In probability theory and time series analysis, a stochastic process is called a white noise process if it is a sequence of random variables that are independent and identically distributed with a mean of zero and a constant variance. In other words, a white noise process is a random sequence of uncorrelated observations with equal variance.
On the other hand, an autoregressive (AR) process is a stochastic process in which each observation is a linear combination of the previous observations and a random error term. The order of an AR process specifies how many previous observations the current observation is dependent upon. The simplest form of an AR process is the first-order AR process, or AR(1):
Xt = aXt − 1 + et
where Xt is the current observation, Xt − 1 is the previous observation, a is a constant coefficient, and et is a random error term.
The main difference between a white noise process and an AR process is that a white noise process has no dependence between the observations, whereas an AR process has dependence between the observations through the autoregressive relationship. Additionally, the autocorrelation function of a white noise process is zero for all lags except zero, while the autocorrelation function of an AR process decays exponentially as the lag increases.
For example, consider two time series: one generated from a white noise process and the other from an AR(1) process. The white noise process would have randomly fluctuating observations with no pattern, while the AR(1) process would have smoother observations that are dependent on the previous observation and the random error term. The autocorrelation function of the AR(1) process would also decay exponentially as the lag increases, while the autocorrelation function of the white noise process would be zero for all lags except zero.
In quantitative trading or investment, it is essential to understand the underlying process that generates the data to make the right predictions and investment decisions. A better understanding of white noise and AR processes can help in developing forecasting models and trading strategies that take into account the underlying dynamics of the time series data.