1. Materials for Lecture 13
Chapter 2 pages 6-12, Chapter 6, Chapter 16 Section 3.1 and 4
Lecture 13 Probability of Revenue.xlsx
Lecture 13 Flow Chart.xlsx
Lecture 13 Farm Simulator.xlsx
Lecture 13 Uniform.xlsx
Lecture 13 Theta UPES.xlsx
Lecture 13 View Distributions.xlsx

2. What is a Simulation Model?
A Model is a mathematical representation of any system of equations
When you think through the many steps to solve a problem you are constructing a model
When you think or plan your way through a complex situation you are making a virtual model
Computer games are models
Econometric equations can be part of a model
We build models so we do not have to experiment on the actual economic system
Will the business be successful if we change management practices, etc.?

3. Outline for the Lecture
Organization of a model in an Excel Workbook
Steps for model development
Parts in a simulation model
Generating random variables from uniform distributions
Estimating parameters for other distributions
Parameters are the numbers that define the center and the dispersion about the center of the random variable
For a Normally distributed random variable, the parameters are the Mean & Std Dev
For Empirical ….

4. Organization of Models in Excel
Input Data:
Costs, inflation & interest rates
Production functions
Assets & liabilities
Scenarios to analyze, etc.
Historical Data for Stochastic Variables:
Prices
Production levels
Other variables not controlled by management
Equations to calculate variables:
Production, Receipts, Costs, Amortize Loans, Update Asset values, etc.
Tables to report financial results:
Income statement, cash flow, balance sheet, financial ratios
KOV Table
List all output variables of interest
Model Outputs:
Statistics for KOVs
Probability charts
Decision summary
Final report tables

5. Organization of Models in Excel
Sheet 1 (Model)
Assumptions and all Input Data
Control variables for managing the system
Logical flow of all calculations
Table of intermediate results
Pro Forma financial tables of results
Key Output Variables (KOVs) Table to send to SimData
Sheet 2 (Stoch)
Historical data for all random variables
Calculations to estimate the parameters for random variables
Simulate all random values to be mapped to the Model
Sheets 3-N (SimData, Stoplite, SERF, STODOM, etc)
Simulation results and charts

6. Equations and Calculations to
Model Design Steps
KOVs
Intermediate Results
Tables and Reports
Equations and Calculations to
Get Values for Reports
Stochastic Variables
Exogenous and Control Variables
Design
Build
Model development is like building a pyramid
Design the model from the top down
Build from the bottom up

7. Steps for Model Development
Determine the purpose of the model and KOVs
Draw a sketch of how data will interact to calculate the KOVs
Determine the variables necessary to calculate the KOVs
For example to calculate Net Present Value (NPV) we need:
Annual net cash withdrawals which are a function of net returns
Ending net worth which is a function of assets and liabilities
This means you need a balance sheet and a cash flow statement to calculate annual cash reserves
An annual income statement is needed as input into a cash flow
Annual net returns are calculated from an income statement

8. Flow Chart for Simulating NPV

9. Steps for Model Development
Write out the equations by hand
This organizes your thoughts and the model’s structure
Avoids problem of forgetting important sections
Example of equations to simulate receipts at this point:
Output/hour = a stochastic variable
Hours Operated = management control value (scenario)
Production = Output/hour * Hours Operated
Price = forecast mean each year with a risk component
Receipts = Price * Production
Define input variables
Exogenous variables are out of the control of management and are deterministic; usually policy driven
Stochastic variables management can not control and are random in nature: weather or market prices, interest rates
Control variables the manager can manipulate and are usually used for sensitivity and/or scenario analyses

10. Steps for Model Development
Stochastic variables (most time is spent here)
Identify key random variables that affect the system
Estimate parameters for the assumed distributions
Normality – means and standard deviations
Empirical – sorted deviates and probabilities
Other distributions should be tested
Use the best possible econometric model to forecast deterministic part of stochastic variables – reduce risk
Model validation starts here
Use statistical tests of the simulated stochastic variables to insure that random variables are simulated correctly
Correlation tests, means tests, variance tests
CDF and PDF charts to compare history to simulated values
Key to validating model are statistical tests

11. Stochastic Variables? What are Stochastic Variables?
Random variables we can not control, such as:
Prices, yields, interest rates, rates of inflation, sickness, etc.
Represented by the residuals from regression equations as this is the part of a variable we did not predict
Why include stochastic variables?
To get a more robust simulation answer
Draw random values from a PDF rather than a single or deterministic value
The result is that we can assign probabilities to KOVs
We can incorporate risk in our decisions of selecting between scenarios

12. Simple Economic Model Supply and Demand Model
You learned there is one Demand and one Supply
But there are many, due to the risk on the equations
Qx = a + b1Px +b2Y + b3Py gives a single line for Demand
Qx = a + b1Px +b2Y + b3Py + ẽ gives infinite Demands
After harvest Supply is a constant, so we get an infinite number of Prices as we draw ẽ values at random
Price/U
Supply
Demand is stochastic so we can have an infinite number of Demand functions passing through the QD distribution
Demand
Quantity/UT

13. The Basic Business Model
Profit is generally our Key Output Variable of interest
𝜋 = Total Receipts – Variable Cost – Fixed Cost
𝜋 = ∑(Pi * Ỹi ) - ∑(VCi * Ỹi * Qi ) – FC
Where Pi is the stochastic price for product i, as $/bu.
Ỹi is stochastic production level as yield or bu./acre
VCi is variable cost per unit of production for i, or $/bu.
Qi is the level of resources committed to i, as acres ~ ~

