4.1.1. ARIMA

ARIMA is probably the most popular method when it comes to time series forecasting, initially developed by Box and Jenkins (1976). Typically, an ARIMA model has two components: an autoregressive (AR) component and a moving average (MA) component. The AR component models association between the value of a variable at a specified time with its value in previous time(s), and the MA component models association between values of error term of a variable at a specified time with its error term value in previous time(s). The integrated (I) component comes into consideration when the time series becomes stationary after the first (or second) difference. An ARIMA (*p,d,q*) model can be represented by Equation (1).

$$
\Delta z\_t = \sum\_{i=1}^p \mathcal{Q}\_i \Delta z\_{t-i} + \sum\_{i=1}^q \mathcal{Q}\_i \varepsilon\_{t-i} + \varepsilon\_t \tag{1}
$$

Here, Δ*zt* = *zt* − *zt*−1; *zt* is the Bitcoin price in USD at time *t*, *zt*−*i* is the Bitcoin price in USD of all previous periods until lag *p*, ∅*i* is the parameter for *zt*−*i*, ε*t* is the error term in time *t*, ε*t*−*i* is the error term of all previous periods until lag *q* and θ*i* is the parameter for ε*t*−*i*.
