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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 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.? 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 …. 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: Model Outputs: • Production, Receipts, Costs, Amortize Loans, Update Asset values, etc. • Statistics for KOVs • Tables to report financial results: • Probability charts • Decision summary • Final report tables • Income statement, cash flow, balance sheet, financial ratios • KOV Table • List all output variables of interest 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 Model Design Steps KOVs Design Intermediate Results Tables and Reports Build Equations and Calculations to Get Values for Reports Stochastic Variables Exogenous and Control Variables • Model development is like building a pyramid – Design the model from the top down – Build from the bottom up 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 Control Variables for Manager such as: Levels of Production, Debt Levels, Market Share Macro Data as inflation rates interest rates Sections and Equations for the Model Generate the Stochastic Values Use Projected Means and Historical Data for Random Variables Use the Stochastic Values in the Equations for the Model Equations for the System to model Production = f( scale of the farm and stochastic values) Price = f( stochastic values) Revenue = Price * Production for each enterprise Variable Costs by Enterprise = Production * Unit Cost Costs = Variable Costs for each enterprise + Fixed Costs Net Returns = Revenue - Costs Balance Sheet Information Asset Valuation Liabilities Net Worth Annual Projected Mean Prices Key Output Variables Net Present Value Probability of Net Returns > 0 Probability of NPV > 0 ( or Prob of Success) Probability of Increasing Real Net Worth Analyze KOVs Budgets for each of the Enterprises Stochastic Variables -- need the historical data to estimate parameters for random variables 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 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 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 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 The Basic Business Model • Profit is generally our Key Output Variable of interest 𝜋 = Total Receipts – Variable Cost – Fixed Cost 𝜋 = ∑(P~i * 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 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 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: min max 2 max min 2 2 12 PDF and CDF for a Uniform Dist. Probability Density Function f(x) min max X Cumulative Distribution Function F(x) 1.0 0.0 min max X 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 1.0 USDi 0.8 0.6 0.5 0.4 0.2 - 3 0 SNDi + 3 Std. Normal Dev. Inverse Transform for Generating a SND from a USD 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 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 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 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 USD = UNIFORM(0,1) Prob CDF for Uniform(0,1) 0.12 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.08 0.06 0.04 0.02 0 0.2 0.4 0.6 0.8 1 0 0.00 0.13 0.25 0.38 0.50 0.62 0.75 0.87 • Equal chance of observing a number in each of the intervals; both charts are for the same output 1.00 USD Output in SimData • Simetar saves the 500 samples in SimData and calculates summary statistics Simetar Simulation Results for 500 Iterations. 