Monte CarloΒΆ

The basic idea behind using the Monte Carlo method is to run simulations over and over to get a probability distribution of an unknown probabilistic entity. Numerical methods such as Monte Carlo are often helpful when analytical methods are too difficult to solve or don’t exist.

  • In this section we will use a Monte Carlo method to explore the bias of an AR(1) process. First, suppose an AR(1) process of the form:
\[y_t = \mu + \phi y_{t-1} + \epsilon _t\]
  • Now suppose that \(\mu = 1\) and \(\phi = 0.95\) and \(\epsilon_t \sim N(0,4)\)

  • In Matlab, we want to simulate data for this process and then estimate \(\phi\). Then, we want to repeat the simulation and estimation process several times, each time saving our estimate of \(\phi\).

  • Once we have done this enough times, we will have a distribution of the estimated \(\phi\). Then we can compare the distribution to the true value.

  • In this example we will show that our selected parameters result in a biased estimate of \(\phi\).

  • Now, simulate the data and estimate \(\phi\) using this code:
_images/mc_code.png
  • The second to last line of code plots the distribution of \(\phi\)‘s relative to a normal distribution. As you can see, the disrtibution is not normal
_images/mc_hist.png
  • The last line of code tells you the percentage of estimated \(\phi\)‘s that are smaller than the actual value. Running this code will show you that there is a lot more probability mass to the left of the actual value, exposing a bias in our estimation