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# Materials for Lecture 08 - PowerPoint PPT Presentation

Materials for Lecture 08. Chapters 4 and 5 Chapter 16 Sections 3.2-3.7.3 Lecture 08 Bernoulli . xlsx Lecture 08 Normality Test.xls Lecture 08 Simulation Model with Simetar.xlsx Lecture 08 Normal.xls Lecture 08 Simulate a Reg Model.xls. Stochastic Simulation.

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• Chapters 4 and 5

• Chapter 16 Sections 3.2-3.7.3

• Lecture 08 Bernoulli .xlsx

• Lecture 08 Normality Test.xls

• Lecture 08 Simulation Model with Simetar.xlsx

• Lecture 08 Normal.xls

• Lecture 08 Simulate a Reg Model.xls

• Purpose of simulation is to estimate the unknown probability distribution for a KOV so decision makers can make a better decision

• Simulate because we can not observe and measure the KOV distribution directly

• Want to test alternative values for control variables

• Sample PDFs for random variables, calculate values of KOV for many iterations

• Record KOV

• Analyze KOV distribution

• Any variable the decision maker can not control is thought to be stochastic

• In agriculture we think of yield as stochastic as it is subject to weather

• For most businesses the prices of inputs and outputs are not directly controlled by management so they are stochastic.

• Production may be random as well.

• Include the most important stochastic variables in simulation models

• Your model can not include all random variables

• In economics we use simulation because we can not experiment on live subjects, a business or the economy without injury

• In other fields they can fabricate an experiment

• Health sciences they feed/treat multiple rats on different chemicals

• Animal science feed multiple pens of steers, chickens, cows, etc.

• Engineers run a motor under different controlled situations (temp, RPMs, lubricants, fuel mixes)

• Vets treat different pens of animals with different meds

• Agronomists set up randomized block treatments for a particular seed variety

• All of these are just different iterations of “models”

Specify the number of iterations in the Simetar simulation engine

Specify the output variables’ names and location

• Change the number of iterations based on nature of the problem -- 500 is adequate.

• Some studies use 1,000’s because they are using a Monte Carlo sampling procedure which is less precise than Latin hypercube

• Simetar uses a Latin hypercube so 500 is an adequate sample size

• Stochastic Model – means the model has at least one random variable

• Monte Carlo simulation model – same as a stochastic model

• Two ways to simulate random values

• Monte Carlo – draw random values for the variables purely at random

• Latin Hyper Cube – draw random values using a systematic approach so we are certain that we sample ALL regions of the probability distribution

• Monte Carlo sampling requires larger number of iterations to insure that we sampled all regions of the the probability distribution

• For a U(0,1) CDF is straight line

• MC has bias from straight line

• LHC is the straight line

• This is with 500 iterations

• Simetar default is LHC

• Normal distribution – a continuous random variable that produces a bell shaped distribution with set probabilities

• Parameters are

• Mean

• Standard Deviation

• Normal distribution reaches to + and - infinity.

• Can produce negative values so be careful

• Can produce extremely high values

• Most of us have memorized several probabilities for the normal distribution:

• 66% of observation within +/- 1 of the mean

• 95% of observation within +/- 2 of the mean

• 50% of observations lie above and below the mean.

Probability Density Function

Cumulative Distribution Function

f(x)

F(x)

-

+

-

+

• Normal distribution used frequently, particularly when simulating residuals for a regression model

• Parameters for a Normal distribution

• Mean expressed as Ῡ or Ŷ

• Standard Deviation σ (or SEP from a regression model)

• Assume yield is a random variable and have production function data, such as:

• Ỹ = a + b1Fert + b2 Water + ẽ

• Deterministic component is: a + b1Fert + b2 Water

• Stochastic component is: ẽ

• Stochastic component, ẽ, is assumed to be distributed Normal

• Mean of zero

• Standard deviation of σe

• See Lecture 8 Simulate a Reg Model.XLS

• Use the Normal distribution if you have lots of observations and have tested for normality

• Watch for infeasible values from a Normal distribution (negative yields and prices)

• It is easy to use, so it often used when it is not appropriate

• It does not allow for extreme events (Black Swans)

• No way to account for record breaking outliers because the distribution is defined by Mean and Std Dev.

• Std Dev is the “average” deviation from the mean and averages out BS’s

• Market outliers are washed away in the average

• It is the foundation for Sigma 6

• So Sigma 6suffers from all of the problems above

• Creates a false sense of security because it never sees a record braking outlier

• Simetar provides an easy to use procedure for testing Normality that includes:

• S-W – Shapiro-Wilks

• A-D – Anderson-Darling

• CvM – Cramer-von Mises

• K-S – Kolmogornov-Smiroff

• Chi-Squared

• Simetar’s Hypothesis Testing Icon provides a tab to “Test for Normality”

• Normal Distribution

=NORM( Mean, Standard Deviation)

=NORM( 10,3)

=NORM( A1, A2)

• Standard Normal Deviate (SND)

=NORM(0,1) or =NORM()

• SND is the Z-score for a standard normal distribution allowing you to simulate any Normal distribution

• SND is used as follows:

Ỹ = Mean + Standard Deviation * NORM(0,1)

Ỹ = Mean + Standard Deviation * SND

Ỹ = A1 + (A2 * A3) where a SND is in cell A3

• General formula for the Truncated Normal

=TNORM( Mean, Std Dev, [Min], [Max],[USD] )

• Truncated Downside only

=TNORM( 10, 3, 5)

• Truncated Upside only

=TNORM( 10, 3, , 15)

• Truncated Both ends

=TNORM( 10, 3, 5, 15)

• Truncated both ends with a USD in general form

=TNORM( 10, 3, 5, 15, [USD])

The values in [ ] are optional

- Stochastic Variables -- Yield and Price

- Management Variables -- Acreage and Costs (fixed and variable)

- KOV -- Net Returns

- Write out the equations and exogenous values

Equations and their order

Program a Simulation Model in Excel/Simetar -- Input Data Section of the Worksheet

• See Lecture 08 Simulation Model with Simetar.XLS

Program Model in Excel/Simetar -- Generate Random Variables and Simulate Profit

CDF for Bernoulli B(0.75)

1

.25

.25

.75

0

0

1

X

1

X

PDF and CDF for a Bernoulli Distribution.

Bernoulli Distribution

• Parameter is ‘p’ or the probability that the random variable is 1 or TRUE

• Simulate Bernoulli in Simetar as

• = Bernoulli(p)

• = Bernoulli(0.25)

• Lecture 8 Bernoulli.XLSX examples follow

Bernoulli Distribution Application

Bernoulli Distribution Application