1. About the Exam No cheat sheet Bring a calculator
You may NOT use the calculator on your phone or iPad or computer or other media device
Short essay answers
Math problems to be solved
Know all materials covered including last Thursday’s lecture on simulation
Materials presented in Lab will also be include in the exam

2. Materials for Lecture 14 Chapters 4 and 5
Chapter 16 Sections
Lecture 14 Bernoulli.xlsx
Lecture 14 Normality Test.xlsx
Lecture 14 Simulation Model with Simetar.xlsx
Lecture 14 Normal.xlsx
Lecture 14 Simulate a Reg Model.xlsx
Lecture 14 View Distributions.xlsx
Lecture 14 Theta UPES.xlsx

3. Steps for Simulating Random Variables
Must assume a probability distribution (shape)
Normal, Beta, Empirical, GRKS, etc.
Estimate parameters required to define and simulate the assumed distribution
Here are the parameters for selected distributions
Normal ( Mean, Std Deviation )
Beta ( Alpha, Beta, Min, Max )
Uniform ( Min, Max )
Empirical ( Si, F(Si) )
GRKS (Min, Middle, Max)
Often times we assume several distribution forms, estimate their parameters, simulate them, and pick the one which best fits the data

4. Steps for Parameter Estimation
Step 1: Check for the presence of a trend, cycle or structural pattern
If present remove it & work with the residuals (ẽt)
If no trend or structural pattern, use actual data (X’s)
Step 2: Estimate parameters for several assumed distributions using the X’s or the residuals (ẽt)
Step 3: Simulate the different distributions
Step 4: Pick the best match based on
Mean, Standard Deviation -- use validation tests
Minimum and Maximum
Shape of the CDF vs. historical series
Penalty function =CDFDEV() to quantify differences

5. Parameter Estimator in Simetar
Use Theta Icon in Simetar
Estimate parameters for 16 parametric distributions
Select MLE method of parameter estimation
Provides equations for simulating distributions

6. Parameter Estimator in Simetar
Results for Theta Estimate parameters for 16 distributions
Selected MLE in this example
Provides equations for simulating distributions based on a common USD

7. Which is the Best Distribution?
Use Simetar function =CDFDEV(History, SimData)
Perfect fit has a CDFDEV value of Zero
Pick the distribution with the lowest CDFDEV

8. Use the “View Distributions.xlsx”
For a random variable with 10 observations can estimate the parameters and view the shape of the distribution

9. Stochastic Simulation
Purpose of simulation is to estimate the unknown probability distribution for a KOV so decision makers can make a better decision
We simulate because we cannot observe and measure the KOV distribution directly
We want to test alternative values for control variables
Business simulation models are basically an equation to calculate profit
Profit = (Price * Quantity) – (VC * Quantity) - FC
Profit = * - VC * FC

10. Stochastic Simulation for Economists
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 create an experiment
Health sciences they feed (or treat) lots of lab rats on different chemicals to see the results
Animal science researchers feed multiple pens of steers, chickens, cows, etc. on different rations
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 with different fertilizer levels
All of these are just different iterations of “models”

11. Iterations, How Many are Enough?
Specify the output variables’ names and location
Specify the number of iterations in the Simetar simulation engine
Change the number of iterations based on the nature of the problem is adequate.
Some studies use 1,000’s because they are using a Monte Carlo sampling procedure which is less precise than the Latin hypercube
Simetar uses a Latin hypercube so 500 is an adequate sample size

12. Definitions
Stochastic Model – means the model has at least one random variable
Monte Carlo simulation model – same as a stochastic simulation model
Two ways to sample or 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 model samples all regions of the probability distribution
For a U(0,1) the CDF is straight line
MC has bias from straight line
LHC is the straight line
This is with 500 iterations
Simetar default is LHC

13. 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
Assume yield is a random variable and we have a production function, such as:
Ỹ = a + b1 Fertilizer + b2 Water + ẽ
Deterministic component is: a + b1 Fertilizer + b2 Water
Stochastic component is: ẽ
Stochastic component, ẽ, is assumed to be distributed Normal
Mean of zero
Standard deviation of σe
See Lecture 14 Simulate a Reg Model.XLSX

