Materials for Lecture

1. Materials for Lecture • Chapter 2 pages 6-12, Chapter 6, Chapter 16 Section 3.1 and 4 • Lecture 7 Probability of Revenue.xlsx • Lecture 7 Flow Chart.xlsx • Lecture 7 Farm Simulator.xlsx • Lecture 7 Uniform.xlsx • Lecture 7 Theta UPES.xlsx • Lecture 7 View Distributions.xlsx

2. Brief Explanation of How Simulation Works • Examples how random numbers are generated • Uniform Distribution • Inverse Transform to simulate any distribution • Standard Normal Distribution • Normal distribution for simulating a forecast • Forecasts can come from Mean or OLS

3. Step 1 for Simulation • Generate a Uniform Standard Deviate (USD) =UNIFORM(0,1) Simetar simulates 500 values These are called iterations They are 500 samples or draws • Equal chance of observing a number in each of the intervals; both charts are for the same output

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

5. Step 2 in Simulation • 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.

6. Step 2 Inverse Transform • The 500 USDs are converted from 0 to 1 scale to the Y scale by direct interpolation • Each random USD is associated with a unique “random” Y value to get 500 Ỹs

7. Step 2 Inverse Transform Cont. • Results of 500 iterations for Y using Inverse Transform • USDs and their resulting Ỹs

8. Step 3 Simulate Normal Distribution • Parameters for a Normal Distribution • Mean or Ŷ from OLS • StdDev or σ of residuals • Simulated using the formula Ỹ = Ŷ + σ * SND Where the SND is a “standard normal deviate” We generate 500 SNDs and thus simulate (calculate) 500 random Y’s

9. Step 3 Generating SNDs • Generate 500 USDs and transform them to SNDs using the Inverse Transform • SND’s have mean of Zero and range from ≈ +/- ∞ • SNDs are the “number of standard deviations from the mean” or the number of σ’s Ỹ is from the Ŷ or Ῡ

10. Step 3 Simulate Normal Distribution • Next apply the random SNDs in 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) • Next problem is where to get Ŷ and σ ? • In forecasting we estimate Ŷ = a + bX1 +bX2 or another OLS equation σ = Std Deviation of residuals

11. Normal Distribution: Simetar Code and Output • The USD is used to calculate the SND • The SND is used to simulate Y-Tilda • Simetar gives same result in one step

12. Simulation Models • 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 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.?

13. Developing Simulation Models • 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 ….

14. Organization of Models in Excel Input Data, such as – Costs, inflation & interest rates, Production functions Assets & liabilities Scenarios to analyze, etc. Historical Data for Random Variables, such as – 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 KOV Table – List all output variables of interest Model Outputs: Statistics for KOVs Probability charts Decision summarys Final report tables

15. 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 • Table of final results – the Key Output Variables (KOVs) • Sheet 2 (Stoch) • Historical data for random variables • Calculations to estimate the parameters for random variables • Simulate all random values • Sheets 3-N (SimData, Stoplite, CDF) • Simulation results and charts

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

17. Steps for Model Development • Determine the purposeof 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

18. Example Flowchart of a Model

19. Flow Chart for Simulating NPV

20. 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 for a model at this point: • Output/hour = stochastic variable • Hours Operated = management control value • 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 driven • Control variables the manager can manipulate

21. Steps for Model Development • Stochastic variables (most time is spent here) • Identify all random variables that affect the system • Estimate parameters for the assumed distributions • Normality – means and standard deviations • Empirical – sorted deviates and probabilities • Use the best model possible econometric model to forecast deterministic part of stochastic variables to 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

22. 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 • PDF rather than a single value • We can assign probabilities to KOVs • We can incorporate risk in our decisions

23. Simple Economic Model • A 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 • Now if Supply is a constant we get an infinite number of Prices as we draw ẽ values at random Price/U Supply Demand Quantity/UT

24. Simple Business Model • Profit is generally our Key Output Variable of interest = Total Receipts – Variable Cost – Fixed Cost = ∑(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 ~ ~

25. 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

26. Uniform Distribution • A continuous distribution where each range has an equal probability of being observed • Parameters for the uniform are minimum and maximum values and the domain includes all real number’s =UNIFORM(min,max) • The mean and variance of this distribution are:

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

28. Uniform Deviate 1.0 USDi 0.8 0.6 0.5 0.4 0.2 3 3 - Std. Normal Dev. 0 + SNDi Inverse Transform for Generating a SND from a USD 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 For example USD is used to simulate a Normal Distribution

29. Uniform Standard Deviate (USD) • In Simetar we simulate the USD as: =UNIFORM(0,1) or =UNIFORM() • Produces a Uniform Standard Deviate (USD) • 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

30. Simulate a Uniform Distribution • Alternative ways to program the Uniform( ) distribution function =Uniform(Min, Max,[USD]) = Uniform(10,20) = Uniform(A1,A2) = Uniform(A1,A2,A3) where a USD is calculated in cell A3

31. 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, [USD]) =TRIANGLE(min, middle, max, [USD]) = EMP( Si, F(Si), [USD]) =EMP(values , , [USD]) • Note the [USD] means that USD is optional

32. Simulating Random Variables • Must assume a probability distribution shape • Normal, Beta, Empirical, etc. • Estimate parameters required to define 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

33. 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

34. 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

35. 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

36. 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

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