐defined subjective probability distribution e.g. farmers considering a new market like native grass seeds, yields and market prices of a are ambiguous.
› OR farmers may have enough experience (traditional crops) or technical information available from agricultural extension, to form a reasonably well‐defined subjective probability distribution about future yields and prices.

Probabilities
› Probabilities are used to specify uncertainties – Simply, the likelihood of some event occurring
› Two schools of thought on identifying probabilities:
– ‘Frequentists’ argue/assume that recorded data is sufficient to elicit probabilities
of outcomes from the observed system
– ‘Subjectivists’ argue that recorded data may not fully capture the range of possible outcomes of the system
› Komarek et al. (2020) Production and market risk data more available (frequentist distributions?) data scarcer for institutional, personal, and financial risks (subjectivist?) may be more appropriate to generate probability distributions.
› Combine the frequentist and subjectivist views here probability distributions are generated based on scarce data and expert judgments. Bayesian decision theory.

Subjective probabilities
› Not just “any number that comes to mind”
› Must obey the laws of probability.
› Probabilities for clients? groups of people?
› Risk perceptions as a socially constructed phenomena. › Rational?

Psychology of judgement about probabilities
› Kahneman and Tversky (1982): People use heuristics when accessing probabilities of specific events- can lead to bias and error. Examples of heuristics:
› Availability – people think of previous times when the event occurred-works well if you have lots experience of the event occurring, e.g. random breath testing on roads – previous experience likely to be well correlated with actual frequency of RBT, but..
› .. some sources bias can occur, e.g. some events easier to recall than others. Experiment: played recorded list of 39 names of people some well known, some famous. Some lists had 19 women+ more famous women than men, & some 19 men+ more famous men than women. 80/99 people when asked whether there were more man than women, chose the sex that involved the larger number of famous individuals. So bias can arise from the availability heuristic if imagination is enhanced in some way.
› Framing – the way that questions about uncertain events are framed is a potential source of bias.

Psychology of judgement about probabilities
› Representiveness: – often used in judgements about uncertain events. If judging whether an event is generated by a particular process people expect the fine structure of the event to reflect the actual process e.g. coin toss, people judge HTHTTH more likely than HHHTTT or HTHTHT because they know coin toss process is random-all 3 sequences equally likely, but the first string looks more random.
› Anchoring & adjustment :- a natural starting point or anchor is selected as first approx to the quantity being estimated & then its value is adjusted to reflect supplementary information. Often adjustment is insufficient and the result is biased towards the anchor.

Dealing with bias
› Try to use any available data (&/or examine scope for collecting it). › Evidence training in probability assessment can reduce bias:
– assessors are presented with almanac-based questions and are asked to provide probability distributions for uncertain quantities
– answers are calibrated using proper scoring rules (evaluate how well probabilities con-form to their true uncertainty about the actual risk)
› Large literature exists on elicitation and improvement of evaluating subjective probabilities using proper scoring methods (beyond the scope of AREC3005- see Cerroni. (2020))

Probabilities
› With sufficient data, we might know the frequency of some specified event(s) occurring
– What is sufficient data?
– The likelihood of occurrence is the ratio of the count of
specified events to the total of all events observed
– For example, 100 days of max. daily temperature recording, with 6 days exceeding 30 degrees. What is likelihood of the 101st day exceeding 30 degrees?

Probability theory
› Theory developed to specify probabilities – comes from statistics – Random variable: a variable whose value is unknown or a function that
assigns values to each of an experiment’s outcomes
– A variable whose value is subject to variations due to chance (i.e. randomness, in a mathematical sense)
– e.g. rainfall, yield, prices
– Observation: an observed outcome of the random variable
– Sample space: the set of all possible outcomes
– Continuous vs. discrete random variables

Probability theory

Probability theory
› Two rules of probability
1. All probabilities are numbers between 0 and 1
particular uncertain prospect must be 1
i Pxi =1
2. The sum of all probabilities across all possible outcomes for a

Example: six-sided die
f Pxi = 36i
Px4= f4 = 7 =0.194 36 36

Probabilities
› Usually, we are faced with situations in which data is insufficient or non-existent, so we have trouble estimating likelihoods
› In other cases, history is no guide to the future:
– One issue with climate change decision making is that we can’t use data
from history to develop likelihoods of future outcomes
– The price history of agricultural commodities includes many previous policy changes
– Really, we are talking about institutional changes i.e. changes to the rules of the game
– Can we use these data to develop likelihoods of the outcomes of future policy changes?

