Understanding Monte Carlo Simulation in Statistics

Monte Carlo simulation is a methodology or tool for understanding the properties of a model or statistic under certain conditions. For instance, we talk about the impact of multicolinearity on a regression model, but we cannot assess the actual impact on our estimates because in real life we almost never know the true value of the parameters. By using Monte Carlo simulation, we can perform the same type of analysis on data with known characteristics. We know these characteristics, because we manufacture the data to have those characteristics.

The process of doing a Monte carlo study is as follows:

1. Decide on the statistic to be tested. (e.g sample Mean, or perhaps regression in the presence of multicolinearity)
2. Decide on the true parameters or conditions under which you wish to test the statistic. (e.g population mean or coefficients from OLS model)
3. Generate data as specified in Step 2 (see below)
4. Test the statistic or model (see below)
5. Save the results
6. Repeat Steps 3-5 the desired number of times called trials (e.g. 100 or 1000 trials)
7. Calculate summary statistics on the stored results. (usually the mean for the parameter(s) you are interested in)

Some Examples of Performing Monte Carlo simulations:

• Some general information on generating distributions
  • The Uniform Distribution
• Generating random numbers in Excel
• Generating random numbers and models in Stata

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Some general information on generating distributions

The Uniform Distribution

The Uniform Distribution, also called a rectangular distribution, is one in which every value has an equally likely chance of occuring. Examples are roll of a die, drawing of a lottery number.

For further information, please see:

• Engineering Distributions Handbook Gallery of Distributions: Uniform Distributions
• Eric Weisstein's World of Mathematics: Uniform Distributions

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Roll of a die

Generating random numbers in Excel

Generating Random Numbers in Exccel is relatively easy. We use the Data Analysis Tools Random number generator.

Tools > Data Analysis > Random Number Generator

(If you do not see the Data Analysis Option, go to Tools > Add-Ins in Windows XP )

Then specify:

• The number of variables,
• The number of random numbers
• The type of distribution
• The distribution parameters
• Specify a seed if you desire to control the random selection process. Specifying the same seed will generate the same numbers each time. If left blank, the seed will be determined by the system clock, and thus is effectively random.
• The location where to put the numbers generated.

See the Excel Spreadsheet on Monte Carlo Simulation for examples:

Generating random numbers and models in Stata

• Open Stata without a data set
• Define how large a data set you want to generate
  • Set obs 100
• Define the model's parameters
  • Set the intercept equal to 2.0 with the scalar command
    • scalar a=2
  • Set the slope equal to 2.0
    • scalar b=2
  • Create a weight to increase or decrease the importance of the error term
    • scalar wt=10
• Generate a uniform X variable on a 0-100.0 scale
  • gen x = uniform()*100
• Generate a random Normal variable
  • gen e = invnorm(uniform())
• Create Y based on a, b, the X variable and the weighted error term.
  • gen y = a + b*X + wt*e
• Examine your model with regression
  • regress y x