9:36:20 AM 2/17/2013 (1 sec.). © 2011. Variable Sheet1!B7 Mean 0.499985 StDev 0.288988 CV 57.79939 Min 0.000895 Max 0.999165 Iteration USD 1 0.512793 2 0.307316 3 0.581277 4 0.787495 5 0.94209 6 0.735971 7 0.048923 8 0.23733 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 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 USD or F(x) 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 55.00 CDF of a Random Variable 60.00 65.00 70.00 75.00 Inverse Transform • Results of 500 iterations for Y using Inverse Transform Simulation Results for 500 Iteratio • USDs and their resulting Ỹs Simetar Variable Sheet1!G33 Sheet1!G34 Mean 0.499985 65.19666 StDev 0.288988 3.136123 CV 57.79939 4.810251 Min 0.000895 56.38011 Max 0.999165 74.43161 Iteration USD Y-Tilda 1 0.512793 65.22607 2 0.307316 63.61534 3 0.581277 65.7939 4 0.787495 67.72464 5 0.94209 70.20308 6 0.735971 67.17892 7 0.048923 60.03664 8 0.23733 62.91843 9 0.955568 70.68873 10 0.634662 66.23654 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 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 1.0 USDi 0.8 0.6 0.5 0.4 0.2 - 3 0 SNDi + Inverse Transform for Generating a SND from a USD 3 Std. Normal Dev. 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 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 Simetar Simulation Results for 500 Iterations. 7:56:32 Variable Sheet1!B47Sheet1!B48Sheet1!B49Sheet1!B50 Mean 0.499985 -0.00015 65.48175 65.48175 StDev 0.288988 1.001471 3.946465 3.946465 CV 57.79939 -650265 6.026817 6.026817 Min 0.000895 -3.12303 53.1755 53.1755 Max 0.999165 3.143506 77.86988 77.86988 Iteration USD SND Y Tilda Simetar 1 0.512793 0.032072 65.60874 65.60874 2 0.307316 -0.50347 63.49834 63.49834 3 0.581277 0.20516 66.29082 66.29082 4 0.787495 0.797758 68.62605 68.62605 5 0.94209 1.572561 71.6793 71.6793 6 0.735971 0.630975 67.96882 67.96882 7 0.048923 -1.65539 58.95901 58.95901 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 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 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 Parameter Estimation, 2/28/2016 2:35:06 PM Maximum Likelihood Estimates (MLEs) Dis tribution Param eter Random Variable Beta α ;α>0, A≤x≤B 1.194356 β ;β>0 1.323942 Double Exponential μ ; -∞<μ<∞, -∞<x<∞ 2.27 σ ; σ>0 0.28625 Exponentialα ; -∞<α<∞, ≤x<∞ 1.5 β ; β>0 0.74375 Gam m a α ; α>0, 0≤x<∞ 36.56802 β ; β>0 0.061358 Invers e Gaus μs ; μ>0, ian 0≤x<∞ 2.24375 σ ; σ>0 0.112584 Logis tic μ ; -∞<μ<∞, -∞<x<∞ 2.242846 σ ; σ>0 0.208322 Log-Log μ ; -∞<μ<∞, -∞<x<∞ 2.061776 σ ; σ>0 0.35179 Log-Logis ticμ ; -∞<μ<∞, 0≤x<∞ 10.58852 σ ; σ>0 2.235521 Lognorm al μ ; -∞<μ<∞, 0≤x<∞ 0.794413 σ ; σ>0 0.16745 Norm al μ ; -∞<μ<∞, -∞<x<∞ 2.24375 σ ; σ>0 0.366774 Pareto α ;α>0, α≤x≤∞ 1.5 β ;β>0 2.571038 Uniform A ; A<B, A≤x≤B 1.5 B ; B>A 3.05 Weibull α ; α>0, 0≤x<∞ 6.554942 β ; β>0 310.5307 Binom ial n ; x=0,1,2,...,n 4 p ; 0≤p≤1 0.560938 Geom etric p ; x=1,2,...; 0≤p≤1 0.308285 Pois s on λ ; 0≤λ<∞, x=0,1,... Negative Binom s ; x=1,2,...; ial 0 p ; 0≤p≤1 2.24375 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 Distributions Beta Double Exponential Exponential Gamma Logistic Log-Log Log-Logistic Lognormal Normal Pareto Uniform Weibull Binomial Geometric Poisson CDFDEV Formula 0.02 =CDFDEV(Sheet1!$A$2:$A$21,SimData!B9:B108) 0.37 =CDFDEV(Sheet1!$A$2:$A$21,SimData!C9:C108) 2.66 =CDFDEV(Sheet1!$A$2:$A$21,SimData!D9:D108) 0.07 =CDFDEV(Sheet1!$A$2:$A$21,SimData!E9:E108) 0.16 =CDFDEV(Sheet1!$A$2:$A$21,SimData!F9:F108) 0.43 =CDFDEV(Sheet1!$A$2:$A$21,SimData!G9:G108) 0.34 =CDFDEV(Sheet1!$A$2:$A$21,SimData!H9:H108) 0.10 =CDFDEV(Sheet1!$A$2:$A$21,SimData!I9:I108) 0.05 =CDFDEV(Sheet1!$A$2:$A$21,SimData!J9:J108) 79.04 =CDFDEV(Sheet1!$A$2:$A$21,SimData!K9:K108) 0.03 =CDFDEV(Sheet1!$A$2:$A$21,SimData!L9:L108) 0.08 =CDFDEV(Sheet1!$A$2:$A$21,SimData!M9:M108) 1.04 =CDFDEV(Sheet1!$A$2:$A$21,SimData!N9:N108) 33.81 =CDFDEV(Sheet1!$A$2:$A$21,SimData!O9:O108) 4.19 =CDFDEV(Sheet1!$A$2:$A$21,SimData!P9:P108) Use the “View Distributions.xlsx” • For a random variable with 10 observations can estimate the parameters and view the shape of the distribution