#### Background

This tool simulates a stochastic process that is autoregressive of order 1 or AR(1). A random variable \(X_t\) that follows an AR(1) process can be written as: $$X_t = (1-\rho) \mu + \rho X_{t-1} + \epsilon_t,$$ where \(\mu\) is the unconditional mean of the process, \(\rho\) is the coefficient of autocorrelation, and \(\epsilon_t\) is a white noise (WN) process with standard deviation \(\sigma_{\epsilon}\). The coefficient of autocorrelation \(\rho\) determines the extent to which the previous — or *lagged*— value of \(X\) affects the current value.

#### Instructions

Choose the values of \(\mu\), \(\rho\), \(\sigma_{\epsilon}\), and \(X_0\) for the process that you want to simulate. Then choose the number of periods for the simulation and select which type of simulation that you want to perform:

**Stochastic simulation**: a new random value for the white noise process \(\epsilon_t\) will be drawn for each period.
**Impulse response**: \(\epsilon_t\) will be set to \(\sigma_{\epsilon}\) in period 1 and to 0 in all subsequent periods.

Use the dropdown menu in the upper-right corner of each plot to download the image in png format. Click the "Download csv" button to download the data in both figures in a single csv file.