ARIMA models are a class of statistical models that are used for analyzing and forecasting time series data. They aim to do so by describing the autocorrelations in the data. ARIMA stands for Autoregressive Integrated Moving Average and is an extension of a simpler ARMA model. The goal of the additional integration component is to ensure stationarity of the series, because, in contrast to the exponential smoothing models, the ARIMA class requires the time series to be stationary. In the next few paragraphs, we briefly go over the building blocks of ARIMA models.
AR (autoregressive) model:
- This kind of model uses the relationship between an observation and its lagged values.
- In the financial context, the autoregressive model tries to account for the momentum and mean reversion effects.
I (integration):
- Integration, in this case, refers to differencing the original time series (subtracting the value from the previous period from the current period's value) to make it stationary.
- The parameter responsible for integration is d (called degree/order of differencing) and indicates the number of times we need to apply differencing.
MA (moving average) model:
- This kind of model uses the relationship between an observation and the white noise terms (shocks that occurred in the last q observations).
- In the financial context, the moving average models try to account for the unpredictable shocks (observed in the residuals) that influence the observed time series. Some examples of such shocks could be natural disasters, breaking news connected to a certain company, and so on.
All of these components fit together and are directly specified in the commonly used notation known as ARIMA (p,d,q).
- ARIMA (0,0,0): White noise
- ARIMA (0,1,0) without constant: Random walk
- ARIMA (p,0,q): ARMA(p, q)
- ARIMA (p, 0, 0): AR(p) model
- ARIMA (0, 0, q): MA(q) model
- ARIMA (0,1,2): Damped Holt's model
- ARIMA (0,1,1) without constant: SES model
- ARIMA (0,2,2): Holt's linear method with additive errors
One of the known weaknesses of the ARIMA class models in the financial context is their inability to capture volatility clustering that is observed in most of the financial assets.
In this recipe, we go through all the necessary steps to correctly estimate an ARIMA model and learn how to verify that it is a proper fit to the data. For this example, we use Google's weekly stock prices from 2015-2018.