Probabilities
› In the absence of good historical data, we can estimate probabilities through a subjective elicitation process
– Probability is the degree of belief an individual has in a particular outcome occurring
– It is possible that individuals are reconciling more data from their surroundings than we might set out to measure, so they might be more able to assign probabilities in a changing world

Subjective probabilities and the reference lottery
› Want to elicit a subjective probability about the chance of rain tomorrow:
› 1. offer subject a bet that pays $100 if it rains tomorrow and $0 otherwise
› 2. offer subject a second bet with same payoff based on a random spin of a fair wheel – this is the reference lottery – $100 of the pointer land on “W” and $0 otherwise.
› 3. Determine what size the segment W needs to be to make the subject indifferent between the 2 bets.

Eliciting and describing probabilities
› Asking people engaged in risky activities about their beliefs might be a good way to elicit probabilities
– Farmers are engaged in all kinds of risky activities
– They might be best placed to reconcile various streams of data and estimate
probabilities of particular outcomes
› Example [HHA2004]: In the coming harvest period, what is the probability that a farmer experiences less than two harvester breakdowns versus two or more breakdowns?
– Ask the farmer a series of questions

Eliciting probabilities [HHA2004] example
› Results of questions:
– Less than two breakdowns: Probability = 0.91 – Two or more breakdowns: Probability = 0.09
› Follow-up questions:
– If the breakdowns during the harvest are fewer than two, what are the
probabilities that it will break down once or not at all?
– If the breakdowns are two or more, what are the probabilities that there will be two or three breakdowns?

Eliciting probabilities [HHA2004] example
› Results of follow-up questions: – If less than two breakdowns:
– Probability of zero breakdowns = 0.76
– Probability of one breakdown = 0.24 – If two or more breakdowns:
– Probability of two breakdowns = 0.78
– Probability of three breakdowns = 0.22

Eliciting probabilities [HHA2004] example

Alternative elicitation method using counters: [HHA2004] example

Alternative elicitation method using counters: [HHA2004] example

Elicitation methods
› These kind of methods, where we ask people about their opinions, can become more important where psychology is an important input to system outcomes
– Commodity markets, for example, may be strongly driven by trader psychology (expectations)
– We could ask a range of traders their opinion about a particular commodity price tomorrow/next week/next year, and develop a probability distribution
– If we ask enough traders, we might be able to use this probability to guide our buying/selling strategy

Example: What will the AUD-USD exchange rate be tomorrow?
› What is the exchange rate today (approximately)?
› Let’s count the number of people who think it will be:
Probability
+0.76% to +1%
+0.51% to +0.75%
+0.26% to +0.5%
+0% to +0.25%
-0% to -0.25%
-0.26% to -0.5%
-0.51% to -0.75%
-0.76% to -1%

Example: What will the AUD-USD exchange rate be
Probability
+0.76% to +1%
+0.51% to +0.75%
+0.26% to +0.5%
+0% to +0.25%
-0% to -0.25%
-0.26% to -0.5%
-0.51% to – 0.75%
-0.76% to -1%

Example: What will the AUD-USD exchange rate be
› Represent probability as a probability density graph and, if sample size is large enough, you could fit a probability density function
› Can also represent probability using a cumulative probability graph – Again, if there is enough data, we could represent probability using a
cumulative distribution function
› In both cases, estimation of the function allows for estimation of un- measured outcomes within the sample

