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## Materials for Lecture 08

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**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**• 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**Stochastic Variables**• 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**Stochastic Simulation**• 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”**Iterations, How Many are Enough?**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**Definitions**• 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**• 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.**PDF and CDF for a Normal Dist.**Probability Density Function Cumulative Distribution Function f(x) F(x) - + - +**Simulating Random Variables**• 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 When:**• 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)**Problems with the Normal**• 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**Test for Normality**• 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”**Simulating a Normal Distribution**• 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**Truncated Normal Distribution**• 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**Example Model of Net Returns for a Business Model**- 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**PDF for Bernoulli B(0.75)**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