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Materials for Lecture 13

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- Purpose summarize the selection of distributions and their appropriate validation tests
- Explain the use of Scenarios and Sensitivity Analysis in a simulation model
- Chapter 10 pages 1-3
- Chapter 16 Sections 7, 8 and 9
- Lecture 13 Scenario.xls
- Lecture 13 Scenario & Sensitivity.xls
- Lecture 13 Sensitivity Elasticity.xls

- Validation of simulated distributions is critical to building good simulation models
- Selection of the appropriate statistical tests to validate the simulated random variables is essential
- The appropriate statistical test changes as we change the method for estimating the parameters

- If the data are stationary and you want to simulate using the historical mean
- Distribution
- Use Normal as =NORM(Ῡ, σY) or
- Empirical as =EMP(Historical Ys)

- Validation Tests for Univariate distribution
- Compare Two Series tab in Simetar
- Student-t test of means as H0: ῩHist = ῩSim
- F test of variances as H0: σ2Hist = σ2Sim
- You want both tests to Fail to Reject the null H0

- Compare Two Series tab in Simetar

- If the data are stationary and you want to simulate using a mean that is not equal tothe historical mean
- Distribution
- Use Empirical as a fraction of the mean so the Si = Sorted((Yi - Ῡ)/Ῡ) and simulate using the formula
Ỹ = Ῡ(new mean) * ( 1 + EMP(Si, F(Si), [CUSDi] ))

- Use Empirical as a fraction of the mean so the Si = Sorted((Yi - Ῡ)/Ῡ) and simulate using the formula
- Validation Tests for Univariate distribution
- Test Parameters
- Student-t test of means as H0: ῩNew Mean = ῩSim
- Chi-Square test of Std Dev as H0: σHist = σSim
- You want both tests to Fail to Reject the null H0

- Test Parameters

- If the data are non-stationaryand you use OLS, Trend, or time series to project Ŷ
- Distribution
- Use =NORM(Ŷ , Standard Deviation of Residuals)
- Use Empirical and the residuals as fractions of Ŷ calculated for Si = Sorted((Yi - Ŷj)/Ŷ) and simulate each variable using
Ỹi = Ŷi * (1+ EMP(Si, F(Si) ))

- Validation Tests for Univariate distribution
- Test Parameters
- Student-t test of means as H0: ŶNew Mean = ῩSim
- Chi-Square test of Std Dev as H0: σê = σSim
- You want both tests to Fail to Reject the null H0

- Test Parameters

- If the data have a cycle, seasonal, or structural pattern and you use OLS or any econometric forecasting method to project Ŷ
- Distribution
- Use =NORM(Ŷ, σê standard deviation of residuals)
- Use Empirical and the residuals as fractions of Ŷ calculated for Si = Sorted((Yi - Ŷ)/Ŷ) and simulate using the formula
Ỹ = Ŷ * (1 + EMP(Si, F(Si) ))

- Validation Tests for Univariate distribution
- Test Parameters tab
- Student-t test of means as H0: ŶNew Mean = ῩSim
- Chi-Square test of Std Dev as H0: σê = σSim
- You want both tests to Fail to Reject the null H0

- Test Parameters tab

- If the data are stationary and you want to simulate using the historical means and variance
- Distribution
- Use Normal =MVNORM(Ῡ vector, ∑ matrix) or
- Empirical =MVEMP(Historical Ys,,,, Ῡ vector, 0)

- Validation Tests for Multivariate distributions
- Compare Two Series for 10 or fewer variables
- Hotelling T2 test of mean vectors as H0: ῩHist = ῩSim
- Box’s M Test of Covariances as H0: ∑Hist = ∑Sim
- Complete Homogeneity Test of mean vectors and covariance simultaneously
- You want all three tests to Fail to Reject the null H0

- Check Correlation
- Performs a Student-t test on each correlation coefficient in the correlation matrix: H0: ρHist = ρSim
- You want all calculated t statistics to be less than the Critical Value t statistic so you fail to reject each t test (Not Bold)

- Compare Two Series for 10 or fewer variables

- If you want to simulate using projected means such that Ŷt ≠ Ῡhistory
- Distribution
- Use Normal as = MVNORM(Ŷ Vector, ∑matrix) or
- Empirical as = MVEMP(Historical Ys ,,,, Ŷ vector, 2)

- Validation Tests for Multivariate distribution
- Check Correlation
- Performs a Student-t test on each correlation coefficient in the matrix: H0: ρHist = ρSim
- You want all calculated t statistics to be less than the Critical Value t statistic so you fail to reject each t test

- Test Parameters, for each j variable
- Student-t test of means as H0: ŶProjected j = ῩSim j
- Chi-Square test of Std Dev as H0: σê j = σSim j

- Check Correlation

- Now lets change gears
- Assume we have a working simulation model
- The Model has the following parts
- Input section where the user enters all input values that are management control variables and exogenous policy or time series data
- Stochastic variables that have been validated
- Equations to calculate all dependent variables
- Equations to calculate the KOVs
- A KOV table to send to the simulation engine

- Simetar simulation engine controls
- Number of scenarios
- Sensitivity analysis
- Sensitivity elasticities

Scenario loop

IS = 1, M

Change management variables (X) from one scenario to the next

Iteration loop

IT = 1, N

Next scenario

- Base scenario – complete simulation of the model for 500 or more iterations with all variables set at their initial or base values
- Alternative scenario – complete simulation of the model for 500 or more iterations with one or more of the control variables changed from the Base
- All scenarios must use the same random values

Use the same random values for all random variables, so identical risk for each scenario

- All values in the model are held constant and you systematically change one or more variables
- Number of scenarios determined by analyst
- Random number seed is held constant and this forces Simetar to use the same random values for the stochastic variables for every scenario
- Use =SCENARIO() Simetar function to increment each of the scenario control variables

- 5 Scenarios for the risk and VC
- Purpose is to look at the impacts of different management scenarios on net returns

- Create as big of table as needed
- Add all control variables into the table

- Sensitivity analysis seeks to determine how sensitive the KOVs are to small changes in one particular variable
- Does net return change a little or a lot when you change variable cost per unit?
- Does NPV change greatly if the assumed fixed cost changes?

- Simulate the model numerous times changing the “change” variable for each simulation
- Must ensure that the same random values are used for each simulation

- Simetar has a sensitivity option that insures the same random values used for each run

- Simetar uses the Simulation Engine to specify the change variable and the percentage changes to test
- Specify as many KOVs as you want
- Simulate the model and 7 scenarios are run

- Change the Price per unit as follows
- + or – 5%
- + or – 10%
- + or – 15%

- Simulates the model 7 times
- The initial value you typed in
- Two runs for + and – 5% for the control variable
- Two runs for + and – 10% for the control variable
- Two runs for + and – 15% for the control variable

- Collect the statistics for only few KOVs
- For demonstration purposes collect results for the variable doing the sensitivity test on
- Could collect the results for several KOVs

- Test Sensitivity of the price received for the product being manufactured on Net Cash Income

- Sensitivity of a KOV with respect to (wrt) multiple variables in the model can be estimated and displayed in terms of elasticities, calculated as:
SEij = % Change KOVi

% Change Variablej

- Calculate SE’s for a KOVi wrt change variablesj at each iteration and then calculate the average and standard deviation of the SE
- SEij’s can be calculated for small changes in Control Variablesj, say, 1% to 5%
- Necessary to simulate base with all values set initially
- Simulate model for an x% change in Vj
- Simulate model for an x% change in Vj+1

- The more sensitive the KOV is to a variable, Vj, the larger the SEij
- Display the SEij’s in a table and chart