Auto-Regressive and Moving Average Models

Definitions

Auto-regressive integrated moving average (ARIMA) models are a battle-tested linear model used extensively in financial applications.
Auto-regressive (AR) models use a linear combination of past values of the time-series variable (lag variables) to forecast future values. Specifically, AR models take historical price time series as inputs and output price forecasts.
Moving average (MA) models use a linear combination of past error terms to forecast future values. They also allow for the modeling of stochastic shocks.
Basic ARIMA models are generally ineffective at forecasting in financial contexts without modifications to account for a variety of market factors/conditions. The particular ARIMA variation Concrete employs an ARFIMA (add fractionalized) model.

ARIMA models are well-established as effective in financial forecasting contexts when correctly calibrated for particular market and asset conditions (including crypto).
ARIMA models have been demonstrated to be particularly effective at short-to-medium range forecasting (15-30 days). This time frame is well-suited to Concrete’s use case.

Formally: where epislon is white noise (usually a Gaussian distribution), φ terms are lag variables, and y_k represents the time series data value, y, at time, k. Y_t is the time series value at time t and the AR model has and order of p equal to the number of lag variables it utilizes.

Formally: where variables are equivalently defined as AR models save the which represent the number of error terms utilized by the MA model.

Auto-regressive moving average (ARMA) models combine AR and MA models thereby minimizing the order of the model, and thereby the computational load required by extraneous variables.
ARMA models assume a constant probability distribution, which renders them impractical in financial contexts. To be specific – financial time series exhibit stochastic behavior, but ARMA models are only able to effectively analyze stationary data.

ARIMA models are constructed as an amalgamation of auto-regressive (AR) and moving average (MA) models. Both are foundational to statistics, financial context-agnostic.
- ARIMA models expand on ARMA models via differencing procedures. Differencing procedure allow for the transformation of non-stationary data to stationary data.
- Differencing is difficult to exhaustively express in a generalized form, but can be understood by where d represents the degree of differencing.