Chapter 9: Monte Carlo Simulation
The chapters in the first part of this book make clear that regression analysis can be used to describe data. The remainder of this book is dedicated to understanding regression as a tool for drawing inferences abouthowvariables are related to each other. The central idea in inferential statistics is that the data we observe are just one sample from a larger population. The goal of inference is to determine what evidence the sample provides about the relationship between variables in the population.
This chapter explains how we will use the computer to draw random samples to evaluate the performance of a variety of sample-based statistics. We will review basic theory behind random number generation with computers, offer a simple example of Monte Carlo simulation, and introduce a Monte Carlo simulation Excel add-in.
Like regression analysis, Monte Carlo simulation is a general term that has many meanings. The word “simulation” signifies that we build an artificial model of a real system to study and understand the system. The “Monte Carlo” part of the name alludes to the randomness inherent in the analysis:
The name “Monte Carlo” was coined by [physicist Nicholas] Metropolis (inspired by [Stanislaw] Ulam's interest in poker) during the Manhattan Project of World War II, because of the similarity of statistical simulation to games of chance, and because the capital of Monaco was a center for gambling and similar pursuits. Monte Carlo is now used routinely in many diverse fields, from the simulation of complex physical phenomena such as radiation transport in the earth's atmosphere and the simulation of the esoteric subnuclear processes in high energy physics experiments, to the mundane, such as the simulation of a Bingo game or the outcome of Monty Hall's vexing offer to the contestant in “Let's Make a Deal.” (Drakos, 1995)
Monte Carlo simulation is a method of analysis based on artificially recreating a chance process (usually with a computer), running it many times, and directly observing the results.
We will use Monte Carlo simulation to understand the properties of different statistics computed from sample data. In other words, we will test-drive estimators, figuring out how different recipes perform under different circumstances. Our procedure is quite simple: In each case we will set up an artificial environment in which the values of important parameters and the nature of the chance process are specified; then the computer will run the chance process over and over; finally the computer will display the results of the experiment.
The next section explains the fundamental principles behind random number generation, which is the engine that drives a Monte Carlo simulation. Section 9.3 is a practical guide to generating random numbers in Excel. Section 9.4 demonstrates Monte Carlo via a simple example, and the last section introduces an Excel add-in that can be used to run a Monte Carlo simulation in any Excel workbook.