14. Univariate Random Variables
More than 50 Univariate Distributions in Simetar
Uniform Distribution
Normal and Truncated Normal Distribution
Empirical, Discrete Empirical Distribution
GRKS Distribution
Triangle Distribution
Bernoulli Distribution
Conditional Distribution
Excel probability distributions have been made Simetar compatible, e.g.,
Beta, Gamma, Exponential, Log Normal, Weibull
See Chapter 16 in Sections 3.1 and 4

15. Uniform Distribution
A continuous distribution where each range has an equal probability of being observed
20% chance of seeing a value between 0 and 0.2 or between 0.8 and 1.0
Parameters for the uniform are minimum and maximum values and the domain includes all real number’s
=UNIFORM(minimum, maximum)
The mean and variance of this distribution are:

16. PDF and CDF for a Uniform Dist.
Probability Density Function
Cumulative Distribution Function
f(x)
F(x)
1.0
0.0
min
max
min
max X X

17. When to Use the Uniform Distribution
Use the uniform distribution when every range of length “n” between the minimum and maximum values has an equal chance of occurrence
Use this distribution when you have no idea what type of distribution to use
Uniform distribution is used to simulate all random variables via the Inverse Transform procedure and USD
An example of how USD is used to simulate a Standard Normal Distribution
Uniform Deviate
Std. Normal Dev. - + 3
0.5
1.0
0.8
0.6
0.4
0.2
SNDi
USDi
Inverse Transform for Generating a SND from a USD

18. Uniform Standard Deviate (USD)
In Simetar we simulate the USD as:
=UNIFORM(0,1) or =UNIFORM()
Produces a Uniform Standard Deviate (USD) 0 to 1
Special case of the Uniform distribution
USD is building block for all random number generation using the Inverse Transformation method for simulation. Inverse Transform uses a USD to simulate a Uniform distribution as:
X = Min + (Max-Min) * Uniform(0,1)
X = Min + (Max-Min) * USD

19. Simulate a Uniform Distribution
Alternative ways to program the Uniform( ) distribution function
= Uniform(Min, Max,[USD])
= Uniform(10,20) Not recommended method
= Uniform(A1,A2) This is the preferred method
= Uniform(A1,A2,A3) where a USD is calculated in cell A3

20. Uses for a Uniform Standard Deviate
USD can be used in all random number formulas in Simetar to facilitate correlating random variables
For example in Simetar we can add USDs:
=NORM(mean, std dev, [USD1])
=TRIANGLE(min, middle, max, [USD2])
= EMP( Si, F(Si), [USD3])
=EMP(values , , [USD4])
NOTE: every variable has its own unique USD
Note the [ ] means that USD is optional

21. Generating Random Numbers
Generate a Uniform Standard Deviate (USD)
=UNIFORM(0,1)
Simetar defaults to simulate 500 values (can be changed to 1,000s)
These are called iterations or draws
Iterations are separate, uncorrelated draws of random variables
Equal chance of observing a number in each of the intervals; both charts are for the same output

22. USD Output in SimData
Simetar saves the 500 samples in SimData and calculates summary statistics

23. Inverse Transform
Use the 500 USDs to simulate random variables for your Ŷ variable
This involves translating the USDs from a 0 to 1 scale to the scale for your random variable
This is done using the Inverse Transform method shown on the next slide.
NOTE: you must have a separate USD for every random variable Y

24. Inverse Transform
The 500 USDs converted from the 0 to 1 scale on the Y axis by direct interpolation
Each random USD is associated with a unique “random” Y value to get 500 Ỹs

25. Inverse Transform
Results of 500 iterations for Y using Inverse Transform
USDs and their resulting Ỹs

26. Simulate the Normal Distribution
Parameters for a Normal Distribution
Mean or Ŷ from OLS
Std Dev or σ of residuals
Simulated using the formula for a Normal
Ỹ = Ŷ + σ * SND
Where the SND is a “standard normal deviate”
We generate 500 SNDs and thus simulate (calculate) 500 random Y’s

27. Simulate the Standard Normal Deviate (SND)
SND is a random value between ±∞
SND has a mean of zero
SND has a standard deviation of one
SND is simulated by =NORM(0,1)
SNDs are the “number of standard deviations from the mean” or the number of σ’s Ỹ is from the Ŷ or Ῡ
Uniform Deviate
Std. Normal Dev. - + 3
0.5
1.0
0.8
0.6
0.4
0.2
SNDi
USDi
Inverse Transform for Generating a SND from a USD

28. Simulate Normal Distribution
Next apply the random SNDs to the Normal distribution formula
Ỹ = Ŷ + σ * SND
In Simetar all of these steps are done for you: = NORM(Ŷ, σ) or
= NORM(Ŷ, σ, USD)
Remember where to get Ŷ and σ ?
In forecasting we estimated
Ŷ = a + bX1 +bX2
σ = Standard Deviation of residuals

29. Normal Distribution: Simetar Code and Output
The USD is used to calculate the SND
The SND is used to simulate Ỹ
Simetar gives same result in one step

30. Steps for Simulating Random Variables
Must assume a probability distribution (shape)
Normal, Beta, Empirical, 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) )
Often times we assume several distribution forms, estimate their parameters, simulate them and pick the one which best fits the data

31. 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

32. 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

33. 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

34. 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

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