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## Monte Carlo Simulation of Conditional Variance 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 variance models specify the dynamic evolution of the variance of a process over time. Perform Monte Carlo simulation of conditional variance models by:

1. Specifying any required presample data (or use default presample data).

2. Generating the next conditional variance recursively using the specified conditional variance model.

3. Simulating the next innovation from the innovation distribution (Gaussian or Student’s t) using the current conditional variance.

For example, consider a GARCH(1,1) process without a mean offset, ${\epsilon }_{t}={\sigma }_{t}{z}_{t},$ where zt either follows a standardized Gaussian or Student’s t distribution and

`${\sigma }_{t}^{2}=\kappa +{\gamma }_{1}{\sigma }_{t-1}^{2}+{\alpha }_{1}{\epsilon }_{t-1}^{2}.$`

Suppose that the innovation distribution is Gaussian.

Given presample variance ${\sigma }_{0}^{2}$ and presample innovation ${\epsilon }_{0},$ realizations of the conditional variance and innovation process are recursively generated:

• ${\sigma }_{1}^{2}=\kappa +{\gamma }_{1}{\sigma }_{0}^{2}+{\alpha }_{1}{\epsilon }_{0}^{2}$

• Sample ${\epsilon }_{1}$ from a Gaussian distribution with variance ${\sigma }_{1}^{2}$

• ${\sigma }_{2}^{2}=\kappa +{\gamma }_{1}{\sigma }_{1}^{2}+{\alpha }_{1}{\epsilon }_{1}^{2}$

• Sample ${\epsilon }_{2}$ from a Gaussian distribution with variance ${\sigma }_{2}^{2}$

$⋮$

• ${\sigma }_{N}^{2}=\kappa +{\gamma }_{1}{\sigma }_{N-1}^{2}+{\alpha }_{1}{\epsilon }_{N-1}^{2}$

• Sample ${\epsilon }_{N}$ from a Gaussian distribution with variance ${\sigma }_{N}^{2}$

Random draws are generated from EGARCH and GJR models similarly, using the corresponding conditional variance equations.

### 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.