14. When to Use the Normal Distribution
Use the Normal distribution if you have lots of observations and have tested for normality
BUT watch for infeasible values from a Normal distribution (negative yields and prices)

15. Problems with the Normal
It is easy to use, so it is 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 6 suffers from all of the problems above
Creates a false sense of security because it never sees a record braking outlier

16. How to 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”

17. Simulating a Normal Distribution
Review the 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, [USD])
Ỹ = Mean + Standard Deviation * SND
Ỹ = A1 + (A2 * A3) where a SND is in cell A3

18. 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 the [ ] are optional but the positions are fixed and critical

19. 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

20. Program a Simulation Model in Excel/Simetar -- Input Data Section of the Worksheet
See Lecture 14 Simulation Model with Simetar.XLSX

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

22. Bernoulli Distribution1 X
PDF for Bernoulli B(0.75)
.25
.75
CDF for Bernoulli B(0.75)
PDF and CDF for a Bernoulli Distribution.
Parameter is ‘p’ or the probability that the random variable is 1.0 or TRUE
Simulate Bernoulli in Simetar as
= Bernoulli(p)
= Bernoulli(0.25)
Lecture 14 Bernoulli.XLSX examples follow

23. Bernoulli Distribution Application

24. Bernoulli Distribution Application

25. Equilibrium Displacement Model Forecasting
Elasticities of demand and supply can be used to forecast changes in price, quantity demanded and quantity supplied
Small changes in the exogenous variables allows one to forecast the dependent variable
This method is simple and reliable for small changes from equilibrium
Information needed:
A Baseline of equilibrium quantities and prices
Own and cross elasticities for demand and supply
Residuals from trend (or a structural model) for the dependent variable if the forecast is to be stochastic

26. Equilibrium Displacement Model Forecasting
Baseline prices and quantities are available from FAPRI, USDA, and some private consulting firms
Here is an example of the corn S&U from the FAPRI March 2013 Baseline

27. Equilibrium Displacement Model Forecasting
To forecast a price change given a change in quantity supplied
Price1 = P0 * [1+ {Price Flex *( Q1 - Q0) / Q0) } ] + ẽ
P0 and Q0 are baseline values Q1 is assumed change
Price
?P1 P0
Demand Q1 Q0
Q/UT

28. Equilibrium Displacement Model Forecasting
To forecast the Q Supplied given a change in price
Qt Supplied1 = Qs0 * [1 + {Es *( P1 - P0) / P0) } ] + ẽ
P0 and Q0 are baseline values P1 is assumed change
Price
Supply P1 P0 Q0
?Q1
Q/UT

29. Equilibrium Displacement Model Forecasting
We can expand the supply response equation by including cross elasticities
Qt Supplied x = Qsx0 * (1 + {[Exs *( Px1 - Px0) / Px0)] + [Exs,yp *( Py1 - Py0) / Py0)]
+ [Exs,zp *( Pz1 - Pz0) / Pz0)] } ) + ẽ
Where x is the own crop (say, corn), y is the price of soybeans, and z is the price of wheat
The supply response equation can be expanded to contain cross elasticities for all other crops
Note: Ex,yp is the elasticity of corn supply with respect to the price of soybeans

30. Equilibrium Displacement Model Forecasting
The general form for an elasticity of demand equation can be used to forecast quantity exported, or quantity demanded for ethanol or any other quantity demanded
All we need is the Baseline quantity demanded, corresponding baseline price and the own and cross elasticities
QD1 = QD0 * (1 + {ED for exports * ( P1 - P0) / P0)}) + ẽ
Price P0 P1
Demand for Exports Q0
?Q1
Q/UT

31. Equilibrium Displacement Model Forecasting
This method of forecasting is widely used in agricultural economics
Particularly useful for policy analysis and consulting
This is why economists place so much emphasis on estimating unbiased elasticities

32. Probabilistic Forecasting (PF)
- PF are inter-temporal forecasts of a random variable with stochastic components
· Determ forecast
· Prob forecast
- PF involves simulating an econometrically estimated forecast equation with the appropriate risk
- Purpose is to evaluate the risk in a forecast
- Readings: Chapter 15
- Lecture 16 Probabilistic Forecasting.XLS
- Lecture 16 Probabilistic Time Series.XLS
Lecture 16