Brief Explanation of How Simulation Works Examples how

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 • 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 Sim. Data • Simetar saves the 500 samples in Sim. Data and calculates summary statistics

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.

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

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

Step 3 Simulate 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

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

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 + b. X 1 +b. X 2 or another OLS equation 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 Y-Tilda • Simetar gives same result in one step