time-seriesarimaforecastingstationaritydickey-fullerequity

ARIMA Forecast Model (ARIMA 预测模型)

The Autoregressive Integrated Moving Average (自回归积分滑动平均) model is the canonical linear baseline for short-horizon stationary time-series forecasting. In the concepts/Filtered Probability Trend Hypothesis construction it plays the role of the continuous forecasting distribution that lives on the larger σ-algebra \mathcal{F}_B of the filtration.

General form

With backshift operator B Y_t = Y_{t-1} and \Delta = 1 - B, an ARIMA(p, d, q) is

\varphi(B)\, \Delta^d Y_t = \theta_0 + \theta(B)\, \epsilon_t, where - \varphi(B) = 1 - \phi_1 B - \dots - \phi_p B^p, - \theta(B) = 1 - \theta_1 B - \dots - \theta_q B^q, - \epsilon_t \sim \text{i.i.d. white noise}, - d is the number of differences needed for stationarity. The two limit cases are the autoregressive AR(p) (\theta \equiv 0) and the moving average MA(q) (\varphi \equiv 1, d=0). d>0 handles a unit-root / non-stationary process: differencing d times turns Y_t into a stationary ARMA process. ## Fitting pipeline 1. Stationarity test. Run an Augmented Dickey–Fuller (ADF) on Y_t and on \Delta Y_t, \Delta^2 Y_t, \dots until the null of a unit root is rejected. In Achury Arredondo Sanchez Ascanio 2024 Filtration ARIMA Random Walk the SAN series passes ADF at d=2 (Dickey–Fuller stat -6.5939). 2. Order selection. Inspect the sample ACF for the MA order q and the sample PACF for the AR order p. Cross-validate with AIC / BIC; do not differentiate beyond d=2 in practice — over-differencing injects noise and inflates variance. 3. Fit and predict. Estimate \phi, \theta by MLE. Produce h-step ahead forecasts \hat Y_h and the in-sample RMSE, which is the model's own estimate of the standard deviation of the one-step innovation. ## The "ARIMA as a probability distribution" step Observation 1 of Achury et al.: under CLT, the in-sample RMSE \sigma is an unbiased estimator of innovation std-dev, and the h-step forecast distribution is approximately Y_h \;\sim\; \mathcal{N}\!\bigl(\hat Y_h,\; \sigma^2 h\bigr).

This is the part of the model that lives on the Borel σ-algebra \mathcal{F}_B and gets combined with the random-walk mass on \mathcal{F}_A \subseteq \mathcal{F}_B inside the Filtered Probability Trend Hypothesis.

Worked example (SAN)

  • Series: 130 daily closes of Banco Santander ending 2023-05-31.
  • ADF requires d=2. PACF picks p=3; ACF picks q=1.
  • Model: ARIMA(3, 2, 1), RMSE = 0.0676, \hat Y_5 = 2.9775 EUR.
  • See Random Walk Price Pathwise for the binomial mass on the same horizon.

Where it fails (and what the wiki flags)

  • Variance of volatility is not captured. ARIMA is linear and homoscedastic. For series with regime switches (volatility clustering, jumps, gaps) prefer GARCH-family or state-space models — flagged as a future-work direction in the source paper.
  • No exogenous regressors. ARIMA is a univariate model. The future-work section of Achury Arredondo Sanchez Ascanio 2024 Filtration ARIMA Random Walk calls out VAR / VECM / GARCH as the natural replacements, and hybrid ARIMA + LSTM / RF / WNN.
  • Small samples are dangerous. 130 observations is barely enough to fit an ARIMA(3,2,1) reliably; the 62.5% directional hit rate reported in the source paper has no bootstrap confidence band.
  • No signal-to-position layer. ARIMA is a forecaster, not a strategy. Pair it with Volatility Targeting or Ferson-Siegel Optimal Weighting to get a tradeable system.

Cross-references