Probability density graph
0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00
Category or Value
Probability

Towards a probability density function
0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00
Category or Value
Probability

Cumulative probability graph
1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00
Category or Value
Cumulative probability

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00
Probability

Towards a cumulative probability function
1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00
Poly. (Cum. Prob)
y = 0.0094×2 + 0.0662x – 0.0611 R2 = 0.9767
Category or Value
Cumulative probability

Elicitation of continuous distributions using fractiles

Description of the distributions by moments
› We can estimate the moments of distributions to help describe them – these moments include:
– The first moment, which is the mean or average of the distribution
– The second moment, which is the variance of the distribution, and its square
root is the standard deviation
– The third moment, which is used to calculate the skewness of the distribution
– The fourth moment, which is used to calculate the kurtosis or peakedness of the distribution

Description of the distributions by moments
› Moments are probability-weighted averages of deviations from a specified point, with the deviations raised to some power
– The power they are raised to gives us the number of the moment – For example, all raised to the power of two is the second moment

The first moment
M x=  Px (x − 0) =  Px  x = E(x)

The second moment
M x=Px(x −E(x)) =V(x)
SD(x)=V(x)

The coefficient of variation
› The coefficient of variation (CV) is given by dividing the standard deviation by the mean:
V (x) SD(x) CV(x)= E(x) = E(x)
› CV tells us the dispersion of the distribution, relative to the mean

The third moment

› A negatively skewed distribution has a longer tail to the left, with a concentration of probability mass on the right-hand side
› A positively skewed distribution has a longer tail to the right, with a concentration of probability mass on the left-hand side

The coefficient of skewness
› The coefficient of skewness (CS) is given by dividing the skewness by the standard deviation:
CSx= (x) = (x)
SD(x) V (x)

The fourth moment

Mesokurtic, leptokurtic and platykurtic

Probability

Example: 600 rolls of a six-sided die

A final suggestion, if very little is known about the distribution
› Triangular distribution
– Can be fully specified from only 3 pieces of information

Agricultural Risks
Event Likelihood
Losses 0 Profits

Agricultural Risks
Event Likelihood
Total area under curve sums to 1

Agricultural Risks
Event Likelihood
Area under curve is “risk” of a loss
Losses 0 Profits

Agricultural Risks
Event Likelihood
Area under curve is “risk” of profits
Losses 0 Profits

Agricultural Risks
“risk” of loss might equal 0.3
Event Likelihood
“risk” of profits might equal 0.7
Losses 0 Profits

More distributions

Risk in the production function

Risk in the production function

Risk in the production function
Concavity in production function said to promote “down-side” risk
– Probability mass tails downwards in direction of lower output

Risk in the production function
Convexity in production function said to promote “up-side” risk
– Probability mass tails upwards in direction of greater output

› [HHA2004] Hardaker, Huirne and Anderson (2004) Coping with Risk in Agriculture, CAB International. Ch 3
› Cerroni, S. (2020) Eliciting farmers’ subjective probabilities, risk and uncertainty preferences using contextualised field experiments. Agricultural Economics, 51 (5) pp 707-724. https://doi.org/10.1111/agec.12587
› Kahneman, D & Tversky, A. (1982). The psychology of preferences. Scientific American. 246(1)
› Komarek, A., De Pinto, A. & Smith, V.H. (2020) A review of types of risks in agriculture: What we know and what we need to know. Agricultural Systems, (178) https://doi.org/10.1016/j.agsy.2019.102738
›, F.,, Y., Lauwers, L.,, S., Vancauteren, M., & Wauters, E. (2016). Determinants of risk behaviour: Effects of perceived risks and risk attitude on farmer’s adoption of risk management strategies. Journal of Risk Research, 19(1), 56–78.
› Zhao, S. & Yue, C. (2020) Risk preferences of commodity crop producers and specialty crop producers: An application of prospect theory. Agricultural Economics, 51 (3) pp 359-372 doi/abs/10.1111/agec.12559

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