Materials for Lecture 09

1
Materials for Lecture 09

• Chapters 4 and 5
• Chapter 16 Sections 3.2-3.7.3
• Lecture 09 Bernoulli Empirical.xls
• Lecture 09 Normality Test.xls
• Lecture 09 Parameter Est.xls
• Lecture 09 Normal.xls
• Lecture 09 Simulate a Reg Model.xls

2
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

3
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

4
Stochastic Simulation

• In economics we use simulation because we can not
  experiment on live subjects 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

5
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 1,000s because they are using
  a Monte Carlo sampling procedure which is
  less precise than Latin hypercube
• Simetar defaults to a Latin hypercube so 500 is
  an adequate sample size

6
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.

7
Simulating Random Variables

• Normal distribution is used frequently,
  particularly when simulating a regression model
• Parameters for a Normal distribution
• Mean expressed as ? or Y
• Standard Deviation s (or SEP from a regression
  model)
• Assume yield is a random variable and have
  production function data, such as
• ? a b1 Fert b2 Water ?
• Deterministic component is a b1 Fert b2
  Water
• Stochastic component is ?
• Stochastic component, ?, is assumed to be
  distributed Normal
• Mean of zero
• Standard deviation of se
• See Lecture 9 Simulate a Reg Model.XLS

8
PDF and CDF for a Normal Dist.
Probability Density Function
Cumulative Distribution Function
f(x)
F(x)
-?
?
-?
?
9
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)

10
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 (BSs)
• 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 BSs
• 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

11
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
• Simetars Hypothesis Testing Icon (Ho Hi)
  provides a tab to Test for Normality

12
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
• ? Mean Standard DeviationNORM(0,1)
• ? Mean Standard DeviationSND
• ? A1 (A2 A3) where a SND is in cell A3

13
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)

14
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
15
Program a Simulation Model in Excel/Simetar? --
Input Data Section of the Worksheet
 
A
C
B
1
VC / acre
150.0
2
VC / Y
0.25
3
Acre
100
4
Fixed Cost
10
5
Yield Mean Std. Dev.
150
30
Price Mean Std. Dev.
6
2
0.40

• See Lecture 09 Simulation Model with Simetar.XLS

16
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
 
 
 
35
Net Returns
24359.26
B27 B33
17
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)

18
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