## Monte Carlo Simulation of Conditional Mean Models

### What Is Monte Carlo Simulation?

Monte Carlo simulation is the process of generating independent, random draws from a specified probabilistic model. When simulating time series models, one draw (or realization) is an entire sample path of specified length N, y1, y2,...,yN. When you generate a large number of draws, say M, you generate M sample paths, each of length N.

Note

Some extensions of Monte Carlo simulation rely on generating dependent random draws, such as Markov Chain Monte Carlo (MCMC). The `simulate` function in Econometrics Toolbox™ generates independent realizations.

Some applications of Monte Carlo simulation are:

• Demonstrating theoretical results

• Forecasting future events

• Estimating the probability of future events

### Generate Monte Carlo Sample Paths

Conditional mean models specify the dynamic evolution of a process over time through the conditional mean structure. To perform Monte Carlo simulation of conditional mean models:

1. Specify presample data (or use default presample data).

2. Generate an uncorrelated innovation series from the innovation distribution that you specified.

3. Generate responses by recursively applying the specified AR and MA polynomial operators. The AR polynomial operator can include differencing.

For example, consider an AR(2) process,

`${y}_{t}=c+{\varphi }_{1}{y}_{t-1}+{\varphi }_{2}{y}_{t-2}+{\epsilon }_{t}.$`

Given presample responses y0 and y–1, and simulated innovations ${\epsilon }_{1},\dots ,{\epsilon }_{N},$ realizations of the process are recursively generated:

• ${y}_{1}=c+{\varphi }_{1}{y}_{0}+{\varphi }_{2}{y}_{-1}+{\epsilon }_{1}$

• ${y}_{2}=c+{\varphi }_{1}{y}_{1}+{\varphi }_{2}{y}_{0}+{\epsilon }_{2}$

• ${y}_{3}=c+{\varphi }_{1}{y}_{2}+{\varphi }_{2}{y}_{1}+{\epsilon }_{3}$

$⋮$

• ${y}_{N}=c+{\varphi }_{1}{y}_{N-1}+{\varphi }_{2}{y}_{N-2}+{\epsilon }_{N}$

For an MA(12) process, e.g.,

`${y}_{t}=c+{\epsilon }_{t}+{\theta }_{1}{\epsilon }_{t-1}+{\theta }_{12}{\epsilon }_{t-12},$`

you need 12 presample innovations to initialize the simulation. By default, `simulate` sets presample innovations equal to zero. The remaining N innovations are randomly sampled from the innovation process.

### Monte Carlo Error

Using many simulated paths, you can estimate various features of the model. However, Monte Carlo estimation is based on a finite number of simulations. Therefore, Monte Carlo estimates are subject to some amount of error. You can reduce the amount of Monte Carlo error in your simulation study by increasing the number of sample paths, M, that you generate from your model.

For example, to estimate the probability of a future event:

1. Generate M sample paths from your model.

2. Estimate the probability of the future event using the sample proportion of the event occurrence across M simulations,

`$\stackrel{^}{p}=\frac{#\text{\hspace{0.17em}}\text{\hspace{0.17em}}times\text{\hspace{0.17em}}\text{\hspace{0.17em}}event\text{\hspace{0.17em}}\text{\hspace{0.17em}}occurs\text{\hspace{0.17em}}\text{\hspace{0.17em}}in\text{\hspace{0.17em}}\text{\hspace{0.17em}}M\text{\hspace{0.17em}}\text{\hspace{0.17em}}draws}{M}.$`

3. Calculate the Monte Carlo standard error for the estimate,

`$se=\sqrt{\frac{\stackrel{^}{p}\left(1-\stackrel{^}{p}\right)}{M}}.$`

You can reduce the Monte Carlo error of the probability estimate by increasing the number of realizations. If you know the desired precision of your estimate, you can solve for the number of realizations needed to achieve that level of precision.