**Materials for Lecture 08** • Chapters 4 and 5 • Chapter 16 Sections 3.2-3.7.3 • Lecture 08 Bernoulli & Empirical.xls • Lecture 08 Normality Test.xls • Lecture 08 Parameter Est.xls • 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

**Normal Distribution** • Normal distribution a continuous distribution 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.

**Simulating Random Variables** • Normal distribution is used frequently, particularly when simulating 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

**PDF and CDF for a Normal Dist.** Probability Density Function Cumulative Distribution Function f(x) F(x) - + - +

**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 (BS’s) • 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 it suffers from all of the problems of the Normal • 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 (Ho Hi) 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)

**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 A B C 1 VC / acre 150.0 2 VC / Y 0.25 3 Acre 100 10 4 Fixed Cost 5 Yield Mean & Std. Dev. 150 30 Price Mean & Std. Dev. 6 2 0.40 • See Lecture 08 Simulation Model with Simetar.XLS

**Program Model in Excel/Simetar -- Generate Random** Variables and Simulate NR A B C 13 Stochastic Yield Formulas in Column B 14 Mean 150 = B5 15 Std. Dev. 30 = C5 16 SND 0.362 = NORM ( ) 17 Random Yield 160.86 = B14 + B15 * B16 18 Stochastic Price 19 Mean 2.00 = B6 20 Std. Dev. 0.40 = C6 21 SND -0.216 = NORM ( ) 22 Random Price 1.9136 = B19 + B20 * B21 23 Receipts from Market 24 Yield 160.86 = B17 25 Price 1.9136 = B22 26 Acres 100 = B3 27 Receipts 30782.16 = B24 * B25 * B26 28 29 Calculate Costs 30 Fixed Cost 10 = B4 31 VC/acre 4000 = B1 * B3 32 VC/Y 2412.9 = B2 * B17 * B4 33 Total 6422.9 = Sum (B30 : B32) 34 Net Returns 24359.26 = B27 – B33 35

**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 variable is 1 or TRUE • Simulate Bernoulli in Simetar as • = Bernoulli(p) • = Bernoulli(0.25)

**Bernoulli Distribution** • Use Bernoulli in a conditional distribution as demonstrated: • It rains 20% of time during June and if it rains, the amount is distributed U(0.1, 0.9) Cell A2 =BERNOULLI(0.20) Cell A3 =UNIFORM(0.1, 0.9) * A2 • Probability of mechanical failure is 5%, cost of repair is $10,000, $20,000, or $30,000 Cell A4 =BERNOULLI(0.050) Cell A5 =DEMPIRICAL(10000, 20000, 30000) Cell A6 = A4 * A5