# AR(1) Process Simulation Tool

#### 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:

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