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

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 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.?

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 ….

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

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

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

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

Flow Chart for Simulating NPV

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 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 Steps for Model Development

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 Stochastic Variables?

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 Q x = a + b 1 P x +b 2 Y + b 3 P y gives a single line for Demand Q x = a + b 1 P x +b 2 Y + b 3 P y + 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 Simple Economic Model Supply Demand Quantity/UT Price/U

The Basic Business Model ~ ~

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

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: Uniform Distribution

PDF and CDF for a Uniform Dist. minmax X f(x) min max X F(x) Probability Density FunctionCumulative Distribution Function

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 When to Use the Uniform Distribution Uniform Deviate Std. Normal Dev SND i USD i Inverse Transform for Generating a SND from a USD For example USD is used to simulate a Normal Distribution

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 Uniform Standard Deviate (USD)

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

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( S i, F(S i ), [USD]) =EMP(values,, [USD]) Note the [USD] means that USD is optional

Generating Random Numbers 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

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

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

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

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

Simulate The Normal Distribution Parameters for a Normal Distribution –Mean or Ŷ from OLS –Std Dev 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

Generating SNDs Generate 500 USDs and transform them to SNDs using the Inverse Transform (=NORMSDIST(USD)) 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

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) Remember where to get Ŷ and σ ? –In forecasting we estimated Ŷ = a + bX 1 +bX 2 σ = Std Deviation of residuals

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

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 ( S i, F(S i ) ) Often times we assume several distribution forms, estimate their parameters, simulate them and pick the one which best fits the data

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 (ẽ t ) –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 (ẽ t ) 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

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

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

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

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