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Brief Explanation of How Simulation Works

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. Step 1 for Simulation.

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Brief Explanation of How Simulation Works

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  1. 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

  2. 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

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

  4. 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.

  5. 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

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

  7. 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

  8. 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

  9. 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

  10. 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

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